Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4: Proceedings of the 2022 Annual Conference on Experimental and Applied Mechanics 3031174704, 9783031174704

Table of contents :
Preface
Contents
Chapter 1: Innovations in Super-Resolution Microscopy
1.1 Introduction
1.2 Methodologies that Make the Nano-Microscope Operation Feasible
1.3 Acousto-Optic Effect
1.4 Numerical Data Processing Complementing the Information Decoding
1.5 Metrology of Spherical Nano-Objects
1.6 Further Analysis of Nanosphere Images
1.7 Fields of Applications of the Nano-Microscope
1.8 Conclusions
References
Chapter 2: Measuring Strain Distribution Around Inclusions and Matrix Interface Using Global Digital Image Correlation
2.1 Introduction
2.2 Simulated Tensile Test Method
2.3 Mesh for Global DIC
2.4 Displacement and Strain Measurement Result
2.5 Conclusion
References
Chapter 3: Evaluation of Stress State and Fracture Strain of High-Strength Steel Using Stereo Image Correction
3.1 Introduction
3.2 Stress Evaluation Using VFM
3.3 Experiment
3.4 Result
3.5 Conclusions
References
Chapter 4: Bistability and Irregular Oscillations in Pairs of Opto-Thermal Micro-Oscillators
4.1 Introduction
4.2 Experimental Methods
4.3 Experimental Results
4.4 Numerical Analysis
4.5 Conclusions
References
Chapter 5: Tympanic Membrane Shape Measurement by Miniaturized High-Speed Fringe Projection Shape Measurement Using MEMS Scanning Mirror
5.1 Introduction
5.2 Method
5.3 Results
5.4 Conclusion and Future Work
References
Chapter 6: High-Speed Optical Extensometer for Uniaxial Kolsky Bar Experiments
6.1 Introduction and Background
6.2 Methodology
6.3 Experimental Results
6.4 Conclusion
References
Chapter 7: On the Miura Ori Modal Response: A Look Throughout the Experimental Side
7.1 Introduction
7.2 Background
7.3 Analysis
7.4 Conclusion
References
Chapter 8: Using Digital Image Correlation to Characterize the Static and Dynamic Behavior of Structures: Industrial Applications and Lessons Learned
8.1 Introduction
8.2 Lattice Structure Characterization
8.3 Scaled Wind Turbine Blade Model Validation
8.4 Rotating fan Operational Vibrations
8.5 CNC Machine Deformation Measurement
8.6 Conclusion
References
Chapter 9: Enabling Digital Image Correlation with High-Resolution Microscopic Optics via Working Distance Automation: Advancing Resolution and Accuracy Limits
9.1 Introduction
9.2 Experimental
9.3 Numerical
9.4 Results and Discussion
9.5 Conclusion
References
Chapter 10: Characterization of Bioengineered Tissues by Digital Holographic Vibrometry and 3D Shape Deep Learning
10.1 Introduction
10.2 Methods
10.3 Results
10.4 Discussion
References
Chapter 11: Coordinated Twinning Bands in Magnesium at the Existence of Stress Raisers via In Situ Microscopic Image Correlation
11.1 Introduction
11.2 Materials and Methods
11.3 Results and Discussion
11.4 Conclusion
References
Chapter 12: Determining the Onset of Transverse Cracking in a Woven Composite Using Digital Image Correlation
12.1 Introduction
12.2 Material
12.3 Tensile Experiments
12.4 Digital Image Correlation
12.5 Combined Visual Inspection and Digital Image Correlation
12.6 Conclusion
References

Citation preview

Conference Proceedings of the Society for Experimental Mechanics Series

Ming-Tzer Lin · Cosme Furlong · Chi-Hung Hwang · Mohammad Naraghi · Frank DelRio   Editors

Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4 Proceedings of the 2022 Annual Conference on Experimental and Applied Mechanics

Conference Proceedings of the Society for Experimental Mechanics Series Series Editors Kristin B. Zimmerman Society for Experimental Mechanics, Inc., Bethel, CT, USA

The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society's two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society's Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research.

Ming-Tzer Lin • Cosme Furlong • Chi-Hung Hwang Mohammad Naraghi  •  Frank DelRio Editors

Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4 Proceedings of the 2022 Annual Conference on Experimental and Applied Mechanics

Editors Ming-Tzer Lin National Chung Hsing University Taichung, Taiwan Chi-Hung Hwang National Applied Research Laboratories Taiwan Instrument Technology Institute Hsinchu, Taiwan

Cosme Furlong Department of Mechanical Engineering Worcester Polytechnic Institute Worcester, MA, USA Mohammad Naraghi Texas A&M University College Station, TX, USA

Frank DelRio National Institute of Standards & Technology Gaithersburg, MD, USA

ISSN 2191-5644     ISSN 2191-5652 (electronic) Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-031-17470-4    ISBN 978-3-031-17471-1 (eBook) https://doi.org/10.1007/978-3-031-17471-1 © The Society for Experimental Mechanics, Inc 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Advancement in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics represents one of six volumes of technical papers to be presented at the SEM 2022 SEM Annual Conference & Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics scheduled to be held during June 13–16, 2022. The complete Proceedings also includes volumes on: dynamic behavior of materials; challenges in mechanics of time-dependent materials and mechanics of biological systems and materials; fracture, fatigue, failure, and damage evolution; mechanics of composite, hybrid, and multifunctional materials; and thermomechanics and infrared imaging, inverse problem methodologies, and mechanics of additive and advanced manufactured materials. Each collection presents early findings from experimental and computational investigations on an important area within experimental mechanics, optical methods and digital image correlation (DIC) being important areas. With the advancement in imaging instrumentation, lighting resources, computational power, and data storage, optical methods have gained wide applications across the experimental mechanics society during the past decades. These methods have been applied for measurements over a wide range of spatial domain and temporal resolution. Optical methods have utilized a full-range of wavelengths from X-Ray to visible lights and infrared. They have been developed not only to make two-dimensional and three-dimensional deformation measurements on the surface but also to make volumetric measurements throughout the interior of a material body. The area of digital image correlation has been an integral track within the SEM Annual Conference spearheaded by Professor Michael Sutton from the University of South Carolina. The contributed papers within this section of the volume span technical aspects of DIC. The micro- and nanomechanics segment of this volume focuses on specialized scientific areas that involve miniaturizing conventional scale components and systems to take advantage of reduced size and weight and/or enhanced performance or novel functionality. These fields also encompass the application of principles ranging from the micron scale down to individual atoms. Sometimes these principles borrow from conventional scale laws but often involve new physical and/or chemical phenomena that require new behavioral laws and impart new properties to exploit. Studying how mechanical loads interact with components of these scales is important in developing new applications, as well as assessing their reliability and functionality. Establishing this symposium at the Annual Meeting of the Society for Experimental Mechanics provides a venue where state-of-the-art experimental methods can be leveraged in these endeavors. The 2022 International Symposium on Micro-and Nanomechanics (ISMAN) is the 23rd in the series and addresses pertinent issues relating to design, analysis, fabrication, testing, optimization, and applications of micro-and nanomechanics, especially as these issues relate to experimental mechanics of microscale and nanoscale structures. The conference organizers thank the authors, presenters, and session chairs for their participation, support, and contribution to this very exciting area of experimental mechanics. Taichung, Taiwan Worcester, MA, USA Hsinchu, Taiwan College Station, TX, USA Gaithersburg, MD, USA

Ming-Tzer Lin Cosme Furlong Chi-Hung Hwang Mohammad Naraghi Frank DelRio

v

Contents

1

 Innovations in Super-Resolution Microscopy������������������������������������������������������������������������������������������������������������ 1 C. A. Sciammarella, L. Lamberti, L. Santoro, F. M. Sciammarella, and E. Sciammarella

2

Measuring Strain Distribution Around Inclusions and Matrix Interface Using Global Digital Image Correlation���������������������������������������������������������������������������������������������������������������������11 Yuki Tsujii, Natsuha Iketa, Keisuke Iizuka, and Satoru Yoneyama

3

Evaluation of Stress State and Fracture Strain of High-Strength Steel Using Stereo Image Correction�����������������������������������������������������������������������������������������������������������������������������������15 Ryo Sugimoto, Sota Ikoma, Keisuke Iizuka, Satoru Yoneyama, Kuniharu Ushijima, and Shota Chinzei

4

Bistability and Irregular Oscillations in Pairs of Opto-Thermal Micro-Oscillators ���������������������������������������������19 Aditya Bhaskar, Mark Walth, Richard H. Rand, and Alan T. Zehnder

5

Tympanic Membrane Shape Measurement by Miniaturized High-­Speed Fringe Projection Shape Measurement Using MEMS Scanning Mirror�����������������������������������������������������������������������������25 Haimi Tang, John Rosowski, Cosme Furlong, and Jeffrey Tao Cheng

6

High-Speed Optical Extensometer for Uniaxial Kolsky Bar Experiments�������������������������������������������������������������31 Richard Leonard III and Wilburn Whittington

7

On the Miura Ori Modal Response: A Look Throughout the Experimental Side�������������������������������������������������37 Antonio Baldi, Pietro Maria Santucci, Giorgio Carta, Michele Brun, Gianluca Marongiu, and Daniele Lai

8

Using Digital Image Correlation to Characterize the Static and Dynamic Behavior of Structures: Industrial Applications and Lessons Learned�������������������������������������������������������������������43 Simone Manzato, Davide Mastrodicasa, Emilio Di Lorenzo, Guven Ogus, and Pascal Lava

9

Enabling Digital Image Correlation with High-Resolution Microscopic Optics via Working Distance Automation: Advancing Resolution and Accuracy Limits��������������������������������������������������49 Olcay Türkoğlu and C. Can Aydıner

10 Characterization  of Bioengineered Tissues by Digital Holographic Vibrometry and 3D Shape Deep Learning �������������������������������������������������������������������������������������������������������������������������������������57 Colin Hiscox, Juanyong Li, Ziyang Gao, Dmitry Korkin, Cosme Furlong, and Kristen Billiar 11 Coordinated  Twinning Bands in Magnesium at the Existence of Stress Raisers via In Situ Microscopic Image Correlation ���������������������������������������������������������������������������������������������������������������63 S. Can Erman and C. Can Aydıner 12 Determining  the Onset of Transverse Cracking in a Woven Composite Using Digital Image Correlation���������������������������������������������������������������������������������������������������������������������������������71 Christopher S. Meyer, Bradley D. Lawrence, and Bazle Z. Haque

vii

Chapter 1

Innovations in Super-Resolution Microscopy C. A. Sciammarella, L. Lamberti, L. Santoro, F. M. Sciammarella, and E. Sciammarella

Abstract  Viruses are organisms that invade cells of living beings to reproduce. They consist of nucleic acids, RNA, underlying proteins, and a protective membrane. Their life cycle comprises three main stages: (1) penetration of a cell, (2) introduction of their genome generating new viruses, and (3) release of replicated viruses to the external cellular space for further infection propagation. Imaging techniques provide an important tool for understanding these mechanisms. Transmission electron microscopy (TEM) is one of the main tools utilized for this purpose. TEM investigations impose environmental limitations on the observation conditions. To get images of viruses, a TEM requires freezing the virus at extremely low temperatures. The bulk of TEM images are limited to 2D; for 3D images, TEM holography is available but poses additional difficulties and costs. A nano-microscope is being developed by the authors with resolution limits in the same range as a TEM. The nano-microscope can be utilized to observe viruses under environmental conditions in the range of biological entities and enables 3D dynamic observations. Keywords  Nano Microscopy · Super resolution · Sub-wavelength observations at the scale of one nanometer · Metrology of nano crystals · Metrology of spherical nano objects · Application to the observation of viruses

1.1 Introduction The word holography was coined by D. Gabor to label an imaging technique with the capacity of encoding 3D spatial information in 2D by recording both the light intensity and the phase of the captured wavefront. Moiré-Holography was introduced as a method that records intensity and phase utilizing the moiré technique [1]. This chapter presents a nano-microscope using moiré-holography as the tool to encode spatial information in the nanometric range. Gratings are introduced in the optical system to produce moiré patterns that contain the in-plane and the out-­of-­ plane information. In [2], a methodology to obtain images of viruses with spatial resolutions in the same order of magnitude as TEM, utilizing light in the range of frequencies that go from violet to visible red (i.e., wavelengths from 475 nm up to 632.8 nm), was introduced. Figure 1.1 shows the device that supports the proposed methodology. The nano-microscope consists of (1) digital microscope, (2) illumination and light conditioning system, (3) desktop processor, and (4) display monitor. The digital microscope is connected to a desktop processor that contains the software required to do (1) image acquisition, (2) image preprocessing, (3) artificial intelligence neural network to classify images, and enables the (4) presentation of the results. The illumination system consists of a laser diode that will generate evanescent illumination wavefronts, a prism that will steer the beam to produce evanescent wavefronts. The observed objects are contained in depression well slides utilized in biosciences. The illumination system contains two important components. A grating that is at the top of the prism and supports the depression well slide, and a ball lens that forms part of the microscope optical circuit as shown in Fig. 1.1.

C. A. Sciammarella (*) Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, Chicago, IL, USA e-mail: [email protected] L. Lamberti · L. Santoro Dipartimento Meccanica, Matematica e Management, Politecnico di Bari, Bari, Italy F. M. Sciammarella MXD Corporation, Chicago, IL, USA E. Sciammarella General Stress Optics Corporation, San Diego, CA, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_1

1

2

C. A. Sciammarella et al.

Fig. 1.1  Schematic representation of the nano-microscope setup proposed in this research

Figure 1.2a shows the ball lens that forms part of the microscope circuit. In the experiments leading to the design of the microscope, the ball lens was a polystyrene sphere of diameter 6 microns. The ball lens was fixed to the microscope slide surface to ensure that it will not move. The ball lens has a dual role in the microscope; it captures the wavefronts emitted by the observed object and at the same time generates Bessel wavefronts. The Bessel wavefronts propagate through the optical circuit to the image plane without experiencing diffraction, which enables observing the near optical field in the far field. The grating indicated in Figs. 1.1 and 1.2 encodes the metrological information in the formed images. Figure 1.2a shows an expanded view of the region of the microscope that contains the observed objects. Figure 1.2b (1) shows a prismatic nanocrystal; Fig. 1.2b (2) shows wavefronts entering and emerging from the ball lens acting as a relay lens; Fig. 1.2b (3) shows wavefronts arriving at the focal plane of the spherical lens; and Fig. 1.2b (4) shows the wavefronts arriving at the image plane of the CCD. In [2], the basic foundations of the data gathering, and processing of the nano-microscope were outlined and illustrated with examples of the work done by the authors in previous papers utilizing prismatic nanocrystals and nanospheres. In the following sections, important additional information is introduced, providing a broader and more comprehensive information on the proposed methodology.

1.2 Methodologies that Make the Nano-Microscope Operation Feasible When holographic moiré is utilized for metrological purposes, the phase distribution of a wavefront that goes through an object is made to interfere with a reference wavefront. The interferogram displays the change of optical phase of the transmitted beam with the reference wavefront. In our earlier work [3–6] we developed a super-resolution methodology that for objects of nano-dimensions arrived to interferograms with spatial resolutions of sub-nanometric sensitivity. To increase the sensitivity of holographic moiré, two phenomena were utilized: (1) the self-emission of light by nano-­ objects due to the acoustic-optic effect [7] and (2) Bessel wavefronts’ evanescent diffraction orders of gratings [8]. The observed objects’ geometry is encoded by the Bessel beams’ wavefronts. To clarify the role of each of these phenomena in the metrology of nano–objects, let us consider the phase change of the interfering wavefronts, reference wavefront, and modulated wavefront:

δ op ( z ) = ∫

h ( x,y )

0

[ n i − n o ] dz

(1.1)

1  Innovations in Super-Resolution Microscopy

3

Fig. 1.2 (a) Expanded view of the region of the microscope that contains the observed objects; (b) schematic representation of the optical circuit forming the image

In Eq. (1.1), h(x,y) is the depth of an object with respect to a reference plane, optically represented by a plane wavefront. In Eq. (1), the coordinates x, y are in the reference plane, and z is of the direction of the normal to the reference plane. In Eq. (1.1), no is the index of refraction in the region where the object is not present and is a constant. The index of refraction of the object is a constant ni, the same for all the values of x of the object. From Eq. (1.1),



δop ( z= )

h ( x,y )

∫

h ( x,y )

n i dz −

0

∫ 0

n 0 dz

(1.2)

Equation (1.2), as a consequence of the fact that no and ni are constants, becomes.

δop ( h ( x,y ) =n i h(x,y) − n o h(x,y) )

(1.3)



Converting the difference of optical path into a phase difference,



2π 2π ∆φ  h ( x,y )  = h ( x,y ) − h ( x,y ) = φm − φr pm pr

(1.4)

In Eq. (1.4), ϕm is the modulated phase of the wavefront going through the object, ϕr is the phase of the reference wavefront, pm is the pitch of the modulated carrier, and pr is the pitch of the reference carrier. From this equation, we can get h(x,y). An example illustrating this procedure is shown in Fig. 1.3. In Fig. 1.3, the pitches of the reference and modulated carriers are constants, which means that they are plane wavefronts; the upper surface of the crystal is a plane. Experimentally it is difficult to get the face of the crystal parallel to the sensor, and phase corrections are introduced to compensate for this problem. The example of Fig. 1.3 involves two fundamental steps: (1) selection of the pitch pr and determination of pm, and (2) application of gradient filters to determine Lo and Wo. In a more general case, h(x,y) is a variable quantity and the shape of the observed object can be obtained from the display of contour lines of the surface. It is possible to display the phase difference in steps of the selected values, thus getting all the dimensions of the crystal in one single display. Figure 1.4 is an example of this approach; it shows a region of the image displaying contour lines of the deposits of sodium chloride. The blue background corresponds to the reference surface, and the steps of the contours are 2.82 nm that correspond to 5 atoms. The image of Fig. 1.3 was extracted from a region of 1153 × 1153 nm from the same area as Fig. 1.4. The higher resolution obtained in the image analysis brings details that were not visible in Fig. 1.3. It is interesting to notice that the deposits of sodium chloride that have prismatic shapes are few compared to the total deposited mass. Figure 1.5 shows a crystal of Lo = 10.15 nm, Wo = 5.64 nm, Ho = 5.64 that is adjacent to a sodium chloride deposit 22.5 nm in depth. The emission of light by the observed objects is a necessary condition arising from the super-resolution requirement of the proposed methodology. However, in Eq. 1.4, the wavelength of the light is not explicitly included. This means that monochromatic images can be utilized in processing of data recorded by the nano-microscope.

4

C. A. Sciammarella et al.

Fig. 1.3  Phase determination for a sodium chloride nanocrystal of length L = 120 nm: (a) reference phase of the carrier fringes pc = 5.53 nm; (b) phase of carrier fringes modulated by the nanocrystal, a constant because the surface is a plane; (c) phase difference in the region of the nanocrystal represented in levels of gray; (d) dimensions of the crystal

Fig. 1.4  Depth contour lines of a region of 51 × 51 nm

Fig. 1.5 (a) Sodium chloride crystal of dimension Lo = 10.15 nm, Wo = 5.64 nm, Ho = 5.64 nm; (b) sodium chloride deposit of 22.5 nm maximum depth

1  Innovations in Super-Resolution Microscopy

5

1.3 Acousto-Optic Effect Although the emission of light of different wavelength is not required for the proposed metrological applications, it is of interest to point out some predictions arising from this phenomenon. The acousto-optic model of the light emission makes three important predictions: (1) the color of the light emitted by an object depends on the geometry of the observed objects, different geometries will emit different wavelengths; (2) in the case of prismatic objects, the color is a function of the length Lo of the crystal; and (3) the light emission involves a limited number of atoms. In the case of prismatic crystals, the interval of emission confirmed by experimental measurements goes from the number of atoms Na = 2 atoms to about Na = 225. These Na correspond to wavelengths that go to visible red 632.8 for Lo = 0, wavelength of the illuminating laser, to the ultraviolet λ = 475 nm. In [2], a correlation between the color of the emitted light and the length of the observed crystal was presented. The new higher-resolution images obtained in this work were utilized to get additional points to the correlation between wavelength and Lo. In Fig. 1.6, the graph to the left represents the correlation between the wavelength and crystals’ (prismatic) lengths in the range Lo = 0 nm to Lo = 12 nm; the graph to the right includes the new measurement together with the measurements presented in [2].

1.4 Numerical Data Processing Complementing the Information Decoding A numerical method to add to the preceding developments is derived from a property of the FT, the Fourier transform of the Fourier transform [4, 5]. This method allows the determination of the intensity distribution of the observed objects in a reference plane. A function f(x) that provides the intensity distribution in an image in the x-coordinate has a Fourier transform FT[f(x)] = F(fex), where fex = 1/x. Suppose that one wants to find F(x), a function that is related to f(x) that has additional information on the analyzed signal. F(.) is a function that can be either in the frequency space or in the physical space, and the dot in parenthesis stands for x or fx. In the physical space, F(x) gives the carrier fringes that provide the intensity distribution in the x,y plane. In, [2, 4, 5], a detailed description of the process of application of the FT is provided. Figure 1.7 integrates the signals that provide intensity distribution in the x,y plane in the image of a crystal of Lo = 86 nm, Wo = 86 nm, and a stepped upper face. The upper plot represents three rectangular pulses of different intensities. A central rectangular pulse of width Wo = 86 nm, and two shifted pulses with the shift Δxs = 26 nm. In [2, 4, 5], the shifted images are explained in detail. The lateral plot represents one rectangular pulse of width 86 nm. Because of the orientation of the laser illumination, the shifts take place only in the x-direction and there are no shifts in the y-direction. The pulses are encoded in carrier fringes of pitch p = 2.91 nm. Figure 1.7b demonstrates that the shift can be obtained utilizing the FT of the FT, independently of the edge detection methods referred to in [2, 4, 5]. Although in the given references the numerical process of FT of the FT is illustrated for prismatic bodies, it is valid for all geometries. Further in this chapter, it will be applied to spherical objects, leading to an important conclusion concerning the mode of vibration of spherical objects.

