Advances in Structures Analysis [1 ed.] 9783038135333, 9783037851593

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Advances in Structures Analysis

Edited by Moussa Karama

Advances in Structures Analysis

Special topic volume with invited peer reviewed papers only.

Edited by

Moussa Karama

Copyright  2011 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of the contents of this publication may be reproduced or transmitted in any form or by any means without the written permission of the publisher. Trans Tech Publications Ltd Kreuzstrasse 10 CH-8635 Durnten-Zurich Switzerland http://www.ttp.net Volume 61 of Applied Mechanics and Materials ISSN 1660-9336 Full text available online at http://www.scientific.net

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Preface

Structural materials are fundamental in human history. Their use in airframe structures has steadily increased since the 1970s. Currently, the applications have expanded to include empennage, fuselage, wing and dynamic components of small airplanes, transport aircraft and rotorcraft. Even though, the aircraft industry is very conservative in the adoption of new designs and technologies and when new aircraft are introduced, they tend to build heavily upon past designs, introducing only incremental updates in technology. However, the global world market has accelerated the race over the last two centuries, with progress in structural materials pacing improvements in living. Advances in aluminum alloys for airframes and super alloys for engines led to the current era of mass international travel by jet aircraft. Advances in high temperature structural materials are leading the way to better engines and materials processing. Structural composites have revolutionized pleasure boating and impacted many types of sports equipment. Advanced structural composites have been essential to space vehicles and offer promises of lighter, more efficient materials for many applications.

Various research subjects fall under the theme of Structure: material damage leading to crack growth and/or fatigue of structures under dynamic loading or impact, durability and reliability of structures, numerical simulations and experimental work involving large deformations and impact associated with suitable experimental techniques. Concerning experimental studies, many non destructive tests have been developed in the past recent years in order to deliver more accurate data. Therefore, papers dealing with processing, characterization and determination of physical properties of materials and structures are welcome in this topic. Finally, this also includes considerations about computing softwares, numerical models and advanced algorithms for simulation. Specific applications of modeling and simulation in science and engineering, with relevant applied mathematics are also included in this topic.

Prof. Moussa KARAMA Guest Editor

Table of Contents Preface Towards an Expert System that Aids in the Diagnosis of Concrete Structures A. Messabhia and N. Hassounet A FEM Model to Analyze the Structural Mechanical Problem in an Electrostatically Controlled Prestressed Micro-Mirror D. Popovici, V. Paltanea, G. Paltanea and G. Jiga Optimization by the Reliability of the Damage by Tiredness of a Wire Rope of Lifting A. Meksem, M. El Ghorba, A. Benali and A. El Barkany Thermal Buckling of Simply Supported FGM Square Plates M. Bouazza, A. Tounsi, E.A. Adda-Bedia and A. Megueni Analytical and Finite Element Analysis for Short Term O-Ring Relaxation M. Diany and H. Aissaoui A New Methodology for an Optimal Shape Design W. El Alem, A. El Hami and R. Ellaia A Multi-Scale Analysis of Materials Reinforced by Inclusions Randomly Oriented in the Ply Plane E. Lacoste, S. Fréour and F. Jacquemin Experimental Study of the Short-Term Creep Behavior of CFRP Strengthened Mortar under Compressive Loading F. Khadraoui and M. Karama Contribution of AFM Observations to the Understanding of Ni3Al Yield Stress Anomaly J. Bonneville, D. Charrier and C. Coupeau The Non Destructive Testing Methods Applied to Detect Cracks in the Hot Section of a Turbojet S. Bennoud and Z. Mourad Simulation of Thermo Mechanical Behavior of Structures by the Numerical Resolution of Direct Problem B. Aouragh, J. Chaoufi, H. Fatmaoui, J.C. Dupré, C. Vallée and K. Atchonouglo Low Temperature Sintering and Characterization of MgTiO3 M. Aliouat, B. Itaalit, N. Amaouz and A. Chaouchi

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Applied Mechanics and Materials Vol. 61 (2011) pp 1-8 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.1

Towards an Expert System that Aids in the Diagnosis of Concrete Structures Ali Messabhiaa, Naceur Hassounetb Route de Constantine, 12002 Tebessa, Algeria a

[email protected], [email protected]

Key words: Diagnosis, Concrete degradation, Knowledge base, Expert system,

Abstract. The concrete works, during their exploitation are exposed to operation or environmental conditions which can inflict certain degradations to them. Establishing a good diagnosis needs a particular knowledge of the behavior of concrete when exposed to aggressive agents, especially the mechanical behavior. The determination of the causes of degradation is a complex matter and the interaction between different pathologies makes it more difficult. The choice of the materials and techniques used to repair the damages is also very important to succeed in the intervention. Our objective is the development of a knowledge base which will be used as a base to formulate the rules used to organize, rationalize and optimize the diagnosis process of these pathologies. The knowledge base gathers the majority of known phenomena related to the degradation of the concrete works. It is organized according to a reasoning which enables to describe or to identify most damages and degradations of chemical or mechanical origins or of implementation. Used with an expert system it will enable us to evaluate the importance of the damage, the need to intervene and finally to propose recommendations about the appropriate procedures and materials needed to repair the damage. Introduction Durability is one of some recent demands of works owners. Diagnosis and renovation take an increasingly important part of the activities in the construction sector. They represent the recent orientations which allow the preservation of works [1]. Establishing a diagnosis needs a big experience and a multidisciplinary approach in the fields of control and pathology of materials. This present approach aims to gather the knowledge, the experience and the knowhow of experts in a tool which enables to answer these questions. The main objective is the elaboration of a knowledge base which will be used as a basis for formulating the process optimization rules of the establishment of a degradation diagnosis of reinforced concrete works. The knowledge base contains most known phenomena linked to degradation of concrete works. It is organized referring to a reasoning allowing the description or the identification of most damages and degradation phenomena of any origin: chemical, mechanical or of implementation. Identification and Classification of Degradations The specific determination of causes of an unspecified deterioration of the concrete is a complex subject. This is explained by the lack of knowledge and the complexity of the phenomena which affect this material, to their evolution in time, as well as their concomitance. On this subject, the majority of the related literature is limited to quote and define briefly the various damages of the concrete [2]. Indeed, the first stage is to define the principal families of degradations which affect the concrete, including: Cracking, Distortion and movement, Infiltrations and deposits, Irregularities of surface, Losses of mass These degradations, according to their origin are gathered into families, presented in Table 1.

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Table 1 - Family of degradation Degradation Family Included Degradations Thermal withdrawal, Withdrawal by auto-desiccation, Drying Chemical Withdrawal, Carbonation, Chlorides attack, Sulfuric attack, Alkalireaction, Efflorescence, Discoloration, Exudation, Incrustations Settlement, Drying of the ground water, fire, vibrations, earth quake, Mechanical Distortion and movement, Infiltration and deposits Bug Holes, Honeycombs, apparent cold Joints, Irregular surfaces, Dust Surface Irregularities and debris (waste) It is clear that the evaluation of the relevance of an intervention is not set according to a uniform and homogeneous approach, but quite rises as case by case. The effective diagnosis passes by a good interpretation of the pathology of deterioration or degradation, as well as by taking into consideration the conditions to which the structure is subjected. The interaction between various pathologies makes the judgment sometimes more difficult. The symptoms observed often lead to several conclusions. It is thus necessary to define criteria of characterization for each type of degradation [1]. These criteria associated with other data can quickly suspect the origin of the disorder or to exclude definitively some pathology. Criteria of classification of damage and type of measurement:A first stage of the approach of rationalization that is proposed consists in classifying damage according to their form and their value. This approach requires on one hand the definition of the type of measurement according to each damage and on the other hand to establish intervals of values. These criteria will allow to unify measurements and to give a gravity index of damage [3] (see the Table 2). Table 2 - Example of characterization criteria of degradation Type of Classifications of Damage Form measurement Damage Vertical, horizontal, - < 2 [mm] Cracking tilted, polygonal, form Openings - 2 to 5 [mm] of grid - >5 [mm] - 0 to 30 [mm] Chipping or Erosion - 30 to 60 [mm] Linear, surface Average depth of Surface - 60 to 120 [mm] - >120 [mm] Criteria of classification of degradations: From these criteria of characterization, according to the same reasoning, we propose to establish a gravity index of damage allowing the classification of degradation and to fix intervals of judgment according to their importance (see Table 3). These intervals will constitute symptoms enabling to lead to suspect certain types of degradation and thus delimiting the investigation field and focusing research on a quite particular family of pathologies [4,5].

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Degradation

Chipping

Cracking

3

Table 3 - Example of degradation classification Gravity Index Description Loss of mortar on the surface until a depth of 6 to 10 [mm], Light leaving some coarse aggregates discovered Loss of mortar on the surface until a depth of 6 to 10 [mm], Medium leaving some coarse aggregates discovered Loss of mortar on the surface until a depth of 11 to 20 [mm], Important leaving coarse aggregates discovered which some detached aggregates Very important Loss of mortar on the surface until a depth of 21 to 25 [mm] Light Skinny cracks: width < 0.1 [mm] Medium Narrow cracks: width from 0.1 to 0.3 [mm] Important Average cracks: width from 0.3 to 1 [mm] Very important Broad cracks: width > 1 [mm]

Classification Criteria of the degradations aspect: The aspect of the damage can constitute an additional index of recognition. It depends closely on the origin of degradation. The damage caused with the concrete always does not have the same aspects; each family of pathologies causes quite particular forms and effects on the works (see Table 4). The aspect of degradation will constitute, therefore a complementary print which allows to limit the investigation fields and to quickly converge towards a solution. Table 4 - Examples of pathology and aspect of degradation Pathology Aspect of degradation Deep and open cracks, directed and short cracks, damaged horizontal Sweating surfaces Cracks and or bursting of the coating concrete, efflorescence, spills, or Carbonation tasks of rust. Chlorides Cracks and or bursting of the coating concrete, efflorescence, spills, or attack tasks of rust, shot (pitting) corrosion Cracks in networks, of depth higher than 10 [cm], cracks directed according to the distribution of the reinforcements, deplanation of the Alkali-reaction facings with exudations, visible deformations in the mass works, bursting located in small cones shape. Sulfuric attack Cracks, Chipping, Disintegration Influence criterion of the ambient environment: The ambient environment is generally the source of the damage of chemical origins which the works can undergo. It can cause, or at least favors failing of a quite particular phenomenon of degradation. in addition ,It can prevent other phenomena to occur. We cannot talk, for example, about efflorescence phenomenon in a hot and dry atmosphere; on the other hand it is evident to suspect an attack of chloride in marine or saline environment. The knowledge of the type of environment is a key information to carry out a coherent reasoning and to establish a fast diagnosis. Table 5 gathers certain pathologies of chemical origin and describes their favorable environment.

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Pathology

Carbonation

Chlorides attack

Efflorescence

Table 5 - Effect of the ambient environment. Favorable environments Description Zones with strong density of population, without Urban atmosphere industrial merger, moderate, and contamination. Industrial zones or their surroundings located Industrial atmosphere under the governing wind, strong contamination. Low rainfall, works safe from the rain. Moisture < Dry atmosphere 80%. Marine and maritime zones strongly being subject to the influence of the sea depending on the direction of the governing wind. More or less Maritime atmosphere strong contamination. Sheltered surfaces of the rain where the chlorides can stick. Zones with strong chloride concentration. Saline atmosphere Surfaces exposed to salts of deglazing. Mixed atmosphere Zone near coasts and zones of industrial merger or their surroundings located under the governing wind. Cold and wet atmosphere Presence of rain, snows, fog etc.

Age criterion of the appearance of the first degradation: The age criterion of the work or the age of the appearance of degradation plays an important role in the investigation of the causes of the damage. Certain phenomenon appear only shortly after the implementation of the work, others can appear only after a well-known duration. The evolution of degradation also depends on the phenomenon type. Some degradation evolves in time and others remain stable. It is thus relevant to classify pathology according to the age of the work while noting their evolution in order of being better informed to pronounce a judgment [3] (See Table 6). Table 6 - Extract influences of the ambient environment. Pathology Age Evolution in time Sweating 1 to 3 hours No Plastic withdrawal 1/2-hour with a few hours after casting No Carbonation 2 years Yes Chlorides Attack 2 years Yes Alkali-reaction 2 up to 10 years Yes Sulfuric attack 2 years Yes Choice of material and repair techniques. Repair materials: For each pathology, it is needed to define the most appropriate material or materials used to repair the damage. This material must be compatible with the damaged concrete and durable with regards to the environmental conditions. The different repair products of damaged concrete are: either mortars or concretes based on hydraulic binders or modified hydraulic binders and synthetic resins. Repair methods: Repairing the surface of concrete in order to apply a repair product includes all the steps that follow the elimination of the damaged concrete. It depends on the type of repair and must be followed with or without the elimination of damaged concrete. A good repair gives a dry flat and clean surface with no dirt, dust, oil nor fat. The elimination of these surface contaminants allows a direct contact between the surface and the repair products and thus enables a better adhesion of the applied product (Table 7).

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Table 7 - Types of repair depending on the degradation. Type of repair Repair methods Pickling, mechanical cleaning, scarification, blast cleaning, Surface repair sandblasting, etching. Elimination of dirt opening the cracks and removal of the non Cracks repair adherent parts. For each type of pathology, the most appropriate remedy must be defined as well as that depending on the materials or repair (Table 8) [6]. Disorders Bubbling Flaking

Cracks

Table 8 - Types of repair depending on the degradation. Treatment and repair Refit the surface with a suitable product Treating corroded reinforcement, compensate for the reduction of the diameter of reinforcement, refill with mortar or concrete made of modified hydraulic binders. Treating crack open with a soft seal Treat corroded frames. Refill with mortar and additional frames An injection into the crack and strengthen where necessary with TFC

The Knowledge Base The knowledge base consists of a set of rules, called production rules. They define knowledge and reasoning relating to the identification of pathology or the causes of the damage. From the criteria defined above, descriptive memos of the whole pathologies are established. These memos gather information necessary to recognize each type of pathology. Associated to the data collection, these memos allow rules to establish a deductive reasoning of recognition of the pathologies responsible for degradation. It sometimes happens, not to give a conclusion of 100% of certainty or to ask for complementary tests to confirm the deductions. The reasoning allows guiding towards a final choice in a homogeneous and coherent way. The knowledge base represents, therefore the whole of knowledge and reasoning relative to each degradation type. It consists of readable and independent rules of the system into which it will be integrated. Each rule is compound of two parts: • Premises which represent the tests of conditions (given or deduced) • Conclusion which can be an intermediate result or a final conclusion. Example of rule IF Cracks Aspect degradation = And A grid Crack-form = THEN Withdrawal; Degradation-origin = The premises and the conclusions handle objects representing the work or the degradation. These objects are either data taken from sites or intermediate deduction engendered from rules. These objects represent the data structure of fact base (See Table 9).

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Object

Age - work

Disorder - aspect

Environment type

Origin-disorder

Table 9 - Extract of the data structure Type Values - After dismantling (3 mn to 6 hrs) - 1 day - 3 weeks - Few weeks Data - Few months - > 2 years - Fissure, Bug Holes, - Localized accumulation of fine gravel, - Visible drawing of the reinforcements (steel), Data - Spills, break, bursting of coating concrete , - Efflorescence, - Urban, rural, Data - Industrial, mixed, - Saline, marine - Withdrawal, bug holes, - Nest rock, spectrum of - Reinforcements, bad, Deduced - Handling, carbonation, - Tackle chlorides, insufficient resistance

The writing of the rules obligatorily passes by a preliminary analysis of collected knowledge then the formulation of the reasoning by crossing the following stages: - Writing a rule for each type of degradation; - Extracting the handled objects; - Determining the possible attributes and values for each object; - Gathering pathologies according to certain common symptoms; - Gathering pathologies in families; - Finding the common premises for each family; - Simplifying the rules according to these families; - Supplementing the reasoning by convergence rules towards a result; - Reformulating the structural scheme of the data. Reasoning The reasoning analysis with the diagnosis presents the difficulty of treating algorithmic consequences. The difficulty lies in the diagnosis operation which is a smart act (logical reasoning) and which does not correspond to any mathematical algorithm: the expert has a bunch of information which, combined with his knowledge of the process, will enable him to lead to a diagnosis. This task is thus hard to automate by traditional techniques [3]. The reasoning is based on the steps followed usually by the experts. These steps were based on the analysis of classification and identifying criteria of degradations quoted above and formalization of knowledge. The reasoning must be able to establish diagnoses from the facts that can be detected in the description of a problem. The writing of the rules is established in two stages. The first stage consists to lay down specific rules to each pathology without any reasoning. Each rule describes the various signs characterizing the pathology. Under this aspect we developed about thirty rules gathering the principal degradations known for the concrete works. The second stage consists in converting this information, in a base of rules based on a deductive reasoning proceeding by elimination or delimitation of the investigation fields. It consists in gathering, by families, pathologies characterized by common symptoms. This regrouping makes it possible, from the data, to focus on a well defined family of pathologies.