Fig. 1.6  The graph to the left represents prismatic crystal of the correlation between the wavelength and crystals lengths, and the graph to the right includes the new measurement together with the measurements presented in [2]

6

C. A. Sciammarella et al.

Fig. 1.7 (a) Image of a crystal of Lo = 86 nm, Wo = 86 nm and stepped upper face. Upper plot carriers on the x-direction. Lateral plot carrier in the y-direction; (b) Δxs measured with edge detection vs. information obtained from the Ft of the FT

1.5 Metrology of Spherical Nano-Objects This section describes how to apply the previously described metrology procedures to spherical objects with the nano-­ microscope. Different aspects of these procedures were illustrated utilizing simple geometries generated by monomers of sodium chloride precipitated on a surface to form crystals that were predicted theoretically and compared with measured values [2, 4, 5]. One of the many possible uses of the nano-microscope is its application to biomechanics and in biomechanics to a very important field, the study of coronaviruses. For this reason, we will analyze the images of nanospheres that roughly define the shape of coronaviruses. Nanospheres were present in the same images that contained the prismatic crystals and were recovered from these images. While prisms are easy to recognize in the observed images, nanospheres are not. In the authors’ early work, spheres were detected through intensity distribution patterns corresponding to the vibration of nano-shaped spheres. These intensity distribution patterns of spheres are also observed in spherical shells. The intensity patterns are the result of whispering gallery modes of vibration (WGMs). WGMs result from light confinement due to total internal reflection inside a high refraction index spherical surface or shell immersed in a lower refraction index medium. In the WGM, the light travels a round trip within the outer surface of the sphere or shell with phase matching. The WG modes are included in Mie’s family of solutions for resonant modes in light scattering by dielectric spheres or shells. Figure 1.8a shows the standing wave along the equatorial line of a nanosphere; (b) shows the gray level image of a nanosphere; (c) shows the filtered image with indication of the nanosphere’s diameter; and (d) shows the color image of the same sphere. The first thing to note is that the image of the nanosphere is noisier than the observed images of nanocrystals. The nanocrystals precipitate from a liquid solution in the microscope slide and are attached to the slide surface. The nanospheres were injected into the fluid, hence are free to move and are subjected to the Brownian motion. The color camera sensor that captured the image of Fig. 1.8d was sensitive to λ = 386 nm, peak value of the light emitted by the polystyrene material of the sphere that is a fluorescent substance at this wavelength. Since the nanosphere is transparent to the wavelengths present in the image, one can see green hues corresponding to sodium chloride precipitates present in the saline solution where the nanosphere is immersed. In the literature [9], a numerical solution for the excitation of a nanosphere of polystyrene for the same wavelength λ = 386 and in the same conditions of excitations prevailing in our experimental observations is available. Figure 1.9a displays the correlation between the equatorial wavelengths of five spheres; four are experimentally measured values, and the fifth sphere corresponds to reference [9]. Figure 1.9b illustrates the procedure to measure the wavelengths of the WGM of the observed spheres in this case utilizing the information coming from Fig.  1.8d. Maximum nodes are

1  Innovations in Super-Resolution Microscopy

7

Equatorial wavelength (nm)

Fig. 1.8 (a) The standing wave along the equatorial line of a nanosphere; (b) gray-level image of a nanosphere; (c) filtered image with indication of the nanosphere’s diameter; (d) color image of the same sphere

160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

l eq = 0.1051Dp + 78.612, R 2 = 0,9976

0

50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

(a)

Radius of the nano-sphere (nm)

(b)

Fig. 1.9 (a) Relationships between nanosphere radius R and equatorial wavelength λeq of the WGM; (b) image of the nanosphere of diameter 150 nm, nodes (red ellipses), and antinodes (black ellipses) corresponding to WGM standing wave

represented by red ellipses, and antinodes are represented by black ellipses. Due to the sphere motion, the maxima and minima are not sharp, but despite this fact can be counted, in this case a total of 10 nodes and antinodes. The angle subtended by successive nodes and antinodes is 36°, that is, 0.62832 radians. The subtended arc is, 0.62832 × 75 = 47.124 nm, and since we have a total of 10 arcs, we get λeq = 471.24 nm that matches the computed value using the correlation equation shown in Fig. 1.9a for R = 75 nm. This mode is relevant because it is based on the mechanical excitation of the nanospheres or shells that causes light emission, the foundation of the proposed methodology.

1.6 Further Analysis of Nanosphere Images Utilizing the FT of the FT of images in a similar way to what is displayed in Fig. 1.7 for a prismatic crystal one gets the profiles of Fig. 1.10. These profiles display the intensity distribution of the sphere of D = 150 nm in the x and y coordinates. In Fig. 1.10, it is possible to verify that both profiles are identical. Utilizing the circular symmetry of the profiles, one can get Fig. 1.11a, the loci of maxima intensities, full red circles, and the loci of minimum intensities, dashed circles. In Fig. 1.11a, the horizontal coordinate is in nm and the vertical coordinate gives the intensity levels of the image. Figure 1.11b shows the intensity distribution of Mie’s solution for a sphere of D = 150 nm. The horizontal scale is in nm, the vertical scale represents relative intensities. From the analysis of Fig. 1.11a and 1.11b based on the circular symmetry of the pattern, it is possible to conclude that the profile of intensities of Fig. 1.11a corresponds to a super-resolution pattern of a nanosphere. The formation of the pattern of interference fringes for a sphere in super-resolution illumination was explained by an approximate method in [4, 5] and formally confirmed in [10].

280 240

X -direction

200 160 120 80 40 0

-150 -120

-90

-60

-30

0

30

60

90

120

150

Normalized light intensity (GL)

C. A. Sciammarella et al.

Normalized light intensity (GL)

8 280 240

Y -direction

200 160 120 80 40 0

-150

Position X [nm]

(a)

-120

-90

-60

-30 0 30 Position Y [nm]

(b)

150

60

90

120

150

150

Fig. 1.10 (a) Sphere of 150 diameter cross section of the intensity profile of the sphere in the x-direction; (b) cross section of the intensity profile of the sphere in the y-direction

280

0

240

1.04

Y -direction

200 160

20

120

1.02

80 40

40 -150 -120

-90

-60

-30

0

30

60

90

120

150

y

-y

1.00

x

0

60 0.98

(a)

(b)

80

0.96

150 nm 0

20

40

y

60

80

Fig. 1.11 (a) Sphere of D = 150 nm, loci of the points of maximum and minimum intensities; (b) Mie’s solution loci of the points of maximum and minimum intensities for the sphere of D = 150 nm

The relationship of the diameter of the super-resolution central circular fringe in dimensionless coordinates corresponding to the zero order of the Bessel function to Mie’s solution is rcm = 2.05/3.83 = 0.5325. Measuring the diameter of Fig. 1.10a first ring, the value is 14 nm, and the corresponding value for Fig. 1.10b is 26.2 nm, resulting in the ratio rcm = 14/26.2 = 0.534; the experimental and theoretical values are in good agreement. It is very interesting to point out that the same intensity distribution is observed in the image of the ball lens [2, 4, 5]. Another way to get metrological evaluations of spheres is to use methods to measure curvatures of surfaces. Figure 1.12 shows fringe patterns filtered from the FT of the sphere of Fig. 1.8b. The analysis of the patterns was performed by applying the methodology of the Ronchi test to determine curvatures of lenses. The two patterns gave an average curvature corresponding to the diameter D = 150 nm.

1.7 Fields of Applications of the Nano-Microscope From the preceding developments, one can conclude that the same methodologies applied to prismatic crystals can be employed in nanospheres. There is a caveat in this statement that the methods utilized for prismatic crystals are based on the acousto-optic effect model of Brillouin. In this model, the medium is made of regular rows of equally spaced atoms that form

1  Innovations in Super-Resolution Microscopy

9

Fig. 1.12  Fringe patterns filtered from FFT of the image of Fig. 1.8d

Fig. 1.13 (a) 3D view of the COVID-19 virus and cross section. (b) Finite element model developed in ABAQUS; one frame of a numerically generated 3D movie of the vibration of the COVID-19 virus as a result of the excitation of the virus by light radiation. (c) The vibration mode predicted by ABAQUS is primarily located at the equator of the sphere and gives the vibration frequency for the WGM

a crystalline space. The material of the nanospheres is polystyrene. Polystyrene is a hydrocarbon polymer made from the monomer styrene. It is a long-chain organic hydrocarbon. The properties of polystyrene are determined by quite different mutual potentials than the analyzed prismatic nanocrystals. From the results of the performed experiments, these differences do not change the decoding techniques that can be applied to the two groups of nano-objects. The reasons for this fact need an explanation; this topic is beyond the scope of this publication. One important potential application of the microscope is the monitoring of viruses. Figure 1.13 provides an example of this application. Figure 1.13a shows a 3D representation of the COVID-19 virus utilized in this chapter as an application of virus analysis and its cross section. The cover membrane of the virus can be represented as a thin shell, a hollow spherical resonator. The shell interior can be represented as a fluid; the spherical resonator is assumed to be in a saline solution. The virus spikes are added to the thin shell through a hinge type of connection. A finite model of the virus based on the described model was developed to help in the process of building the microscope. Experiments with viruses must address general protection concerns for researchers’ safety and guidelines for the security of the facilities. In the preliminary design steps, we opted for a numerical model to simulate the light emission process of a virus. The development of finite element model provides information concerning wavelengths and frequencies to be utilized in the microscope. A linear eigenvalue analysis was performed in order to detect natural frequencies and eigenmodes compatible with WGM that will be associated with the lasers to be utilized in the microscope setup and connected to the transformation of mechanical energy into light emission. The WGM is important because it corresponds to the vibration mode that causes the emission of light in the experimental conditions prevalent in the geometrical configuration of the proposed nano-microscope. The finite element model is shown in Fig. 1.13b. The FE model of the virus includes about 800,000 elements and nodes. Convergence analysis was carried out in order to have a mesh-independent solution. It was verified that the spikes do not have a large influence on the energy spectrum of the virus. Figure 1.13c shows that the vibration mode predicted by ABACUS is primarily located in the equator and is within the order of the magnitude predicted by the observed nanospheres in our earlier experiments. Figure 1.13b is one frame of a movie that shows that spikes move in all space directions, independently of the shell vibrations. This phenomenon has been observed in TEM holography movies. The proposed nano-microscope has a resolution of the order of magnitude of the TEM microscope. This fact can be added to the capability of digital microscopes to deliver movies of the observed events. These two capacities offer an alternative to accomplish similar tasks to those achieved by TEM at a much lower cost, effort, and in environmental conditions that correspond to the normal conditions in living tissues without loss of spatial resolution.

10

C. A. Sciammarella et al.

1.8 Conclusions For humans, visual information plays a fundamental role in the understanding of complex processes. In biomechanics, experimental observations deal with multiple dimensional spatial scales that go from atomic distances to the ensembles of molecules in complex organizations. Microscopy is a paramount tool to access these multiple scales that are present in biological events. In the analysis of the mechanisms of infections of living organisms by viruses, imaging technologies have played a particularly important role in the process of understanding the virus behavior and the developments associated with the mitigation of the viruses’ effects on humans. Transmission electron microscopy (TEM) has been and is the main provider of static and dynamic sequences showing the mechanisms of interaction of cells and the invading viruses. A TEM has a spatial resolution limit in ranges that produces high-resolution images of the viruses interacting with cells. Currently, to observe the images of viruses with TEM requires freezing the viruses at extremely low temperatures. Dynamic TEM holography is a very complex procedure that requires a large sequence number of steps and a very expensive and complex equipment. The development of the proposed nano-microscope with a resolution of the order of magnitude of the TEM microscope and with its dynamic capability offers an alternative to accomplish similar tasks to those achieved by TEM at a lower cost, effort, and in environmental conditions that correspond to the normal environments in living tissues without reducing the spatial resolution. The utilization of FE, together with experimental observations, proves once more to be a valuable complementary tool that together with artificial intelligence can provide a priori knowledge in helping to understand these complex molecular mechanisms.

References 1. Sciammarella, C.A., Dichirico, G., Wang, T.C.: Moiré-holographic technique for three-dimensional stress analysis. J.  Appl. Mech. 37(1), 180–185 (1970) 2. Sciammarella C.A., Lamberti L, Sciammarella F.M.  Super resolution optical microscopy to detect viruses (SARS-CoV-2) in real time. Proceedings of the Virtual SEM Annual Meeting 2021 3. Sciammarella, C.A.: Experimental mechanics at the nanometric level. Strain. 44, 3–19 (2008) 4. Sciammarella, C.A., Lamberti, L., Sciammarella, F.M.: The equivalent of Fourier holography. Exp. Mech. 49, 747–773 (2009). https://doi. org/10.1007/s11340-­008-­9189-­2 5. Sciammarella, C.A., Lamberti, L., Sciammarella, F.M.: Optical holography. In: Rosen (ed.) Reconstruction of Nano Objects. Intechopen (2011) 6. Sciammarella, F.M., Sciammarella, C.A., Lamberti, L.: Nano-holagraphy interferometry for in-vivo observations. In: Shaked, N.T., Zalesvsky, Z., Satterwhite, L.L. (eds.) Biomedical optical phase microscopy and nanoscopy. Elsevier (2013) 7. Brillouin, L.: Wave propagation in periodic structures. McGraw-Hill Book Company, New York (1946) 8. Toraldo di Francia, G.: La Diffrazione Della Luce. Edizioni Scientifiche Einaudi, Torino (1958) 9. Pack, A. Current Topics in Nano-Optics. PhD Dissertation, Chemnitz Technical University, Chemnitz (Germany), 2001 10. Ayyagari, R.S., Nair, S.: Scattering of P-polarized evanescent waves by a spherical dielectric particle. J. Opt. Soc. Am. Part B Opt. Phys. 26, 2054–2058 (2009)

Chapter 2

Measuring Strain Distribution Around Inclusions and Matrix Interface Using Global Digital Image Correlation Yuki Tsujii, Natsuha Iketa, Keisuke Iizuka, and Satoru Yoneyama

Abstract  A method for evaluating displacement and strain distributions around inclusion and matrix interface is studied in this chapter. Global digital image correlation (global DIC) is used for this method. Global DIC can obtain the displacement in inclusions and matrix parts separately. Strains are also computed using a mesh. The results show that strain distributions of rapid change around interface can be obtained by global DIC. Then, by comparing the measurement results of local DIC and global DIC, it is found that global DIC has the validity to measure strain around the interface. Keywords  Global DIC · Finite element mesh · Inclusion · Matrix · Interface · Strain distribution

2.1 Introduction Recently, in the automotive field, reducing the weight of vehicles has become an issue in terms of environmental problems and cruising range. In particular, battery-powered vehicles such as hybrid vehicles and electric vehicles are heavy because of the battery itself. For this reason, the use of CFRP for major components is being promoted. In order to reduce the weight and increase the safety of the vehicle, one of the possible solutions is to increase the ratio of CFRP, but it is essential to study the deformation and fracture behavior of CFRP. The mechanical properties of FRP composites are affected not only by the mechanical properties of the constituent fibers and resins but also by the fiber–resin interface. Therefore, the failure of FRP composites is caused by microstructural damage such as fiber fracture, kink, interface delamination, and microcrack propagation. Thus, multiscale studies are necessary to understand the fracture behavior, and microscale strain measurement, which is the source of crack propagation, is extremely important. Canal et al. [1] and Mehdikhani et al. [2] pointed out that the subset-based DIC (local DIC) cannot accurately obtain the strain at the interface because the discontinuous strain at the fiber–resin interface is smoothed during strain calculation. Besides local DIC, there is mesh-based DIC (global DIC) [3–5]. The mesh is created overlapping the image, and the nodal displacements are determined from the brightness value information. This method is considered to provide accurate strain distribution for discontinuous areas. In this study, a measurement method for displacement and strain distributions is investigated to observe strain concentration around the fiber/resin interface of CFRP. In order to observe the fiber–resin interface of FRP material, it is necessary to test and photograph it using a laser microscope. However, due to various problems in a test with CFRP, we try to establish a displacement and strain measurement method using a test with a specimen that is simulated the cross section of CFRP. Then, comparing the measurement results of local DIC and global DIC, it is found that global DIC has the validity to measure strain of CFRP.

Y. Tsujii (*) · N. Iketa · K. Iizuka · S. Yoneyama Department of Mechanical Engineering, Aoyama Gakuin University, Sagamihara, Kanagawa, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_2

11

12

Y. Tsujii et al.

2.2 Simulated Tensile Test Method The images taken in the simulated experiment are shown in Fig. 2.1. The test machine is Instron Limited, the camera is XiQ MQ042MG-CM, and the lens is Micro NIKKOR. The mock-up specimens are made with epoxy adhesive (Hysol EA 9363) and three S45C specimens are embedded. The difference in Young’s modulus between epoxy adhesive and S45C is used to simulate resin and carbon fiber of CFRP. The surface of the specimen is painted with a random pattern of white and black spray paint. The tensile rate is set at 1.0 mm/min. The images are taken at 2048 × 2048 pixels at a rate of 1 fps, with a color depth of 8 bits and, a length of about 0.02 mm/pixel. The analysis area is set to 1180 × 1200 pixels.

2.3 Mesh for Global DIC The mesh used for global DIC is shown in Fig. 2.2. The element geometry used in the mesh is four-node quadrilateral elements, and the mesh is created with approximately 400 nodes. The strains are calculated from the measured nodal displacements based on the displacement–strain relationship of the finite element method.

2.4 Displacement and Strain Measurement Result Figure 2.3 shows (a) y-direction displacement and (b) y-direction strain obtained by global DIC. Subset size and gauge length are 31 × 31 pixels and 41 pixels, respectively. Fig. 2.4 shows (a) y-direction displacement and (b) y-direction strain obtained by global DIC. The result shows that the displacement distribution obtained by local DIC has more variation in value than the one by global DIC. In strain measurements, small strains are obtained in the S45C part and large strains in the epoxy part for both distributions. In addition, the strain map by local DIC shows continuous strain at the interface, while the one by global DIC shows discontinuous strain near the interface. This suggests that global DIC is more effective than local DIC for measuring strain near the interface.

Fig. 2.1  Random pattern for FRP simulated specimen

2  Measuring Strain Distribution Around Inclusions and Matrix Interface Using Global Digital Image Correlation

13

Fig. 2.2  A mesh with 4 node quadrilateral elements for global DIC

Fig. 2.3  Displacement and normal strain distributions to the loading direction by local DIC (a) uy, (b) εy

Fig. 2.4  Displacement and normal strain distributions to the loading direction by global DIC (a) uy, (b) εy

2.5 Conclusion In this study, in order to establish a strain calculation method that can measure strain concentration, a tensile experiment is conducted using a simulated FRP specimen before the actual test on CFRP.  Global DIC is used for the measurement. Comparing the distributions obtained by local DIC and global DIC, the results show that there is no significant difference in the displacement distributions. On the other hand, the strain distributions show that the strain concentration is smoothed in local DIC, whereas global DIC can calculate discontinuous strains without smoothing at the interface. This demonstrates the effectiveness of global DIC in strain measurement with discontinuous strain distribution such as CFRP.

14

Y. Tsujii et al.

References 1. Canal, L.P., Gonzalez, C., Molina-Aldareguia, J.M.: Application of digital image correlation at the microscale in fiber-reinforced composites. Compos. Part A. 43, 1630–1638 (2012) 2. Mahoor, M., Mohammadali, A., Baris, S.: Full-field strain measurements at the micro-scale in fiber-reinforced composites using digital image correlation. Compos. Struct. 140, 192–201 (2016) 3. Yoneyama, S., Koyanagi, J., Arikawa, S.: Measurement of discontinuous displacement/strain using mesh-based digital image correlation. Adv. Compos. Mater. 25, 329–343 (2016) 4. Hild, F., Roux, S.: Comparison of local and global approaches to digital image correlation. Exp. Mech. 52, 1503–1519 (2012) 5. Wang, B., Pan, B.: Subset-based local vs. finite element-based global digital image correlation: a comparison study. Theor. Appl. Mech. Lett. 6, 200–208 (2016)

Chapter 3

Evaluation of Stress State and Fracture Strain of High-Strength Steel Using Stereo Image Correction Ryo Sugimoto, Sota Ikoma, Keisuke Iizuka, Satoru Yoneyama, Kuniharu Ushijima, and Shota Chinzei

Abstract  A method for evaluating the stress state and fracture strain of the thin sheet of high-strength steel is studied in this chapter. Displacements and strains on the surface of high-strength steel plates are measured using stereo image correlation (DIC). The stress–strain relation after the necking is obtained from the load and strains by inverse problem analysis using the virtual field method. The stresses obtained by the proposed method are well balanced with the external forces. Keywords  Stereo image correlation · Necking · Stress triaxiality · High-tensile strength steel · VFM

3.1 Introduction In the automotive industry, weight reduction and collision safety have been required, and high-strength steel plates have been actively used in automobile body frames. High-strength steels are more brittle with increasing strength. Fracture is highly likely to occur during a collision. So, it is important to predict deformation and fracture phenomena during a collision. Therefore, it is necessary to evaluate the fracture strain, which depends on the stress state of the material. The fracture strain is affected by stress triaxiality [1]. Stress triaxiality depends on the stress. For geometries with stress gradients, such as notched specimens, the stress–strain relationship within the yield region is complex and it is very difficult to evaluate it accurately. Dunaud et al. [2] showed the error of the equivalent plastic strain in the numerical analysis just before fracture. This shows modeling is difficult after the necking. After the necking, it is considered that the evaluation of the experiment is appropriate. Virtual field method (VFM) [3] is an inverse problem analysis method. VFM applies the principle of virtual work to the measured displacement strain and load to obtain the material properties that are in balance with the external force. In this study, VFM is used to obtain stresses. The purpose of this study is to evaluate the stress triaxiality from stress–strain relation obtained by the proposed method. In this study, the strain distribution of 980 MPa high-strength steel sheet was measured by stereo image correlation. It is proposed to obtain the equivalent stress-equivalent plastic strain relationship at the neck just before yielding and fracture using VFM from the strain distribution and load data.

3.2 Stress Evaluation Using VFM Since the exact stress–strain relation is unknown at the neck, the stress is determined using inverse problem analysis. Based on the principle of virtual work, the internal energy is balanced by the external energy, and the following equation is established: R. Sugimoto (*) · S. Ikoma · K. Iizuka · S. Yoneyama Department of Mechanical Engineering, Aoyama Gakuin University, Sagamihara, Kanagawa, Japan e-mail: [email protected] K. Ushijima Department of Mechanical Engineering, Tokyo University of Science, Tokyo, Japan S. Chinzei Kobe Steel Ltd, Kobe, Hyogo, Japan © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_3

15

16

R. Sugimoto et al.

 

 ij ij





d    Ti u i d  q

(3.1)

where σij is the stress, ε∗ij is the virtual strain component, u∗i is the virtual displacement component, Ti is the external force, Ω is the inside an object, and Γ is the load boundary. In this study, the load, strain, and virtual displacement obtained from experiments are used in Eq. (3.1). The stresses in the plastic deformation region are calculated point by point from just before plastic deformation by solving the equation using the Newton–Raphson method. The incremental strain theory is used here to obtain the stresses. The stresses are obtained to be balanced with the external forces, displacements, and strains obtained from the experiment.

3.3 Experiment In this experiment, the specimen shape is shown in Fig. 3.1. The tensile strength of the specimen is 980 MPa, thickness is 1.4 mm, and notch radius is 5 mm. The analysis range of the stereo image correlation method is set within the red frame in Fig. 3.1. The shutter speed is 1 fps, and the crosshead speed is 0.5 mm/s. After the tensile test, the camera calibration is performed to obtain the internal and external parameters and unify the coordinate system.