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Example: Rule 1: Aspect – degradation = Cracks IF Crack-form And = A grid THEN Degradation-origin = Withdrawal; That means: if the aspect of degradation is cracks and these cracks are in the shape of grid we can conclude that degradation belongs to the family of the withdrawals. What also means that from two criteria we delimited the investigation fields to only one family of pathology? The reasoning is carried out forward in-depth by laying down the rules allowing to break up the families to differentiate these pathologies based on the criteria characterizing subgroups of pathologies. Rule 2: Degradation-origin IF Crack-depth Crack-opening And THEN Degradation-origin

= = =

Withdrawal; Deep; Not plastic-withdrawal.

It means that if degradation belongs to the family of the “withdrawals” and observed cracks are deep and broad then the withdrawal is not of plastic origin (thermal withdrawal or by autodesiccation). By following the same reasoning (rule 3 or 4) until the convergence of the result. Rule 3: IF And And THEN

Rule 4: IF And And And And THEN

Degradation-origin age-work Type-element Degradation-origin

= = = =

Gradient Temperature

=

Degradation-origin Environment-type Exposure Component Concrete Disorder

= = = = = =

Not plastic withdrawal; (1day-3 weeks); thick; Disorder thermal withdrawal. Complementary criteria to check. Raised (high), Climate= Very hot

In (chloride attacks, carbonation); Urban, industry; Safe /sheltered; Protected from the rain; without chloride; Carbonation 100% (To confirm phenolphthalein).

by

test

with

the

In this second stage the number of rules is increased, but each rule represents less premises and consequently more facility in reasoning and execution time, the Knowledge base is thus composed of a number of facts (predicates) and of some inference rules (production rules) which constitute a sequence of reasoning allowing the establishment of a coherent diagnosis. This diagnosis is based on the available knowledge to build a correspondence table enabling to effectively associate the observations to the corresponding diagnoses. In automatic mode the inference engine connects the rules either by front Chaining or Back chaining: Front chaining: In the mode of front chaining, the inference engine starts from facts to arrive to the aim, i.e. it selects only the rules of which the conditions of the left part are checked, then applies one of these rules which adds facts to the base. This process is reiterated until there are no more applicable rules or the aim is reached. The effectiveness of the inference engine lies in the relevance of the decision taken (elected rule) during the phase of choice. Front chaining proceeds by irrevocable strategy.

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Back chaining: The system starts from the goal and tries to go up to the facts to demonstrate them. The selected rules are those whose right part corresponds to the required aim. The unknown conditions (left part) of these rules become as many sub-goals to demonstrate. This process is repeated until all the sub-goals are demonstrated. The aim is then reached or until there are no more appropriate rules to be selected. In this case, the system can ask the user to solve one or more subgoals and the process starts again. The failure occurs when the system cannot select any more rules nor raise any question to the user. The back chaining proceeds in strategy by attempts. The inference engine operates then a backtracking to contest the application of a rule which leads to a failure and to test a rule drawn aside previously. Conclusion The determination of the degradation causes of the concrete is essential to succeed during the interventions in terms of quality, durability and profitability. The process leading to the source of the problem is included in a tool of adapted diagnosis. This way of proceeding enable thereafter to improve the choice of methods and products the most able to fill deficiencies caused by the damage on the work. The developed knowledge base contains the main part of the elements relating to the degradation of the concrete works. This organization allows systematic for diagnostic concrete works and consequently a better interpretation of the pathology of degradation since the various conditions to which the component is subjected are taken into account. It allows the experts of concrete to quickly find a coherent answer on the origin of degradation. It especially allows less qualified technicians to come to a conclusion about the more common diagnoses of the concrete works. It can also constitute a technical and didactic support. This knowledge base will confirm these assets once coupled to an inference engine to constitute an expert system of diagnosis and assistance to the improvement of durability of the concrete works. The possibilities of these systems allows the graphic exploitation of pathologies prints and the extension by a repair module suggesting solutions and recommendations relating to the specification of materials as well as methods of repairs, rehabilitation and maintenance of concrete. It is about developing qualitative methods which help to analyze the degradations of the concrete works, intended for the building owners. The objective is also providing operational tools to the specialized engineers, to assist them in their missions of curative diagnosis [2] [5], evaluation of the works state and forecast of necessary reparations. References [1] Fargeot B., Mathieu G., Sari J., Pathologie, diagnostic et réparation des ouvrages en béton de stockage et de transport des liquides, 14eme rencontres universitaires de GC (Clermont Ferrand, 9-10 mai 1996. 1, COS'96 [2] Bruno Godart, Les Techniques d’auscultation des ouvrages en Béton Armé, Laboratoire Central des Ponts et Chaussées, Colloque sur les OA, LE PONT, Toulouse, 19 & 20 Octobre 2005. [3] Mosser, André, Outil d'aide à la gestion des interventions sur les barrages en béton, Thèse université de Laval (Canada) ,2004. [4] Saleh, K., Mosser, A., Chekired, M., Gagnon, J., Rivest, M. Indice d’Endommagement: Classification et priorisation des travaux de réfection des barrages en béton, Rapport IREQ2003-112C, Hydro-Québec, 2003. [5] Peyras, L. , Royet, P. , Boissier, D. , Vergne, A. Diagnostic et analyse de risques liés au vieillissement des barrages - revue Ingénieries - E A T, 38,3-12, 2004. [6] CNRC du Canada, solution constructive n24, ISSN1206-1239, Déc 1998.

Applied Mechanics and Materials Vol. 61 (2011) pp 9-14 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.9

A FEM model to analyze the structural mechanical problem in an electrostatically controlled prestressed micro-mirror D. Popovici1, a, V. Paltanea1, b, G. Paltanea1, b, G. Jiga2, c 1

Faculty of Electrical Engineering, POLITEHNICA University from Bucharest, 313 Splaiul Independentei, Bucharest, Romania 2

Faculty of Engineering and Management for Technological Systems, POLITEHNICA University from Bucharest, 313 Splaiul Independentei, Bucharest, Romania a

[email protected], [email protected], [email protected]

Key words: numerical analysis, prestressed micro-mirror, deflection angle, MEMS device

Abstract. In many industrial and biomedical applications (laser scanning displays, optical switch matrices and biomedical imaging systems) the sensing and actuation components are realized using micro-mirrors fabricated by MEMS technology. In this paper is evaluated, through numerical methods, the structural mechanical properties of the actuation mechanism of a ring shape micro-mirror. For the lift-off of the structure there are used four springs simulating a prestressed cantilever beam. Introduction In the last decades the development of Micro Electro Mechanical Systems (MEMS) technology determined a huge step forward in automotive, communication and medical industries. By reducing size and mass of devices through MEMS technology, the performances of accelerometers, massflow sensors, bio-chips, RF devices have been improved [1]. Popular MEMS devices for optical applications are optical switch arrays for communications [2, 3], optical coherence tomography (OCT) for an endoscope [2, 4], confocal laser scanning microscopy (CLSM) and digital micro-mirror devices for Digital Light Process (DLP) projection [2, 5]. The actuation of the MEMS micro-mirrors is done by electrostatic, magnetic, thermal or piezoelectric mechanism. The electromagnetic actuation [2, 6, 7] imposes that the mirror should be deflected by using Lorentz force, in one case by moving a patterned coil placed in the excitation area of an external magnetic field, or by attracting/repelling an magnetic material attached to the mirror. Through this method, a scan angle of 8° at an intensity of 0.75 mA applied current could be obtained [8]. The thermal actuation main advantage consists in a simple manufacturing method but the presence of a high power consumption and slow response time represents a great disadvantage [9, 10]. The micro-actuator uses thermal expansion, due to Joule effect. In piezoelectric actuation the physical deformations are obtained by applying an electrical voltage. This method uses of very low operation voltages (3 to 20 V), has a good linearity and allows quick switching time (0.1 to 10 ms) [11, 12]. The fastest response time (less than 0.1 ms) is achieved with the help of an electrostatic actuation. This method has also low power consumption and easiness in integration with the electrical control systems. Despite suffering from the pull-in effect, nonlinear behavior and high operating voltages, the electrostatic actuation is one of the preferred choices for micro-mirror actuation [2, 13]. The operational voltage of the micro-mirror can be reduced when the spring stiffness is reduced and the angular deflection increases. This implies also that the natural frequency of the mirror decreases, reducing thereby operational bandwidth [2, 14]. One method of creating spring-like structures or inducing curvature in plate structures is to cover materials onto a substrate such that the layer should have a residual stress after the coating process. This process can control the different prestressed levels, which can be compressive or tensile. Using these approaches the

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device manufacturers can create spring-loaded micromechanical structures that can lift-off the substrate when deliberately undercut by an etchant. Geometry description of the numerical problem The size and geometry of the micro-mirror are determined by the diameter of the optical beam as well as its applications. In this paper the micro-mirror, subjected to a 3D structural numerical analysis, has two layers of cylindrical shape of a radius r = 0.5 mm and the height h = 20 µm as presented in Fig. 1a. In diametrically opposing direction there are four prestressed plated cantilevers (red areas in Fig. 1b) that act as a spring-loaded micromechanical structures.

a)

b)

Figure 1. The 2D (a) and 3D (b) representations of the geometrical model of the micro-mirror (blue area) with four prestressed springs (red layers) (the torsion bars have a length of 0.7 mm and a width of 0.1 mm). Fixed boundary constraints are placed on the terminal areas of the cantilevers. On the four bottom domains of the springs there are imposed a negative initial normal stress, on x-axis and yaxis, of σ = - 0.8 GPa. The superior layers are subjected also to a positive initial normal stress of σ = 0.8 GPa (see Fig. 2).

Figure 2. Fixed boundary conditions on the far edge of the springs-like structures (black areas) with the imposed negative (blue layers) and positive (red layers) initial normal stress. The micro-mirror is assumed to be made of polysilicon, that has a Young’s modulus of E = 160 GPa, Poisson’s ratio of v = 0.22 and density of ρ = 2330 kg/m3 [2]. The spring cantilevers are made of aluminum, for the first part of the analysis, with the following parameters: E = 69 GPa,

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v = 0.33 and ρ = 2730 kg/m3 and, in the second part, there are made of structural steel (E = 205 GPa, v = 0.28 and ρ = 7850 kg/m3). To avoid obtaining a very large unstructured tetrahedral mesh, because of the very thin 3D layered structure, it was generated, as first step, a 2D triangular mesh and then, as second step, it was extruded into 3D to produce 900 prism elements. Displacement and deflection static analysis It is analyzed the lift-off of the polysilicon micro-mirror that can be obtained by using prestressed cantilevers that are made of aluminum or structural steel. The results are presented in Fig. 3.

a)

b)

Figure 3. Total displacement (lift-off) of the micro-mirror in case of aluminum (a) and steel (b) cantilever springs for 0.8 GPa initial normal stress. The aluminum made springs produce a lift-off of the mirror much higher than in the case of the steel cantilevers by approximately 1.7 µm. This is due because the steel it is harder than the aluminum and deforms less.

Figure 4. Comparison between the micro-mirror total displacement obtained in the case of aluminum and steel springs (the lines are ploted along a diametral line between two springs).

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One can observe that there is almost no curvature of the micro-mirror surface. By relaxing one of the plates of the springs structure the angle of orientation of the micro-mirror can be modified (Fig. 4). The use of prestressed spring-like structures made of steel generates a higher angle of deflection than in the case of aluminum ones.

Figure 5. Colour map of the von Mises stress distribution in the aluminum (left) and structural steel (right) springs prestressed at 0.8 GPa.

Figure 6. Von Mises stress values obtained at the separation zone between the spring cantilevers and the suport arms of the micro-mirror (at 0.8 GPa initial normal stress). In the case of steel cantilevers the maximum strees value in the micro-mirror is smaller (1.17 GPa) that in the case of aluminum ones (1.37 GPa). This observation is further developed through the representation in Fig. 6, when one can notice that, in the separation zone between the spring cantilevers and the suport arms of the micro-mirror, for the steel case, reduced mechanical tensions are presented. In comparison, for the aluminum case, the stress is maximum at the joining zone between the actual micro-mirror and the suport arms.

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Parametric stress analysis of the micro-mirror surface During the lift-off procedure the micro-mirror is subjected to different levels of stress, in order to set the necesary position to reflect the incident beam. A parametric stress analysis was performed in order to determine the maximum stress value that can be applied. The initial normal stress put on the spring-like structures were set on between 0.1 GPa to 1.5 GPa.

a) aluminum springs

b) steel springs

Figure 7. Mirror curvature along the surface plane for several values of the initial normal stress that were set on the cantilevers. In the case of aluminum structure there is no major modification of the displacement along the surface of the micro-mirror and the lift-off for the maximun stress is almost 25 µm. For the steel case, because of the hardness of this material, from 0.8 GPa, some curvature of the micro-mirror horizontal plane is induced, which can damage the reflected beam (Fig. 7). In this case one can also notice a reduced of the lift-off distance, of approximately 21 µm (Fig. 8).

Figure 8. Vertical deflection of the center of the micro-mirror for different prestress levels.

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Conclusions The two structures presented in this article have their positive and negative influences over the mirror’s mechanics. In the case of aluminum spring cantilevers it could be obtained a better lift-off displacement and no deformation of the micro-mirror horizontal plane for different prestress levels. As a negative output there are the high loads that appear in the suport arms of the mirror. For the steel prestress plates the positive outputs are the following: a higher angle of deflection and a smaller value of stress induced in the coupling zone between the springs and the actual mirror. References [1] S. D. Sentruria: Microsystem Design, (Kluver Academic Publishers, 2001). [2] Sangtak Park, So-Ra Chung and John T.W. Yeow: Int. Journal on Smart Sensing and Intelligent Systems, vol. 1 (2008), pp. 480-497. [3] T. W. Yeow, K. L. Law and A. A. Goldenberg: IEEE Journal of Selected Topics in Quantum Electronics (2003), pp. 603-613. [4] J. T. W. Yeow, V.X.D. Yang, A. Chahwan, M. L. Gordon, B. Qi, I. A. Vitkin, B. C. Wilson and A. A. Goldberg: Sensors and Actuators A: Physical, vol. 117 (2005), pp. 331-340. [5] Information on www.ti.com/dlp/ [6] T. Iseki, M. Okumura and T. Sugawara: Optical Review, vol. 13 (2006), pp. 189-194. [7] O. Cugat, J. Delamare and G. Reyne: IEEE Tran. On Magn., vol. 39 (2003), pp. 3607-3612. [8] J. Bernstein, W. P. Taylor, J. D. Brazzle, C. J. Corcoran, G. Kirkos, J. E. Odhner, A. Pareek, M. Waelti and M. Zai: Journal of Microelectromechanical Systems, vol. 13 (2004), pp. 526-535. [9] J. Singh, T. Gan, A. Agarwal and S. Liw: Sensors and Actuators A: Physical, vol. 123-124 (2005), pp. 468-475. [10] A. Amarendra: J. of Micromechanics and Microengineering, vol. 16 (2006), pp. 205-213. [11] W. P. Robbins: IEEE Transaction on Ultrasonics, Ferroelectrics and Frequency Control, vol. 38 (1991), pp. 461-467. [12] Y. H. Seo, D-S. Choi, J. H. Lee, T-M Lee, T-J. Je and K-H. Whang: IEEE International Conference on Micro Electro Mechanical Systems, 2005, pp. 383-386. [13] J. F. Rhoads, S. W. Shaw and K. L. Turner: J. of Micromechanics and Microengineering, 2006, pp. 890-899. [14] F. Pan, J. Kubby, E. Peeters, A. T. Tran, and S. Mukherjee: Int. Conf. on Modeling and Simulation of Microsystems, 1998, pp. 474-479.

Applied Mechanics and Materials Vol. 61 (2011) pp 15-24 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.15

OPTIMIZATION BY THE RELIABILITY OF THE DAMAGE BY TIREDNESS OF A WIRE ROPE OF LIFTING A. Meksema, M. El Ghorba, A. Benali, A. El Barkany Team of Control and Mechanical Characterization of Materials. ENSEM, B.P. 8118, Oasis, Casablanca, Morocco. Email: [email protected] Key words: metalic cables, life expectancy, reliability, models probabilistic - damage tirednessreliability.

Abstract: The metallic cables are used for various applications in many industrial fields, such as the aircraft industry, the systems of lifting, the electric lines… In addition, according to the application considered and the conditions of use, the metallic cables undergo degradations whose direct consequences are the strong modifications of the geometrical and mechanical characteristics of the components. What induces a notable reduction of the resistant capacity of the cable according to time, able to bring to failure. In particular at the time of the cyclic requests of loading and unloading where the cable undergoes a phenomenon of tiredness. For safety reasons and an optimal use, it is important to anticipate any brutal rupture. Our work consists in finding a method which allows the optimization of the critical damage and the prediction of its useful life expectancy to be able to change it at convenient time. An analysis making it possible to evaluate the effect of the factors affecting the performance of the long-term cable constitutes the principal work in our lab. It consists in developing a modelling making it possible to envisage the capacity resistant of a cable to various levels of damage of its components, the estimate of the residual life expectancy, the evaluation of the risk of rupture for a level of request given, a mechanical model describing the state of damage by tiredness and another mathematics describing the reliability and finally to an analytical modeling of the relation DamageReliability to predict the phenomenon of tiredness of the hoisting cables. The adopted approach is an approach multi-scales with a total decoupling between the scale of the wire and that of the cable. The criterion of the failure in fatigue for the cable is more complex than that applied to the continuous structures, where the measures of length of the crack or a simple observation of the loss of integrity can be enough. These criteria are based on a mixture of former experiment, personal preferences, and of damage, for each particular type of application of cable. The occurrence of the unacceptable number of the cuts of wire is, by far, the most common action adopted for the evaluation of the damage in fatigue of the cable, which justifies our choice. This relation makes it possible to connect the reliability to the damage through the fraction of life expectancy; this led to associate at each stage of damage corresponding reliability. In fact, the theory of the damage considers that the damage reached its maximum value 1 when there is appearance of a macroscopic crack; but the cable keeps a resistance translated by a no null reliability. The latter becomes it when the cable is completely broken. Optimization by the reliability of the damage is a technique which supports knowledge and a follow-up of the state of an entity requested during time. Thing was being able to have interest for a possible application in industrial maintenance. In this context, this work and other works of the same tendency were worked out to manage to establish the bond between reliability as being a statistical size, and the damage by tiredness observed and caused by the cyclic requests. 1. Introduction The metallic cables constitute the essential element carrying the lifting devices. Their mechanical properties change during their uses. So the safety of the people and the goods depends directly on their states [1, 2].

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Advances in Structures Analysis

A cable (Fig.1) generally consists of several regularly helical strands laid out around a central core, in one or more superimposed layers. The strand itself is composed of several regularly helical steel wire laid out around a central core, in one or more superimposed layers. A cable can be made up of only one strand; one will speak then about a cable mono strand or a helicoidal cable.

Fig. 1: Structural components of a wire rope The industrial experiment shows that the rupture of most of the hoisting cables in service is generally due to the phenomenon of tiredness. This last is particularly insidious because of its masked character (evolution of the damage), which will be able to induce of serious accident. The parameters which make it possible to encircle holding them and the outcomes of this phenomenon are: the damage which describes In spite of many research, the problem of the cables remains rather little known for the researchers mechanics. The cable thus represents a particularly rich field of investigations, according to the nature of the required study. In this study, our contribution is primarily reliability engineer by the means with mathematical model based on a probabilistic approach. Thus our objective is to be able to describe the fatigue behavior of each wire to deduce that from it from the cable. 2. Reliability of the made up systems Many technical systems belong to the class of the complex systems because of the great component count which built them and their processes of complicated operations. In general they are systems series made up of great component count. Sometimes the systems series have components parallel and then they become structures of series- parallel reliability or parallel- series [3, 4, 5]. Considering the components of a system Ei i = 1,2,..., n, n ∈ N having functions of reliability : Ri(t) = P (Ti > t), t ∈ (−∞, ∞), where Ti, i = 1,2,..., n, are the random variables independent representing the life expectancy of the components with functions of distribution: Fi(t) = P (Ti ≤ t), t ∈ (−∞, ∞). We define initially the functions of reliability of the simplest systems [3, 4]: * For a series system: R t

∏

R t

1

R t , t ϵ

* For a parallel system: ∏

∞, ∞

F t , t ϵ

(1)

∞, ∞

(2)

* For a series -parallel system: R

, , ……

,

t

1

∏

1

∏

R t , t ϵ

: the parallel number of subsets and

∞, ∞

(3)

the component count series.

* For a logical majority system m/n (m material among n functional), the function reliability is: R t

∑

∏∈

R ∗ ∏∈

1

R

With A any arrangement of minimal cut (1,2….m) including at least k material in service.

(4)

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When they are the homogeneous regular systems, the function reliability will be: - For a series system: R t = R t

, t ∈ −∞, +∞

- For a parallel system: R t =1− F t

, t ∈ −∞, +∞

- For a series- parallel system: R

,

,

(6)

= 1 − {1 − R t

} , t ∈ −∞, +∞

(7)

= {1 − 1 − R t

} , t ∈ −∞, +∞ .

(8)

- For a parallel-series system: R

(5)

- For a logical majority system: R t =∑ C R . 1−R ! With: C = ! !

(9)

3. Probabilistic models of the behavior of a cable 3.1. Approaches multi-scales of a cable A system is a whole of inter-connected or interdependent elements so that the state of the system depends on the states of its components. What means that any approach of modeling of the cables will be an approach multi-scales: According to the study carried out by Al Achachi [1]. A cable of suspension can be regarded as a system, made up of a whole of strands laid out in parallel. A strand itself consists of a whole of twisted wire. The study of the behavior of a cable is consequently a study multi-scale where one can distinguish the scale from the wire, the scale of the strand and the scale of the cable. This distinction is justified by the concerned mechanisms and the various interactions which must be treated in a distinct way. The systemic diagram of a suspended cable is thus a system: parallel (k strands) - series (m sections of strands) - parallel (N wire). The choice can be justified as follows: • The behavior of a wire governs the behavior of the whole of the cable; • When the wire are twisted and rolled up between them, a wire broken can recover of over a given length, called recovery length (being worth from 1 to 2,5 times the step of twisting) and which defines the dimension of the section of the strand; • The behavior of a strand is deeply related to the behavior of the weakest section (the system is in series); • The strands being laid out in parallel, the resistance of the cable depends on their individual resistances and on inters distribution on the request. This approach makes it possible to go up individual damages of wire to total degradation and thus to the effective resistance of the cable. By adopting the system presented on Fig.1, a cable consists of (k.m.N) wire. 3.2. Study of the reliability of a wire rope of lifting The study carried out by Kolowrocki [2] consists with the development of a modeling allowing the estimate of the life expectancy of a cable. The study is multi-scales where we can distinguish the scale from the wire, the scale of the strand and that of the cable. So that the cable can be consider series/parallel system. The choice of the system series parallel is justified by:

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Advances in Structures Analysis

• The external layer of the cable is consisted strands having diameters higher than those of the internal layer; • The failure of the one of these strands involves that of the cable (system series). The latter are assembled in parallel with the interior layer (parallel system). One can thus say that the whole of the cable constitutes a parallel system series. 3.3. Modeling suggested We adopt here the same approach multi-scales here where one can distinguish the scale from the wire, the scale of the strand and that of the cable. We can consider that the cable is an analog and digital system (series/ majority logical). The choice of this system is justified by: • A broken wire does not involve the failure of the cable, on the other hand starting from a certain number of broken wires, the cable can be declared failing. The system is thus logical majority; • On the level of the strand, the system is in series, with l/lr sections in series; • The strands are laid out in series; the rupture of only one strand involves that of the whole cable, which exempts our study to go until the scale of the layers. And consequently the function reliability is: R t = ∑

C R . 1−R



(10)

With p: number of strands x number of sections; N: number of total of wire and m: number of not broken wire. In order to allow a more reliable comparison between the three models and to be able to put forward the criterion of failure based on the unacceptable number of cuts of wire, let us consider a mono strand T 2.4 (1+6) with step of twisting of 31mm (Fig.2, Fig.3, Fig.4). According to these curves, one notices that the first two probabilistic models do not hold in account of the criterion of failure. However the model suggested appears well adapted to the real situation of use of the cables and takes into account the degradation of the state of the cable under operation. 4- Modeling of the plain damage axial The behavior of materials in fatigue is studied in general under cyclic requests with constant amplitude. The damage by tiredness is described by the deterioration of the mechanical properties of material following cyclic requests. There is thus one cumulates damage since the beginning until the end of the lifetime of the requested cable. This damage is characterized by: * Evolution of the crack and plastic energy absorption. * Loss of resistance in static traction: The original cable (virgin) has a resistance in static traction this one decreases gradually as the number of cyclic loadings increases. *Reduction of the limit of endurance: The original cable has also a limit of endurance σ0; this limit decrease with the increase in the number of cycle applied; finally it takes a criticized value σe* at the time where the rupture occurs.

Applied Mechanics and Materials Vol. 61

Fig. 2 : The criterion of failure is one wire broken

Fig. 3: The criterion of failure is three broken wire

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Advances in Structures Analysis

Fig. 4: The criterion of failure is six broken wire 5-General principles of the theories of the damage The damage of a material being progressive, its variation according to the number of cycles is influenced mainly by the level of loading. The various theories representative of this damage is unified theory [5, 9]: The request of a material in fatigue generally induces a degradation of its physical properties. The unified theory is precisely based on the reduction of the stress limit and the loss of the resistance of material. According to this theory and in the case or the average constraint is null, the expression of the rate of variation of the limit of endurance according to the number of cycles applied N, is written:

= −

γα γ − γe

(11)

γ= ∆σ/σ0 ; γe = σe/σ0 ; ∆σ : Amplitude of request, σ0: Limit of endurance of virgin material and kf, a : Constants for material. The integration of this expression must hold in account of the boundary conditions defines by : *

γe = 1

*

=

∆

if

n=0

if

n=N

The instantaneous value of the limit of endurance is used to define the damage:

D=

(12)

The expression of the damage is obtained:

D=

β β

β





(13)

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6– Reliability: 6- Reliability and failure [6]: The reliability of a material is a statistical parameter, it represents the probability of survival of this material, i.e. the probability of not having a failure (achievement of the necessary function) under given conditions of use during a given operation life, one notes it R (t). One also speaks about size characterizing or the measurement operational safety of the probability of operation of equipment according to prescribed standards. 6-1 mathematical Modeling of reliability: a) - Function of distribution A part requested as a whole for the first time, will fall inevitably broken down not at one moment T defines a priori. T is a random variable (in term of probability) of function of distribution F(t), this function defines the probability that in the random variable discrete T to be lower or equal to the value fixed and : F(t) = P(T≤ t). This function of distribution has the following properties: F (-∞) = 0; F (+∞) =1. b) - Instantaneous rate of failure: The instantaneous rate of failure describes in general the probability of conditional failure between t and (t+dt). Probability of failure between T and (t+dt) = F(t+dt) – F(t) Thus the conditional probability of failure is:

=

(14)

Poses:

Z t =

(15)

And one calls Z(t) : the instantaneous rate of failure From where the function of distribution:

F t =1−e



f t = Z t .e



And the density of probability (Probability between t and t+dt): (16)

Finally the Function Reliability is:

R t =e



(17)

6-2 Use of the laws or statistical distributions for reliability [7-9] Generally, there are three statistical laws used by the reliability engineers to adjust the phenomena of appearance of failures. It with the law of Gauss from where the distribution of failure is seen appear there centered around a median value in phase three of their life, phenomenon little running. The exponential law: corresponds to a constant failure rate, frequent in phase 2 of the life of very many components and the failures occur with causes independent between them, and independent of time. The law of Weibull which is used in mechanics makes it possible to adjust failure rates increasing and decreasing. * Density of probability: according to a law of Weibull. F t =

γ β

.e



for t > γ

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Advances in Structures Analysis

7 - Analytical modeling of the relation damage Reliability [10, 11] Two parameters are likely to follow the evolution of the deterioration of material until the total ruin: a variable continues, which is the damage D(t) and a statistical parameter of nature, which is reliability R(t); it is thus obvious to think of connecting these two quantities between them. To establish this bond, one must initially express the reliability, which is an explicit function of time according to the fraction of life that one notes (β). β = n/Nf Thus one will regard time as a succession of increments of period (τ). T = η. τ η = Nf. τ

Thus:

. n : number of instantaneous cycle. . τ : Time between two successive cycles of loading. . η : Spreading out of the distribution. . Nf : Number of cycle cumulated with the rupture. By exploiting this discretization of time T= n. τ; η =Nf. τ with γ = 0 and by replacing it in the model which appeared to us most likely to adjust the phenomena of appearance of failure, which is the law of Weibull [8], we will obtain like expression of reliability:

R t = e



(18)

We will note the factor of form β by λ not to confuse it with the fraction of life (β =n/Nf) We will write:

R t = e



From where

R t = e

β

The selected mechanical model to translate the damage of the cable by tiredness, following a thorough synthesis of the various theories which describe the damage under request of tiredness, is the unified theory which was established. The expression of the theory unified according to the fraction of life β is: Poses :

α





(19)

So, the expression of reliability according to the damage:

R t = exp

∝ .

∝

(20)

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23

Fig. 5 : Reliability according to the damage γ = ∆σ/σ0 = (1.2) ; λ= (1.5) γu = σu/σ0 = 2 This curve describes well decreasing reliability during operation. It is also noticed that for a fraction of life β=1 reliability is not by null but is worth (1/e), this non null value, can be attributed to a residual reliability, right before the rupture of material. One can make the same thing if one introduces the reliability through the fraction of life, in the expression of the damage:

R t So:

exp −β

β = log

One introducing the form of the fraction of life β, we will find: D =

1

1

1 .α

1 log R

With α

; 1/e ≤ R < 1 and D = 0 for R = 1

This relation also makes it possible to connect reliability to the damage. One will thus have: D

(21) λ

∑



λ η

.

λ η

.α

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Advances in Structures Analysis

8. Conclusion The criteria of the failure in fatigue for the cable are more complex than that are applied to the continuous structures, where the measures of length of the crack or a simple observation of the loss of integrity can be enough. Criteria of failure are based on a mixture of former experiment, personal preferences, and of prejudice, for each particular type of application of cable. The occurrence of the unacceptable number of the cuts of wire is, by far, the most common action adopted for the evaluation of the damage in fatigue of the cable, which justifies our choice. This relation makes it possible to connect the reliability to the damage through the fraction of life, this lead associate at each stage of damage corresponding reliability. In fact, the theory of the damage considers that the damage reached its maximum value 1 when there is appearance of a macroscopic crack; but the cable keeps a certain resistance translated by a non null reliability. The latter becomes it when the cable is completely broken. Optimization by the reliability of the damage is a technique which supports knowledge and a follow-up of the state of an entity requested during time. It’s interesting for application in plan maintenance. In this framework, this work and other work of the same tendency were worked out to manage to establish the bond between reliability as being a statistical size, and the damage by tiredness observed and caused by the cyclic requests. References: [1] S.M. Elachachi, D. Breysse, S. Yotte, C. Crémona: Analysis multi-scale of the resistance of the cable carrying a suspended bridge (European review of civil engineering, vol. 9, N° 4, pp.455496, 2005). [2] K. kolowrocki: Asymptotic approach to reliability evaluation of piping and rope transportation systems (Exploitation Problems of Machines 2 (122), pp. 111-133. 2000). [3] E. CASTILLO, A.FERNANDEZ-CANTELI, J. R. TOLOSA and J.M.SARABIA : Stastical Models for Analysis of Fatigue Life of Long Elements (Journal of Engineering Mechanics, ASCE, Vol 116, N° 5, pp. 1036-1049, 1990. [4] LEFEVRE R., Metallic cables, volume 2, Technip Edition, Paris, 1975. [5] A. Meksem, M. El Ghorba, A. Benali1, and A. El barkany: Model probabilistic of the life expectancy of a wire rope of lifting (9th Congress of Mechanics – Marrakech, Morocco, 2009). [6] P.Chapouille and P. De Pazzis: Reliability of the systems, edited by Masson (1968) [7] A. Pagès and Michel Gondran: Reliability of the systems, edited by Eyrolles 1980. [8] H. Procaccia and L. Piepszownik: Reliability of the equipment and frequential statistical decision theory and bayésienne, edited by Eyrolles (1992). [9] H. Procaccia and P. Morilhat: Reliability of the structures of the industrial facilities, edited by Eyrolles (1996). [10] A.MEKSEM, M.EL GHORBA, A. BENALI and M.CHERGUI : model-building the relation damage/reliability (Review of the Composites and Advanced Materials, Vol 20, N°1, pp103-113, France, 2010). [11] E. CASTILLO and A. FERNANDEZ-CANTELI : A Dependent Fatigue Lifetime Model (Communications in Statistics : theory and methods, Vol 16, N° 4, pp.1181-1194, 1987).

Applied Mechanics and Materials Vol. 61 (2011) pp 25-32 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.25

Thermal buckling of simply supported FGM square plates M. Bouazza1, 2,a, A. Tounsi 2, E.A. Adda-Bedia 2, A. Megueni3 1 2

Department of Civil Engineering, University of Bechar, Bechar 08000, Algeria.

Laboratory of Materials and Hydrology (LMH), University of Sidi Bel Abbes, Sidi Bel Abbes 2200, Algeria. 3

Department of Mechanical Engineering, University of Sidi Bel Abbes, Sidi Bel Abbes 2200, Algeria. a

[email protected]

Key words: Thermal buckling; functionally graded material; classic plate theory.

Abstract: Thermal buckling behaviour of FGM square plates with simply supported edges has been studied in this note using the classic plate theory (CPT). It is assumed that the nonhomogeneous mechanical properties of the plate, graded through thickness, are described by a power-law FGM (simply called P-FGM) and sigmoid FGM (S-FGM). The plate is assumed to uniform temperature rise. Resulting equations are employed to obtain the closed-form solutions for the critical buckling temperature change of FGM. The results are compared with the results of the first order shear deformation theory.