3.4 Result The time history of the load in the tensile test is shown in Fig. 3.2. The maximum load was 16.0 kN at around 291 s. The fracture occurred at about 342 s and 14.6 kN. The load obtained is applied to the principle of virtual work. The displacement at the maximum load and just before fracture is shown in Fig. 3.3. The strain is obtained from the displacement. The load, displacements, and strain obtained are applied to the principle of virtual work. The virtual displacement is determined by stress sensitivity from the DIC displacement distribution [4]. In this study, the proposed method is used to compute the stress–strain relationship up to the strain at maximum load. The stresses obtained by the proposed method are well balanced with the external force. The true stress–true strain relationship by uniaxial tension and the stress–strain relationship by the proposed method and the results by the proposed method are shown in Fig. 3.4. It is observed that the stress–strain relations of the proposed method and uniaxial tension are almost in agreement. From this result, the proposed method is considered to be appropriate. Fig. 3.1  Shape of the specimen

3  Evaluation of Stress State and Fracture Strain of High-Strength Steel Using Stereo Image Correction

17

Fig. 3.2  Load time history

Fig. 3.3  Displacement distribution (a) ux Maximum load (b) uy Maximum load (c) ux Before fracture (d) uy Before fracture

Fig. 3.4  Equivalent plastic strain- equivalent stress

3.5 Conclusions Tensile tests were conducted on specimens of 980 MPa class high-strength steel with notches. The load time history, displacement, and strain were obtained using load measurement and stereo image correlation. The equivalent stress-equivalent plastic strain relation was obtained by performing the proposed method. The results are consistent with the results of uniaxial tensile tests, indicating the validity of the proposed method.

18

R. Sugimoto et al.

References 1. Bai, Y., Wierzbicki, T.: Forming severity concept for predicting sheet necking under complex loading histories. Int. J.  Mech. Sci. 50, 1021–1022 (2008) 2. Dunand, M., Mohr, D.: Hybrid experimental–numerical analysis of basic ductile fracture experiments for sheet metals. Int. J. Solids Struct. 47, 1130–1143 (2010) 3. Grediac, M., Pierron, F.: Applying the Virtual Fields Method to the identification of elasto-plastic constitutive parameters. Int. J. Plastic. 22, 602–627 (2006) 4. Aleksander, M., Frances, M.D., Fabrice, P.: Sensitivity-based virtual fields for the non-linear virtual fields method. Comput. Mech. 60, 409–431 (2017)

Chapter 4

Bistability and Irregular Oscillations in Pairs of Opto-Thermal Micro-Oscillators Aditya Bhaskar, Mark Walth, Richard H. Rand, and Alan T. Zehnder

Abstract  In this work, we experimentally and numerically investigate the nonlinear dynamics of pairs of coupled opto-­ thermal micro-oscillators. The oscillators are driven into limit cycle oscillations using an external constant power laser source and coupled via mechanical linkages. As the input laser power is increased, the oscillators transition from a state of synchrony to a state of bistability that manifests as irregular oscillations in the experiments. The experiments are performed on two sets of devices with different coupling strengths, which reveal that the laser power required for the transition to irregular oscillations increases with an increase in the coupling strength. A numerical parameter sweep in the coupling strength and laser power parameter space reveals a similar trend. Keywords  Limit cycle oscillations · Laser drive · Mechanical coupling · Bistability · Irregular oscillations

4.1 Introduction Micro-oscillators have been extensively used to study nonlinear phenomena in devices. Synchronization of nonidentical oscillators [1], frequency entrainment of oscillators by an external sinusoidal drive, and appearance of various exotic states of oscillations in simple networks [2] have been experimentally demonstrated. In this work, we report the experimental observation of irregular oscillations in pairs of micro-oscillators and the influence of the input laser power and coupling strength on the onset of this phenomenon. We observe that the minimum laser power required for irregular oscillations increases with an increase in the coupling strength. A third-order lumped-parameter model is used to describe the nonlinear oscillator, and numerical simulations are used to study the phenomenon of bistability. The experimental method for laser drive and detection is also briefly explained.

4.2 Experimental Methods Pairs of micro-oscillators are fabricated on a silicon-on-insulator (SOI) chip using a general photolithography process. The oscillators are clamped–clamped silicon beams that are L1 = 40 and L2 = 38 μm long and w = 2 μm wide. The out-of-plane device thickness is 205 nm. The devices are supported by the 400 nm silicon dioxide layer underneath and are released to vibrate out of plane. The silicon dioxide layer rests on a thick silicon substrate. The devices are driven into limit cycle oscillations using opto-thermal heating provided by a helium–neon laser source at 633  nm wavelength and the Fabry–Perot interference cavity between the silicon device layer and silicon substrate. The modulation of the absorbed energy due to the motion of the devices sets up a feedback loop, resulting in limit cycle oscillations [3]. The modulation of the reflected energy A. Bhaskar · A. T. Zehnder (*) Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA e-mail: [email protected] M. Walth Department of Mathematics, Cornell University, Ithaca, NY, USA R. H. Rand Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA Department of Mathematics, Cornell University, Ithaca, NY, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_4

19

20

A. Bhaskar et al.

due to the motion of the devices is directed to a photodetector and analyzed for its frequency content. A schematic for this method for drive and detection is shown in Fig. 4.1a. Mechanical coupling is introduced in the pair using mechanical linkages near the anchor points. We study devices that have links at l = 2 and l = 3 μm from the anchor points, and they correspond to lower and higher coupling strengths, respectively. A schematic with the dimensions of the micro-oscillators is shown in Fig. 4.1b. A single-laser spot is focused and centered to span both the oscillators that are g = 3 μm apart such that both oscillators are driven simultaneously at the same input power. The frequency responses of both oscillators are superposed in the reflected spectrum. The peaks in the recorded spectrum indicate the frequencies of the oscillators and their dynamic response. A microscope image of the two oscillators taken at 30 × magnification illuminated simultaneously by a single-laser spot is shown in Fig. 4.1c. The laser power entering the microscope body that supports the stage for the chip is controlled using an optical setup. The combination of the linear polarizers and the half-wave plate allows for the control of the polarization and power of the light striking the chip. A polarizing beam splitter and a quarter-wave plate are used as an optical isolator to redirect the reflected light to the photodetector. The photodetector is connected to a spectrum analyzer and the spectral response is recorded in the experiments. Phase data are not recorded or reported. The chip is held inverted on a chip holder in a vacuum chamber at around 10−7 mBar. The vacuum environment ensures low damping and protects the device from contamination. Figure 4.2a shows a simplified schematic of the optical setup. Figure 4.2b shows the SOI chip mounted on the device holder before being loaded into the vacuum chamber.

4.3 Experimental Results For the two levels of coupling strength studied in this work, the laser power increased continuously and the spectral response from the oscillators is plotted for four different laser power values. A schematic of the response is shown in Fig. 4.3. The responses shown in (a) and (e) are at 3.7 mW of laser power entering the setup, (b) and (f) are at 4.8 mW, (c) and (g) are at 5.6 mW, and (d) and (h) are at 6.1 mW. Note that the laser power reaching the devices is lower than the laser power entering the microscope setup supporting the vacuum chamber. At a lower coupling strength, as the laser power was increased, synchronized oscillations with a single prominent peak in the spectrum were produced in a Hopf bifurcation and transitioned to irregular oscillations between (a) and (b). For higher laser powers, at (b–d), irregular oscillations indicated by a broadband spectral response were recorded. At a higher coupling strength, synchronized oscillations were seen for various laser powers as shown in (e–g) and irregular oscillations were observed at a higher laser power (h). This indicates that increasing the coupling strength results in a higher laser power requirement for irregular oscillations. The irregularity in oscillations

Fig. 4.1 (a) Schematic of the drive and detection of oscillations in a pair of micro-beams. The multiple reflections within the interference cavity between the silicon device layer and silicon substrate sets up a feedback loop, resulting in stable limit cycle oscillations. The modulation of the total reflected light is recorded in a high-speed photodetector and analyzed for its frequency content. (b) Schematic of the top view of a pair of coupled nonidentical clamped–clamped micro-beams. The position of the coupling bridges from the anchor determines the coupling strength. (c) Microscope image of oscillators of length 38 and 40 μm illuminated by a single-laser spot

4  Bistability and Irregular Oscillations in Pairs of Opto-Thermal Micro-Oscillators

21

Fig. 4.2 (a) Schematic of the optical setup for driving the micro-oscillators opto-thermally into limit cycle oscillations. The half-wave plate and linear polarizers control the power and polarization of the laser light. The reflected light is collected in the photodetector and recorded using a spectrum analyzer. (b) SOI chip mounted on the chip holder prior to loading in the vacuum chamber

manifests as a bistable oscillation with the oscillators switching between the synchronized state and a quasi-periodic state of oscillations due to noise in the experiments, as shown in Fig. 4.4.

4.4 Numerical Analysis The pair of oscillators was numerically modeled using a third-order lumped-parameter model given by Eqs. (4.1)–(4.4) [4]: 

¨



z1  z1 / Q  1 1  CT1  z1   z13    z1  z2   DT1 , 

¨





T2   BT2  HPlaser    sin 2  2  z2  z   .



(4.1)

(4.2)



z2  z2 / Q   2 1  CT2  z2   z23    z2  z1   DT2 , 





T1   BT1  HPlaser    sin 2  2  z1  z   ,









(4.3) (4.4)

The displacement of the center of the oscillator is denoted by z(t) and the average temperature of the oscillator by T(t), where t denotes time. The model parameters are fixed as follows: quality factor Q = 1240 AU, linear stiffness values κ1 = 1 AU and κ2 = 0.81 AU, thermal coefficient for stiffness C = 2 × 10−2 1/K, nonlinear stiffness β = 15.5 AU, static displacement per unit change in temperature D = 2.84 × 10−3 1/K, thermal constants B = 0.112 AU and H = 6780 K/W, minimum laser absorption α = 0.035 AU, contrast in absorption γ = 0.011 AU, and equilibrium position of the oscillator with respect to the absorption curve z = 0.18 AU. All parameters except the thermal parameters are non-dimensionalized, and AU stands for arbitrary units. Above a certain threshold of laser power, the oscillators exhibit stable limit cycle oscillations. In the parameter sweep, the laser power Plaser is varied from 3.5 to 21.5 mW, and the linear coupling strength ζ is varied from 0.01 and 0.02 and is shown in Fig. 4.5. The model is numerically integrated, and the Fourier transform of the resulting time traces is computed. The numerical spectra are used to determine synchrony and quasi-periodic behavior of the oscillators. Twenty-­ five different initial conditions are used for every combination of laser power and coupling strength, and the probability of synchronization is plotted as a heat map. For the range of coupling strength shown, the oscillators are synchronized at low laser powers. As the laser power is increased, the system transitions to a state of bistability where the oscillators may exhibit either synchronous oscillations or quasi-periodic oscillations depending on the initial conditions. The numerical results showing the transition to a state of bistability at high laser powers for fixed coupling strength support the experimental results. The numerical model does not include noise terms, and thus, the switching behavior between the two states is not captured.

22

A. Bhaskar et al.

Fig. 4.3  A schematic of the experimental results of the response of the oscillators in the laser power vs. coupling strength parameter space. Responses denoted by (a), (e), (f), and (g) correspond to synchronized oscillations resulting in a single peak in the spectrum, and those denoted by (b), (c), (d), and (h) correspond to irregular oscillations resulting in a broadband spectrum

4.5 Conclusions The dynamics of a pair of coupled, nonidentical, opto-thermally driven micro-oscillators was analyzed experimentally and numerically. In the presence of fixed coupling, the oscillators transition from a state of certain synchrony to a state of bistability; coexistence of synchronized and quasi-periodic oscillations. In the experiments, the oscillators switch between these two states in the presence of noise, resulting in a broadband spectral response. In the numerical analysis, bistability is shown as a dependence of the response on the initial conditions.

4  Bistability and Irregular Oscillations in Pairs of Opto-Thermal Micro-Oscillators

23

Fig. 4.4  Experimental results showing irregular oscillations exhibited by oscillators switching between the synchronized state (top) and quasi-­ periodic state (bottom) of oscillations in the presence of noise Fig. 4.5  Probability of synchronization of a pair of oscillators numerically calculated in the laser power vs. coupling strength parameter space. For a fixed coupling strength, as the laser power is increased the oscillators transition from a state of synchrony to a state of bistability

Acknowledgments  This chapter is based upon the work supported by the National Science Foundation under Grant No. CMMI-1634664. This work was performed in part at the Cornell NanoScale Facility, a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation (Grant NNCI-2025233). This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. Specifically, it used the Bridges-2 system, which is supported by NSF award number ACI-1928147, at the Pittsburgh Supercomputing Center (PSC). This work made use of the Cornell Center for Materials Research Shared Facilities, which are supported through the NSF MRSEC program (DMR-1719875).

References 1. Shim, S.-B., Imboden, M., Mohanty, P.: Synchronized oscillation in coupled nanomechanical oscillators. Science. 316(5821), 95–99 (2007) 2. Matheny, M.H., Emenheiser, J., Fon, W., Chapman, A., Salova, A., Rohden, M., Li, J., de Badyn, M.H., Posfai, M., Duenas-Osorio, L., et al.: Exotic states in a simple network of nanoelectromechanical oscillators. Science. 363(6431) (2019) 3. Aubin, K., Zalalutdinov, M., Alan, T., Reichenbach, R.B., Rand, R., Zehnder, A., Parpia, J., Craighead, H.: Limit cycle oscillations in cw laser-­ driven NEMS. J. Microelectromech. Syst. 13(6), 1018–1026 (2004) 4. Zehnder, A.T., Rand, R.H., Krylov, S.: Locking of electrostatically coupled thermo-optically driven MEMS limit cycle oscillators. Int. J. Non-­ Lin. Mech. 102, 92–100 (2018)

Chapter 5

Tympanic Membrane Shape Measurement by Miniaturized High-­Speed Fringe Projection Shape Measurement Using MEMS Scanning Mirror Haimi Tang, John Rosowski, Cosme Furlong, and Jeffrey Tao Cheng

Abstract  Mammals possess an auditory system that carries out a number of functions that allow for sound perception from the surroundings. The hearing process is initiated when the tympanic membrane (TM) transfers acoustic waves from the air into mechanical vibration. The information pertaining to TM shape is essential to the expansion of our understanding of the middle ear. This will allow for better hearing protection, clinical diagnosis, and treatment of damaged or diseased ears. This chapter proposes a shape measurement approach with the use of a miniaturized fringe projection system having a scanning microelectromechanical systems (MEMS) mirror. This method utilizes a single fringe to scan the entire field of view while a series of images with spatially varied single fringe is captured by the high-speed camera. An individual multifringe image is then formed and reconstructed by summing up the single-fringe images numerically. Individually scanning with one fringe boosts optical efficiency and image quality during high-speed image capture. A National Institute of Standards and Technology (NIST)-traceable gauge helps validate shape measurements obtained using our methodologies. This shape measurement method is then applied to determine the shape of human eardrums. Keywords  Tympanic membrane (TM) · Fringe projection shape measurements · Microelectromechanical systems (MEMS)

5.1 Introduction The human ear is typically sectioned into three parts: outer, middle, and inner ears. The middle ear is an air-filled space separated from the outer ear by the tympanic membrane (TM), or eardrum. During the normal hearing process, sound travels in an acoustic wave in the air and enters the external auditory ear canal to vibrate the TM [1]. The vibration is then transmitted to the inner ear through the ossicles to trigger the sensory hair cells sending nerve signals received by the brain [2]. The thin semitransparent TM has unique anatomical and physical features critical to the hearing process [3, 4]. In particular, the adult human TM has an elliptical shape, measuring 9–10 mm horizontally and 8–9 mm vertically. The TM has characteristics of inhomogeneous, multilayered, and anisotropic [5] with a 2–3 mm depth flattened conical shape. Characterizing H. Tang (*) Center for Holographic Studies and Laser micro-mechaTronics, Worcester Polytechnic Institute, Worcester, MA, USA Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA Eaton-Peabody Laboratory, Massachusetts Eye and Ear, Boston, MA, USA e-mail: [email protected] J. Rosowski · J. T. Cheng Eaton-Peabody Laboratory, Massachusetts Eye and Ear, Boston, MA, USA Department of Otolaryngology-Head and Neck Surgery, Harvard Medical School, Boston, MA, USA C. Furlong Center for Holographic Studies and Laser micro-mechaTronics, Worcester Polytechnic Institute, Worcester, MA, USA Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA Eaton-Peabody Laboratory, Massachusetts Eye and Ear, Boston, MA, USA Department of Otolaryngology-Head and Neck Surgery, Harvard Medical School, Boston, MA, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_5

25

H. Tang et al.

26

the TM’s morphology is essential to understand middle-ear structure and functions in normal ears and pathological ears such as otitis media with effusion, traumatic TM perforation, and retraction pocket formation tympanosclerosis [6–17]. This chapter reports a new high-speed shape measurement approach based on fringe projection techniques to improve the optical efficiency and image quality during high-speed image acquisition for TM measurements. The shape measurement results using this method of a gauge traceable to the National Institute of Standards and Technology (NIST) are presented to show the accuracy of the fringe projection setup.

5.2 Method The fringe projection method projects well-defined fringe patterns onto the sample. The optical geometry of a general projection and imaging system is shown in Fig. 5.1. Points P and E represent the locations of projection and imaging optics, respectively. The phase (∆ϕ) and height h of the sample are related using the triangulation method [18] in Eq. (5.1): h  x ,y  

L L  AC  d Dd 2f

(5.1)

where L is the distance between the sensor and the reference plane; d is the distance between the sensor and projector; |AC| is the distance between points A and C; f is the spatial frequency of the projected fringes in the reference plane; and ∆ϕ is the phase of the projected fringe between points A and C. Thus, the height information h(x, y) is only a function of phase ∆ϕ, whereas the other parameters in Eq. 5.1 are fixed based on the optical configuration. The high-speed fringe projection (HSFP) system is configured to scan a single fringe across the sample at one camera exposure and take a sequence of high-speed images. Once a series of images of single-fringe projections of different locations on the sample have been recorded, they are numerically summed to form a fringe image with four-step phase-shifting for shape measurement. The high-speed camera captures the 4n single-fringe images sequentially: I11 , I12 , I13 , I14 , I 21 , I 22 , I 23 , I 24 … I n1 , I n2 , I n3 , I n4 .



The optical phase ∆ϕ is obtained by Eq. 5.2 [20, 21]:   arctan

Fig. 5.1  Optical geometry for fringe projection [19]

I I

4 1 1 1

 I 24  I n4    I12  I 22  I n2 

 I 21  I n1    I13  I 23  I n3 

(5.2)

5  Tympanic Membrane Shape Measurement by Miniaturized High-Speed Fringe Projection Shape Measurement Using MEMS…

27

5.3 Results The surface of a NIST-traceable gauge was analyzed using high-speed fringe projection (HSFP) to demonstrate its working principle. During image acquisition, a single fringe was projected and scanned across the sample’s surface by the scanning micro-electromechanical system (MEMS) mirror. Figure 5.2a shows a single projected fringe. The single-fringe images were grouped to form the four-step phase-shifted images based on the phase-shifting orders, as shown in Fig. 5.2b. In Fig. 5.2c, the optical phase of the sample’s shape was obtained from the four reconstructed phase-shifted images. The dimensions of a NIST gauge are shown in Fig. 5.3a. The optical phase and 3D reconstructed shape measured by the HSFP are shown in Fig. 5.3b. In Fig. 5.3b, the red line traces the height (in the z-axis) of the middle section of the sample, where the lowest middle segment of NIST gauge is aligned to zero (z = 0). The average measured height profiles of the two end segments of the NIST gauge estimated are 0.24 mm and 1.52 mm, respectively. The standard deviations (σ) for the three segments are 0.042 mm, 0.053 mm, and 0.113 mm. The averaged results agree with the dimensions provided by the manufacturer of 0.25 mm and 1.50 mm, respectively. Figure 5.3c is the optical phase and 3D reconstructed shape of a cadaveric human TM. A smoothing filter based on a moving kernel of 5 × 5 pixels size was applied, and the standard deviation between the raw and the smoothed shape result was computed. Figure 5.3c shows the wrapped optical phase of the human TM sample obtained by the HSFP and the smoothed 3D shape profile. The standard deviation between the raw and smoothed shapes obtained with HSFP is 0.11 mm, and the measurement was done by capturing 72 images in 25 ms.

5.4 Conclusion and Future Work This chapter presents a newly developed shape measurement method using a miniaturized fringe projection system with a microelectromechanical systems (MEMS) mirror. We showed results of the shape measurements of a NIST target and a cadaveric human TM using this method. The fringe projection method requires fewer captured images (less than 100), making it a more practical shape measurement for live animal study. However, the fringe projection method requires triangulation volume, posing challenges to measurements made through an intact ear canal. In the future, we will continue developing

Fig. 5.2  Fringe projection measurement of a flat surface. (a) Raw single-fringe images ( I qm ) captured by a high-speed camera, where m indicates the phase-shifting order and q indicates the fringe order. (b) Reconstruction of four-step phase-shifted images; each multifringe image is the summation of I qm of the same phase order. (c) Optical phase of the flat surface obtained by the four-step phase-shifting method

28

H. Tang et al.

Fig. 5.3  Representative shape measurement results: (a) NIST gauge dimensions; (b) reconstructed optical phase image and 3D shape measurement result of NIST gauge; (c) optical phase and 3D shape measurement of NIST of a cadaveric human TM sample

shape measurement to utilize the different optical methodologies for this application. The future implementation will be combined with the HSFP system with an endoscopic configuration to allow measurements of the eardrum shape and responses through the intact ear canal in live ears. Acknowledgments  Grant support from the US National Institute on Deafness and Other Communication Disorders (NIDCD R01 DC016079) is gratefully acknowledged. We would also like to acknowledge partial support by the Center for Holographic Studies and Laser micro-mechaTronics (CHSLT) at WPI. We also thank all of the reviewers of this chapter for their insightful suggestions and input.