1. Introduction In recent years studies on new performance materials have addressed new materials known as functionally graded materials FGM. These are high performance, heat resistant materials able to withstand ultra high temperature and extremely large thermal gradients used aerospace industries. FGM are microscopically in homogeneous in which the mechanical properties vary smoothly and continuously from one surface to the other (Suresh and Mortensen [1]; Yamanouchi and Koizumi [2]). Typically, these materials are made from a mixture of ceramic and metal. It is apparent from the literature survey that most research on FGM has been restricted to thermal stress analysis, fracture mechanics, vibration, and optimization. Generally, there are two ways to model the material property gradation in solids: (1) assume a profile for volume fractions (Fukui [3]) and (2) use a micromechanics approach to study the nonhomogeneous media (Reddy and Cheng [4]). For composition profile modeling, polynomial representations including quadratic (Fuchiyama et al [5]) variations are used. At the microstructural level, an FGM is characterized by transition from a dispersion phase to an alternative structure with a network structure in between. Nan et al. [6] directly address the constitutive relations of FGM and specifically, used an analytical approach to describe the uncoupled thermomechanical properties of metal/ceramic FGM. These novel materials were first introduced by a group of scientists in Sendai, Japan (Koizumi [7]) and then rapidly developed by the scientists. The nonlinear equilibrium equations and associated linear stability equations were expressed for bars, plates, and shells by Brush and Almroth [8]. The subject matter of this book is the buckling behavior of structural members subjected to mechanical loads. Subsequently, many researchers developed equilibrium and stability equations for plates and shells made of composite layered materials and used them to determine the buckling and vibrational behavior of structures. A review of recent developments in laminated composite plate buckling was carried out by Leissa [9]. Considerable research has focused on the buckling analysis of composite plates under mechanical and thermal loads based on the classical plate theory (Birman and Bert [11]; Pandey and Sherbourne, [12]). Using the classical plate theory, which neglects the effects of transverse shear deformation, the calculations of the buckling loads are rather simple and generally may result in closed-form solutions.

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Advances in Structures Analysis

The present article, the equilibrium and stability equations for FGM are obtained on the basis of classic plate theory. Resulting equations are employed to obtain the closed-form solutions for the critical buckling temperature. The plate is assumed to uniform temperature rise. The results are compared with the results of the first order shear deformation theory. 2. Theoretical Formulations 2.1. FGM material properties In this study, square FGM plates having sides axa and thickness h, as shown in Fig. 1, are considered. Plates are made of a mixture of ceramics and metals; and it is assumed that their composition is gradual and that they are smoothly varied from the ceramic-rich top surface of the plate (z=+h/2) to the metal rich bottom surface (z= - h/2). As a consequence, the material properties of FGM plates, Peff (z), such as the Young’s modulus E, and the shear modulus G are functions of depth z, measured from the middle plane of plate. There are many analytical and computational models that discuss the issue of obtaining suitable functions for material properties of FGMs. Also, there are several criterions for selecting the most suitable function. These functions are meant to be simple and continuous, and should have the ability to exhibit curvatures, both ‘‘concave upwards” and ‘‘concave downwards” [13]. The effective material properties Peff (z), position dependent, can be obtained in the following form according to the linear rule of mixtures [14]. Peff ( z ) = Pm + Pcm V f ( z ) (1) Pcm = Pc − Pm where Pm and Pc are material properties of the metal and ceramic, respectively. The material properties are assumed to be temperature-independent

Fig. 1. Typical FGM square plate. P-FGM, volume fraction function is defined by [14] k V f ( z ) = (z / h + 1 / 2) (2) Sigmoid FGM, volume fraction function is defined by [15]  1  2 z k h 0 ≤z ≤  1 − 1 − h  2  2 (3) V f ( z) =  k 1 2 z h    − ≤z ≤ 0  2 1 + h  2  where volume fraction index k indicates the material variation profile through the thickness direction and is non-negative real number. The variation of Young's modulus in the thickness direction of the P-FGM plate is depicted in Fig. 2, which shows that the Young's modulus changes rapidly near the lowest surface for k >1 , and increases quickly near the top surface for k < 1 . Fig. 3 shows that the variation of Young's modulus of sigmoid distributions, and this FGM plate is thus called a sigmoid FGM plate (S-FGM plates).

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400 k=0.3 k=1 k=3 k=5 k=10

350

Young’s modulus

300 250 200 150 100 50 -0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

z/h

Fig.2. Young’s modulus variation associated with different exponent indexes for a P-FGM plate. 400 350

k=0.3 k=1 k=3 k=5 k=10

Young's Modulus

300 250 200 150 100 50 -0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

z/h

Fig.3. Young’s modulus variation associated with different exponent indexes for a S-FGM plate. 2.2. Governing equations This theory is based on the Cauchy, Poisson and Kirchhoff assumptions which maintain that the normal to the midplane before deformation remains normal after deformation. Then the displacement field in the (x; y; z) reference frame has the following form [10]: U = u − zw, x V = υ − zw, y W =w (4) where U , V and W are displacement components of a typical point in the plate, and u ;υ are inplane displacements at a point of the mid-plane. Using the strain–displacement equations of the classical plate theory: ε x = U , x ε y = V , y γ xy = V , x + U , y (5) these kinematical equations can be written as o ε x = ε 0x − zk x ; ε y = ε 0y − zk y ; γ xy = γ xy − zk xy (6) o where ε 0x , ε 0y and γ xy are the mid-plane strains, and k x , k y and k xy are the curvatures of the midplane during deformation. These are given by ε x0 = u, x , ε y0 = υ, y , γ xy0 = u, y + υ, x (7) k x = w, xx , k y = w, yy , k xy = 2 w, xy

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Advances in Structures Analysis

Hooke’s law for a plate is defined as E (ε x +νε y − (1 +ν )α T ) σx = 1 −ν 2 E (ε y +νε x − (1 +ν )α T ) σy = (8) 1 −ν 2 E τ xy = γ xy 2 (1 + ν ) The forces and moments per unit length of the plate expressed in terms of the stress components through the thickness are h/2

∫σ

N ij =

h/2 ij

dz

M ij =

−h / 2

∫σ

ij

z dz

(9)

−h / 2

The equations of equilibrium for the plate are in the following form N x , xx + 2 N xy , xy + N y , yy = 0 M x , xx + 2 M xy , xy + M y , yy = 0

(10)

q + N x w, xx + N y w, yy + 2 N xy w, xy = 0 Using Eqs.(5) (6) and (8), and assuming that the temperature variation is constant, the equilibrium Eq. (10) may be reduced to a set of one equation as E (1 −ν 2 ) ∇4w− 1 ( N x w, xx + N y w, yy + 2 N xy w, xy + q ) = 0 (11) E1 E3 − E 22 where h/2

( E1 , E2 , E3 ) =

∫ (1, z, z

2

) E ( z ) dz

−h / 2 h/2

(Φ , Θ ) =

(12)

∫ (1, z ) E ( z )α ( z )T ( x, y, z ) dz

−h / 2

To establish the stability equations, the critical equilibrium method is used. Assuming that the state of stable equilibrium of a general plate under thermal load may be designated by w0 . The displacement of the neighboring state is w0 + w1 , where w1 is an arbitrarily small increment of displacement. Substituting w0 + w1 into Eq. (11) and subtracting the original equation, results in the following stability equation E1 (1 −ν 2 ) 4 ( N x0 w1, xx + N y0 w1, yy + 2 N xy0 w1, xy ) = 0 (13) ∇ w1 − 2 E1 E3 − E2 where, N x0 , N y0 and N xy0 refer to the pre-buckling force resultants. To determine the buckling temperature difference ∆Tcr , the pre-buckling thermal forces should be found firstly. Solving the membrane form of equilibrium equations, gives the pre-buckling force resultants Φ Φ N x0 = − , N y0 = − , N xy0 = 0 (14) 1 −ν 1 −ν Substituting Eq(14) into Eq. (13), one obtains E (1 −ν 2 ) Φ 2 ∇ 4 w1 + 1 ∇ w1 = 0 (15) E1 E3 − E22 1 −ν

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The simply supported boundary condition is defined as w1 = 0 , M x1 = 0 , φ y 1 = 0 on x = 0 , a

(16)

w1 = 0 , M y1 = 0 , φ x 1 = 0 on y = 0 ,b

The following approximate solution is seen to satisfy both the governing equation and the boundary conditions w1 = c sin (mπ x / a) sin (n π y / b) (17) where m, n are number of half waves in the x and y directions, respectively, and c is a constant coefficient. Substituting Eq. (17) into Eq. (15), and substituting for the thermal parameter Φ from Eq. (12), yields ( E1 E3 − E 22 )(1 −ν )π 2 (m 2 + n 2 Ba2 ) E1 (18) ∆T = P E12 a 2 (1 −ν 2 ) where h/2

P=

∫ E ( z )α ( z )dz

; Ba = a / b

(19)

−h / 2

The critical temperature difference is obtained for the values of m, n that make the preceding expression a minimum. Apparently, when minimization methods are used, critical temperature difference is obtained for m= n=1, thus ( E1 E3 − E 22 )(1 −ν )π 2 (1 + Ba2 ) E1 (20) ∆Tcr = P E12 a 2 (1 −ν 2 )

2. Numerical results and discussion First, Based on the derived formulation, a computer program is developed to study the behavior of FGM plates in thermal buckling to validation checks against the results available in the literature. The critical temperatures of simply supported, isotropic square plates subjected to constant temperature distributions obtained using classic plate theory are verified against the energy method based results of Gowda and Pandalai [16] and solution of Kri et al [17] based on finite element method using semiloof element, in Table1. Both results are in excellent agreement.

Table1. Critical temperature for isotropic square plates subjected to different forms of temperature distribution (a / h = 100, α = 2 ⋅ 10−6 ,ν = 0.3) Temperature distribution

Analytical [16]

FEM [17]

Present

Uniform temperature rise

63.27

63.33

63.27

Numerical results are presented for two types of functionally graded materials, P-FGM and S-FGM. The properties of these materials are given in Table 2.Using the expression (20), the buckling temperatures have been evaluated for square plates made of these materials and are given in fig. 4.

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Table 2. Material properties of FGM plate. Property Material

E (GPa)

ρ ( kg / m 3 )

ν

α (1 / °C )

k (W / mk )

Aluminum

70

2707

0.3

23e-6

204

Alumina

380

3800

0.3

7.4e-6

10.4

The variation of the critical temperature change ∆Tcr of ceramics and metals P-FGM plates under uniform temperature rise for different geometric parameters and volume fraction index are plotted in Fig. 4. While the other cases, k =0, 0.3; 1 and 5 are for the graded plates with two constituent materials. The aspect ratio of the plate is set as a/b=1. In Figs. 4, the critical temperature change increases as volume fraction index k is decreased. This is because for P-FGM, as the volume fraction index is decreased, the contained quantity of ceramic increases. In all material cases, the critical temperature change increases, when the geometric parameter a/h is increased.

16000

P-FGM

k=0 k=0.3 k=1 k=5

14000 12000

Tcr

10000 8000 6000 4000 2000 0 -2000 0.00

0.05

0.10

0.15

0.20

0.25

0.30

h/a

Fig.4. Variation of critical temperature change with h/a under uniform temperature rise; using classic plate theory.

Table 3. Comparison of critical temperature with different theories for S-FGM h/a=0.01 h/a=0.1 Theories

k=0

k=0.2

k=5

k=0

k=0.2 k=5

CPT

8.3246

8.2900

7.6990

832

829

FSDT

8.3207

8.2862

7.6965

795.1

792.5 746.1

770

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Table 3 represents the results of thermal buckling analysis for the S-FGM based on the CPT compared to FSDT are presented. The critical temperature change with respect to volume fraction index k and geometric parameters h/a; under uniform temperature rise, this table show that the buckling temperature increases by the increases of the ratio h/a. Also, based on the table, the results obtained by classic plate theory coincide with the results of first order shear deformation theory (h/a=0.01). This is well explained by the short plate aspect ratio h/a=0.01 or the small plate thickness h.

Conclusion The critical buckling temperatures of FGM plate have been obtained using a first order shear deformation theory. The results of the sample problem show good agreement with the literature values as seen from the validation checks. Based on the results reported here for various parameters of FGM plates, the following conclusions may be drawn: (1) The critical buckling temperature differences for functionally graded plates are generally lower than corresponding values for homogeneous plates. It is very important to check the strength of the functionally graded plate due to thermal buckling, although it has many advantages as a heat resistant material. (2) The critical buckling temperature difference for a functionally graded plate is increased when the plate aspect ratio or the thickness to span ratio increases. However, it is decreased when the power law index k increases. (3) Transverse shear deformation has considerable effect on the critical buckling temperature difference of functionally graded plate, especially for a thick plate or a plate with large aspect ratio.

References [1] Suresh, S., Mortensen, A,. Fundamentals of Functionally Graded Noble, New York,(1998).

Materials. Barnes and

[2] Yamanouchi, A., Koizumi, M.,. Functionally gradient materials. In: Proceeding of the First International Symposium in Japan,(1990). [3] Fukui, Y,. Fundamental investigation of functionally gradient material manufacturing system using centrifugal force. Int. J. Jpn Soc.Mech. Eng. 3 (34), 144–148,(1991). [4] Reddy, J.N., Cheng, Z.Q,. Three dimensional trenchant deformations of functionally graded rectangular plates. Eur. J.Mech. A Solids 20,(2001). [5] Fuchiyama, T., Noda, N., Tsuji, T., Obata, Y,. Analysis of thermal stress and stress intensity factor of functionally gradient materials. Ceramic. Trans. Funct. Gradient Mater. 34, 425– 432. 841–855,(1993). [6] Nan, C.W., Yuan, R.Z., Zhang, L.M,. The physics of metal/ceramic functionally gradient materials. Ceramic. Trans. Funct. Gradient Mater. 34, 75–82,(1993). [7] Koizumi, M.,. FGM activities in Japan. Composites Part B 28 (1–2), 1–40,(1997). [8] Brush, D.O., Almroth, B.O,. Buckling of Bars, Plates and Shells. McGraw-Hill, New York, (1975). [9] Leissa, A.W.,. Review of recent developments in laminated composite plate buckling analysis. Composite Mat. Tech. 45, 1–7, (1992). [10] Leissa AW. Vibration of plates. NASA, SP-160, (1969).

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Advances in Structures Analysis

[11] Birman, V., Bert, C.W, . Buckling of composite plate and shells subject to elevated temperature. Trans. ASME J. Appl. 60, 514–519, (1993). [12] Pandey, M.D., Sherbourne, A.N,. Buckling of anisotropic composite plates under stress gradient. J. Engrg. Mech. 117 (2), 260–275,(1991). [13] Markworth AJ, Ramesh KS, Parks Jr WP. Modeling studies applied to functionally graded materials. J Mater Sci (1995); 30:2183–93. [14] Huang X-L, Shen H-S. Nonlinear vibration and dynamic response of functionally graded plates in thermal environment. International Journal of Solids and Structures (2004); 41:2403–27. [15] Bouazza. M, Tounsi. A, Adda-Bedia.E.A, Megueni.A,. Thermal buckling of sigmoid functionally graded plates using first order shear deformation theory. MAMERN09: 3rd International Conference on Approximation Methods and Numerical Modeling in Environment and Natural Resources Pau (France), June 8-11, (2009). [16] R. M. S. Gowda and K. A. V. Padalai, Thermal buckling of orthotropic plates. In Studies in Structural Mechanics (Edited by K. A. V. Padalai), pp. 9-44. IIT, Madras (1970). [17] Kari R. Thangaratnam, Palaninathan and j. Ramachandran,Thermal buckling of composite laminated plates. Computers & Structures vol.32, no. 5. pp. 1117-1124, (1989) Rimed in Great Britain.

Applied Mechanics and Materials Vol. 61 (2011) pp 33-42 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.33

Analytical and Finite Element Analysis for Short term O-ring Relaxation Mohammed Diany a, Hicham Aissaoui b Faculté des Sciences et Techniques de Béni Mellal, BP 523 Béni Mellal, MAROC a

[email protected], [email protected]

Key words: O-ring, contact pressure, analytical modeling, relaxation, FEA.

Abstract. The like-rubber O-ring gaskets are widely used in hydraulic and pneumatic equipments to ensure the sealing of the shaft, the pistons and the lids. The correct operation is due to the good tightening of the joint that generate a sufficient contact pressure able to confine the fluids inside rooms or to prevent their passage from one compartment to another. Several studies are carried out to model the O-ring behavior but without taking in account the effect of the relaxation and creep phenomena. In this article, an axisymmetric finite element model is proposed to study the O-ring relaxation during the first hours of its installation in the unrestrained axial loading case. The results of the numerical model are compared with an analytical approach results based on the classical Hertzian theory of the contact. The effects of the o-ring mechanical and geometrical characteristics are examined. The contact stress profiles and the peak contact stresses are determined versus the time relaxation in order to specify the working conditions thresholds. Nomenclature b

the contact width between the gasket and plat (mm)

C

the ratio e/d

d

the O-ring cross-section diameter (mm)

D

the O-ring mean diameter (mm)

e

initial O-ring axial displacement (mm)

Erelax

relaxation modulus (MPa)

Ej

O-ring elastic modulus (MPa)

E∞

O-ring elastic modulus at permanent state (MPa)

F

total compression load (N)

R

the axial compression ratio

x

radial position compared to the vertical axis of the O-ring cross-section (mm)

po

maximum contact pressure value or peak contact stress (MPa)

αj

relative’s relaxation coefficients

τj

relaxation time (s)

ν

Poisson's ratio

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Advances in Structures Analysis

Introduction The elastomeric O-ring gaskets, presented in Fig. 1, are widely used in hydraulic and pneumatic equipments to ensure the sealing of the shaft, the pistons and the lids. The correct operation is due to the good tightening of the joint that generate a contact pressure able to confine the fluids inside rooms or to prevent their passage from one compartment to another.

Fig. 1 : O-rings examples The equations developed until today to determine analytically the distribution and the values of the contact pressure are deduced from the conventional Hertzian theory of the contact [1]. The correct operation of the O-ring is conditioned, on the one hand, by the maximum value of the contact pressure created during the O-ring compression and on the other hand by the preservation in operating stage of a minimal threshold value below which the sealing of the joint is blamed. So the evaluation of the maximum value of contact pressure evolution in time has a primary importance to ensure the correct O-ring function during its nominal lifespan. Several teams were interested in O-ring assembly used in various industrial services. A temporal reading of published works on this subject can be classified on three shutters. An analytical approach based in all cases on the Hertzian classical theory, an experimental part using various assemblies allowing to characterized the O-ring itself in traction and compression loads and to model his real behavior. In the third shutter, finite elements models are developed to numerically simulate assemblies with the O-ring. George and al. [2] used a finite elements model to study the behavior of the O-ring compressed between two plates. The gasket characteristics were introduced into the program according to parameter defining the total deformation energy or by using the Neo-Hookean model. The results of this analysis were compared with those of several experimental studies and analytical approaches based on the Hertzian theory. Dragoni et al. [3] propose an approximate model to study the O-ring behavior placed in rectangular grove. The influence of the grove dimensions variation and the friction coefficient was treated. The work of Green and al. [4] reviews the majority of used O-rings configurations. A finite elements Models were developed considering hyperelasticity behavior. The results of these models were confronted with those of empirical studies. New relations expressing the maximum contact pressure and the width of contact were proposed. Rapareilli and al. [5] present a validation of the experimental results by a numerical model which regarded the joint as an almost incompressible elastic material. The effects of the fluid pressure as well as the friction effect between the gasket and the shaft are studied. The two shutters of the study were in perfect agreement. In an experimental study [6], the authors tried to determine the influence of the fluid pressure on the contact pressure which ensures of sealing as well as the ageing deterioration of the joint. Kim and al. [7, 8] tried to

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find an approximate solution for the mechanical behavior of the O-ring joints in several configurations. The influence of the friction coefficient is highlighted. An experimental study was carried out to find more realistic elastic modulus values for elastomeric O- ring. They compared their results with those obtained in experiments and by the finite element analysis. They found that the values given by Lindley [9, 10] to calculate the compressive force are similar to those determined by the finite elements model. The O-ring relaxation was treated by the reference [11] where the degradation is caused by oxidation or nuclear irradiation. The authors describe several improvements to the methods used in there previous studies.




In this article, we study the O-ring relaxation during the first hours of its installation in the unrestrained axial loading case, Fig. 2. A 2D axisymmetric model is developed to simulate the Oring relaxation when it is axially compressed by initial tightening between two rigid plans. The effect of the temporal variation of the longitudinal elasticity modulus as well as the influence of the axial compression ratio will be analyzed. The model of the classical contact theory will be confronted with the results of the numerical study.

Fig. 2 : Unrestrained axial loading assembly (a) Initial position; (b) loaded case

Conventional analytic Theory Most of the previous work dedicated to study the O-ring gasket behavior use the same analytical model based on the Hertzian pressure contact theory. By adopting this classical theory, Lindley [9, 10] developed, for the case presented in Fig.1, a simple approximate formula Eq. 1, expressing the compressive force F, according to the ratio of initial compressed displacement by the cross-section O-ring diameter, C=e/d. 3 2

F = πDdE (1.25.C + 50.C 6 )

(1)

The same theory allowed finding out the contact width, b, and the maximum value of the contact stress po , according to the formulas (2) and (3).

b = d.

6

π

3 2

(1.25.C + 50.C 6 )

(2)

3 2

(1.25.C + 50.C 6 ) p o = 4 .E . 6π

(3)

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Advances in Structures Analysis

The contact pressure distribution according to the radial position on the gasket is given by the Eq. 4.

 2x  p ( x ) = po 1 −    b 

2

(4)

These formulas do not use the mechanical characteristics of the plates in contact with the joint. Only the O-ring longitudinal elasticity modulus E is used. By consequence, the same equations remain valid for the evolution study of the O-ring behavior according to time but using a variable Young modulus according to time, called relaxation modulus Erelax. The viscoelastic behavior of the gasket is given by the modified Maxwell model [12] presented in Fig. 3. σ

Einf

E1

E2

η1

η2

E3

η3

E4

Ej

η4

ηj

ε

Fig. 3 : A generalized Maxwell model according to Eq.5 The relaxation modulus is defined by the following equation: Erelax (t ) = E∞ + ∑ E j e

−

t

τj

j

(5)

With

αj =

Ej E∞

and E0 = E∞ +

∑E

j

(6)

j

The relaxation modulus of the Eq. 5 becomes: t −   τj E relax (t ) = E 0 1 − ∑ α j (1 − e )    j

(7)

The initial elasticity modulus E0 and the coefficients αJ, called Prony series coefficient, are deduced from the experimental data of the reference [13]. The time variation of this relaxation modulus is presented in Fig. 4 The relaxation study consists to evaluate the variation of the contact stress versus time, when an initial axial displacement e, characterized by an axial compression ratio R, given by Eq. 8, is imposed to the gasket. For each axial compression ratio R, the variation of the contact pressure distribution as well as the change of the contact surface width are determined with Eqs. 2, 3 and 4.

R = 100 × 2.e / D = 200.C

(8)

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1.2

Erelax/E01(MPa)

1

0.8

0.6

0.4

0.2

0 0

20000

40000

60000

80000

100000

Time (sec)

Fig. 4 : Relaxation modulus ratio, Erelax/E01

Finite element analysis The study of the O-ring relaxation, placed between two rigid plates and compressed initially by the application of a constant displacement, consists in following the evolution in time of the contact pressure. To achieve this goal, axisymmetric finite elements model of the assembly was produced using ANSYS software [14] as shown in Fig. 5.

Fig. 5: O-ring axisymmetric finite elements model Since the problem is axisymmetric and the median horizontal plane cutting the O-ring in two equivalent parts is a symmetry plane, the joint is modeled by a half-disc with four node’s 2D plane elements. The O-ring material is regarded as viscoelastic characterized by the Prony coefficients. The plates are modeled by rigid elements for which the displacements are blocked in all directions. The geometric and mechanic characteristics of the O-ring joint are summarized in Table 1. In order to check the influence of the O-ring rigidity two initial Young modulus values are considered. The mesh refinement is optimized to have the convergence while using less computer memory capacity. The value of the vertical displacement imposed on the upper surface of the joint is calculated by the axial compression ratio R, which varied between 7.5 and 25 % of the O-ring cross-section diameter. Thereafter, the contact pressure distribution is recorded according time.

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Advances in Structures Analysis

Table 1 : O-ring characteristics O-ring Subscript d (mm) D (mm) E0 (MPa) ν

1

2

6.98 123.19 2.82 46.20 0.49967

Results and discussions In a preliminary step, we study the effect of the O-ring mean diameter and the cross section diameter on the pressure contact. This effect can be appreciated from Fig. 6 and Fig. 7. It’s clear that the maximum contact pressure value is the same for all cases and it is independent of the two O-ring diameters. This conclusion brings us to use one O-ring diameters combination shown in Table 1. The suggested analytical model calculates the maximum contact pressure, in the assembly sealing conditions, according to initially imposed displacement. In addition, the finite elements model in the same working conditions studied the effect of several parameters relating to the O-ring material rigidity. Fig. 8 presents the contact pressure distribution according to the radial position for a compression ratio of 15% with various intervals of operating time. It is noticed that the contact pressure is maximum in the average diameter position. All the curves have the same appearance and admit the middle diameter like a symmetrical position. The relaxation speed is more important at the beginning and becomes null after 18 operating hours. 0.495 Inner Diameter = 18 mm 3 4 5 6 8 10

0.445

Contact pressure (MPa)

0.395 0.345 0.295

d (mm)

0.245 0.195 0.145 0.095 0.045 -0.005 -5

-4

-3

-2

-1

0

1

2

3

Relatif radial position (mm)

Fig. 6 : Cross section diameter effect

4

5

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0.495 d = 10 mm

0.445

285

Contact pressure (MPa)

0.395

185 0.345

59

0.295

23 D (mm)

0.245 0.195 0.145 0.095 0.045 -0.005 -5.5 -4.5 -3.5 -2.5 -1.5 -0.5

0.5

1.5

2.5

3.5

4.5

5.5

Relatif radial position (mm)

Fig. 7 : The mean O-ring diameter effect

0.8 Relaxation times

0.7 0.6

Contact pressure (MPa)

E01=2.82 MPa

0 1h

0.5 4h 0.4

8h

0.3

12 h

0.2

18 h

0.1

24 h

0 Percentage compression set : R=15%

-0.1 -0.2 56

58

60

62

64

66

Radial position (mm)

Fig. 8 : Contact pression distribution for R=15% To inspect the effect of the rigidity of the O-ring, two values of the longitudinal modulus of elasticity were used. Fig. 9 compares the contact pressure distributions for the two cases. For a given position, the contact pressure value depends on the value of the corresponding elasticity modulus. When E is larger the contact pressures are higher. When the contact pressure is divided by the initial elastic modulus, the curves depend only on the relaxation time and the compression ratio as illustrated in Fig. 10. Consequently, we can conclude that the effect of the gasket rigidity does not appear when the curve of the ratio p/E0 is represented according to the radial position for several cases.

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Advances in Structures Analysis

8

0.5 E01=2.82 MPa

E02=46.2 MPa

7

Relaxation times 0h 1h

0.2

4h 24 h 0.1

6 5 4 3 2 1

Contact pressure for E02 (MPa)

0.3

Percentage compression set : 7.5%

Contact pressure for E01 (MPa)

0.4

0 0 -0.1

-1 -1

-0.5

0

0.5

1

Relative radial position

Fig. 9 : Contact pressure in the two elastic modulus cases

0.37 E01=2.82 MPa

E02=46.2 MPa

0.32

Axial compression ratio : R=20%

0.27 Relaxation times

p(x)/E0i=1;2

0.22

0h 1h

0.17

4h 24 h

0.12 0.07 0.02 -0.03 -1.5

-1

-0.5

0

0.5

1

1.5

Relative radial position

Fig. 10 : Initial elastic modulus effect During the installation of the joint, the analytical model envisages the same stress distribution as the finite elements model for any imposed axial displacement value as shown in Fig. 11. The contact surface and the maximum contact pressure are larger when the compression ratio or the relaxation time are more significant. The difference between the results of the two models is rather negligible and does not exceed 10%. It can be affirmed that the analytical model deduced from the classical theory of the contact pressure remains valid even for the study of the O-ring relaxation. Fig. 12 compares the influence of the compression ratio on the speed and the values of contact pressure due to the relaxation of the viscoelastic. It is clear that in all the cases the maximum contact pressure loses a great percentage of its initial value with time. This loss is very fast in the first operating hours.

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0.35

At t=0 sec

Relative contact stress, P/E0

0.3

Axial compression ratio R 20%

0.25

15%

Solide line : FE model Dashed line : Analytical model

0.2 0.15 0.1

At t=4 h 0.05 0 -0.05 0

0.2

0.4

0.6

0.8

1

Relative radial position

Fig. 11 : FE and Analytical models comparison

Maximum Contact pressure (MPa)

1.2

1

Axial compression ratio R

0.8

7.50%

15%

20%

25%

Solide line : FE model Dashed line : Analytical model

0.6

0.4

0.2

0 0

20000

40000

60000

80000

100000

Relaxation time (sec)

Fig. 12 : Relaxation of maximum contact pressure Conclusion This study shows that the classical theory of contact, developed initially for steady operation, remains valid for the relaxation case but with some modifications on the O-ring mechanical characteristics. In addition, the finite elements model developed produces the same results as the analytical model. In order to generalize these remarks, other cases might be regarded as the radial loading and the grooves configuration. The analytical and numerical models give an idea about the correct O-ring behavior but they should be supported by experimental tests in a test bench more representative of the operating conditions.

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Advances in Structures Analysis

References [1] Timoshenko and Goodier, in: Theory of elasticity, edited by McGraw-Hill, (1934) [2] George AF, Strozzi A, Rich JI, in: Stress fields in a compressed unconstrained elastomeric Oring seal and a comparison of computer predictions and experimental results, Tribology International, Vol. 20, No. 5, (1987) p. 237-247. [3] Dragoni E, Strozzi A, Theoretical analysis of an unpressurized elastomeric O-ring seal inserted into a rectangular Groove, Wear; Vol. 130, (1989) p. 41-51. [4] Green I, English C, Stresses and deformation of compressed elastomeric O-ring seals, 14th International Conference on Fluid Sealing, Firenze, Italy, 6-8 April (1994) [5] Rapareilli T, Bertetto AM and Mazza L, Experimental and numerical study of friction in an elastomeric seal for pneumatic cylinders, Tribology International Vol. 30, No 7, (1997). p. 547552 [6] Yokoyama K, Okazaki M and Komito T, Effect of contact pressure and thermal degradation on the sealability of O-ring, JSAE 1998; 19: 123-128. [7] Kim HK, Park SH, Lee HG, Kim DR, Lee YH, Approximation of contact stress for a compressed and laterally one side restrained O-ring, Engineering Failure Analysis; 14: (2007) p. 1680-92 [8] Kim HK, Nam JH SH, Hawong JS, Lee YH, Evaluation of O-ring stresses subjected to vertical and one side lateral pressure by theoretical approximation comparing with photoelastic experimental results, Engineering Failure Analysis, doi:10.1016/j.engfailanal. (2008) [9] Lindley PB, Compression characteristics of laterally-unrestrained rubber O-ring, J IRI, 1, (1967) p. 220-13. [10] Lindley PB, Load-compression relationships of rubber units, J Strain Anal, 1(3), (1966) p. 1905. [11] Gillen K.T., Celina M. and Bernstein R, Validation of improved methods for predicting longterm elastomeric seal lifetimes from compression stress-relaxation and oxygen consumption techniques, Polymer Degradation and Stability, 82, (2003) p. 25-35. [12] McCrum,N. G., Buckley C. P. and Bucknall C. B, Principles of Polymer Engineering, Oxford University Press, New York. (2004) [13] Christensen, R. M., in: Theory of Viscoelasticity - An Introduction, 2nd ed., Academic Press, New York. (1982) [14] ANSYS, ANSYS Standard Manual, Version 11.0 (2003).

Applied Mechanics and Materials Vol. 61 (2011) pp 43-54 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.43

A NEW METHODOLOGY FOR AN OPTIMAL SHAPE DESIGN W. El Alem1,2,a, A. El Hami2,b and R. Ellaia1,c 1

Mohammed V university - Engineering Mohammedia School, LERMA BP. 765, Ibn Sina avenue, Agdal, Rabat, Morocco.

2

Laboratory of Mechanics of Rouen, National Institute for Applied Sciences

BP 08, university avenue 76801, St Etienne du Rouvray Cedex, Rouen, France. a

[email protected], b [email protected], c [email protected]

Key words: Structural optimization, Simultaneous perturbation stochastic approximation, Finite element analysis, MATLAB.