5  Tympanic Membrane Shape Measurement by Miniaturized High-Speed Fringe Projection Shape Measurement Using MEMS…

29

References 1. Fastl, H., Zwicker, E.: Information processing in the auditory system. In: Fastl, H., Zwicker, E. (eds.) Psychoacoustics. Springer, Berlin, Heidelberg (2007). https://doi.org/10.1007/978-­3-­540-­68888-­4_3 2. Volandri, G., et  al.: Biomechanics of the tympanic membrane. J.  Biomech. 44, 1219–1236 (2011). https://doi.org/10.1016/j. jbiomech.2010.12.023 3. Rosowski, J.J.: Outer and middle ears. In: Fay, R.R., Popper, A.N. (eds.) Comparative Hearing: Mammals, pp. 172–247. Springer, New York (1994). https://doi.org/10.1007/978-­1-­4612-­2700-­7_6 4. Geisler, C.D.: From Sound to Synapse: Physiology of the Mammalian Ear. Oxford University Press, New York (1998) 5. Lim, D.J.: Structure and function of the tympanic membrane: a review. Acta Otorhinolaryngol. Belg. 49, 101–115 (1995) 6. De Greef, D., et al.: Details of human middle ear morphology based on micro-CT imaging of phosphotungstic acid stained samples. J. Morphol. 276, 1025–1046 (2015) 7. van der Jeught, S., et al.: Full-field thickness distribution of human tympanic membrane obtained with optical coherence tomography. J. Assoc. Res. Otolaryngol. 14, 483–494 (2013). https://doi.org/10.1007/s10162-­013-­0394-­z 8. Aernouts, J., Aerts, J.R.M., Dirckx, J.J.J.: Mechanical properties of human tympanic membrane in the quasi-static regime from in situ point indentation measurements. Hear. Res. 290, 45–54 (2012). https://doi.org/10.1016/j.heares.2012.05.001 9. Rosowski, J.J., et al.: Computer-assisted time-averaged holograms of the motion of the surface of the mammalian tympanic membrane with sound stimuli of 0.4–25 kHz. Hear. Res. 253, 83–96 (2009) 10. Rosowski, J.J.: Models of external- and middle-ear function. In: Hawkins, H.L., et  al. (eds.) Auditory Computation, pp.  15–61. Springer, New York (1996) 11. Lim, D.J.: Human tympanic membrane: an ultrastructural observation. Acta Otolaryngol. 70, 176–186 (1970) 12. Wang, X.L., et al.: Motion of tympanic membrane in Guinea pig otitis media model measured by scanning laser Doppler vibrometry. Hear. Res. 339, 184–194 (2016). https://doi.org/10.1016/j.heares.2016.07.015 13. Fay, J.P., Puria, S., Steele, C.R.: The discordant eardrum. Proc. Natl. Acad. Sci. U. S. A. 103, 19743–19748 (2006). https://doi.org/10.1073/ pnas.0603898104 14. Aernouts, J., et al.: Elastic characterization of membranes with a complex shape using point indentation measurements and inverse modelling. Int. J. Eng. Sci. 48, 6 (2010) 15. Cheng, T., et  al.: Viscoelastic properties of human tympanic membrane. Ann. Biomed. Eng. 35, 305–314 (2007). https://doi.org/10.1007/ s10439-­006-­9227-­0 16. Fay, J., et al.: Three approaches for estimating the elastic modulus of the tympanic membrane. J. Biomech. 38, 1807–1815 (2005) 17. Milazzo, M., et al.: The path of a click stimulus from ear canal to umbo. Hear. Res. 346, 1–13 (2017). https://doi.org/10.1016/j.heares.2017.01.007 18. Su, W.-H.: Color-encoded fringe projection for 3D shape measurements. Opt. Express. 15(20), 13167–13181 (2007) 19. Xu, Y., Jia, S., Bao, Q., Chen, H., Yang, J.: Recovery of absolute height from wrapped phase maps for fringe projection profilometry. Opt. Express. 22(14), 16819–16828 (2014) 20. Jones, R., Wykes, C.: Holographic and Speckle Interferometry, 2nd edn. Cambridge University Press, Cambridge (1989) 21. Kreis, T.: Handbook of Holographic Interferometry: Optical and Digital Methods. Wiley-VCH, Weinheim (2005)

Chapter 6

High-Speed Optical Extensometer for Uniaxial Kolsky Bar Experiments Richard Leonard III and Wilburn Whittington

Abstract  This work studies the implementation of a high-speed linescan camera as a 1D high-speed optical extensometer in gathering strain histories during dynamic strain rate experiments. Aluminum 6061-T6 is tested in tension and compression and recorded using both the high-speed extensometer and a high-speed camera, and the resulting images are analyzed for sample strain and compared to strains found via 1D wave theory calculations. The results show good alignment between the two optical methods with the strain gathered from the wave calculations showing slightly higher strains in both tension and compression. Keywords  Hopkinson bar · Kolsky bar · Dynamic experimentation · High-speed camera · DIC

6.1 Introduction and Background Many areas of experimentation rely on optical solutions to gather data. In the field of dynamic mechanical testing, high-­ speed and ultra-high-speed cameras allow for events to be recorded and analyzed. Kolsky bar experiments utilize these cameras or other optical methods such as laser extensometers or the LORD system to gather strain histories of experiments in a multitude of materials, stress states, and speeds [1–4]. In contrast to using optical methods, one-dimensional wave equations allow for strain histories to be calculated from stress waves during testing. It is known that in compression testing, localized flexure of the typically metal bars at the sample-bar interface causes these measurements to lose validity especially in the testing of brittle materials [5–7]. It is also known that certain gripping methods during tensile experiments can cause the calculated strain histories derived from wave equations to overestimate the strains present during testing [4, 8, 9]. In some visual applications such as quality inspection and industrial machine vision, users elect to use linescan cameras to sequentially record line images [10–13]. These cameras are traditionally used in automated applications where 2D images are generated from a subject passing by the camera. Linescan cameras have also been used for length measurements. Computational methods have even been presented to create subpixel resolution to the images found in these edge-finding applications [14]. This work presents a novel method of obtaining strain histories of tensile and compression Kolsky bar samples using a linescan camera as a 1D optical extensometer. The strain history results from the linescan extensometer are compared to strain data obtained via high-speed camera using digital image correlation to process the images.

6.2 Methodology This work uses an REL Kolsky bar to perform the tensile and compression testing. Each of the bars in the system was made from 0.75-inch-diameter C350 maraging steel and was instrumented using a quarter Wheatstone bridge. Two cameras were used during testing: a high-speed camera and a high-speed linescan camera. The high-speed camera recorded images of the

R. Leonard III (*) Standard Mechanics, LLC, Starkville, MS, USA e-mail: [email protected] W. Whittington Department of Mechanical Engineering, Mississippi State University, Starkville, MS, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_6

31

32

R. Leonard III and W. Whittington

Fig. 6.1  Tensile sample geometry

tests at 100,000 frames per second, and the high-speed linescan camera recorded at 200,000 lines per second. The linescan camera recorded 4080 pixels each being 7.5 um wide. Six tests were conducted on 6061-T6 samples: three in tension and three in compression. Figure 6.1 shows the tensile sample geometries used in testing. Each of the tensile samples was extracted from a 0.125-inch sheet using a water jet. Each of the compression samples was cut from a 0.5-inch-diameter rod into 0.5-inch-long cylinders. Each side of each sample was painted to suit the camera that was recording that side. Figure 6.2 shows the samples that are speckle-coated on one side for 2D DIC analysis and lined for 1D extensometer analysis. Figure 6.3 shows the experimental setup for a tensile test from the side of each camera. DIC analysis of test images was conducted using REL Surepulse software.

6.3 Experimental Results Each of the tensile and compression tests maintained a striker bar impact velocity of 34 fps and 38 fps, respectively. Figure 6.4 shows images gathered from the high-speed camera just before and just after a tensile and compression test, and Figs. 6.5 and 6.6 show the resulting lines gathered from the linescan camera throughout a tensile and compression test, respectively. The output of the linescan camera is shown in a 2D image. As time progresses and the camera records a new line, it is added beneath the previously recorded line. This means that the x-axis indicates position left or right and the y-axis indicates time starting at the top of each figure. In the tensile test shown in Fig. 6.4, there are five sections of each line: gray, white, black, white, and gray. The gray sections on the left and right are images taken from the grip of the tensile experiment. The white sections are the sample on the outside of the gage section, and the black section in the middle is the gage section. The initial sections are all vertical indicating there is no movement of the sample at that time. Then, each of the sections begins to slant left indicating the tensile stress wave reached the section being recorded. The angle of slant correlates to the movement speed of the section. The left grip and left part of the sample are shown moving at a higher speed compared to the right grip and right part of the sample. This shows the sample elongating. After a few more lines, the right side of the sample ceases angling and begins to be vertical again, and a new section of white and gray lines appears in the middle of the gage section. At this point, the sample has fractured, and the fracture surface (which is not painted) is shown in various shades of gray. After this point, each of sections to the left and the right of the fracture surface repeats periods of movement and periods of stagnation as the stress wave moves into and out of the recorded area. In the compression test in Fig. 6.6, the platens on the ends of the bars are shown in white and the sample is shown in black. Similar motion can be seen in this figure but with it moving from left to right instead of right to left, since the direction of testing is opposite. For each test, the recorded images were analyzed to extract the strain history of the tested sample. The raw stress wave data was also analyzed to produce strain history based on one-dimensional wave theory. Figure 6.7 shows the resulting strain vs time data as calculated via the three different methods. The strain histories for the DIC images and the extensometer are

6  High-Speed Optical Extensometer for Uniaxial Kolsky Bar Experiments

33

Fig. 6.2  Prepared tensile and compression samples

Fig. 6.3  Experimental setup for tensile testing

Fig. 6.4  First and last photos taken of a tension and compression test from the high-speed camera

in good agreement for both the tension and compression tests. The strains obtained from the wave equations were generally higher than both sets of optically found strains. For the tensile tests, this was most likely due to displacements of the sample outside the gage section, and for the compression tests, this was likely due to the incident bars slightly cupping around the sample during loading.

34

R. Leonard III and W. Whittington

Fig. 6.5  Lines recorded during tensile test from the linescan extensometer

Fig. 6.7  Strain histories for tensile (T) and compression (C) testing obtained via linescan extensometer, DIC, and wave calculations

Fig. 6.6  Lines recorded during compression test from linescan extensometer

6.4 Conclusion This work introduced a 1D high-speed optical extensometer that could be used in Kolsky bar experiments to record elongation and strains of samples during testing. The strain history results obtained from the extensometer were compared to two other methods of acquiring strain data during testing: high-speed camera with 2D DIC and 1D wave theory calculations. The results showed that the high-speed extensometer can produce strain data comparable to those obtained via 2D DIC.

6  High-Speed Optical Extensometer for Uniaxial Kolsky Bar Experiments

35

References 1. Rubino, V., Rosakis, A.J., Lapusta, N.: Full-field ultrahigh-speed quantification of dynamic shear ruptures using digital image correlation. Exp. Mech. 59(5), 551–582 (2019). https://doi.org/10.1007/s11340-­019-­00501-­7 2. Hijazi, A., Madhavan, V.: A novel ultra-high speed camera for digital image processing applications. Meas. Sci. Technol. 19(8), 085503 (2008). https://doi.org/10.1088/0957-­0233/19/8/085503 3. Haiting, S., Zhaoxiu, J., Beike, W., Chenghua, L., Lili, W., Yonggang, W.: Full field strain measurement in split Hopkinson tension bar experiments by using ultra-high-speed camera with digital image correlation. bzycj. 37(1), 15–20 (2017). https://doi.org/10.11883/1001-­1455(201 7)01-­0015-­06 4. Gilat, A., Schmidt, T.E., Walker, A.L.: Full field strain measurement in compression and tensile Split Hopkinson Bar experiments. Exp. Mech. 49(2), 291–302 (2009). https://doi.org/10.1007/s11340-­008-­9157-­x 5. Ravichandran, G., Subhash, G.: Critical appraisal of limiting strain rates for compression testing of ceramics in a Split Hopkinson pressure Bar. J. Am. Ceram. Soc. 77(1), 263–267 (1994). https://doi.org/10.1111/j.1151-­2916.1994.tb06987.x 6. Chen, W.W., Song, B.: Kolsky compression Bar experiments on brittle materials. In: Chen, W., Song, B. (eds.) Split Hopkinson (Kolsky) Bar: Design, testing and applications, pp. 77–118. Springer, Boston, MA (2011). https://doi.org/10.1007/978-­1-­4419-­7982-­7_3 7. Foster, J.T.: Comments on the validity of test conditions for Kolsky Bar testing of elastic-brittle materials. Exp. Mech. 52(9), 1559–1563 (2012). https://doi.org/10.1007/s11340-­012-­9592-­6 8. Qiu, Y., Loeffler, C.M., Nie, X., Song, B.: Improved experimental and diagnostic techniques for dynamic tensile stress–strain measurement with a Kolsky tension bar. Meas. Sci. Technol. 29(7), 075201 (2018). https://doi.org/10.1088/1361-­6501/aabc9f 9. Li, Y., Ramesh, K.T.: An optical technique for measurement of material properties in the tension Kolsky bar. Int. J. Imp. Eng. 34(4), 784–798 (2007). https://doi.org/10.1016/j.ijimpeng.2005.12.002 10. Yu, S.-N., Jang, J.-H., Han, C.-S.: Auto inspection system using a mobile robot for detecting concrete cracks in a tunnel. Autom. Constr. 16(3), 255–261 (2007). https://doi.org/10.1016/j.autcon.2006.05.003 11. Gong, Q., Zhu, L., Wang, Y., Yu, Z.: Automatic subway tunnel crack detection system based on line scan camera. Struct. Control. Health Monit. 28(8), e2776 (2021). https://doi.org/10.1002/stc.2776 12. Lemstrom, G.F.: True RGB linescan camera for color machine vision applications, pp.  494–502. Boston, MA (1994). https://doi. org/10.1117/12.188921 13. De Grauw, C.J., Otto, C., Greve, J.: Linescan Raman microspectrometry for biological applications. Appl. Spectrosc. 51(11), 1607–1612 (1997). https://doi.org/10.1366/0003702971939587 14. Li, Y.-S., Young, T.Y., Magerl, J.A.: Subpixel edge detection and estimation with a microprocessor-controlled line scan camera. IEEE Trans. Ind. Electron. 35(1), 105–112 (1988). https://doi.org/10.1109/41.3072

Chapter 7

On the Miura Ori Modal Response: A Look Throughout the Experimental Side Antonio Baldi, Pietro Maria Santucci, Giorgio Carta, Michele Brun, Gianluca Marongiu, and Daniele Lai

Abstract  The Miura Ori auxetic geometry has been deeply studied for its capability to reach high negative Poisson coefficient values. This response is associated with some geometrical configuration, and there are several examples in specific literature. Miura Ori metamaterial has been analyzed with static loads, while from the dynamical side, the studies have been conducted with analytical/numerical techniques. The authors analyze this subject and propose an experimental setup (the most suitable technique appears to be an interferometric one, i.e., the time-averaged speckle approach) to get the modal response of a Miura Ori specimen. The results are validated by comparison with FEA simulation to state the reliability of experimental results. Keywords  Miura Ori geometry · Auxetic materials · Interferometry · Speckle technique · Modal analysis

7.1 Introduction In the last 30 years, metamaterials have become very popular in solid mechanics. Metamaterials’ properties depend on their geometrical configuration rather than their chemical composition. Thus, auxetic metamaterials, which show a negative global Poisson’s ratio [1–3], have been deeply studied. Auxetic materials present some challenging aspects for their enhanced mechanical property, such as a higher indentation resilience [2] [4], augmented fracture toughness [5], and better response to fatigue loads [6]. Hence, the curvature of composite sandwiches that use an auxetic core becomes synclastic rather than anticlastic as in standard materials [1–3]. Another crucial aspect of auxetic materials is the energy absorption at a high strain rate [7]. In nature, there are several examples of auxetic behavior, from crystals [8, 9] to some skins [3, 9, 10]. In engineering structures, the auxetic behavior can be reached through a pattern of linear [11] or particularly shaped [12] cuts. Folded materials like the Miura Ori origami [13] are theoretically obtained by folding a paper sheet (Fig. 7.1). The basic idea is that the edges act like hinges while the thin facets are considered rigid. Thus, cell deformation is provided by the relative rotation of each facet around the edge’s axis. The main geometrical characteristics of the Miura Ori cell are the edges length, l1and l2 (Fig. 7.1), and the folding angles, α1 and α2. The work of Schenk et al. [14] provides several parameters set to obtain an auxetic behavior with a kinematic approach (i.e., considering each facet of the cell as a mechanism). The employment of Miura Ori covers several engineering applications, from energy absorption [15] to electronic circuits [16, 17]. Hence, Miura Ori's dynamic properties are investigated with conventional technique in [18] while the work of Patrapa et al. analyzed its acoustical property [19]. The modal response of Miura Ori geometry using a time-averaged speckle technique is studied in this chapter. The experimental results are compared with FE numerical simulation to validate the data. The following sections provide first an overview of the manufacturing process, then a description of the experimental technique and experimental setup, and lastly, a discussion about the results.

A. Baldi (*) · P. M. Santucci · G. Carta · M. Brun · G. Marongiu · D. Lai Department of Mechanical, Chemical and Materials Engineering, Università degli Studi di Cagliari, Cagliari, Italy e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_7

37

38

l

1

A. Baldi et al.

1

2

l2

Fig. 7.1  Miura Ori unit cell in local reference system

7.2 Background Manufacturing Miura Ori specimen is challenging because of its sharp edges and narrow peak-valley structure. This problem has been solved by the Cold Gas Pressure Folding technique [14, 20]. In this chapter, the authors opted for the 3D printing of specimens with FFF technology. The printer is a Prusa i3 MK2s, and the material used for specimen fabrication is polylactic acid (PLA). PLA has a Young’s modulus of 2400 ± 100 MPa [21] and a Poisson’s coefficient of 0.35. Figure 7.2 shows the 3D printing process of a Miura sample. The specimen is designed upon a Miura Ori unit cell with equal edges length (l1 = l2 = 10 mm) and folding angles α1 = 50° and α2 = 125°. Thus, the unit cell is arranged in a 7 × 9 periodic array. Figure 7.3 illustrates the unit cell packaging in a global view of the specimen. Specimen extremities are designed to fit the grip zone on the vice. The experimental technique applied to evaluate the modal response of Miura Ori is the time-averaged speckle interferometry [22–26]. Figure 7.4 shows a schematic of interferometric setup designed to obtain a quasi-out-of-plane sensitivity (the angle between CCD axis and illumination direction is about 6.4 deg). The laser source is an He–Ne laser with an output of 50 mW and a wavelength of 632.8 nm. The CCD is an Allied Vision Pike F421b mono with a resolution of 2048 × 2048 pixels, equipped with a UNIFOC 12 Schneider lens. The lens L is positioned to have the focus plane on the camera lens iris. The Hemo self-centering vice is modified by introducing a PZT stack actuator (Thorlabs PZS001 [27]) as reported in Fig. 7.5. The stack actuator is used for exciting the specimen in the horizontal direction. PZT actuator excites one grip while the other is at rest. The sample has a thickness of about 0.34 mm, and its surface is sprayed with a thin layer of white acrylic paint to prevent cells’ transparency. The intensity field I on the object at a given time t can be expressed as [23] I  x,y,t   I r  I o  2 I r I o cos     



(7.1)



where –– Ir, Io are, respectively, the reference and object wave –– ϕ = ϕ(x, y) is phase field (unknown), –– ∆ = ∆ ( x, y,t ) =

2π 1 + cos ( θ )  Acos ( ωt ) λ

in which θ is the illumination angle, ω is the oscillation frequency, and A is the

oscillation amplitude. The voltage CCD response to the intensity field input described in (7.1) can be written as [23] V CCD  G

t0  

I

t0

r

 I o  2 I r I o cos     dt

(7.2)

where τ is the shutter time while G accounts for the CCD internal gain and sensitivity to the given wavelength [23]. The main 2 hypothesis of the time-averaged technique is that the shutter time interval must contain several oscillation periods ( T  )  [26]. Thus,

39

7  On the Miura Ori Modal Response: A Look Throughout the Experimental Side

Fig. 7.2  3D printer Prusa i3 MK2 while printing a Miura Ori specimen

Fig. 7.3  Miura Ori specimen with a periodic array of 7 × 9 cells

MM

Self Centering Vice

BE 2 L CCD LS

Ir

Io CCD Axis

BS 2 BS 1

BE 1 Specimen

Fig. 7.4  Schematic of interferometric setup. LS, laser source; BS#, beam splitters; BE#, beam expander; MM, mobile mirror; L, lens; Ir, reference wave; Io, object wave

40

A. Baldi et al.

Fig. 7.5  Hemo self-centering vice with a focus on PZT exciter

  nT  



(7.3)

Equation (7.3) splits the contribution to shutter time into an integer number of vibration periods and a remainder η (i.e., n is the integer number of periods contained in the shutter aperture time). Substituting Eq. (7.3) into Eq. (7.2) leads to the definition of the output of CCD as [23] 2 nπ  I r + I o + 2 I r I o J 0 ( k ) cos ( φ )  + ω      ksin (ωη ) k 2 sin ( 2ωη) +G η  I r + I o + 2 I r I o cos ( φ ) − sin ( φ ) − cos ( φ ) +…  ωη ωη    

V CCD G =



(7.4)

where



k

2 A 1  cos     



J0 is the Bessel function of order 0 The second term of Eq. (7.4) becomes progressively smaller when the object’s oscillation frequency increases because it is scaled by η. Thus, we consider the first term only [23]. In addition, the latter is controlled by two terms: k and J0. The former is related to the amplitude of the oscillations, which becomes smaller as the frequency increase, leading to a less sharp fringe field, while the latter modulates the fringe intensity field. In addition, the zeroth-order Bessel function is unitary at nodes (i.e., where vibration amplitude A is zero) and decreases steeply when the vibration amplitude (i.e., the phase) increases [23]. Post-processing of time-averaged analysis focuses on the modulation instead of the phase field. Under the hypothesis of uniform intensity field, Eq. (7.4) suggests that points with the same modulation have the same vibration amplitude. This consideration has a significant impact concerning standard postprocessing because it avoids the usage of a reference phase field and the computation of phase modulo 2π field [23]. The phase-shifting algorithm used for analysis is Hariharan’s fivestep algorithm [26].

7.3 Analysis The experimental campaign consists of testing the Miura Ori specimen at several excitation frequencies. Eigenmodes and eigenfrequencies were calculated before via FE simulations to optimize the search of eigenmodes. A PZT sensor is bonded to the specimen to analyze the response and aid to find resonance conditions. The shutter time is set to 50 ms.

7  On the Miura Ori Modal Response: A Look Throughout the Experimental Side

41

Fig. 7.6  Comparison between numerical and experimental eigenmodes. Note the PZT sensor on the bottom of the specimen 650

Experimental Eigenfrequencies [Hz]

600 550 500 450 400 350 300 250 200 150 100 100

150

200

250

300

350

400

450

500

550

600

Numerical Eigenfrequencies [Hz]

Fig. 7.7  Trend of experimentally observed eigenfrequencies with respect of numerical ones

Figure 7.6 shows a comparison between four experimental and numerical eigenmodes. Experimental data report an asymmetric distribution of fringes, probably due to specimen self-weight (actually, specimen is mounted horizontally, as shown in Fig. 7.5). Nevertheless, results obtained by time-averaged speckle match the numerical eigenmodes quite well.

42

A. Baldi et al.

Figure 7.7 provides a brief comparison between the numerical and experimental eigenfrequencies. The small deviation from the theoretical unitary slope is probably due to the paint sprayed on specimen surface causing a mass increase. In addition, 3D printing introduces some anisotropic effects caused by filament deposition direction and layer contact.

7.4 Conclusion This chapter furnishes an experimental assessment of Miura Ori’s modal response that has been mainly studied from the theoretical/numerical side. The time-averaged speckle interferometry provides results in tight accordance with the numerical simulation. The paint on the specimen and the anisotropy introduced by the printing process might produce minor differences between numerical and experimental data. All the results refer to out-­of-­plane modes because of the geometrical setup configuration.