Abstract. The aim of this paper is to study the implementation of an efficient and reliable methodology for shape optimization problems where the objective function and constraints are not known explicitly and are dependent on the Finite Element Analysis (FEA). It is based on the Simultaneous Perturbation Stochastic Approximation (SPSA) method for solving unconstrained continuous optimization problems. We also propose Penalty SPSA (PSPSA) for solving constrained optimization problems, the constraints are handled using exterior point penalty functions within an algorithm that combines SPSA and exact penalty transformations. This paper presents a new structural optimization methodology that combines shape optimization, geometric modeling, FEA and PSPSA method to successfully optimize structural optimization problems. Several tests have been performed on some well known benchmark functions to demonstrate the robustness and high performance of the suggested methodology. In addition, an illustrative two-dimensional structural problem has been solved in a very efficient way. The numerical results demonstrate the robustness and high performance of the suggested methodology for structural optimization problems. 1- Introduction Structural optimization is a field of research that has experienced noteworthy growth for many years. Researchers in this area have developed optimization tools to successfully design and model structures, for example, genetic algorithms [9, 2, 31], evolutionary algorithms [32, 7], metaheuristic approaches [21, 17, 25], simulated annealing algorithms [19]. In the literature, the above algorithms are applied to general nonconvex functions given only in analytical form. Nowadays, structures have became more and more complex, a design engineer working in the field of research and development has to often design completely new structures, within this field most real-world analysis are carried out using FEA. Through the increase in capacity and speed of modern computers and the theoretical foundations from before, together with modern optimization algorithms, significant progress has been made in the development of finite element-based shape optimization [15, 5, 6, 26]. In more recent works, commercial packages such as ANSYS [14, 8, 20] are commonly used for geometric and finite element modeling, but to successfully implement a robust methodology for structural problems collaboration between high performance Finite Element solver and a robust optimization method should take part. In this paper we have developed an optimum structural design methodology by coupling the ANSYS finite element package with PSPSA optimization method. The PSPSA method is implemented in MATLAB and the reliable finite element package program, ANSYS, is used for structural analysis. The present methodology can help researchers and practitioners devise optimal solutions to countless real-world problems. Numerical examples are provided to demonstrate the capabilities of this structural optimization program. This paper is structured as follows. Formal structural optimization problem is presented in Sect. 2. Optimum structural design program is then presented in Sect. 3. Section 4 presents numerical examples in which the PSPSA algorithm is validated and Sect. 5 reports some conclusions.

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2- Structural optimization Formal structural optimization problems are associated with the specification of a mathematical objective function and a collection of factors (or parameters) that can be adjusted to optimize the objective function. In particular, one can formulate an optimization problem as follows  Minimize f ( x) (1)  Subject to : x ∈ S Where f : IR n  → IR represents some loss function to be minimized, x represents the ndimensional vector and S ⊆ IR n represents a constraint set defining the allowable values for the parameters x. In the present paper we are interested in problems for which x represents a vector of continuous parameters. 2.1 Selection of objective function and constraint functions Generally, there are two types of objective functions for structural problems. One is the stress and the other is the weight of a structure. In this paper, both cases are studied; in the first case the weight is used as the objective function under stress constraints while in the second case we use the von Mises stress as the objective function where the weight and the design variables are taken as constraints. The two cases are discussed in the numerical examples section. 2.2 Optimization method Commonly, structural analysis problems are constrained. To solve the constrained problem, the penalty method approach is adopted for this research. The penalty method enables constrained optimization problems to be converted to unconstrained problems. This method in conjunction with the SPSA algorithm results in a new method called the Penalty Simultaneous Perturbation Stochastic Approximation (PSPSA) method. 2.2.1 Penalty method Penalty method is a procedure for approximating constrained optimization problems by unconstrained ones. The approximation is accomplished in the case of penalty methods by adding to the objective function a term that prescribes a high cost for violation of the constraints, (for further details see [22]). Consider the problem (1), where f is a continuous function on IR n . The idea of a penalty method is to replace problem (1) by an unconstrained problem of the form: Minimize f ( x) + µ P ( x). (2) Where µ is a positive constant and P is a function on IR n satisfying: (i) P is continuous (ii) P ( x) ≥ 0 for all x ∈ IR n (iii) P ( x) = 0 if and only if x ∈ S Suppose that S = {x : g ( x) = 0}, the problem (2) can be replaced by the unconstrained one as following: Minimize f ( x) + µ g 2 ( x) µ is large enough  n x ∈ IR  Suppose now that S is defined by a number of inequality constraints S = {x : g i ( x) ≤ 0, i = 1,2,............., p} A very useful penalty function in this case is p

P( x) =

∑ (max [0, g ( x)])

2

i

i =1

For large µ it is evident that the minimum point of problem (2) will be in a region where P is small. Thus, for increasing µ it is expected that the corresponding solution points will approach the feasible region S and, subject to being close, will minimize f. Ideally then, as µ → ∞ the solution point of the penalty problem will converge to a solution of the unconstrained problem.

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More generally if the subset S is defined as S = {x : g i ( x) ≤ 0, i = 1,2,........., m and h j ( x) = 0, j = 1,2,........., p}. Then the problem (1) is equivalent to that of (2) with a penalty function of the form: m

P( x) =

∑ (max [0, g i ( x)]) 2 + i =1

p

∑

h j ( x)

2

j =1

2.2.2 SPSA method SPSA method has attracted considerable application in many different areas such as air traffic network [18], environmental [1], queuing network [3], classification and biomedical classification problems ([11, 23]), pattern recognition [16], robot arm control [12], image restoration [24] and so on. In the following we will focus on SPSA for the reasons of its relative efficiency. SPSA is based on a highly efficient and easily implemented simultaneous perturbation approximation to the gradient; this gradient approximation uses only two loss-function measurements, regardless of the dimension of the optimization problem (For further details see [4, 30, 29]).

2.2.2.1 Background Consider the problem of minimizing a (scalar) differentiable loss function f (x) , where x ∈ IR n , n ≥ 1 , and where the optimization problem can be translated into finding the minimizing x* such that the gradient g ( x* ) = 0 . It is assumed that measurements of f (x ) are available at various values of x, these measurements may or may not include added noise. No direct measurements of g (x ) are assumed available. The recursive procedure we consider is in the general SA form ⌢ ⌢ ⌢ ⌢ xk +1 = xk − ak g k ( xk ) (3) ⌢ ⌢ ⌢ where g k ( xk ) is the estimate of the gradient ∇f at the iterate xk . The essential part of Eq. (3) ⌢ ⌢ is the gradient approximation g k ( xk ) . We discuss below the form that have attracted the most of attention that is of Simultaneous Perturbation.

Simultaneous perturbation Let y (.) denote a measurement of f (.) at a design level represented by the dot ⌢ [i.e. y (.) = f (.) + noise] , and ck , be some (usually small) positive number. All elements of xk are ⌢ ⌢ randomly perturbed together to obtain two measurements y (.) , but each component of g k ( xk ) is formed from a ratio involving the individual components in the perturbation vector and the difference in the two corresponding measurements. ⌢ ⌢ y ( xk + ck ∆ k ) − y ( xk − ck ∆ k ) ⌢ ⌢ g ki ( xk ) = 2ck ∆ ki where the distribution of the user-specified random perturbations for simultaneous perturbation, ∆ k = (∆ k 1 , ∆ k 2 ,..., ∆ ki ,..., ∆ kn )T , satisfies conditions that will be mentioned later. 2.2.2.2. Convergence of the SPSA algorithm As with any optimization algorithm, it is of interest to know whether the xk will converge to x* as k gets large. In fact, one of the strongest aspects of SA is the rich convergence theory that has been developed over many years. Researchers and analysts in many fields have noted that if they can show that a particular stochastic optimization algorithm is a form of SA algorithm, then it may be possible to establish formal convergence. Note that since we are in a stochastic context, convergence is in a probabilistic sense. In particular, the most common form of convergence established for SA is in the almost sure (a.s.) sense. The SPSA algorithm works by iterating from an initial guess of the optimal x, where the iteration process depends on the above-mentioned simultaneous perturbation stochastic approximation to the gradient g(x). The form of the SPSA gradient approximation was presented

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above. J. Spall (1992) presents sufficient conditions for convergence of the SPSA ⌢ iterate xk → x* a.s. . In particular, we must impose conditions on both gain sequences ( ak and ck ),

(

)

the user-specified distribution of ∆ k and the statistical relationship of ∆ k to the measurements y(.). The main conditions are that ( ak and ck ) both go to 0 at rates neither too fast nor too slow, that f (x) is sufficiently smooth (several times differentiable) near x* , and that the ∆ ki are independent and symmetrically distributed about 0 with finite inverse moments E ( ∆ k ) for all ki . One particular distribution for ∆ ki that satisfies these latter conditions is the symmetric Bernoulli ± 1 distribution; two common distributions that do not satisfy the conditions (in particular, the critical finite inversemoment condition) are the uniform and the normal. Although the convergence result for SPSA is of some independent interest, the most interesting theoretical results in Ref. [28], and those that best justify the use of SPSA, are the asymptotic efficiency conclusions that follow from an asymptotic normality result. It can be shown that ⌢ k β / 2 ( xk − x* ) → N ( µ , ∑ ) as k → ∞ Where β > 0 depends on the choice of gain sequences (ak and ck), µ depends on both the hessian and the third derivatives of f(x) at x * , and Σ depends on the Hessian matrix at x * .

2.2.3 SPSA algorithm Step 1: Initialization and coefficient selection. Set the SPSA gain sequences a k = a / ( A + k ) α and c k = c / k γ . Step 2: Generation of the simultaneous perturbation vector. Generate an n-dimensional random perturbation vector ∆ k where each of the n components of ∆ k is independently generated. Step 3: Loss function evaluations. Obtain two measurements of the loss function y (.) based on the simultaneous perturbation around the current xˆ k : y ( xˆ k + c k ∆ k ) and y ( xˆ k − c k ∆ k ) with c k and ∆ k from Steps 1 and 2. Step 4: Gradient approximation. Generate the simultaneous perturbation approximation to the unknown gradient g ( xˆ k ) :

 ∆−k11    y ( xˆ k + c k ∆ k ) − y ( xˆ k − c k ∆ k )  .  gˆ k ( xˆ k ) =  .  2 ck  −1  ∆ kn  th where ∆ ki is the i component of the ∆ k vector Step 5: Updating x estimate. Use the standard SA form: xˆ k +1 = xˆ k − a k gˆ k ( xˆ k ) to update xˆ k to a new value xˆ k +1 . Step 6: Iteration or termination. Return to Step 2 with k + 1 replacing k. Terminate the algorithm if there is little change in several successive iterates or the maximum allowable number of iterations has been reached.

Fig 1: The basic steps of SPSA algorithm

3- Optimum structural design program In this study, to achieve optimum design, we face three parts of work; geometric modelling, FEA and mathematical programming. Different program files are developed for each part, and communication between these parts is manipulated by an interface. One of the most interesting

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features of ANSYS program is the possibility to use it as a mere subroutine of any other external program. Parameters can be either directly passed or exchanged through external files, this flexibility allows us to build an interface between ANSYS and our external SPSA algorithm, written in MATLAB, where ANSYS is a finite element package able to calculate the objective function and constraints. In the present paper, the methodology schema developed for structural design optimization is shown in Fig 2, in the following the main parts are outlined. 3.1 The ANSYS file Commands for the construction of the geometric model, for meshing it and then applying loads, are incorporated in a command file using the ANSYS Parametric Design Language (APDL). The general procedure for ANSYS file is shown in Fig 3. Upon completion of the pre and postprocessing stages, ANSYS provides a results file which records very detailed information, this information are stored in a files.out and returned to the interface.

Fig 2: The methodology schema for structural design optimization

Fig 3: The general procedure for ANSYS file

3.2 The interface In the present work the design variables, the volume and the stress values are the required results extracted from the result file. These results are stored in separate files and returned to the interface. The aforementioned interface allows the interaction between MATLAB and ANSYS by defining the design variables as a macro, designvar.txt. The interface allows MATLAB to set the value of the design variables, obtained by SPSA program, in designvar.txt and let ANSYS to get these values and to create the volume.out and the stress.out. The SPSA program opens these files.out, run the algorithm and provides the new design variables. Therefore, a new shape will be created and analyzed. This procedure goes on until convergence. In the following the main parts of the MATLAB file are outlined. 3.3 The MATLAB file The basic algorithm for the mathematical programming is shown in Fig. 4 and can be summarized in the following steps 1. Define starting point. 2. Set the non-negative coefficients a, c, A, α and γ in the SPSA gain sequences a k = a /( A + k ) α and c k = c / k γ .

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3. Evaluate the objective function and constraints of the current design variable, by calling the ANSYS file. 4. Generate a new structural shape which satisfies the constraints, using the objective function value. 5. If the new structural shape is not optimum, go to step 3; otherwise STOP. 4- Numerical examples There are four examples in this section. In the first three examples the objective and constraint functions are all expressed as explicit functions of the design variables and the solutions are known; these examples are used to demonstrate how the proposed optimization method is applied and check the validity of the obtained results, these examples have been previously solved using a variety of other techniques, see Table 1. The last example is FEM-based, so the function is not known explicitly and it is rather dependent on the methodology developed for this purpose.

Fig 4: The optimization program.

Table 1: Optimization methods used for performance analysis.

4.1 Square function The square function is expressed as follows: 2  n  i −1  f ( x) = ∑  xi −   +1 n    i =1  where n is equal to the number of design variables. The optimization problem can be formulated as, min f ( x)   subject to − 2 ≤ x ≤ 4 for x = 1,..., n The minimum value is f ( x* ) = 1 and can be found in the point of [0,1,2,..., n − 1] / n . The square function is used with three and eight design variables, this function is one of the typical benchmark

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problems often used for numerical tests. The considered function has been solved by PSPSA and Table 2 reports the results found, where n=3. The solutions obtained are compared with previous solutions [27] for attempting to display the power of PSPSA algorithm.

Table 2: Results from the optimization of the square of the square function, 3 dimensions

Table 3: Results from the optimization function, 8 dimensions

Table 2 shows the optimization results returned by the three methods while solving the square function. It can be observed that the results obtained by SM and RSM are close to the analytical optima, but when compared with our proposed method, PSPSA, it can be clearly seen that the results are much closer to the analytical optima in terms of searching the accuracy of results. The accuracy adopted in this paper is in terms of Mean Square Error (MSE) given as follows: 1 p MSE = ( f i − f Analytical ) 2 where f = ∑ f i , in our case p = 1 . p i =1 The comparison given in Table 2 shows that PSPSA outperforms the others methods, for the three-dimensional square function, and also produces more accurate solution than the others in terms of MSE, it's found that the low MSE is provided by PSPSA. Table 3 provides results for the eight-dimensional square function. The starting point is x0 = [4,4,4,4,4,4,4,4] with f ( x0 ) = 103.1875 . As seen, PSPSA outperforms, once again, the other methods and can reach the global optimum with perfect accuracy, see MSE. We vary the dimensions of the square function from n=20, 50, 100, 200 to 500. In all cases the MSE is equal to 0. The results show that the proposed method reaches the optimum with perfect accuracy regardless of the dimensions of the problem. Contrary to SM and RSM which have some difficulties in achieving the optimum when the dimensions of the problem increase, Table 3 provides results for the eight-dimensional square function, it can be clearly observed that their MSE increases respectively with the increase of the problem's dimensions.

4.2 Sphere function n

f ( x) = ∑ xi2 ,

,−600 ≤ xi ≤ 600

i =1

The minimum value of the sphere function is f ( x * ) = 0 [33] and can be found in the point x * = (0,0,...,0). Where n is the dimensions of the sphere function. Table 4 summarizes the performance of the proposed method and compares the results found with the other methods. It can be observed that PSPSA performs well compared to the other methods and with the function’s dimension up to 150.

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Table 4: The results of PSPSA for the sphere function, n is the dimension of the test function.

Fig. 5:Two-bar problem, height h=1m

4.3 Two bars The third example considered is a structural optimization problem, with a two-bar truss, shown in Fig. 5. The optimization problem has three design variables, two cross-section areas and one geometric variable. The height of the truss is h = 1 m and the length of the bars varies with the base. The objective is to find a feasible set of dimensions Area1, Area2 and Base (denoted by x = [ x1 , x 2 , x3 ] ) that minimize the mass without violating constraints on the stresses in the bars. The optimization problem is formulated as follows: Minimize mass = ( x1 + x 2 ) (h 2 + x32 ) ρ  - 100 ≤ σ 1 ≤ 100 MPa  Subject to  - 100 ≤ σ 2 ≤ 100 MPa  20 ≤ x1 ≤ 400 mm 2   20 ≤ x 2 ≤ 400 mm 2  100 ≤ x3 ≤1600 mm    Where ρ = 0.01 g mm 3 and the stresses, Fx h + Fy x3 σ1 = h 2 + x32 2 h x1 x3 − Fx h + Fy x3 σ2 = h 2 + x32 2 h x 2 x3

The optimum solution is [ x1 , x 2 , x3 ] = [180.78 mm 2 , 20.0 mm 2 , 156.06 mm] the mass is equal to 2.0321 kg and both stresses are equal to 100MPa. In order to examine the capacity of PSPSA in structural optimization, a comparison is made with various prominent algorithms from the literature [27]. Table 5 shows that PSPSA outperforms, once again, the other methods. It's found that the low MSE is provided by PSPSA. The low MSE ensures the degree of consistency in producing the optimal value and hence the proposed method shows better consistency than the other methods. Computational results show that PSPSA reduces the mass value, objective function, about 14.45 % from SM shape and 1.43 % from RSM. One

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sees then, that PSPSA outperforms both of the methods and can reach the global optimum with perfect accuracy, see MSE indicated in Table 5.

Table 5: Optimal results of the two bars design, - means that the values are Unavailable

Fig. 6: The plate

4.4 Plate This last example is presented to demonstrate the capability of the methodology presented, above, for structural design optimization when the objective function and constraints are not known explicitly and are dependent on the FEA. 4.4.1 Problem description The problem considered concerns the design of a plate shape such that the maximum equivalent stress reaches the optimum value with the volume as a constraint, i.e. the goal is to determine the shape of the plate that minimize the σ von with the volume as a constraint. The objective function to be optimized is defined as follows:

Minimize Subject to :

σ von

V ≤ V0 (4) 16 ≤ DVi ≤ 40, i = 1,2 Where σ von is the Von Mises stress, V is the volume, V0 is the initial volume obtained with initial design variables, i is the index number of the design variables and DVi is the design variable number i. In this case, we have two design variables P1 and P2 as shown in the Fig. 6. 4.4.2 Numerical results In order to show the applicability, the efficiency and the consistency of the proposed method a comparison is made with the international software ANSYS, the optimization method used for ANSYS is First Order method (FO). To solve problem (4) we have applied two schemes: Scheme 1: both Structural Analysis and Optimization done using the commercial software FEA code ANSYS. Scheme 2: Structural Analysis done in the commercial software FEA code ANSYS but an external Optimization code is used (PSPSA optimizer).

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Table 6: PSPSA vs. FO

Fig. 7: The Graph of the initial shape

Fig. 8: The Graph of the optimal shape

Table 6 shows the plate results when using FO method implemented in ANSYS and PSPSA method. The optimal Von Mises stress obtained by our method is less than that obtained by the FO method. Calculation shows that the reduction from FO method is 3.91%. 4.4.3 Graphical results Calculations show that the structure considered had an initial shape of σ von,init = 15.2745 , see Table 6, result provided by PSPSA is σ von,opt = 11.5640 . It can be noticed that the Von Mises stress value reduction is about 24.29 % from initial shape (see Fig. 7 and Fig. 8), which leads to economic structure. As evidenced by the computational results given here, the proposed methodology shows a great deal of promise as a robust methodology with higher accuracy, efficiency and robustness in terms of solution quality for structural problems.

5- Conclusion A methodology for structural design optimization, when the objective and constraint functions are not known explicitly, is proposed. Its accuracy is demonstrated by solving a number of examples. It follows from the computations that PSPSA always converges to the optimal solution with the highest accuracy compared to that of other methods from the literature. It can be concluded that PSPSA method is a suitable feasible optimization method which can be used to handle large scale problems. Since it, only, needs two evaluations of the objective function regardless of the dimensions of the design space corresponding to the optimization problem. The present investigation forms the basis for further study to look at more complex real structures and to compare the results found by PSPSA with other optimization methods including Simulated Annealing or Genetic Algorithms to come to more general conclusion.

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Applied Mechanics and Materials Vol. 61 (2011) pp 55-64 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.55

A multi-scale analysis of materials reinforced by inclusions randomly oriented in the ply plane Emmanuel Lacostea, Sylvain Fréourb, Frédéric Jacqueminc Institut de Recherche en Génie Civil et Mécanique (UMR CNRS 6183), Université de Nantes – Centrale Nantes, 37 Boulevard de l’Université, BP 406, 44 602 Saint-Nazaire cedex, France a

[email protected], [email protected], c [email protected]

Key words: composite materials, multiscale analysis, Kröner – Eshelby self-consistent model, multiple inclusions, random orientation, thermoelastic behaviour.

Abstract. The present work aims to investigate the validity of Eshelby-Kröner self-consistent model [1] for thermoelastic behaviour, in the case of a material reinforced by inclusions randomly oriented in the ply plane. The model provides predictive information on the properties and multi-scale mechanical states experienced by the material, accounting for its constituents properties, but also their morphology. However, it cannot reliably account for multiple inclusion morphologies (shape and orientation) in the material [2-4]. A study of the two applicable formulations and their limits leads to suggest a mixed formulation as an acceptable compromise between those alternatives. The results of this original approach are also described in the case of a thermo-mechanical load. Introduction The recent development of composite materials during the last decades opened new prospects to mechanical part engineering, particularly for aeronautical applications, because of their high strength-to-weight ratio as well as corrosion and fatigue resistance. However, the inherent heterogeneity of these materials induces internal stresses during the curing process, and complex mechanical behaviour under service loads. The so-called “scale transition models” answer the necessity to predict the distribution of stresses between the constituents of the composite plies. Among these, the self-consistent model for the polycrystals [1] suggests a realistic and interesting approach; it enables one to calculate the homogenized properties of the material, but also the mechanical states experienced by the constituents, accounting for their morphology and properties. The present application is focused on an in-plane isotropic composite material [5] made of unidirectional reinforcing strips with rectangular shape (60x8x0.15 mm) and randomly disposed in the ply plane (see Fig. 1 below). The strips themselves are composed of an unidirectional (UD) carbon-organic ply. An N5208 epoxy matrix and T300 carbon fibers were considered for this study.

2 y

Matrix

Reinforcing UD strip

1 Fibers + Matrix

Θ x

Figure 1: Schematic representation of the microstructure of the composite

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The self-consistent model is used twice in our approach, in order to perform a two-steps scale transition procedure (Fig. 2). First, the effective properties of the reinforcing strip are estimated from those of the intra-reinforcements matrix and the carbon fibers. These effective properties are then used for the second homogenization step, in order to find those of the whole composite.

Figure 2: Schematic representation of the two steps scale transition procedure The self-consistent model Hill’s formalism and Eshelby’s inclusion. Scale transition models are based on a representation of the material at several scales: on the one hand, the “local” scale denoted by the superscript i, where one observes the behaviour of each constituent, considered as an ellipsoidal and homogeneous inclusion (also called Base Volume or BV). On the other hand, the macroscopic scale, denoted by the superscript I, defines the behaviour of the Effective Medium (EM). A linear thermoelastic law (Eq. 1) expresses those behaviors. In this relation, the stiffness is represented by the 4thorder tensor L, and the Coefficients of Thermal Expansion (CTE) by the 2nd-order tensor α. The temperature increment is denoted by T, whereas σ and ε stand respectively for the stress and strain. σ k = Lk : (ε k − α k T ), k = i, I .

(1)

Hill [6] demonstrated, in a very general way, the equivalence between volume integrals and set (i.e. volume fraction weighted) averages, denoted here by angle brackets . The semi-statistical (“mean-field”) approach studied here uses Hill’s volume average relations over the mechanical states, written as in the equations below (Eq. 2, Eq. 3): εI = εi ,

(2)

σI = σi .

(3)

In a fundamental work, Eshelby [7] determined the behavior of an inclusion embedded in a homogeneous medium, loaded at the infinite. He demonstrated that, if the inclusion had an ellipsoidal shape, the local stresses and strains were homogeneous inside the BV. Using this work, Hill [8] proposed the following relation (Eq. 4) between local and overall states, where L* is widely known as Hill’s constraint tensor. σ i − σ I = − L* : (ε i − ε I ) , with

(

L* = LI : S I

−1

) (

− I (4) = E I

−1

(4)

)

− LI .

(5)

This tensor can be obtained from the Eshelby tensor SI or the Morris tensor EI [1, 9] (also referred to “influence tensor” PI), thanks to Eq. 5 (with I(4) the 4th-order Identity tensor). In the general case, the computation of the Morris tensor is a key-point of the model, as it implies several tensorial inversions and a numerical integration. Yet, analytical expressions can be found in the literature for

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a few specific configurations [10]. Morris’ tensor depends on the stiffness of the embedding medium, and most importantly, on the shape ratios and orientation of the elementary inclusions. This dependency allows one to take into account the morphology of the constituents, i.e., to represent the microstructure of the considered material. Thus, the material anisotropies yielded by the microstructure (in glass fiber unidirectional plies, for instance) can be taken into account straightforwardly, which is a great advantage over to the more classical “rules of mixtures”. Several models are derived from these equations. The first and simplest one is the “dilute approximation”, where one inclusion interacts with an embedding medium. The Mori-Tanaka scheme is an extent of this approximation to non-dilute inclusions, where one of the constituents (often called the “matrix phase”) is considered as the embedding medium; the model is mainly adapted for materials whose matrix phase volume fraction dominates all the others. On the contrary, within the self-consistent model, the embedding medium is given the properties of the EM, which makes the expressions of the effective stiffness implicit. Yet, this approach is more adapted than Mori-Tanaka’s for materials where all the phases have comparable volume fractions, as the EM yields a better emulation of the inter-particle interactions than the resin [11]. At high reinforcement ratios, the self-consistent model gives more reliable information than the Mori-Tanaka model upon the effective properties, particularly over the shear moduli. For this reason, the self-consistent model should be preferred to the Mori-Tanaka model to perform the second homogenization step (mesomacro), where one cannot define a dominating medium. Eshelby’s equations were first used by Kröner for estimating the elastic moduli and the plastic behaviour of polycrystals [12, 13], from the properties of its constitutive crystallites. This elastoplastic model inspired similar studies on the thermoelastic behaviour of heterogeneous materials [14], then on the time-dependent creep and relaxation of polycrystals [15]. They were also found interesting for composite materials with organic [10, 13] or metallic matrix [16], owing to the strong heterogeneity of their constituents. Formulation with stresses and strains. If the inclusions constituting the material do not present a single morphology in the macroscopic coordinate system RI, the tensor L* is not purely macroscopic anymore but related to each inclusion (and in consequence, denoted L*i). Then, using (Eq. 1) and (Eq. 4), one can express the local stresses and strains as:

[

]

(6)

]

(7)

Ri

ε i = (Li + L * i ) : (LI + L * i ) : ε I + (Li : α i − LI : α I ) T = A i : ε I + a i . T , and

Ri

σ i = Li : (Li + L * i ) : (LI + L * i ): LI : σ I + L* i : (αI − αi ) T = Bi : σ I + b i . T,

−1

−1

[

−1

where Ai is the elastic strain localization tensor, Bi is the elastic stress concentration tensor, and ai and bi are the thermal strain and stress polarization tensors, respectively. Of course, Hill’s averages principles imply and the following relation over the averages of Ai, Bi, ai and bi (Eq. 8, Eq. 9), where 0(2) represents the 2nd-order null tensor: A i = I (4) and a i = 0 (2)

(8)

B i = I (4) and b i = 0 (2)

(9)

Hill’s averages principles are expressed over both strains and stresses. Nevertheless, in the classical self-consistent scheme, only one relation is needed in order to obtain the effective properties, which leads to two alternate expressions. Actually, using Hill’s average principle over the stresses (formulation denoted Hσ), one obtains the following stiffness and CTE given in

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(Eq. 10) and (Eq. 11). On the contrary, if one uses Hill’s average principle over the strains (formulation denoted Hε), one would obtain the effective properties given in (Eq. 12) and (Eq. 13). σ LI = Li : A i ,

H

(10)

σ α I = Li : (Li + L* i )−1 : L* i

−1

ε LI = LI : A i

−1

H

H

−1

−1

= Li : B i

ε α I = LI −1 : (Li + L* i )−1

H

−1

: Li : (Li + L* i ) : L* i : α i ;

(11)

,

(12)

−1

: (Li + L* i ) : Li : α i . −1

(13)

Self-consistency of the model. Several authors have shown that the two above formulations are equivalent, but with some restrictions over the materials microstructure: either the inclusions must have the same morphology (shape and alignment), or the material and inclusions must be isotropic. Analytical expressions of the thermo-elastic macroscopic properties of the EM have been determined for these specific cases in a series of paper from Benveniste, Dvorak and Chen [3, 4]. These configurations match many industrial applications, and in particular polycrystals and unidirectional composites plies. Similar models have also been applied to new industrial materials that present a microstructure containing anisotropic inclusions of various morphologies or geometrical orientations [17]. However, for such microstructures, Benveniste, Dvorak and Chen [2-4] demonstrated that the MoriTanaka and self-consistent approximations may lead to two distinct sets of effective properties, and violate some rigorous bounds. At the present time, the computation of thermo-elastic properties for materials exhibiting this kind of microstructure still seems to constitute an open question. Computing the effective properties of the composite Effective properties of the reinforcing strips. The homogenization of the reinforcing strip corresponds to the case, treated in a recent paper [18], of an unidirectional fiber-reinforced composite ply. As a unique morphology is considered for every constituent of the reinforcing strip, the two above formulations of the self-consistent model lead to the same results. Consequently, one will only give the effective properties of the reinforcing strip (Table 1), accounting for a fiber volume ratio of 63 %. Mechanical moduli Ex Ey, Ez νxy, νxz Gxy, Gxz νyz [GPa] [GPa] [GPa]

CTE Gyz αx αy, αz -6 [GPa] [10 /K] [10-6/K]

Reinforcing strip 146.8 10.2 0.274 7.0 0.355 3.8 -0.620 48.0 T300 fibers [18] 230 15 0.20 15 0.07 7 -1.5 27 N5208 matrix [19] 4.5 4.5 0.4 1.61 0.4 1.61 60 60 Table 1: Thermomechanical properties of a reinforcing strip and its constituents. Effective properties of the composite. The second homogenization step is more problematic, as it involves inclusions of ellipsoidal shapes presenting a random geometrical orientation, combined with a strong anisotropy of the constituents and the effective medium. As a consequence, the two formulations detailed above (Eq. 10-13) will be used to perform this second homogenization step.

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(Eq. 10) and (Eq. 12) provide the effective stiffness of the composite. The averaging operations are achieved onto the two previously described constituents (reinforcing strips with 95 % volume ratio and extra-reinforcement matrix with 5 % ratio). A set of 10 orientations, uniformly distributed on 180° in the 1-2 plane (ply plane, see Fig. 1), is considered in order to account for the random orientation. The effective elastic moduli obtained are summed up in the Table 2 below. The notation Hmixed denotes the direction-dependent formulation, described below. E1, E2 E3 G12 ν12 ν13, ν23 G13, G23 [GPa] [GPa] [GPa] [GPa] Hσ 55.48 11.24 0.290 21.49 0.266 3.97 Hε 16.63 9.92 0.121 7.42 0.337 4.45 Hmixed 55.41 11.20 0.289 21.49 0.279 4.496 Voigt bound 55.46 11.28 X 21.49 X 5.20 Reuss bound 15.44 9.66 X 6.65 X 4.46 Table 2: Estimated elastic moduli of the effective composite. One may first notice that the two methods lead to drastically different stiffnesses, particularly for the components that govern the in-plane behaviour (E1, ν12 and G12). On the contrary, the “out-ofplane” components (E3, ν13 and G13) do not vary very much from a homogenization procedure to the other. Moreover, these moduli respect, within the prescribed 10-3 accuracy, the Reuss and Voigt bounds (see Table 2). The elastic moduli obtained with the strain-based formulation are close to the Reuss bound; whereas those obtained with the stress-based formulation are practically merged with the Voigt bound. Yet, previous works achieved on composite laminates (which have a similar structure) suggested that extreme direction-dependent homogenization procedures gave satisfying results on both in-plane and out-of-plane behaviour. A mixed homogenization scheme, inspired from the Vook-Witt model [20, 21] is thereby proposed. The in-plane behaviour is modeled using the stress-based formulation, whereas the strainbased formulation is used for the out-of-plane behaviour. With this mixed formulation, the stiffness tensor of the composite satisfies the form given in (Eq. 14) below. The results given by this formulation are also summed up in Table 2 above.  Hσ LI 11  Hσ I  L12  Hε I L 31 Hmixed I L =  0   0   0

σ LI

H

σ LI

H

ε LI

H

H

ε LI

0

0

ε LI

0

0

0 ε LI

0

12

H

22

H

32

13 23

ε LI

33

H

0

0

0

0

0

0

0

0

0

44

ε LI

H

44

0

  0   0  . 0  0   Hσ I L 66  0

(14)

The computation of the averages for localization and concentration tensors, upon the two constituents and all the orientations, is a good indicator of the relevance of each formulation (see Table 3 below). For the two “pure” formulations (Hσ and Hε), the in-plane components (11, 12 and 66 components in Eq. 14) exhibit very important errors regarding Hill’s averages principles; for them, the stress-based formulation (Hσ) gives the lowest errors. On the contrary, for the out-of-plane components, the strain-based formulation (Hε) is the most reliable. The direction-dependent formulation combines the advantages of those two and guarantees a relative error lower than 3 % on every term.