References 1. Alderson, A., Alderson, K.L.: Auxetic materials. Proc. Inst. Mech. Eng. Part G J. Aeros. Eng. 221(4), 565–575 (2007). https://doi.org/10.124 3/09544100JAERO185 2. Evans, K.E., Alderson, A.: Auxetic Materials: Functional Materials and Structures from Lateral Thinking! vol. 12, p. 617. Adv. Mater. (2000) 3. Evans, K.E., Alderson, K.L.: Auxetic Materials: the Positive Side of Being Negative, vol. 9, p. 148. Eng. Sci. Educ. J. (2000) 4. Chan, N., Evans, K.E.: Indentation resilience of conventional and Auxetic foams. J. Cell. Plast. 34(3), 231–260 (1998). https://doi.org/10.117 7/0021955X9803400304 5. Choi, J.B., Lakes, R.S.: Fracture toughness of re-entrant foam materials with a negative Poisson’s ratio: experiment and analysis. Int. J. Fract. 80(1), 73–83 (1996). https://doi.org/10.1007/BF00036481 6. Francesconi, L., Baldi, A., Dominguez, G., Taylor, M.: An investigation of the enhanced fatigue performance of low-porosity Auxetic metamaterials. Exp. Mech. 60(1), 93–107 (Jan. 2020). https://doi.org/10.1007/s11340-­019-­00539-­7 7. Scarpa, F., Ciffo, L.G., Yates, J.R.: Dynamic properties of high structural integrity auxetic open cell foam. Smart Mater. Struct. 13(1), 49–56 (Feb. 2004). https://doi.org/10.1088/0964-­1726/13/1/006 8. Baughman, R.H., Shacklette, J.M., Zakhidov, A.A., Stafströ, S.: Negative Poisson’s Ratios as a Common Feature of Cubic Metals, vol. 392, p. 362 (1998) 9. Rao, N.R., et al.: Do Zeolites Have Negative Poisson’s Ratios?**. Wiley (2000) 10. Lees, C., Vincent, J.F.V., Hillerton, J.E.: Poisson’s Ratio in Skin. Biomed. Mater. Eng. 1, 19–23 (1991). https://doi.org/10.3233/BME-­1991-­1104 11. Carta, G., Brun, M., Baldi, A.: Design of a porous material with isotropic negative Poisson’s ratio. Mech. Mater. 97, 67–75 (2016). https://doi. org/10.1016/j.mechmat.2016.02.012 12. Francesconi, L., Taylor, M., Bertoldi, K., Baldi, A.: Static and modal analysis of low porosity thin metallic Auxetic structures using speckle interferometry and digital image correlation. Exp. Mech. 58(2), 283–300 (2018). https://doi.org/10.1007/s11340-­017-­0345-­4 13. Miura, K.: Method of Packaging and Deployment of Large Membranes in Space The Institute of Space and Astronautical Science report, no. 618, pp. 1–9 (1985) [Online]. Available: https://ci.nii.ac.jp/naid/120006832687/en/ 14. Schenk, M., Guest, S.D.: Geometry of Miura-folded metamaterials. Proc. Natl. Acad. Sci. U. S. A. 110(9), 3276–3281 (Feb. 2013). https://doi. org/10.1073/pnas.1217998110 15. Tolman, S.S., Delimont, I.L., Howell, L.L., Fullwood, D.T.: Material selection for elastic energy absorption in origami-inspired compliant corrugations. Smart Mater. Struct. 23(9) (2014). https://doi.org/10.1088/0964-­1726/23/9/094010 16. Y. Li, W. Liu, Y. Deng, W. Hong, and H. Yu, “Miura-ori enabled stretchable circuit boards,” npj Flex. Electron., vol. 5, no. 1, 2021, doi: https:// doi.org/10.1038/s41528-­021-­00099-­8 17. Y. Hou, Z. Li, Z. Wang, and H. Yu, “Miura-ori structured flexible microneedle array electrode for biosignal recording,” Microsyst. Nanoeng., vol. 7, no. 1, 2021, doi: https://doi.org/10.1038/s41378-­021-­00259-­w 18. Liu, B., Sun, Y.: Modal response of carbon-fiber-reinforced Miura-ori core sandwich panels. Mech. Adv. Mater. Struct. 27(5), 364–372 (2020). https://doi.org/10.1080/15376494.2018.1473536 19. Pratapa, P.P., Liu, K., Paulino, G.H.: Geometric mechanics of origami patterns exhibiting Poisson’s ratio switch by breaking mountain and valley assignment. Physic. Rev. Lett. 122(15) (2019). https://doi.org/10.1103/PhysRevLett.122.155501 20. M. Schenk and S. D. Guest, Geometry of Miura-Folded Meta-Materials-Supplementary Information 21. Prusament PLA by Prusa Polymers. https://prusament.com/media/2018/07/PLA_TechSheet_ENG22052020.pdf. Accessed 21 Feb 2022 22. Romero, G., Alvarez, L., Alan, E., Nallim, L., Grossi, R.: Study of a Vibrating Plate: Comparison between Experimental (ESPI) and Analytical Results, vol. 40, p. 81 (2003) 23. A. Baldi, P. Jaquot, and F. Ginesu, “Una applicazione della tecnica speckle all’analisi dinamica di uno statore per piezo-motors” 2004 24. Joenathan, C.: Vibration fringes by phase stepping on an electronic speckle pattern interferometer: an analysis. Appl. Opt. 30(32), 4658–4665 (Nov. 1991). https://doi.org/10.1364/AO.30.004658 25. Tiziani, H.J., Klenk, J.: Vibration analysis by speckle techniques in real time. Appl. Opt. 20(8), 1467–1470 (1981). https://doi.org/10.1364/ AO.20.001467 26. Robinson, D.W., Reid, G.T.: Interferogram Analysis, Digital Fringe Pattern Measurement Techniques. CRC Press (1993) 27. PZS001-Manual. [Online]. Available: https://www.thorlabs.com/thorproduct.cfm?partnumber=PZS001. Accessed 22 Feb 2022

Chapter 8

Using Digital Image Correlation to Characterize the Static and Dynamic Behavior of Structures: Industrial Applications and Lessons Learned Simone Manzato, Davide Mastrodicasa, Emilio Di Lorenzo, Guven Ogus, and Pascal Lava Abstract  Digital image correlation has recently seen a growing interest in both the research and industrial community thanks to the possibility to measure full-field information, with high confidence, and with a very limited instrumentation. Furthermore, advances in camera technology, particularly on resolution and data transfer rate, are now opening the door to new application, such as modal analysis and testing on rotating components. In this chapter, we give an overview of industrial applications of digital image correlation, ranging from the more classical characterization of material samples up to modal analysis and dynamic characterization of several components in stationary as well as operating conditions and show how the same instrumentation can be reused to cover multiple scenarios at the same time, without the need for changing instrumentation or data acquisition as is the case for other experimental techniques. We also show how DIC is used to characterize the static behavior of lattice structures, characterize both statically and dynamically mechanical components to validate numerical models, extract the modal behavior of rotating components, such as tires and fans, and finally understand the behavior of huge machines where the size and stiffness pose great limitations to the use of optical techniques. Keywords  DIC · Static testing · Modal · Optical methods

8.1 Introduction Digital image correlation (DIC), in combination with digital cameras, is gaining a lot of interest in the experimental mechanics community. The steady increase in performance of digital cameras, as well as the development of more performant image processing algorithm which can be run on standard computers, is making this technique more and more accessible and expanding its range of applications [1, 2]. While the technique has been originally developed to study in-plane deformations of material coupons under static loading, it can nowadays be applied to a much wider range of applications, including the analysis of lattice structures, study of vibration response of rotating structures, such as fans and tires, and even to extract modes on an F16 during a GVT measurement campaign. While DIC is a very powerful and flexible solution, it has also some limitations, the first being the fact that it measures displacements, so it is always important to verify the actual applicability of DIC before embarking on an experiment. In this chapter, we will report some applications of DIC to both static and dynamic testing on challenging situations, and we will critically review the results and actual applicability of the methods to these scenarios. In particular, we first will show how DIC was successfully used to measure the structural response of a lattice structure cell with a relatively complex geometry for model validation. Second, DIC was used to identify the modal behavior of a scaled clamped-free wind turbine blade and validate a numerical model. In the third application, DIC is used to extract the vibration of a PC fan while rotating. Finally, the use of DIC to monitor deformations and extract modes on a big CNC machine is presented.

S. Manzato (*) · D. Mastrodicasa · E. Di Lorenzo · G. Ogus Siemens Digital Industries Software, Simulation and Testing Solution, Leuven, Belgium e-mail: [email protected] P. Lava MatchID NV, Ghent, Belgium © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_8

43

44

S. Manzato et al.

Fig. 8.1  Compression test on lattice structure (left) with the reconstructed shape (center) and the full-field strain (right)

8.2 Lattice Structure Characterization Historically, DIC has been, and still is, extensively applied to characterize the mechanical properties of material, particularly in those situations where the sensors embedded in the testing machine or traditional strain gauges are not sufficient. With modern manufacturing technology, such as additive manufacturing, not only the material properties are important, but also the way lattice structures behave as individual cell as well as when joined. In this example, together with the Additive Manufacturing Research group of the KU Leuven, we explored the use of DIC to get local deformation and strain on a back-­ centered cubic (BCC) cell of 30 × 30 × 30 millimeters under compressive loading. The cell was placed on a uniaxial loading machine and hand-speckled using a black marker. Two 5 megapixel cameras with 25 mm lenses were used to maximally fit the cell in the field of view and maximize the pixel size, resulting in approximately 0.003-millimeter in-plane resolution. Despite the complex shape of the cell and the difficulties in uniformly speckling the surface, both the geometry of the object and the strain field during the test could be extracted and are now being used to compare with simulation model predictions (Fig. 8.1).

8.3 Scaled Wind Turbine Blade Model Validation In a recent paper [3], the use of DIC to identify the modes of a scaled wind turbine blade and use the results to validate a numerical model has been presented and the results are here summarized. The blade, shown in Fig. 8.2, has been instrumented on the back with accelerometers for comparison. The under-sampling technique, described in the paper, has been used here to capture the dynamic response up to 400 Hz, while acquiring images at only 50 frames per second. By aligning the measured response with the excitation signal, DIC frequency response could be derived and used for modal analysis. The plot at the center of Fig. 8.2 shows the overlaid FRF complex sum for the accelerometer and DIC FRFs, which shows that the under-sampling techniques in combination with DIC provide one such structure as a reliable alternative to traditional accelerometer testing. The MAC matrix in Fig. 8.2 (right) compares the DIC modes with those obtained from a finite element model of the blade after model updating has been performed to improve material properties and boundary conditions to better match the experimental data. DIC offers a very local characterization of the structure close to the clamping area, which can then be used to update the numerical model. With traditional sensors, this information would not typically be available, and additionally the additional mass of the sensor would need to be added to the model before performing the correlation.

8  Using Digital Image Correlation to Characterize the Static and Dynamic Behavior of Structures: Industrial Applications…

45

Fig. 8.2  Scaled wind turbine blade setup (left) with comparison of accelerometer and DIC FRFs (center) and MAC matrix between the DIC modes and those from the finite element model

8.4 Rotating fan Operational Vibrations One of the most interesting applications of DIC in the structural dynamics community is the possibility to extract the dynamic behavior of rotating components in operating conditions. This is extremely challenging with other techniques because of cabling in case of traditional sensor or because of the need of synchronizing the scanning laser with the rotational speed of the component in case of laser Doppler vibrometry. In case of DIC, however, we only need to ensure that the scene is sufficiently bright so that the exposure time can be kept to a minimum and avoid motion blur. When processing the images, a special approach has been developed; while traditional methods always correlate the reference image to the reference one, in case of rotating structure this would cause some problem as the subset are rotating with the risk of losing the correlations. On the other hand, updating continuously the reference image would cause errors to propagate and increase continuously. Our solution offers a dedicated algorithm to solve these problems by monitoring the correlation and error propagation and only updates the reference image when required. To test the algorithm, a PC cooling fan has been speckled and damaged at the root of one of the blades with a small indent. The analysis focused on the damaged blade, where it was possible to extract the strain at the root, but also to analyze the frequency content and extract some of the operational deflection shapes up to approximately 350 Hz, as shown in Fig. 8.3.

8.5 CNC Machine Deformation Measurement The final application is the measurement of static and dynamic deformations on the arm and tool tip of a CNC machine by Soraluce, which was done in collaboration with the IK4 Ideko Research Center. Static and dynamic measurements on these machines are performed by installing several inductive displacement probes at critical locations to monitor the static deformation under loads as well as accelerometers to analyze the dynamic and modal response that are then used to improve control strategies and validate finite element models. The tested item is shown in Fig. 8.4. The customer wanted in particular to understand whether DIC could be used as a more flexible, yet equally accurate, alternative to traditional sensing techniques. A static test was then devised where the tip of the machine arm was pushed against a steel block (the motors of the arm are behind the wall on the right of the picture). Different elements of the machines were speckled using sticky-back paper, and the cameras were positioned inside the chamber with wide lenses to capture the complete region of interest of approximately 1 m × 0.5 m. The results are shown in Fig. 8.5. The load was progressively increased from 0 to 4660 N, with two proximity probes measuring the displacement close to the tip and in the middle of the arm. With this particular setup, the accuracy was estimated to be around 0.03  mm, which proved to be sufficient to capture the deformation distribution, as shown in the

46

S. Manzato et al.

Fig. 8.3  PC fan rotating experiment showing the geometry of the area of interest (left), the Eyy strain component during rotation (center), and torsional mode of the damaged blade at 165 Hz (right)

Fig. 8.4  Test setup in the Soraluce machine as well as overview of the region of interests

comparison with the probe data. However, because the actual deformations are relatively close to the noise floor of the setup, the resulting time traces are extremely noisy. The accuracy could be improved by reducing the size of the area of interest, but that would take away the advantage of DIC or capturing full-field information. Besides, this is only a small part of the machine, with the control tower behind the wall also of interest for the measurement. Again, using a multicam setup could be a solution, but the setup time and complexity would also increase.

8.6 Conclusion In this chapter, we have shown how digital image correlation could be used to measure deformation and strain, under static and dynamic loadings, and for objects of very different sizes. It is also important to stress that the same pair of 5 megapixels camera has been used in all experiments, with 12.5 or 25 mm lenses depending on the distance from the object and the size of the region of interest. To increase the measurable frequency bandwidth, the under-sampling algorithm has been used.

8  Using Digital Image Correlation to Characterize the Static and Dynamic Behavior of Structures: Industrial Applications…

47

Fig. 8.5  Out-of-plane deformation at max load and comparison between measured responses from DIC and the proximity probes at the highlighted locations

For the lattice cell and rotating fan, DIC could provide information that would have been extremely difficult to measure with traditional sensors. In the case of the wind turbine blade, DIC provides comparable accuracy but with a significantly higher spatial resolution and without the risk of mass loading the structure. In the final case, while DIC proved to be able to measure the required displacement level, the size of the region of interest limited the accuracy and thus results in very nosy data. In this case, traditional sensors, despite the lengthy instrumentation time and limited spatial resolution, still provide a more flexible and accurate solution than DIC. Acknowledgments  The authors gratefully acknowledge the Flanders Innovation & Entrepreneurship for its support of the Baekeland project “Digital Image Correlation for Structural Dynamics Full-field Analysis” [nr/HBC.2019.2595].

References 1. Zanarini, A.: Full-filed optical measurements in experimental modal analysis and model updating. J. Sound Vib. 442, 817–842 (2019) 2. Witt, B.L., Rohe, D.P.: Digital image correlation as an experimental modal analysis capability. Exp. Tech. 45(3), 273–286 (2021) 3. Mastrodicasa, D., Ferreira, C., Di Lorenzo, E., Peeters, B., Pires Vaz, M.A., Guillaume, P.: DIC Using Low Speed Cameras on a Scaled Wind Turbine Blade. In: Proceedings of the 40th International Modal Analysis Conference (IMAC). Orlando, Florida (2022)

Chapter 9

Enabling Digital Image Correlation with High-Resolution Microscopic Optics via Working Distance Automation: Advancing Resolution and Accuracy Limits Olcay Türkoğlu and C. Can Aydıner

Abstract  Optical microscopy (OM) implementation of the digital image correlation (DIC) technique stands out with high practicality (e.g., no vacuum requirement and high amenability to be combined with other measurement channels). Despite its limited intragrain resolution compared to scanning electron microscopy (SEM) variants, OM-DIC provides critical identification at the interaction length scale of polycrystalline aggregates. The OM-DIC variant that is considered here (Shafaghi N, Kapan E, Aydıner CC, Exp Mech 60:735–751, 2020; Özdür NA, Üçel IB, Yang J, Aydıner CC, Exp Mech 61:499, 2020), however, further attempts to minimize the comparative intragrain resolution deficiency of OM-DIC by utilizing high-­ resolution [high numerical aperture (NA)] objectives. Images with high-NA objectives will typically immediately suffer defocusing for a deforming sample, given the extremely limited depth of fields (in the μm order). The technique employs continual automated working distance (WD) adjustment to fight off defocusing through custom instrumentation that also implements area scanning to expand field coverage to the mm scale. The precise WD adjustments also help to minimize WD error (a biaxial strain error in DIC measurements). While the technique has been formerly used to study highly strained polycrystalline fields (Shafaghi N, Kapan E, Aydıner CC, Exp Mech 60:735–751, 2020; Özdür NA, Üçel IB, Yang J, Aydıner CC, Exp Mech 61:499, 2020), the purpose of this study is to investigate and advance its accuracy limits. For this purpose, 40× microscopy with a high-NA objective (Özdür NA, Üçel IB, Yang J, Aydıner CC, Exp Mech 61:499, 2020) is utilized over a pure FCC nickel polycrystal with enlarged grains (average 70  μm), yielding about 5000 DIC grid points per grain. Regardless of the length scale, however, DIC offers limited sensitivity for pointwise strains (about 0.1%). While this is deemed sufficient for plasticity, elastic strains are not locally resolved. Here, we will employ small load increments in the initial elastic ramp of nickel to test the sensitivity limits of the method in a grain-resolved setting, i.e., the strain fields of individual grains are separately considered. The crystallographic orientations will be known thanks to pre-experiment electron backscatter diffraction. The trade-off between accuracy and resolution will be tested by local averages inside the grains to see whether regional DIC accuracy can be pushed toward elastic strain levels. Results of multiple grains will be compared and consistency with finite element predictions that account for the crystallite orientations will be presented. Keywords  DIC · Polycrystal · Microscopy · High resolution

9.1 Introduction Digital image correlation (DIC) has become the prominent strain measurement technique in recent years both in industry and academia. The method offers very high spatial resolution and does not have an innate length scale, the latter leading to its scientific application with all forms of microscopy [1–4]. When it comes to the accuracy of its pointwise strain output, however, DIC is typically mediocre, with uncertainties on the order of 0.1%. This is 2–3 orders inferior to a careful electrical resistance strain gauge application (e.g., that can reveal minute residual stresses [5]). The DIC strain accuracy, however, can be improved by local averaging (effectively increasing the radius of the region where strain is calculated) at the cost of spatial resolution. The comparison with a strain gauge can be reevaluated in this contest since a strain gauge also innately averages over the material patch under its resistance element (typically of several millimeters size). A high-resolution DIC application can be fathomed over the same material patch such that the DIC field average nears the strain gauge accuracy. O. Türkoğlu · C. C. Aydıner (*) Department of Mechanical Engineering, Boğaziçi University, Istanbul, Turkey e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_9

49

50

O. Türkoğlu and C. C. Aydıner

In terms of calculated DIC strain fields, one can produce multiple plies that represent different trade-off points between accuracy and resolution, e.g., a ply of noisy strain results with the highest spatial resolution and a ply of high-accuracy local averages that are smeared out spatially. A consideration of these layers is particularly interesting for the optical microscopy application of digital image correlation (OM-DIC) over polycrystalline aggregates [6, 7]. These most typical engineering materials are extremely complex composites at the crystallite scale, and the intricacy of local strain fields (now mapped with microscopic applications of DIC) with patterning and sharp localizations has been typically ignored. Conversely, however, a new generation of full-field models [8, 9] that will account for microstructure in mechanical behavior (rather than the usual practice of relying on further experimentation for each variation in microstructure) precisely needs such measurements to achieve high fidelity at the material length scales. The variant of OM-DIC considered here [1, 2, 10] attempts to get the highest resolution offered by light microscopy, utilizing high-numerical aperture (NA) optics. This is difficult since images with high-NA objectives will typically immediately suffer defocusing [11] for a deforming sample, given the extremely limited depth of fields (in the μm order). This OM-DIC variant employs working distance (WD) adjustment to fight off defocusing through custom automation that also implements area scanning to expand field coverage to the mm scale. In this method, the spatial resolution of the strain fields is pushed toward the 1–2 μm scale [2] with acceptable (around 0.1%) strain uncertainty, the latter clearly contingent upon optimal patterning. This resolution nears the coarse side of the resolution range that is achieved by DIC with scanning electron microscopy (SEM-DIC). Thus, with OM-DIC with WD automation, model-­ validating data sets with adequate intragrain resolution can be generated with the speed, convenience, and practicality of the optical measurement. For polycrystals, a scientifically valuable target in combining the idea of DIC accuracy advancement (by sacrificing resolution) is reaching the sensitivity to ascertain elastic strains. Each grain in a polycrystalline aggregate is an elastically anisotropic chunk that will compete with each other and will incur different levels of elastic strains (load portioning) as the aggregate is loaded. Since elastic strains are typically too small to be resolved in DIC fields, in this study, we test the idea of averaging over the grains to see whether the signal starts to show sensible variations among grains. A large-grained pure nickel aggregate is elected as the model material. The large grains (average 70 μm) yield a large DIC field per grain (about 5000 DIC grid points for the average-sized grain) with the utilized 40× microscope. Pure FCC nickel is chosen to avoid complications in microstructure (e.g., lack of precipitates) and micromechanisms (e.g., lack of twinning). In addition, anisotropy ratio (AR) for nickel is about 2.5, a high value should precipitate large elastic strain differences among the differently oriented grains. The DIC strain fields are resolved over individual selected grains. The orientations of the grain neighborhood that contain the selected grains are predetermined by electron backscatter diffraction and are fed to a polycrystal finite element (FE) engine (PRISMS) to conduct purely elastic FE runs. Numerical results are evaluated for the selected grains and compared to the experimental results.

9.2 Experimental Nickel at 99.5% purity has been acquired from Alfa Aesar in the form of a rod with 6.35-mm-diameter radius. The raw material has been subjected to grain growth by high-temperature annealing (4 h at 800 °C) and the final average grain size is approximately 70 μm. A dog-bone sample with nominal 3 × 3 mm2 square gauge section and 8-mm gauge length has been machined by wire electric discharge machining such that the load axis coincides with the rod axis. Orientations are mapped with electron backscatter diffraction (EBSD) after careful metallography (progressive grinding with final polishing with colloidal silica) on the sample surface using an AMATEK EDAX detector in an FEI Quanta 400 FEG SEM. The surface of the sample is then etched with Marble’s reagent (CuSO4 + HCl + H2O) and then a judicious application of paint with an Iwata Custom-B airbrush is introduced to augment the features that enable DIC tracking. The main elements of the custom DIC setup shown in Fig.  9.1a are (i) stationary optical lines: macroscopic (FLIR Grasshopper 3 camera, 2.3 MP Sony IMX174 monochrome sensor, Edmund Optics 0.5× telecentric lens) and microscopic (FLIR Grasshopper 3 camera, 5 MP Sony IMX250 monochrome sensor, Navitar Ultrazoom 6000 tube lens, Mitutoyo M Plan APO 20× objective at 0.1 μm/pixel resolution), (ii) a small-form factor tension-compression load frame (10 kN, Kammrath & Weiss), and (iii) an X-Y-Z positioning array (Newport M-ILS series of X-Y, GTS30V for the vertical Z positioner). The entire setup is positioned on a vibration isolation table (Newport RS4000). An in-house instrument software is used to facilitate the automated coordination of cameras and the motion of the stages. During the load application in the experiment, continuous strain monitoring is implemented under the macroscopic line (with live macro-DIC analysis). At selected loads, the sample is sent under the micro-DIC line and micro-DIC area scans (X-Y) are conducted. These record images of the surface over a predefined grid of overlapping frames as shown in Fig. 9.1b. The image overlaps allow continuous mergers of both image and data (e.g., strain) fields. In Fig. 9.1b, one of the 5 megapixel frames is shown, selected from the image matrix

9  Enabling Digital Image Correlation with High-Resolution Microscopic Optics via Working Distance Automation: Advancing…

51

Fig. 9.1 (a) Sketch of the multiscale digital image correlation setup with area scanning (X-Y) and autofocusing (via Z-scans) capabilities; (b) a depiction of the area scan that covers the field with overlapping microscopic images; an actual individual image in the scan is shown on the right (recorded in the reference configuration)

of the reference (undeformed) state. In this experiment, the frame matrix is 17  ×  11, encompassing an overall area of 2.5 × 2 mm2. For the purposes of this treatise, we will focus on a 0.35 × 0.4 mm2 region of interest. Given the utilized high-numerical aperture (NA  =  0.42) Mitutoyo objective with a very limited depth of field (DOF = 1.6 μm), autofocusing is vital to avoid defocusing. Each frame in the automated micro-DIC scan is autofocused individually. Autofocusing entails the computerized selection of the image with best focus measure value from an image stack that is recorded with fine increments in Z-direction. Regarding motion along the Z-direction (either due to out-of-plane deformation of the sample or due to motion imposed by the Z-positioner), a direct concern is the working distance (WD) error. A change in WD would cause a perspective error that would manifest as an artificial biaxial strain field in the DIC results. This strain error goes by δZ/WD, where δZ is the error between the WDs of the reference and the deformed states [12]. In this scheme, the small (1.6 μm) DOF is used as an advantage and autofocusing maintains WD within a factor of the DOF. With the 20 mm WD of this particularly long WD objective, δZ/WD is dropped below the 10−4 order. While this is an order below the typical DIC uncertainty, its prominence as an error source will be elevated by local averaging (that increases accuracy in lieu of spatial resolution). DIC is conducted with first-order deformation mapping using 61 × 61 pixel subsets (6 × 6 μm2) and a 10 pixel (1 μm) grid spacing. Local strains are recalculated from the displacement grid by central difference differentiation (details in, e.g., [2]). In the experiment, tensile load is applied with very fine increments (approximately every 8 MPa in engineering stress) in position control to collect multiple micro-DIC scans in the elastic region. In total, 10 micro-DIC scans are considered in this treatise. The first nine are located in the roughly proportional part of the stress–strain curve (full-field average under 0.03% = 300 microstrain); the tenth marks the first point of clear yielding and shows much higher strain (around 0.4%).

9.3 Numerical Figure 9.2a shows the grain neighborhood we will focus on in this effort with the four neighboring grains selected for detailed analysis numbered accordingly. This shown field has contributions from four image frames [recall a singular frame from Fig. 9.1b]. The crystal plasticity finite element (FE) package PRISMS [13] is utilized in this effort. It is fed an approximation of a much larger polycrystal domain (as identified by EBSD mapping) by the sequential use of OIM Analysis™ and Dream3D [14]. This ~2000 crystallite domain contains grains 1–4 though with a coarser representation due to the lower resolution of EBSD. Nevertheless, the actual neighbor relations of these grains are better represented than, e.g., a model where the selected orientations are surrounded by random neighbors. Aside from grains 1–4, other markers “*,…” are used to match large grains in Fig. 9.2a, b with the aim of guiding the eye.

52

O. Türkoğlu and C. C. Aydıner

Fig. 9.2 (a) The micro-DIC domain shown in reference configuration that contains the selected neighboring grains 1–4. (b) A regional zoom-in from the FE analysis whose domain is constructed from (the visibly coarser) electron backscatter diffraction results with grains 1–4 marked. The contour plot variable derived from grain orientations is Ehkl, a measure of local stiffness in the loading direction y. Other markers (*,..) are provided in both (a) and (b) to guide the eye for some matching large grains

The corresponding PRISMS structure is shown in Fig. 9.2b using an Ehkl map over the grains. Ehkl given by



 

1 / Ehkl  s11   2 s12  2 s11  s44  k 2 l 2  h 2 l 2  k 2 h 2 / h 2  k 2  l 2



2



(9.1)

is the effective Young’s modulus of each grain in the loading direction, and it is a useful measure to evaluate the relative stiffness of grains. Note, however, that Ehkl corresponds to tensile test boundary conditions (stress-free in transverse directions) that discounts grain interactions in the aggregate. Compliances s11, s12,and s44 take values of 7.34 × 10−3, −2.74 × 10−3, and 8.02 × 10−3 GPa−1, respectively, for nickel. The current PRISMS model is a 3D slab with stress-free transverse boundary conditions (the four faces normal to x and normal to z are stress-free; note that Fig. 9.2b is a top view of this slab). The bottom face normal to y is held at zero displacement while the other face is extended uniformly. Since this treatise considers the elastic region, there is a unique solution for the problem. Plasticity is blocked by artificially exaggerated slip resistance values (i.e., critical-resolved shear stress). Due to apparent numerical precision issues, better numerical resolution is obtained when the applied loads are higher, and the histograms considered are rescaled from a 1200 MPa analysis. (As noted, linear elastic fields are unique after a load normalization).

9.4 Results and Discussion Figure 9.3a shows the local DIC axial strains εyy, overlaid over the grain neighborhood with a level of transparency for load point 8. Point 8 corresponds to nominal stress σnom = 61.4 MPa and full-field axial strain average 〈εyy〉F = 3.65 × 10−4 as shown in Table 9.1. The nominal stress is simply load divided by nominal sample cross section. This table also contains individual grain averages that will simply be denoted by 〈εyy〉, i.e., without the subscript “F,” which stands for “full field”. No trends are evident in the experimental DIC map of Fig. 9.3a, where speckles of error dipoles are frequently observed. [Error dipoles in strain are a common occurrence when strains are differentiated from displacement fields over the DIC grid. Essentially, a displacement error/outlier leads to artificial extension (compression) in its wake and artificial compression (extension) ahead]. This lack of trends is entirely consistent with the original prediction that local DIC strain sensitivity is significantly below the strain signal at this point.

9  Enabling Digital Image Correlation with High-Resolution Microscopic Optics via Working Distance Automation: Advancing…

53

Fig. 9.3  Plotted at load point 8 (61.4 MPa engineering stress; 3.65 × 10−4 full-field average strain): (a) DIC local strains overlaid over the microstructure, (b) DIC grain-averaged strains overlaid over the chosen four grains, and (c) FE results Table 9.1  Selected grain and full-field averages for DIC strain at all load points considered 1–10 Averaged axial strains Load Pt

stress (MPa)

Grain 1

Grain 2

Grain 3

Grain 4

full field

1 2 3 4 5 6 7 8 9 10

8.5 15.9 23.4 30.7 38.4 46 53.8 61.4 68.2 75.8

1.07e-4 1.85e-4 2.77e-4 2.22e-4 3.94e-4 4e-4 4.8e-4 5.5e-4 5e-4 4.78e-3

8.02e-5 1.19e-4 2.82e-4 2.32e-4 3.72e-4 4e-4 5.06e-4 6.02e-4 5.16e-4 5.15e-3

7.07e-5 1.14e-4 2.64e-4 2.29e-4 3.42e-4 3.8e-4 4.64e-4 5.92e-4 4.72e-4 4.5e-3

5.5e-5 8.45e-5 2.14e-4 2.29e-4 2.88e-4 3.2e-4 3.8e-4 4.88e-4 3.6e-4 3.93e-3

3.37e-5 9.69e-5 1.58e-4 1.69e-4 1.97e-4 2.3e-4 2.99e-4 3.65e-4 2.61e-4 3.97e-3

Note: The second stress column corresponds to the nominal stress over the entire sample. The last row is shaded to designate that the material has shown clear plastic yielding by load point 10

The insufficient sensitivity of local DIC in the elastic region is presented in two more types of presentation. First, Fig. 9.4a shows εyy of singular grid points inside four grains (chosen as points about the center of mass of each grain) plotted against σnom. There is a very dominant scatter for each point and the expected physical strain trend (a proportional strain increase at each point) is completely obscured by measurement uncertainty/noise, δεyy. The fact that these points are selected from each grain is in fact immaterial; any DIC point in the field shows a very large δεyy scatter when plotted against σnom. Second, Fig. 9.5a shows the strain histograms that are produced from the DIC data in each selected grain 1–4. The spread is easily ±0.005, and though the intragrain strain distribution has a physical component, the existence of a large negative strain population under tensile load already corroborates that much of this broad distribution stems from δεyy. Figure 9.3b shows 〈εyy〉 for each selected grain at load point 8, and Fig. 9.3c shows local εyy fields over the FE analysis subdomain shown in Fig. 9.3c presented for the same load, σnom = 61.4 MPa. At this point, note from Table 9.2 that the number of elements in each grain is easily kept comparable to DIC subsets, i.e., while the geometric detail is coarse due to the EBSD import, FE analysis itself is finely discretized. While any comparison to the noisy experimental data in Fig. 9.3a is futile, the spread in Fig. 9.3c is likely representative of the intragrain and intergranular strain spread for a polycrystalline cubic symmetry aggregate of AR = 2.5. The intergranular spread is well understood and can be estimated with, e.g., mean field models [15] as well as that assume homogeneity for intragrain fields. To the best of the authors’ knowledge, the intragrain spread for the mere elastic region is rarely considered, but stress concentrations around grain boundaries and triple junctions surely call for a degree of strain heterogeneity. To detail the intragrain elastic strain distribution that is ideally calculated from an FE analysis, the histograms of Fig. 9.5b can be consulted. These show PRISMS-CPFE strain distributions inside each selected grain. Spreads up to ±20% are noted.

54

O. Türkoğlu and C. C. Aydıner

Fig. 9.4 (a) Nominal stress vs. local DIC strain, εyy, for the selected grid points in each grain 1–4, (b) nominal stress vs. DIC grain averaged strain, 〈εyy〉, for each grain 1–4, and (c) curves of part (b) individually expressed for each grain superimposed with the corresponding FE results (computed grain averages) in the elastic region

For the 〈εyy〉 results in Fig. 9.3b, the homogeneously tensile results are promising for the sensibility of the grain averages and the premise that averaging over a grain with thousands of data points (see Table 9.2, column 4, for the number of DIC subsets in each grain) will increase the accuracy perhaps to the level of resolving grain-averaged elastic strains. Fig. 9.4b indeed corroborates this hypothesis through a plot of σnom versus 〈εyy〉. The curves here for each grain by and large follow a sensible proportional increase for the first nine points and then each show clear yielding at load point 10. At this point, a comparison to the noisy individual data point behavior in Fig. 9.4a is informative. There is still a zigzag in the 〈εyy〉 curves of Fig. 9.4b, but the behavior of all grains is largely consistent in this regard as well, e.g., a negative strain step from point 8 to 9 (61.4–68.2 MPa) is realized by each grain (see also Table 9.1 for full numerical details). This unrealistic strain step as well as other zigzags is estimated to be due to the WD error, a systematic error surfacing now that the random error is averaged out. All four grains are in the same image frame (Fig. 9.1b), and it makes sense that their systematic WD error stays largely consistent. This error changes from load to load since the Z-error takes a new value in each, leading to the zigzag. Nevertheless, the results for the grain averages of Fig. 9.4b look very promising from the standpoint of accuracy. Fig. 9.4c shows the FE results for 〈εyy〉 superposed over the experimental 〈εyy〉 for each grain. The agreement can be deemed reasonable for most grains, and the FE results with columnar grains should also not be taken as absolutely correct. The final aspect that we will delineate is the variations among each selected grain with different orientations and a considerable variety in Ehkl (Table 9.2). Hence, they should all take different amounts of strain in the elastic region. To bring a rigorous numerical comparison, we define a grain effective modulus, E , as E

 nom  yy

(9.2)

The calculation of E is trivial for FE results (where σnom is the field average of stress, and 〈εyy〉 is calculated over the domain of each grain) and is listed in the final column of Table II. For calculation over the experimental data, we choose a linear fit to all nine pairs of {σnom, 〈εyy〉} in Table 9.1 and report the slope. These experimental values of E are shown in the

9  Enabling Digital Image Correlation with High-Resolution Microscopic Optics via Working Distance Automation: Advancing…

55

Fig. 9.5  Intragrain strain histograms populated from each selected grain for (a) DIC axial strains (population: DIC grid points inside the grain) and (b) FE axial strains (population: FE integration points inside the grain) Table 9.2  Specifications and results for each selected grain 1–4 Grain number 1 2 3 4

Grain size (μm) 66.7 45.2 55.3 80

Ehkl (GPa) 242.2 148.5 262 282.5

# of subsets in OM-DIC 4644 2238 3144 5472

# of CPFE elements 8447 3071 2815 9727

DIC grain effective modulus (GPa) 131.3 79.5 118.7 170.8

FE grain effective modulus (GPa) 212.3 259.8 257.9 243.6

“DIC grain effective modulus” column of Table 9.2. The comparison of EFE and EDIC does not show an agreement neither numerically nor in terms of trends. However, it must be said that EDIC makes more sense in light of the Ehkl column. The soft grain 2 behaves compliant and the hard grain 4 behaves stiffly according to EDIC but not according to EFE . EDIC surely behaves softer than nickel elasticity suggests, but part of this can likely be brushed off to unavoidable plastic activity in the supposed elastic region of pure nickel. An important future work is to check the appropriateness of boundary conditions and geometry for the columnar FE analysis. The coarseness of the initial grain morphology is also suspect and will be amended.

9.5 Conclusion Grain-resolved results are derived from a custom optical microscopy-DIC instrument with intragrain resolution over a pure nickel polycrystalline aggregate. The focal point of the study is to see how far DIC accuracy can be improved by field averaging and, in particular, whether the elastic strains can be identified by grain averages. The results of the selected grains are promising. Grain averages in the elastic region show a near proportional and sensible trend before yielding, much improved from individual data points that simply cannot resolve elastic strains by at least an order. There is a zigzag in individual results that is interpreted as a working distance error in this instrument where working distance can be corrected within a factor of the depth of field. Corresponding FE analysis has been conducted with a columnar grain model, but the particular agreement with experimental results is at best mediocre. FE results are also under scrutiny since DIC results show better agreement with physical expectations of how grains should share the load based on their orientation. These studies will be expanded with much higher grain statistics and more careful complementary FE analysis in the future. Acknowledgments  This work was supported by the B.U. Research Fund at Bogazici University under contract 14A06P4.

56

O. Türkoğlu and C. C. Aydıner

References 1. Shafaghi, N., Kapan, E., Aydıner, C.C.: Cyclic strain heterogeneity and damage formation in rolled magnesium via in situ microscopic image correlation. Exp. Mech. 60, 735–751 (2020). https://doi.org/10.1007/s11340-­020-­00612-­6 2. Özdür, N.A., Üçel, I.B., Yang, J., Aydıner, C.C.: Residual intensity as a morphological identifier of twinning fields in microscopic image correlation. Exp. Mech. 61, 499 (2020). https://doi.org/10.1007/s11340-­020-­00672-­8 3. Kammers, A.D., Daly, S.: Digital image correlation under scanning electron microscopy: methodology and validation. Exp. Mech. 53, 1743–1761 (2013). https://doi.org/10.1007/s11340-­013-­9782-­x 4. Orozco-Caballero, A., Lunt, D., Robson, J.D.J.D., et al.: How magnesium accommodates local deformation incompatibility: a high-resolution digital image correlation study. Acta Mater. 133, 367–379 (2017). https://doi.org/10.1016/j.actamat.2017.05.040 5. Can Aydiner, C., Üstündag, E., Prime, M.B., Peker, A.: Modeling and measurement of residual stresses in a bulk metallic glass plate. J. Non-­ Cryst. Solids. 316, 82–95 (2003). https://doi.org/10.1016/S0022-­3093(02)01940-­3 6. Efstathiou, C., Sehitoglu, H., Lambros, J.: Multiscale strain measurements of plastically deforming polycrystalline titanium: role of deformation heterogeneities. Int. J. Plast. 26, 93–106 (2010). https://doi.org/10.1016/j.ijplas.2009.04.006 7. Aydıner, C.C., Telemez, M.A.: Multiscale deformation heterogeneity in twinning magnesium investigated with in situ image correlation. Int. J. Plast. 56, 203–218 (2014). https://doi.org/10.1016/j.ijplas.2013.12.001 8. Lebensohn, R.A.: N-site modeling of a 3D viscoplastic polycrystal using fast Fourier transform. Acta Mater. 49, 2723–2737 (2001). https:// doi.org/10.1016/S1359-­6454(01)00172-­0 9. Roters, F., Eisenlohr, P., Bieler, T.R., Raabe, D.: Crystal Plasticity Finite Element Methods. Wiley-VCH (2010) 10. Üçel, İ.B., Kapan, E., Türkoğlu, O., Aydıner, C.C.: In situ investigation of strain heterogeneity and microstructural shear bands in rolled magnesium AZ31. Int. J. Plast. 118, 233–251 (2019). https://doi.org/10.1016/j.ijplas.2019.02.008 11. Padilla, H.A., Lambros, J., Beaudoin, A.J., Robertson, I.M.: Relating inhomogeneous deformation to local texture in zirconium through grain-­ scale digital image correlation strain mapping experiments. Int. J. Solids Struct. 49, 18–31 (2012). https://doi.org/10.1016/j.ijsolstr.2011.09.001 12. Schreier, H., Orteu, J.J., Sutton, M.A.: Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications. Springer, Boston, MA (2009) 13. Yaghoobi, M., Ganesan, S., Sundar, S., et al.: PRISMS-plasticity: an open-source crystal plasticity finite element software. Comput. Mater. Sci. 169, 109078 (2019). https://doi.org/10.1016/j.commatsci.2019.109078 14. Groeber, M.A., Jackson, M.A.: DREAM.3D: a digital representation environment for the analysis of microstructure in 3D.  Integr. Mater. Manuf. Innov. 3, 56–72 (2014). https://doi.org/10.1186/2193-­9772-­3-­5 15. Hutchinson, J.W.: Elastic-plastic behaviour of polycrystalline metals and composites. Proc. R Soc. London A. Math. Phys. Sci. 319, 247–272 (1970). https://doi.org/10.1098/rspa.1970.0177

Chapter 10

Characterization of Bioengineered Tissues by Digital Holographic Vibrometry and 3D Shape Deep Learning Colin Hiscox, Juanyong Li, Ziyang Gao, Dmitry Korkin, Cosme Furlong, and Kristen Billiar

Abstract  One of the critical components of large-scale manufacturing of bioengineered tissues is the sensing of information for quality control and critical feedback of tissue growth. Modern sensors that measure mechanical qualities of tissues, however, are invasive and destructive. The goal of this project is to develop noninvasive methodologies to measure the mechanical properties of tissue engineering products. Our approach is to utilize acoustic waves to induce nanoscale level vibrations in the engineered tissues in which corresponding displacements are measured in full-field with quantitative optical techniques. A digital holographic system images the tissue’s vibration at significant modes and provides the displacement patterns of the tissue. These data are used to train a supervised learning classifier with a goal of using the comparisons between the experimental vibrational modes and the ones obtained by finite element simulation to estimate the physical properties of the tissue. This methodology has the promise of mechanical properties that would allow technicians to noninvasively determine when samples are ready to be packaged, if their growth deviates from expected time frames, or if there are defects in the tissue. It is expected that this approach will streamline several components of the quality control and production process. Keywords  Digital holography · Tissue engineering · Vibration · Pattern recognition · 3D deep learning classification

10.1 Introduction Mechanical properties are critical and often overlooked factors defining the quality of bioengineered tissues. Current tests used to measure the physical properties of engineered tissues, such as uniaxial tensile tests and nanoindentation, tend to be invasive and destructive and thus not applicable for in-process quality control. To measure a tissue destined for therapeutic applications, all devices must be aseptic and not damage the tissue. The goal of this work is to develop a noncontact nondestructive method to measure the properties of bioengineered tissues during production. We use acoustic vibration and digital holography to measure the sample. The tissue this work examines is Apligraf, created by Organogenesis [1] and is a circular tissue consisting of an epidermal and dermal layer laying on a porous plastic membrane. Circular vibrations induced by noninvasive acoustic waves create natural modes as can be predicted with Bessel functions on a circular plate according to classical plate theory [2]. Holography operates by mathematically interpreting the interference between two beams of light, one of which reflects off the sample, in a digital sensor [3], thus operating noninvasively.

C. Hiscox (*) · C. Furlong Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA e-mail: [email protected] J. Li · K. Billiar Biomedical Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA Z. Gao · D. Korkin Department of Bioinformatics and Computational Biology, Worcester Polytechnic Institute, Worcester, MA, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_10

57

58

C. Hiscox et al.

10.2 Methods Our methodology used three components to measure important information of the sample: optical coherence tomography (OCT), digital holographic vibrometry (DHV), and nanoindentation. All measurements and experiments were performed on samples of Apligraf generously donated by Organogenesis Inc. First a sample was measured with OCT (Telesto SD-OCT, Thorlabs). A b-scan taken at multiple points was used to estimate the thickness of each layer of the Apligraf nondestructively. DHV, the primary tool, uses a speaker and a holographic sensor (Fig. 10.1) to induce and measure vibrations. The sensor uses a 532 nm light to illuminate the sample while it is still in its packaging. A subwoofer placed above the sample outputs single-tone frequencies and sweeps between 70 Hz and 500 Hz. We identified vibrational modes of the 0th order and imaged the sample at 12 equidistant points within a single vibration curve by modifying the phase of camera strobing for image acquisition. This gave a full-field-of-view video with 12 frames that showed the nm-scale waveforms oscillating in the sample. Samples were also measured with nanoindentation at 4 multiple random points with a 51 μm-radii probe on a 3.45 N/m stiffness cantilever (Chiaro, Optics 11). During indentation, the sample was covered by a lid with a small opening to minimize evaporation. The epidermis was measured and then peeled back with a scalpel and forceps, and the tissue’s dermis was measured the same way. This was used to validate the modulus of the epidermal and dermal layers to compare to simulated results and to algorithm predictions. To demonstrate the utility of the method, two experiments were performed. In the first experiment, a single sample was measured with nanoindentation and then with DHV. Four different speaker volumes were measured at each mode, measuring different pressures applied by the acoustic waves to the sample, to test for linearity of the tissue response. For the second experiment, the sample was subjected to controlled drying to change the thickness and/or stiffness of the sample and measure the resulting change in frequency of vibrational modes over time. Due to the destructive nature of the drying, two different samples from the same production batch were used for nanoindentation and DHV. Each sample was dried for 90 minutes in a 23 °C oven (Isotemp, Fisher Scientific) for controlled humidity and temperature, and measured at several time points: 0, 3, 6, 10, 15, 30, 60, and 90 minutes. Each sample was put back into the oven with an open lid after measurement until the next time interval was reached. To relate the mode shapes and frequencies of the complex system, we trained a machine learning algorithm with a finite element (FE) simulation in Ansys APDL.  We developed a composite linear-elastic FE model with thin shell elements (Fig. 10.2b). The model includes three layers: the epidermis, dermis, and porous plastic membrane on which they sit. To emulate the triangular pattern of the Apligraf packaging, we applied fixed supports to 30 degree increments evenly spaced

Fig. 10.1  DHV sensor with inset of tissue during measurement [4]

10  Characterization of Bioengineered Tissues by Digital Holographic Vibrometry and 3D Shape Deep Learning

59

Fig. 10.2 (a) A schematic of the model used in the FE simulation (b) the mesh pattern of the FE model [4]

around the edge (Fig. 10.2a). We then added an elastic support underneath the sample to emulate the agar upon which the system sat. We performed a parametric modal analysis, the stiffness of dermis and epidermis were modified in the range of 10–60 and 3–20, and the thickness was in the range of 50–200 for both layers. For this model, Poisson’s ratio and density were assumed to hold constant at 0.48 and 997 kg/m3 respectively for both layers of tissue. This model outputs the mode shapes and frequencies of several modes for each combination of stiffness and thickness values. A 3D shape classification supervised learning model was built by leveraging Pointnet deep learning architecture [5]. Specifically, the model was trained on the point cloud dataset from the FE simulation to classify the stiffness and thickness of the epidermis and dermis, where each data point was organized as a separate simulation file. Only 0th order modes, such as shown in Fig. 10.4, were selected by finding all modes with triangular symmetry defined by three partial cross sections (Fig. 10.3b). In total, a dataset that included 1440 simulation files was selected, and randomly split by the ratio of 3:1 into the training and testing subsets. We treated the 1763 nodes in the mesh files as point cloud images, with x,y, and z coordinate values of each point as the representative features. The frequencies of FE stimulation were extracted for the corresponding 0th mode. Each property was encoded as either 0 or 1 if the stiffness or thickness value was less or no less than the medium value of all simulation files, respectively. In our supervised learning model, the Pointnet architecture was modified, by adjusting the hidden layer nodes to better fit the FE simulation data (Fig. 10.4). The input transformer network (T-net) linked the input layer with the first primary multilayer perceptron (MLP), and the feature transformer network linked two primary MLPs together. The global feature was generated by maxpooling the second MLP output, and we concatenated the input frequencies as an additional feature. The overall 257 features were processed by the final MLP to generate the output class.

10.3 Results For the first time point of the drying experiment, the fundamental frequency was identified when the speaker played a 149 Hz tone. Inputting this into classical plate theory equations for a thin round plate, we calculated a stiffness of 1.1 GPa. Meanwhile, the nanoindentation measurements indicate a modulus around 36.5 ± 26.2 kPa for the epidermis of the tissue. In the variable pressure experiment, the maximum positive displacement measured at the first four vibration modes for four different acoustic forces applied by the speaker shows roughly linear relationships (Fig. 10.5), indicating that the tissue is within the linear region at these low displacements. For these analyses, the largest displacement is taken from the point in the 12-image set of measurements that reflects the top of the sinusoidal curve. Taking the steepest and flattest slope indicated in Fig. 10.2 and converting to strain using a thickness of 400 μm, we calculate a stiffness of 1.3 kPa or 570 Pa. In the second experiment, OCT measurements of thickness for both epidermal and dermal layers of the Apligraf tissue decreased with drying time (Fig. 10.6), thus establishing a change to the tissue and physical properties for the DHV sensor to measure.

60

C. Hiscox et al.

Fig. 10.3  Fundamental mode shapes calculated by the FE simulation of Apligraf tissue (a) an example of a fundamental mode output with relative z-displacements (b) symmetry lines overlaid onto a fundamental mode image

Fig. 10.4  Architecture of the Pointnet-based 3D classification model combined with frequency input. The Pointnet based classification network for FE simulation pointcloud takes 1763 nodes as input N, and x,y,z coordinate values as the representative features. Input and feature transformations are linked between two primary MLPs. Input frequencies were concatenated as an additional feature to the maxpool global features for the last mlp. The output was densed by softmax as binary classification

The frequencies at which each of the first four modes of vibration occurs decrease monotonically with drying (Fig. 10.7a). The modal FEA in which the layer thickness was decreased to match the measured data shows the opposite trend in frequency of the fundamental mode (Fig. 10.7b). This result indicates that it is more likely that the increase on modulus with drying dominates the vibratory behavior. The 3D deep learning classification method presented the ability to predict the modulus of the epidermis based off of the spatial patterns in the FE simulation. Prediction accuracy of the thickness was 0.7101 for the epidermis and 0.7731 for the dermis, while prediction accuracy of the stiffness was 0.7668 for the epidermis and 0.4937 for the dermis.

10  Characterization of Bioengineered Tissues by Digital Holographic Vibrometry and 3D Shape Deep Learning

61

Fig. 10.5  Amplitude of tissue displacement for multiple applied pressures during vibration [4]

Fig. 10.6  Decrease of tissue thickness during drying. OCT images show the thickness, while a numerical of the epidermis (blue) and dermis (orange) thickness are overlaid [4]

Fig. 10.7 (a) Frequency location for each mode during drying (b) Expected frequency of vibration from FEM [4]

62

C. Hiscox et al.

10.4 Discussion This study demonstrates the ability to measure the full-field surface displacements of engineered skin tissue nondestructively and aseptically within its packaging using stroboscopic holographic vibrometry. By utilizing multiple acoustic pressures, the linearity of the tissue behavior at such low applied pressures was confirmed. Drying the tissue demonstrated the ability of the system to detect changes in the physical properties. After an initial increase in the measured frequencies, a consistent decrease in the modes’ frequencies was found as the tissue thinned. This contrasted expected frequency change from the FE simulation. We believe this is due to the number of factors that change. The FE simulation accounted for only thickness change, while the real tissue thinned by losing water, thus altering its stiffness, density, and possibly even Poisson’s ratio. Due to the complexity of the layered tissue, analytical methods and vibrometry measurements alone cannot currently extract precise mechanical properties of the Apligraf layers. Multiple attempts to calculate the stiffness of the system using different methods resulted in estimates of 1.1 GPa or 1.3 kPa, which were both orders of magnitude of the 36 kPa nanoindentation measurement. The application of the Pointnet-based deep neural network can identify certain mechanical properties when fed an image from the FE simulation. Concatenating the frequency with the mode shape helps increase the accuracy slightly than using the shape alone. It shows less accuracy on dermis stiffness compared with other properties, which can be explained that the 3D shape of FE simulation is less sensitive toward the change of dermis stiffness. The next step after this is to fine-tune the model with experimental data. For several tissue samples, we have collected stiffness with nanoindentation, thickness with OCT, and modes with DHV. We will re-evaluate the fine-tuned model and use it to identify which features can be accurately predicted, and what pieces of information, such as what nodes will not affect the prediction accuracy, to determine whether they need to be collected. Once completed, this noninvasive measurement technique and associated analysis method would likely take around 5 minutes allowing for in-line measurement during production. This QC assay has the potential to enable identification of faulty batches before the full growth time is reached, increasing efficiency of manufacturing. Acknowledgments  This work was supported by the NSF (CMMI 1761432) and ARMI Bio FabUSA (T0137). We would also like to thank Kate Faria and Organogenesis for providing the Apligraf tissue used in this research.

References 1. 2. 3. 4. 5.

Faria, K.: Personal Communications. Organogenesis Inc. (2021) Leissa, Vibration of plates, NASA SP-160, 1969 Kuppers, J., et al.: Proc. SPIE 6293. Interferometry XII: Applications, 629309 (2006) Hiscox, C.: MS thesis. ME Department WPI (2022) Qi, C.R., et al.: Proceedings of the IEEE conference on computer vision and pattern recognition. (2017)

Chapter 11

Coordinated Twinning Bands in Magnesium at the Existence of Stress Raisers via In Situ Microscopic Image Correlation S. Can Erman and C. Can Aydıner

Abstract  Mechanical twinning underlies many of the challenges in the forming and structural utilization of magnesium alloys, the class of engineering metals with the lowest density. Intense research activity has been devoted to the understanding and modeling of the abrupt twinning phenomenon. A particularly challenging aspect of twinning is its abrupt coordinated proliferation across the polycrystalline aggregate. Macroscopically, these events are akin to Lüders banding while the geometry and compactness of the coordinated twinning bands show pronounced dependence on microstructure (most notably on crystallographic texture) in magnesium alloys. A series of systematic studies employed an in situ area scanning variant of digital image correlation with optical microscopy to quantitatively characterize these bands. While the length scale of optical microscopy is typically suited to investigate long-range strain structures over the polycrystalline aggregate, this instrument employs high optical resolution (high numerical aperture) objectives to also provide adequate intra-grain resolution even in medium-sized (~10 μm) grains (Özdür NA, Üçel IB, Yang J, Aydıner CC, Exp Mech 61:499, 2020).The utilization of these objectives that also possess extremely small depth of fields is only possible through continual automated working distance corrections to fight off defocusing. The sharpness of the consequent imaging has further been used to introduce a novel microscopy mode, called residual intensity (Özdür NA, Üçel IB, Yang J, Aydıner CC, Exp Mech 61:499, 2020). This mode isolates and presents the twins with an order higher resolution than the DIC strains, albeit using the same images. It has the further advantage of showing deformation structures (in this case, twins) that are activated in a specific load increment, namely, residual intensity is an imaging mode with a reference state. Here, we will employ this OM-DIC technique over the sharp rolling texture (for which the band strain reaches about one-­ third of the twin transformation strain (Özdür NA, Üçel IB, Yang J, Aydıner CC, Exp Mech 61:499, 2020) in an unnotched sample), but in a notched sample to guide and overlap macroscopic twin bands. The notch locations will be specifically designed for this regime of extreme plastic anisotropy. Microscopic DIC is again implemented in situ to study formation, expansion, and overlap of the coordinated twin bands. Both strain and residual intensity calculations will be performed across loads to investigate the effect of stress raisers on the twin coordination. Keywords  Magnesium · Twinning · Microscopy · Image correlation · Residual intensity

11.1 Introduction Aside from the considerable engineering interest in magnesium as the lowest density structural metal, it also has become the primary material for investigating deformation twinning with a variety of experimental techniques (e.g., [1–4]). The twinning mechanism underlies much of the complexity in the mechanical behavior of the hexagonal close packed (HCP) Magnesium that includes heavy load path dependence and severe plastic anisotropy. Twinning is unipolar (activated only in one sense of the load but not the other) and emerges abruptly, bringing about a large transformation strain [13.1% transformation shear for the most common 10 12 1011 tensile twin]. The abruptness of the twin operation promotes spatially coordinated activity across a polycrystalline aggregate. This coordination is due to the autocatalytic activity [5, 6], crudely, twinning in one grain, instigating twinning in the neighboring grain(s). The close spatial coordination of twinning with autocatalytic activity implies multiscale strain heterogeneity. At the component scale, macroscopic strain localization bands are observed. The geometry and compactness (the degree of average S. C. Erman · C. C. Aydıner (*) Department of Mechanical Engineering, Boğaziçi University, Istanbul, Turkey e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_11

63

64

S. C. Erman and C. C. Aydıner

strain accommodation) of these bands heavily depend on texture [7]. In particular, rolled material with the typical c-axes-­ aligned texture produces very sharp and severely anisotropic twin bands in the profuse twinning regime. These findings on deformation geometry are largely due to strain mapping with digital image correlation (DIC) [7–10]. DIC has been applied at multiple length scales on twin fields, including implementations with optical [11, 12] and electron microscopy [13]. In particular, a heavily automated variant of DIC with optical microscopy (OM-DIC) has led to novel in situ measurements that portray the composition, propagation, and interaction of the sharp twinning bands in rolled magnesium. This technique provides granular resolution as well as sample-scale coverage [1, 11, 12]. Under compression in the rolling direction (RD), the plane of shear of the sample-scale bands is effectively aligned with the RD-ND plane (ND indicates normal direction of the plate). This means extreme plastic anisotropy where ±45° band structures extend all through the thickness in the transverse direction (TD). The resulting volumetric shear bands, hence, essentially obey a 2D shear deformation. This is also evident in the extreme asymmetry of the apparent Poisson’s ratios that take a value of 1 on the RD-ND surface (where the cross section of a volumetric twin band is oriented ±45° to the load) and a value of 0 on the RD-TD surface (where the volumetric band appears horizontally) [7, 12]. In this study, we design a notched sample that aims to guide and ensure the overlap of twin-mediated shear bands for the rolled material. This sample design is based on the knowledge of the acute anisotropy of the shear bands as revealed by the previous studies. The effectiveness of this idea is tested by multiscale DIC with its optical microscopy component applied at 40× magnification. The compactness of the shear bands and their overlap is analyzed quantitatively with strain and residual intensity [1, 14].

11.2 Materials and Methods Magnesium AZ31 notched dog-bone specimens are cut via electrical discharge machining (EDM) along rolling direction (RD) from a 6.35-mm-thick hot rolled plate (procured from Alfa Aesar). The material is in T0 temper (annealed), and the average grain size is approximately 12 μm. Figure 11.1a shows the designed geometry of the sample. The 0.4-mm-deep notches are positioned in a cross-pattern at the center of the sample. The internal width of the sample is 3 mm and nominally equal to the thickness of the sample except for a 0.1 mm allowance that is removed during metallography. The observation surface of the sample is the RD-ND plane in likeness to previous studies where the macroscopic collaborative twinning bands form ±45° patterns. This sample design precisely attempts to guide these bands in a cross-pattern. Figure 11.1b provides unit cell sketches for the statistically abundant orientation. It portrays the physical underpinning of this severely anisotropic deformation that is akin to the deformation in a single crystal. As it is reflected in this unit cell, in the considered rolling texture, c-axes are aligned with ND. Under compression in RD, the two 10 12 1011 tensile twin systems that would be activated (that would undergo maximum resolved shear stress) are shown in the unit cells. The ±45° pattern in these unit cells roughly translates to the ±45° pattern in the macroscopic bands (details in [11, 12]). The orientation of the structures will not be perfect, and the material volume will only partially be converted by twinning across the polycrystal; however, the ±45° orientation of the bands persists in a statistical sense. The surface preparation of the sample starts with metallography. Grinding and polishing stages are conducted with a fully automated machine to control uniformity of material removal. Waterproof grinding paper from 1200 to 4000 grit and appropriate polishing cloth with oil-based colloidal silica are utilized. This is followed by etching (0.56 gr picric acid, 1.5 ml acetic acid, 12 ml absolute ethanol, and 1.5 ml distilled water) to reveal grain boundaries. Lastly, speckle pattern is added to the observation surface by spraying a mixture of polymer-based paint and distilled water via Iwata Custom-B airbrush. Figure 11.2a schematically represents the composition of the utilized setup with multichannel DIC lines in addition to loading and positioning elements. The loading element is a 10 kN Kammrath & Weiss tension-compression module. Macroand micro-images are recorded at macro-DIC and micro-DIC lines, respectively. The macro-DIC line comprises a FLIR Grasshopper 3 camera with 2.3 MP Sony IMX174 monochrome sensor, Edmund Optics low-magnification microscope at 1×. The resulting optical resolution is 5.67 μm/pixel. The micro-DIC line contains FLIR Grasshopper 3 camera with 5 MP Sony IMX250 monochrome sensor, Navitar Ultrazoom 6000 tube lens, Mitutoyo M Plan APO 20× objective. The full magnification of the line is rated 40×, and the optical resolution is 0.099 μm/pixel. All elements of the experimental setup are mounted on vibration isolator Newport RS4000 optical table. The automation of the setup with area scanning operations and vital autofocusing function is described elsewhere [1, 15]. Figure 11.2b shows the schematic view of the imaging grid formed by a combination of area scanning, autofocusing, and imaging functions. The full micro-DIC domain is comprised of 29 × 15 frames covering an area of 4.77 × 3.08 mm2. A more limited zone will be presented in the current treatise as shown in the next section. Each micro-DIC image scan takes about 75 minutes in this configuration. The micro-DIC domain is adjusted to cover all notches to clearly show initiation point of

11  Coordinated Twinning Bands in Magnesium at the Existence of Stress Raisers via In Situ Microscopic Image Correlation

65

Fig. 11.1 (a) Technical drawing of the sample with the design cross-notch pattern; detailed view of a notch is designated with “A”; (b) sketch of hexagonal-close-packed unit cell for the statistically abundant orientation. The two 10 12 1011 tensile twin systems with the highest resolved shear stress are indicated with “red”







Fig. 11.2 (a) Sketch of the multiscale digital image correlation setup with area scanning (X-Y) and autofocusing (via Z-scans) capabilities; (b) a depiction of the area scan that covers the field with overlapping microscopic images; an actual individual image in the scan is shown on the right (recorded in the reference configuration)

twin-driven shear bands. During the in situ DIC experiment, load is increased in fine position increments (~0.05%) over the twin plateau. The results of the macro-DIC line (evaluated concurrently) are used to control the experiment. The DIC methodology and data merger over frames have been described elsewhere [15]. Here, we suffice to note that linear subsets are utilized with strains recalculated from displacement results via

66



S. C. Erman and C. C. Aydıner

 xx 

u v 1  u v  ,  yy  ,  xy     x y 2  y x 

(11.1)

using central difference differentiation. Tied to OM-DIC, a new imaging channel called residual intensity (ΔI) has been proposed recently to reveal surface twins in Özdür et al. [1]. In a nutshell, residual intensity is the material-point-aligned subtraction of reference and deformed images that rely on the displacement output of DIC. Hence, ΔI calculation is a postprocessing step over DIC tracking results and works on the same images. It, however, offers much higher spatial resolution than the DIC strain maps. The reader is referred to Özdür et al. [1] for comprehensive detail and application ideas for the ΔI channel.

11.3 Results and Discussion Figure 11.3a shows the stress–strain (engineering stress versus average axial strain,    yy ) curve of the sample up to −0.71% average strain. The averaging domain is chosen to exclude notches and is a 2.5 × 8 mm2 corridor that follows the center fiber of the sample. Since this is a notched sample, ε is not a rigorous parameter but will depend on the chosen gauge area. It should only be regarded as a load point marker. (To be perfectly rigorous, in the Lüders-like banded strain accommodation, even unnotched samples have the same issue of gauge-area-dependent nominal strains [12]). A and B represent the two load points that are investigated at macroscopic and microscopic length scales. Macro-DIC maps of transverse (εxx) and axial strain (εyy) are presented in Fig. 11.3b at points A and B. At both points, the characteristic of this extremely anisotropic deformation that εxx magnitudes virtually equal εyy over the shear bands (stated before as apparent Poisson’s ratio to be approximately equal to 1) is evident. At point A, the primary deformation element is a strain band that is guided from the top-left notch to the bottom-right notch [see Fig. 11.4a for the sketch of the intended strain accommodation]. When the load is advanced to point B, the opposite band is also triggered, enforcing an overlap at the middle of the sample. Thus, even in the macro-DIC results, it is observed that the intended operation of the macroscopic band structures in the cross-pattern shown in Fig. 11.4a is realized. It is instrumental to note that, in a material that shows (transverse) isotropy along the loading axis, the sample will rather show horizontal strain localizations, linking top (bottom) left notch to the top (bottom) right notch. Hence, the shown pattern is not at all typical of other engineering metals. Figure 11.4b shows micro-DIC strain maps (εxx, εyy), and rotation map (ωxy) at load points A and B. For the brief coverage in this presentation, the data field is limited to a box [shown with “green” in Fig. 11.4a] that surrounds the intended band overlap. Once again, rough magnitude equality of εxx and εyy fields is evident. The use and caveats of rotation as a deformation measure are detailed elsewhere [11, 15]. Here, one can clearly see its utility with a sense-discerning color map [blue (−); red (+)], namely, +45° and − 45° structures are auto-discerned by color. This does not just apply to the macroscopic band structures; the strain network inside the band also contains both elements [11].

Fig. 11.3 (a) Nominal stress–strain curve [axial strain from the gauge section of sample chosen as a 2.5 × 8 mm2 corridor that extends amid the notches shown in part (b)]; (b) macro-DIC strain maps over the gauge section at load points A and B marked in part (a)

11  Coordinated Twinning Bands in Magnesium at the Existence of Stress Raisers via In Situ Microscopic Image Correlation

67

The higher resolution in Fig. 11.4b shows thinner band structures that are not resolved by macro-DIC maps of Fig. 11.3b (e.g., the bands parallel to the major band at load point A). This −45° band (“blue” in ωxy maps) thickens from point A to point B while the +45° band (“red” in ωxy maps; from top right notch to the bottom left notch) that does not exist at point A emerges. The dashed yellow region indicated in Fig. 11.4b is the intersection of the −45° band at point A and the +45° band at point B. In other words, this is the material region that incurs full band strains at both points in time. This polygon contains nearly 500 grains with a size of approximately 0.2 mm2. When the axial strain data inside this region is averaged at points A and B, one finds −2.26% and − 3.78%, respectively. Also, −2.26% agrees very well with the previously reported single-band strain for a similar grain size over maiden material [1, 12]. At this stage, the reciprocal band increased the strain content by about 67% (so shying away from doubling it but significantly increasing it). Figures 11.5a and b show reference and deformed (to point A) images of a single frame in the band intersection zone [indicated with a white box in Fig. 11.4b]. Figure 11.5c shows the residual intensity, ΔI, map that is computed between point A and the undeformed reference. In the originating paper, Özdür et al. [1] have used ΔI to present a finely detailed collaborative twin field with twin lamellae in good positional agreement with the grain boundaries and largely oriented with a distribution about the expected ±43.3° [Fig. 11.1b]. The ΔI map in Fig. 11.5c, however, does not obey the same specifications and rather has smeared out features that seem to imply plasticity elements. Nevertheless, there is no reason to believe that the deformation is not tensile twin dominated with the band strain (−2.26%) in strong agreement with previous results [12]. The difference we believe is a skin effect due to unintended deformation of the surface layers during metallography. This particular sample has shown a dense array of lenticular structures [exemplified with “yellow” arrows in Figs. 11.5a, b] after metallography, i.e., they exist in the undefomed images as well. It is speculated that these are twins on the surface layer [not of the family shown in Fig. 11.1b] that actually altered the twinning behavior on the surface. Hence, this aspect of the experiment will be reeavaluated using a different sample that is part of our future work.

Fig. 11.4 (a) Schematic view of sample gauge section; (b) εxx, εyy, and ωxy maps obtained from micro-DIC

68

S. C. Erman and C. C. Aydıner

Fig. 11.5 (a) Reference image and (b) load point A image of the frame shown in Fig. 11.4b; (c) the computed residual intensity field, ΔI

11.4 Conclusion In this study, the collaborative shear band structures instigated by the abrupt twinning phenomena were investigated on magnesium AZ31 via multichannel OM-DIC. The sample was intentionally designed with a four-cornered notch pattern to guide and overlap shear bands over a rolled sample with load axis coincident with RD. Both macro- and micro-DIC channels show that the sample deforms exactly as designed, i.e., with notch-to-notch strain localization bands in a cross-pattern. One arm of this pattern forms first (with an average of −2.26% strain that comes abruptly) and the other arm forms upon further loading intersecting the first band at the notch pattern center. The average strain at this overlap jumps to −3.78%. It is noteworthy that the observed cross-pattern of strain accommodation is fundamentally different from what would take place for an isotropic metal, which would instead show horizontal localization bands. Normally, the position of shear banding in rolled magnesium over the twin plateau is unpredictable, and this sample allows precise positional control of the shear bands. From an experimentation perspective, controlling the position of this abrupt phenomenon is significant for time-resolved experiments. It can also have component design implications for wrought magnesium with the common rolling texture. Acknowledgments  This work was supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK), grant no. 120N722.

References 1. Özdür, N.A., Üçel, I.B., Yang, J., Aydıner, C.C.: Residual intensity as a morphological identifier of twinning fields in microscopic image correlation. Exp. Mech. 61, 499 (2020). https://doi.org/10.1007/s11340-­020-­00672-­8 2. Agnew, S.R., Brown, D.W., Tomé, C.N.: Validating a polycrystal model for the elastoplastic response of magnesium alloy AZ31 using in situ neutron diffraction. Acta Mater. 54(18), 4841–4852 (2006). https://doi.org/10.1016/j.actamat.2006.06.020 3. AydIner, C.C., Bernier, J.V., Clausen, B., Lienert, U., Tomé, C.N., Brown, D.W.: Evolution of stress in individual grains and twins in a magnesium alloy aggregate. Phys. Rev. B Condens. Matter Mater. Phys. 80(2), 1–6 (2009). https://doi.org/10.1103/PhysRevB.80.024113 4. Drozdenko, D., Bohlen, J., Yi, S., Minárik, P., Chmelík, F., Dobroň, P.: Investigating a twinning–detwinning process in wrought mg alloys by the acoustic emission technique. Acta Mater. 110, 103–113 (2016). https://doi.org/10.1016/j.actamat.2016.03.013 5. Muránsky, O., Barnett, M.R., Luzin, V., Vogel, S.: On the correlation between deformation twinning and Lüders-like deformation in an extruded Mg alloy: in situ neutron diffraction and EPSC.4 modelling. Mater. Sci. Eng. A. 527(6), 1383–1394 (2010). https://doi.org/10.1016/j. msea.2009.10.018 6. Barnett, M.R., Nave, M.D., Ghaderi, A.: Yield point elongation due to twinning in a magnesium alloy. Acta Mater. 60(4), 1433–1443 (2012). https://doi.org/10.1016/j.actamat.2011.11.022 7. Kapan, E., Shafaghi, N., Uc̣ar, S., Aydıner, C.C.: Texture-dependent character of strain heterogeneity in magnesium AZ31 under reversed loading. Mater. Sci. Eng. A. 684(December 2016), 706–711 (2017). https://doi.org/10.1016/j.msea.2016.12.085 8. Mo, C., Kontsos, A.: Twinning contributions to strain localizations in magnesium alloys. Mater. Sci. Eng. A. 722(March), 206–215 (2018). https://doi.org/10.1016/j.msea.2018.03.024 9. Hazeli, K., Cuadra, J., Vanniamparambil, P.A.A., Kontsos, A.: In situ identification of twin-related bands near yielding in a magnesium alloy. Scr. Mater. 68(1), 83–86 (2013). https://doi.org/10.1016/j.scriptamat.2012.09.009

11  Coordinated Twinning Bands in Magnesium at the Existence of Stress Raisers via In Situ Microscopic Image Correlation

69

10. Anten, K., Scholtes, B.: Formation of macroscopic twin bands and inhomogeneous deformation during cyclic tension-compression loading of the Mg-wrought alloy AZ31. Mater. Sci. Eng. A. 746(August 2018), 217–228 (2019). https://doi.org/10.1016/j.msea.2019.01.033 11. Aydıner, C.C., Telemez, M.A.: Multiscale deformation heterogeneity in twinning magnesium investigated with in situ image correlation. Int. J. Plast. 56, 203–218 (2014). https://doi.org/10.1016/j.ijplas.2013.12.001 12. Shafaghi, N., Kapan, E., Aydıner, C.C.: Cyclic strain heterogeneity and damage formation in rolled magnesium via in situ microscopic image correlation. Exp. Mech. 60(6), 735–751 (2020). https://doi.org/10.1007/s11340-­020-­00612-­6 13. Chen, Z., Daly, S.: Deformation twin identification in magnesium through clustering and computer vision. Mater. Sci. Eng. A. 736(July), 61–75 (2018). https://doi.org/10.1016/j.msea.2018.08.083 14. Özdür, N.A., Aydıner, C.C.: Advance of collaborative twinning fields in magnesium AZ31 via the strain and residual intensity channels in microscopic image correlation. Conf. Proc. Soc. Exp. Mech. Ser. 2, 1–9 (2022). https://doi.org/10.1007/978-­3-­030-­86737-­9_1 15. Üçel, İ.B., Kapan, E., Türkoğlu, O., Aydıner, C.C.: In situ investigation of strain heterogeneity and microstructural shear bands in rolled magnesium AZ31. Int. J. Plast. 118, 233–251 (2019). https://doi.org/10.1016/j.ijplas.2019.02.008

Chapter 12

Determining the Onset of Transverse Cracking in a Woven Composite Using Digital Image Correlation Christopher S. Meyer, Bradley D. Lawrence, and Bazle Z. Haque

Abstract  Quasi-static tension testing of a single-layer composite reduces strain field complexity compared to multilayer composites in which strain fields are obscured by nesting, interpenetration, and overlapping of tows and laminas. Digital image correlation (DIC) may be used in single-layer tension tests to accurately reveal the complex strain fields of woven composites, and the time-resolved damage onset and evolution of transverse cracking. However, typical tensile specimens are 25.4 mm wide, which means only five yarns across the width for a standard 24 oz./yd2 fabric composite. Free edge effects can cause shear strain concentrations, which leads to higher edge stresses than toward the specimen center. Also, free edges can result in periodic, out-of-plane warping. To ensure the area of interest for DIC analysis is adequate to capture the strain behavior without including strain concentrations at the free edges, three specimen widths were tested: 25.4, 50.8, and 76.2 mm (1, 2, and 3 inch). The 50.8 mm width was found to be optimal. A partial-coverage DIC speckle pattern was used to observe surface strain and transverse crack evolution simultaneously. Onset of transverse cracking was found to occur at a local strain of 0.45%, which is consistent with experiments using acoustic emission to identify transverse crack onset. During tensile testing, the number of cracks in a transverse yarn unit cell increases sequentially. The first three of these cracks are termed primary, secondary, and tertiary transverse cracks in a unit cell. The onset of the first secondary transverse crack occurred at a local strain of 0.50%, and the onset of the first tertiary transverse crack occurred at a local strain of 0.74%. This chapter reports on tensile testing of three specimen widths and on the local and global strains during the evolution of transverse cracks. Keywords  Free edge effects · Plain weave composite · Transverse cracks · Digital image correlation · Mesoscale

12.1 Introduction In plain weave composites, tension and tension-shear-induced damage modes at the mesoscale include transverse-tow matrix cracking [1, 2]. Here, the mesoscale is defined as the length scale of a tow (or yarn) cross section. Transverse cracks propagate through the tow thickness and in the fiber direction along the length of the tow, which is shown schematically in Fig. 12.1. Under quasi-static tensile loading, transverse cracks occur in tows where the fiber direction is oriented 90° (“transverse”) to the direction of loading. Additional delamination cracks can occur between transverse tows and tows where the fiber direction is normal to the tensile loading (i.e., delamination between 0° and 90° tows). Quasi-static tensile testing of a single-layer composite reduces strain field complexity compared to multilayer composites in which strain fields are obscured by nesting, interpenetration, and overlapping of tows and laminas [3, 4]. Digital image correlation (DIC) can be used in single-layer tension tests to reveal the complex strain fields of woven composites, and high-­ resolution visualization techniques can be used to correlate physical damage with strain fields [4].

C. S. Meyer (*) U.S. Army Combat Capabilities Development Command Army Research Laboratory, Aberdeen Proving Ground, MD, USA Center for Composite Materials, University of Delaware, Newark, DE, USA e-mail: [email protected] B. D. Lawrence U.S. Army Combat Capabilities Development Command Army Research Laboratory, Aberdeen Proving Ground, MD, USA B. Z. Haque Center for Composite Materials, University of Delaware, Newark, DE, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M.-T. Lin et al. (eds.), Advancements in Optical Methods, Digital Image Correlation & Micro-and Nanomechanics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-17471-1_12

71

72

C. S. Meyer et al.

Fig. 12.1  Schematic of single-layer plain weave fabric composite (matrix fill not shown) illustrating tensile damage in tows aligned with tensile loading (0° tows) and transverse to tensile loading (90° tows). Damage includes transverse cracks in 90° tows and tow–tow delamination cracks between 0° and 90° tows

Composites are used in a variety of situations in which the onset and progressive evolution of damage is a critical concern. Therefore, this chapter aims to identify the local strain at the onset and evolution of transverse cracking. To do this, we conduct quasi-static tensile tests on single-layer, plain weave S-2 glass/epoxy composites and use DIC and digital imaging to study the onset and evolution of transverse cracking.

12.2 Material Vacuum-assisted resin transfer molding (VARTM) was used to produce single-ply composite panels from plain-weave (PW) S-2 glass fabric [5 × 5 tows/inch, areal density of one ply is 744 g/m2 (24 oz./yd2), AGY 463-AA-2BL, 30 ends] infused with SC15 epoxy resin (Kaneka Americas Holding, Inc.). Composite panels were produced between two glass tooling surfaces to ensure a uniform thickness and surface texture. The VARTM process is shown in Fig. 12.2. A two-part cure under vacuum was followed, first at 35 °C (95 °F) for 24 hours and then temperature ramped up at 0.5 °C/min to 115 °C (239 °F) and was held for 3 hours. A wet saw was used to machine tensile test specimens that were 25.4 cm (10″) long by either 2.54 cm (1″), 5.08 cm (2″), or 7.62 cm (3″) wide (straight sides, uniform width). Specimens were cut from several different panels, so thickness varied slightly. Dimensions are provided in Table 12.1.

12.3 Tensile Experiments Quasi-static tensile testing was conducted in accordance with ASTM D3039 [5]. Tests used an Instron 5985A test frame with a 250 kN load cell. All specimens were loaded at 2.54 in/in/min. Hydraulic grips enabled test specimens to be up to 7.62 cm (3″) wide. Hydraulic grips also provide controllable uniform clamping pressure, which facilitates testing plain weave specimens without the need for end tabs. End tabs are not required for fabric composites, according to ASTM D3039 (see 8.2.2.2). For example, Bergman et  al. [6] tested plain weave E-glass/epoxy specimens (8-ply, 2.65  mm thick, 25  mm wide, and 230 mm long) without end tabs. Initial testing of untabbed, straight-sided, 25 mm (1″) tensile specimens with hydraulic grips indicated a controlled grip pressure of 2000–2200 psi (13.8–15.2 MPa) was adequate. Post-test observations showed little or no catastrophic fiber damage to specimen gripped sections, though resin-rich surfaces were scored by the surface texture of the hydraulic grips (this will be evident later). Post-test thickness measurements indicated minimal crushing or thickness reduction. Failed specimens were inspected for axial splitting (due to specimen misalignment either in cutting/processing or installation into testing machine), grip failure (due to grip pressure or damage caused by grips), and edge delamination (not expected since single layer). Any tests found with these failure modes were discarded. Quasi-static, tensile stress–strain response of single-layer, plain weave S-2 glass/SC15 epoxy composites are provided in Fig. 12.3. Tensile properties are summarized in Table 12.1. Examining the tensile properties, the average stiffness and average failure stress for each of the three specimen widths can be seen to decrease with increasing size, which may point to the influence of size effect [7, 8].

12  Determining the Onset of Transverse Cracking in a Woven Composite Using Digital Image Correlation

73

Fig. 12.2  Vacuum-assisted resin transfer molding process used to manufacture 30.5 cm by 30.5 cm (12″ × 12″) single-layer, plain weave, S-2 glass/SC15 epoxy composites. Process shown (a) schematically and (b) photographically Table 12.1  Tensile test specimen dimensions and tensile properties Specimen no. P2-S1 P2-S4 P4-S5 P4-S6 P4-S7 P10-S1 P10-S2 P10-S3 P2-S2 P2-S3

Width, inch 1 1 2 2 2 2 2 2 3 3

Width, mm 25.3 25.6 50.8 50.8 50.8 50.4 50.9 50.9 76.2 76.1

Length, mm 254 254 254 254 254 254 254 254 254 254

Thickness, mm 0.83 0.92 0.85 0.86 0.84 0.82 0.84 0.84 0.84 0.87

Elastic modulus, GPa 13.4 13.3 12.2 11.9 12.3 11.7 12.1 12.1 11.6 11.2

Failure stress, MPa 579.5 532.8 525.7 524.0 551.3 541.0 531.9 501.9 528.9 472.8

Failure strain, % 4.3 4.1 4.6 4.4 4.5 4.8 4.5 4.3 4.6 4.6

74

C. S. Meyer et al.

Fig. 12.3  Tensile test engineering stress–strain results

12.4 Digital Image Correlation Free edge effects cause shear strain concentrations at the free edges, leading to potentially problematic DIC readings and possible premature failure. Free edge effects are due to the presence of adjacent layers of differing fiber orientation or material properties and the resulting three-dimensional stress state [9, 10]. The standard 24 oz./yd2 glass fabric used in the subject composites has five tows per inch, so the typical 1-inch-wide tensile specimen has only five tows across the width. The DIC region of interest (ROI) could be influenced by the two outer tows, which are affected by free edge effects. To investigate the influence of free edges, three tensile specimen widths were investigated: 2.54  cm (1″), 5.08  cm (2″), and 7.62  cm (3″). Inspecting the failure of a 1″ specimen, P2-S4, shows that the DIC surface strains can be correlated with high-resolution imaging and provides information about failure mechanisms and local strains. Figure 12.4 is specimen P2-S4, and the DIC shown is the frame immediately preceding failure. Local strains were identified just before catastrophic failure and the corresponding damage is shown. In the case of this 1″ specimen, two half-tows were on the free edges, so four full tows can be identified in the middle of the specimen. However, other specimens only have three full tows in the middle. Typical DIC surface strain fields are shown for the instant before failure for a 1″ wide specimen in Figs. 12.2 and 12.4” wide specimen in Fig. 12.5, and for a 3″ wide specimen in Fig. 12.6. The 2″ specimen width was selected for subsequent investigation of the local strain at transverse crack onset (Fig. 12.7). A single-camera (2D) DIC system was used with Correlated Solutions software. Camera was a Point Grey 2.3 MP with Nikon lens (f/3.7). Care was taken to align the single camera parallel to the specimen face. Single-layer plain weave composites can experience out-of-plane deformation as 0° tows straighten under tensile load, which pushes 90° tows out of the plane. However, the single-layer composites are very thin (see Table 12.1) and surface strains appear reliable with a one-­ camera system given. Nonetheless, uncertainty and error were estimated for strain measurement as follows. Error sources in 2D DIC strain measurement include out-of-plane motion, bias errors due to experimental setup and out-of-plane noise, and measurement errors due to imaging noise, rigid motion, and strain bias [11–13]. A conservative estimate of error attributable to the camera system is ≤  ±  100 με (microstrain) [11]. The maximum out-of-plane motion is estimated as the half-tow thickness 0.2 mm (~0.1875 mm) divided by the stand-off distance between lens and specimen (measured as 46 cm), i.e., = ± 437 με. 457 mm Note that out-of-plane motion does not appear until >2 mm extension, so including it before that extension (as in the following analysis) is conservative. Bias errors due to experimental setup are conservatively estimated as ≤  ±  100 με [12]. Measurement errors are conservatively estimated as follows: due to imaging noise, ≤ ± 1.5 με on 5000 με; due to rigid motion ≤ +0.05, −0.15 με on 5000 με; due to strain bias (stretch ratio) ≤ ± 30 με on 5000 με. Therefore, the total error on 5000 με is (conservatively) estimated as 5000 με ± 667 με or 0.5% ± 0.07% strain, which is a standard deviation in strain of about ±13%.

12  Determining the Onset of Transverse Cracking in a Woven Composite Using Digital Image Correlation

75

Fig. 12.4  Digital image correlation surface strain field (instant before failure) and high-resolution imaging of failed 1″ wide tensile specimen P2-S4

Fig. 12.5  Digital image correlation surface strain field (instant before failure) of 1″ wide tensile specimen P2-S4. Tensile strain, εxx, is applied in the long dimension

76

C. S. Meyer et al.

Fig. 12.6  Digital image correlation surface strain field (shortly before failure) of 2″ wide tensile specimen P4-S5. Tensile strain, εxx, is applied in the long dimension

Fig. 12.7  Digital image correlation surface strain field (instant before failure) of 3″ wide tensile specimen P2-S3. Tensile strain, εxx, is applied in the long dimension

12.5 Combined Visual Inspection and Digital Image Correlation Digital image correlation (DIC) makes use of a speckle pattern, which is used by correlation software to calculate surface strains. Unfortunately, the speckle pattern usually obscures the damage evolving on the specimen surface. To observe the evolution of transverse cracking on the surface of the single-layer, plain weave glass/epoxy composite tensile specimens, a combined visual/DIC speckle pattern was devised, an example of which is shown in Fig. 12.8. Combined visual/speckle marking was applied with a fine point black permanent marker. First, the specimen was backlit to identify the 0° tow

12  Determining the Onset of Transverse Cracking in a Woven Composite Using Digital Image Correlation

77

Fig. 12.8  Combined visual inspection and digital image correlation speckle pattern on 2″ wide tensile specimen of a single-layer, plain weave S-2 glass/SC15 epoxy composite. Speckles applied by fine point black permanent marker. P10-S1 is shown

positions. To avoid edge effects, the outermost 0° tow positions were identified and marked off on both sides, as can be seen in Fig. 12.8. Similarly, two tows were marked off from the planned grip position (specimens were 10″ in length with 1.5″ gripped on each side). Then the remaining eight 0° tows were divided in half in the 0° direction (direction of tensile loading) and in the 90° direction. This approach assumes that the unit cell strains are approximately symmetrical about the 0° center line. For a single-layer plain weave composite, without nesting, interpenetration, and overlapping of adjacent tows and laminas, this is a reasonable assumption. Evidence for this can be seen in the relative uniformity of the strain field in Fig. 12.6. The tensile specimen was subdivided into regions as seen in Fig. 12.8 with the goal of having a region of DIC surface strains corresponding to a region of visual inspection. The tensile test on P10-S1 was conducted as described earlier, and images similar to Fig. 12.8 were recorded at 0.6 second intervals. Then the images were inspected to identify the onset of transverse cracking. The first (visible) transverse crack formed at a time of 10.8 s and is shown in Fig. 12.9. At this time, the strains are still developing toward the middle of the specimen, and some strains are within the error indicated by earlier estimates (i.e., εxx 2.2 mm)

Time (s) 10.8 14.3 15.4 18.5 26.6 45.5 206.3

Local strain (%) 0.45% ± 0.06% 0.50% ± 0.07% 0.74% ± 0.10%

6.34% ± 0.82%

Global strain (%) 0.15 0.23 0.27 0.33 0.52 0.97 4.79

Extension (mm) 0.26 0.41 0.48 0.59 0.93 1.72 8.52

Load (kN) 0.958 1.414 1.538 1.889 2.784 4.877 22.892

12.6 Conclusion The material investigated in this work was single-layer, plain weave S-2 glass/SC15 epoxy composites. Tensile tests were conducted with 1″, 2″, and 3″ widths. A unique combination of visual inspection and digital image correlation was developed in this work. This visual/DIC inspection was used to identify the local strain at the onset of transverse cracking. It was found that transverse cracks initiate at local strains of 0.45 (one transverse crack), 0.5 (two transverse cracks), and 0.74 (three transverse cracks). Acknowledgments  Research was sponsored by the U.S.  Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-12-2-0022. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Army Research Laboratory or the U.S. Government.

80

C. S. Meyer et al.

Fig. 12.12  Visual/DIC of the instant before failure (top) and after failure (bottom). P10-S1 is shown at ~206 s. Local maximum strain at tow failure was εxx = 6.34% ± 0.82%

Fig. 12.13  Engineering stress–strain response shown with local strains corresponding to transverse crack onset

12  Determining the Onset of Transverse Cracking in a Woven Composite Using Digital Image Correlation

81

References 1. Meyer, C.S., Haque, B.Z.G., O’Brien, D.J., Getinet, N., Yu, J.H., Bonyi, E., Aslan, K., Gillespie Jr., J.W.: Mesoscale ballistic damage mechanisms of a single-layer woven glass/epoxy composite. Int. J. Imp. Eng. 113(August 2017), 118–131 (2018) 2. Bonyi, E., Meyer, C.S., Kioko, B., Adesina, O., Lansiquot, C., Onuk, Z., O’Brien, D.J., Haque, B.Z., Gillespie, J.W., Aslan, K.: Assessment and quantification of ballistic impact damage of a single-layer woven fabric composite. Int. J. Damag. Mech. 28(2), 249–269 (2019) 3. Lee, J.R., Molimard, J., Vautrin, A., Surrel, Y.: Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension. Compos. A: Appl. Sci. Manuf. 35(7–8), 849–859 (2004) 4. Lomov, S.V., Ivanov, D.S., Verpoest, I., Zako, M., Kurashiki, T., Nakai, H., Molimard, J., Vautrin, A.: Full-field strain measurements for validation of meso-FE analysis of textile composites. Compos. A: Appl. Sci. Manuf. 39(8), 1218–1231 (2008) 5. D3039M/D3039M-14, Standard test method for tensile properties of polymer matrix composite materials, Annual Book of ASTM Standards, vol. i, no. Reapproved, pp. 1–13, 2013 6. Bergmann, T., Heimbs, S., Maier, M.: Mechanical properties and energy absorption capability of woven fabric composites under ± 45 ° off-­ axis tension. Comp. Struct. 125, 362–373 (2015) 7. Zweben, C.: Is there a size effect in composites? Composites. 25(6), 451–454 (1994) 8. Bazant, Z.P., Daniel, I.M., Li, Z.: Size effect and fracture characteristics of composite laminates. J. Eng. Mater. Technol. Trans. ASME. 118(3), 317–324 (1996) 9. Owens, B.C., Whitcomb, J.D., Varghese, J.: Effect of finite thickness and free edges on stresses in plain weave composites. J. Compos. Mater. 44(6), 675–692 (2010) 10. Heinzlmeier, L., Sieberer, S., Kralovec, C., Schagerl, M.: Implications of free-edge effect at thin plain-woven carbon fiber reinforced plastic laminates with out-of-plane waviness under cyclic loading. J. Compos. Mater. 55(27), 4097–4108 (2021) 11. Reu, P.L., Sweatt, W., Miller, T., Fleming, D.: Camera system resolution and its influence on digital image correlation. Exp. Mech. 55(1), 9–25 (2015) 12. P.  L. Reu, Quantifying errors in image based measurements DIC has changed the way we do experimental, Albuquerque, NM, SAND2015-7021C, 2015 13. Wang, Y., Lava, P., Reu, P., Debruyne, D.: Theoretical analysis on the measurement errors of local 2D DIC: part II assessment of strain errors of the local smoothing method-approaching an answer to the overlap question. Strain. 52(2), 129–147 (2016) 14. Doitrand, A., Fagiano, C., Carrère, N., Chiaruttini, V., Hirsekorn, M.: Damage onset modeling in woven composites based on a coupled stress and energy criterion. Eng. Fract. Mech. 169, 189–200 (2017)