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It is important to notice that these errors are null for a penny-shaped morphology, whichever formulation is used. However, they quickly grow up with an increased stretching or thickness of the reinforcing strips, so the model is satisfying only for a limited range of morphologies. Ai

A11 = A22

A33

A44 = A55

A66

A12 = A21

A13 = A23

A31 = A32

Hε Hσ Hmixed

1.001 1.031 1.029

1 1 1

0.5 0.443 0.500

0.5 0.512 0.511

0 0.007 0.007

0 0 0

0 -0.033 -0.015

Bi

B11 = B22

B33

B44 = B55

B66

B12 = B21

B13 = B23

B31 = B32

Hε 3.465 1 0.5 1.431 0.604 -1.178 0 Hσ 1.000 1 0.5 0.5 0 0 0 Hmixed 1.001 1.001 0.497 0.5 0.001 -0.009 0.001 Expected 1 1 0.5 0.5 0 0 0 Table 3: Averages of the localization and concentration tensors, for the composite. As above, the Coefficients of Thermal Expansion have been computed with respect to the three homogenization approaches previously presented (see Eq. 11 and Eq. 13). With the mixed formulation, the effective CTE satisfies the structure given below (Eq. 15). The stiffness obtained with each formulation was used for computing the corresponding CTE. This leads to the results presented in Table 4 below. As for the stiffness, a significant deviation between the results occurs, depending on the homogenization procedure used. This discrepancy remains if the same stiffness is used for the three formulations. In order to quantify the relevance of these results, the dimensionless errors X and Y were also defined as in (Eq. 16), and detailed in Table 4. I  Hσ α11  Hmixed αI =  0  0 

0 Hσ I α

11

0

 X = a i .(α I T )−1   −1 i I I  Y = b .(L : α T )

0   0 . Hε I  α 33 

(15)

(16)

One can observe that the strain-based formulation satisfies Hill’s averages principle over the strains, but leads to some errors on thermal stresses and mainly in the normal direction. Respectively, the stress-based formulation satisfies Hill’s averages principle over stresses but underestimates the in-plane thermal strains by more than 200%. As observed for the elastic behaviour, the mixed formulation gives the best compromise between these two aspects, and leads to absolute errors lower than 3 % on every term (increasing with the shape ratios of the reinforcing strips). This error vanishes if one prescribes a penny-shaped morphology to the constituents. This demonstrates that the classical self-consistent model is valid if and only if a single morphology is used for all the constituents in the Representative Elementary Volume. This fact is very important as, for this particular case, the property values obtained using a penny-shaped morphology and a stretched one are very close (with Hmixed formulation only), which opens a way for a significant simplification of the model.

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CTE

Error on ε X1 X3

61

Error on σ Y1 Y3

α3 α1, α2 -6 [10 /K] [10-6/K] Hε 24.8 49.9 0 0 -2.088 0 Hσ 3.5 114.9 0.062 -0.424 0 0 Hmixed 3.5 64.3 0.031 0.027 0.004 0.002 Expected [5] 3.5~5 45~50 0 0 0 0 Table 4: Estimated CTEs of the composite and associated errors. A few experimental results can corroborate the estimated effective properties. The in-plane elastic modulus was measured to be approximately 42 GPa [5], and the CTEs were also measured at room temperature (see Table 4). These values are in the same order of magnitude to those estimated with the scale transition procedure. The difference is due to the use of slightly different constituents (AS4 fibers and bismaleimide resin) and fiber ratio (53 %), and to out-of-plane waviness of the reinforcing strips (which deteriorate their in-plane apparent properties). A more realistic estimation is given in [22], for which the predicted properties match the measured ones to within ±5 %. Application to thermo-mechanical loads Response of the composite to purely mechanical load. In order to describe the multi-scale mechanical behaviour of the composite, a macroscopic uniaxial in-plane traction of 100 MPa is considered. (Eq. 7) is used to compute the local stresses in the constituents, dropped in the local coordinate system Ri.

Figure 3: Local stress states in the composite under a 100 MPa uniaxial traction. One can observe (Fig. 3) that these stresses evolve with the orientation angle as π-periodic sinusoids. At the mesoscopic scale, the in-plane stresses are strongly heterogeneous (contrarily to

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the in-plane strains, which are rather homogeneous); the reinforcing strips experience up to 260 MPa in the x-direction, while the organic matrix undergoes less than 10 MPa. This concentration of stresses is a result of the orientation mismatch between the reinforcement strips and the loads. It is increased into the fiber constituting the reinforcement strips, which experience more than 400 MPa while the matrix takes less than 20 MPa. On the other hand, out-of-plane stresses are very low: one will only notice the emergence of complementary stresses between the constituents of the reinforcing strips. Response of the composite to purely thermal load. A similar study was achieved for the case of a macroscopically stress-free -100 °C thermal load, which is typical of the cool-down that occurs after the curing stage of composite materials, and which induces severe residual stresses [23, 24]. Contrarily to the mechanical load studied previously, the thermal expansion respects the in-plane symmetry of the material, thus the shear stresses and strains are null, and the mechanical states are independent on the orientation angle Θ of the reinforcing strip. This result can be generalized to any load respecting the in-plane symmetry. Scale

Medium

Macroscopic

Composite Extra-reinforcement matrix Reinforcing strips Intra-reinforcement matrix Fibers

Mesoscopic Microscopic

σxx 0.0 42.6 -48.1 57.79 -109.9

Stresses (MPa) σyy σzz 0.0 0.0 42.5 0.4 44.1 0.1 55.51 25.39 37.5 -14.7

Table 5: Stress states in the composite and its constituents exposed to a -100 °C thermal load.

The local stresses inside the composite are summed up in the Table 5 above. At the mesoscopic scale, one can observe a marked gap between the σxx stresses in the two constituents: the matrix undergoes traction stresses, although the reinforcing strips are compressed (which implies a risk of meso-buckling). The same scheme appears at the microscopic scale: the fibers are compressed and the matrix stretched. Along the normal direction z, one can also notice the emergence of nonnegligible complementary stresses in the fibers and the matrix, due to the gap of properties. Discussion and perspectives A two-steps scale transition procedure based on the self-consistent model has been introduced to describe the thermo-mechanical behaviour of a composite material reinforced by inclusions randomly oriented in the ply plane. The limits of the self-consistent model for this kind of microstructures have been discussed on the base of Hill’s averages principles, which were used to define error estimators. Those results inspired the use of a mixed direction-dependent formulation, which enables to drastically reduce the error for this particular material and microstructure. This method may also be applicable to other particles-reinforced materials such as short-fibers reinforcements, or nanocomposites (as an interesting alternative to the Krenchel model [25, 26]). More generally, for materials constituted by inclusions with very distinct shapes and/or alignment, a special attention must be paid to the uniqueness of estimated properties and the satisfaction of Hill’s averages principles. In a second time, this mixed formulation has been applied to the simulation of local stresses in the constituents of the material, when subjected to thermal of mechanical loadings. For uniaxial loading, a dependency of the local stresses on the orientation angle between the reinforcing strips and the direction of solicitation was found, along with high stress levels into the rigid elements. On the other hand, thermal loading (cooling) seems to tax mainly the matrix with tensile stresses, while the fibers are subjected to compressive stresses that might induce micro-buckling effects. However,

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as the self-consistent model only provides average values of the local stress fields, its application to the determination of material health is somewhat limited. A direct application of this work is the simulation the development of residual stresses during the manufacturing process of carbon-epoxy composites [22]. The very low computational time cost of the self-consistent method may be an advantage for this kind of simulations, where the very nonlinear behavior of the resin during the fabrication process (due to hardening, thermo-chemical shrinkage, dependency with temperature…), implies repeated computations of the effective properties. References [1] U.F. Kocks, C.N. Tomé and H.R. Wenk: Texture and anisotropy (1998), Cambridge University Press. [2] Y. Benveniste: A new approach to the application of Mori-Tanaka’s theory in composite materials, Mechanics of Materials, Vol. 6 (1987), pp. 147-157. [3] Y. Benveniste, G.J. Dvorak and T. Chen: On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media, Journal of Mechanics and Physics of Solids, Vol. 39 (1991), pp.927-946. [4] T. Chen, G.J. Dvorak and Y. Benveniste: Mori-Tanaka estimates of the overall elastic moduli of certain composite materials, Journal of Applied Mechanics, Vol. 59 (1992), pp. 539-546. [5] Hexcel France: Hextool Datasheet (information on www.hexcel.com). [6] R. Hill: The essential structure of constitutive laws for metals composites and polycrystals, Journal of the Mechanics and Physics of Solids, Vol. 15 (1967), pp. 79-95. [7] J.D. Eshelby: The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems, Proceedings of the Royal Society London, Issue A241 (1957), pp. 376–396. [8] R. Hill: The essential structure of constitutive laws for metals composites and polycrystals, Journal of the Mechanics and Physics of Solids, Vol. 13 (1965), pp. 89-101. [9] R. Morris: Elastic constants of polycrystals, International Journal of Engineering Science, Vol. 8 (1970), pp. 49-61. [10] S. Fréour, F. Jacquemin and R. Guillén: Extension of Mori-Tanaka approach to hygro-elastic loading of fiber-reinforced Composites – Comparison with Eshelby-Kröner self-consistent model, Journal of Reinforced Plastics and Composites, Vol. 25 (2006), pp. 1039-1052. [11] J. Berryman, P. Berge: Critique of two explicit schemes for estimating elastic properties of multiphase composites, Mechanics of Materials, Vol. 22 (1996), pp. 149-164. [12] E. Kröner: Berechnung der elastischen Konstanten des Vielkristalls aus des Konstanten des Einkristalls, Zeitschrift für Physik, Vol. 151 (1958), pp. 504-508. [13] E. Kröner: Zur plastischen verformung des vielkristalls, Acta Metallurgica, Vol. 9 (1961), pp.155-161. [14] J.W. Hutchinson: Elastic-plastic behaviour of polycrystalline metals and composites, Proceedings of the Royal Society London, issue 319 (1970), pp. 247-272. [15] G.J. Weng: A self-consistent relation for the time-dependent creep of polycrystals, International Journal of Plasticity, Vol. 9 (1993), pp. 181-198.

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[16] E. Le Pen, D. Baptiste: Multi-scale fatigue behaviour modelling of Al/Al2O3 short fibre composites by a micro-macro approach, International Journal of Fatigue, Vol. 24 (2002), pp. 205-214. .

[17] D. Baptiste: Non Linear Behavior Micromechanical Multi-scale Modelling of Discontinuous Reinforced Composites, Materials Science Forum, Vol. 426-432 (2003), pp. 3939-3944. [18] F. Jacquemin, S. Fréour and R. Guillén: A hygro-elastic self-consistent model for fiberreinforced composites, Journal of Reinforced Plastics and Composites, Vol. 24 (2005), 485502. [19] A. Agbossou, J. Pastor: Thermal stresses and thermal expansion coefficients of n-layered fiberreinforced composites, Composite Science and Technology, Vol. 57 (1997), pp.249-260. [20] R.W. Vook, F. Witt: Thermally induced strains in cubic metal films, Journal of Applied Physics, Vol. 39 (1968), pp. 2773-2776. [21] U. Welzel, S. Fréour: Extension of the Vook-Witt and inverse Vook-Witt elastic graininteraction models to general loading states, Philosophical Magazine, Vol. 87 (2007), pp. 3921-3943. [22] E. Lacoste, K. Szymanska, S. Terekhina, S. Fréour, F. Jacquemin, M. Salvia: A multi-scale analysis of local stresses during the cure of a composite tooling material, In Press (Composites Part A). [23] J.A. Guemes: Curing Residual Stresses and Failure Analysis in Composite Cylinders, Journal of Reinforced Plastics and Composites, 13 (1994), pp. 408-419. [24] K. Ogi, H.S. Kim, T. Maruyama and Y. Takao: The infuence of hygrothermal conditions on the damage processes in quasi-isotropic carbon/epoxy laminates, Composites Science and Technology, Vol. 59 (1999), p. 2375. [25] H. Krenchel, in: Fibre Reinforcements (1964), Akademisk Forlag, Copenhagen, Denmark. [26] E.T. Thostenson, T.W. Chou: On the elastic properties of carbon nanotube-based composites: modeling and characterisation, Journal of Physics D: Applied Physics, Vol. 36 (2003), pp. 573582.

Applied Mechanics and Materials Vol. 61 (2011) pp 65-69 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.65

Experimental study of the short-term creep behavior of CFRP strengthened mortar under compressive loading KHADRAOUI Fouzia1, a, KARAMA Moussa2, b 1

ESITC Caen; Campus2 – 1 rue Pierre et Marie Curie, 14610 Epron, France 2

ENIT; 47 avenue d’Azeirex BP. 1629, 65016 Tarbes Cedex, France a

[email protected], [email protected]

Key words: mortar; composite material; creep; strengthening; strains.

Abstract. Creep in cementitious materials is an important part of the delayed strains. It is a complex phenomenon in which many physical and chemical parameters are involved. In this paper, an experimental program was conducted to clarify the creep performance of CFRP strengthened mortar. The main parameters under study are the age at the time of loading and the drying. Specimens are tested at a sustained load of 30% of the ultimate strength. These investigations show the interest of the reinforcement by CFRP of the prismatic mortar specimens, this one allows a notable improvement of the creep behavior. Introduction The use of composite materials in the repair or the reinforcement of the concrete structures has undeniable advantages. Several studies were carried out to explain the behaviour of the concrete strengthened with composite materials. Many works was devoted to the study of the mechanical behavior of the concrete structures reinforced with composite materials [1-7], much less investigations was interested in the study of their delayed behavior. However, the prediction of the concrete strains is essential to study the durability of the structures. Indeed, these strains could result in cracking, and even more in the ruin of the structure. The strains are essentially due to creep phenomenon. They depend on the intrinsic properties of the material and on the external conditions such as applied stress. Indeed, after a reinforcement of the concrete with CFRP composites, its behaviour becomes more complex. The redistributions of the stresses due to creep can affect the interface concrete - reinforcement, and of this fact on the longterm behavior of the reinforced structure. In this context, the main aim of this work is to study the influence of the composite reinforcement on the delayed behaviour. We will study the consequences of creep in compression on prismatic mortar specimens. Specimen preparation The experimental study is related to prismatic mortar specimens which are reinforced by using composite material plates, tested at 7 or 28 days for short-term creep. Normalized mortar was prepared with CPA-CEM I 52.5 cement, and a Water/cement ratio of 0.5. Prismatic test specimen measuring 4x4x16 cm were removed from the moulds one day after casting, and then stored at 20°C and 50% of relative humidity. The compressive strength value at 28 days was 50.5 MPa. Unidirectional carbon fibre plates « Sika Carbodur S » were used for the strengthening (see table 1)

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according to the recommendations of SIKA. Bonding is carried out using an adhesive epoxy "Sikadur-30 colle". Table 1 – Composite characteristics

The mortar specimens were strengthened on the four side faces (see figure 1). The samples intended for the creep tests were reinforced one day before the time of testing. Non reinforced specimens were also tested and used as reference samples.

Figure 1 – Specimen reinforced with CFRP plates

Test setup and procedure To measure the creep of the different specimens, we used a machine of compression Lloyd (LR 30 kN) provided with a sensor of force allowing to apply a maximum force of 30 kN. The machine contains a plate high mobile and a fixed lower plate. To measure the longitudinal strains of the samples tested, an inductive sensor of displacement LVDT AX/1.0 allowing a maximum race of 1 mm was used. It was placed in the hole envisaged for this purpose in the plate mobile of compression of the machine and posed on a support. The setting charges some is done in two stages: the first corresponds to the rise in load at a generally quasi-instantaneous speed of 0.5 MPa /s (value resulting from the tests from Chen and Wang [8]). The second phase maintains the load applied constant during time, and thus makes it possible to obtain the creep strains. The specimens were tested at a sustained load of 30 % of the ultimate strength. The load was then maintained for 48 hours (duration of creep test).

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Creep test results The experimental curves giving the short-term evolution of the strains in time show the presence of secondary creep for the reference samples and almost a primary creep for the reinforced ones (see figure 2). When the specimens are reinforced with composite plates, the strains due to creep are reduced and the progression of theses ones depends on the age at loading. At 7 days, the slope of the linear creep evolution between the first and the 2nd day after the beginning of the test represents 4% compared to reference specimens. The slow evolution of the creep strains is due to the slow progression of the cracks which can be allotted to the capacity of the composite to slow down the damage created by the stress and to reduce the hydrous exchange with the external environment at this term. At 28 days, the kinetic, in the same amount of time, represents 18% compared to the reference specimens. The variation of creep kinetic is less important in this term, the evolution of the kinetic remains however weaker in the presence of the composite. Indeed, in this term, the composite makes it possible to slow down the evolution of cracking by slowing down the damage of the specimen. However, it does not influence much the hydrous exchange, considering this last is less after 28 days. (a)

(b)

Figure 2 – Creep strains evolution for non reinforced (a) and reinforced (b) specimens At the end of the test (48 hours), the reduction of the creep strains compared to the reference samples reaches 73% for the tests at 28 days. Figure 3 shows the comparison of creep of the mortar specimens which dried before and during the test, and of the specimens which were protected from the desiccation as of the release from the mould. The results show an important variation of strains between the drying specimens and those protected from the desiccation. At 7 as at 28 days, the strains variation increases with time. At the end of the test, there are 2.5 times more creep on the specimens drying than on the specimens protected from the desiccation. It is noted whereas drying increases the creep strains. Indeed, drying increases the shrinkage strains and can lead to a microcracking thus affecting the macroscopic properties of material. Because of progression of the cracks, the shrinkage of desiccation can be thus at the origin of the increase in the total creep strains for the specimens not protected from the desiccation.

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Figure 3 – Drying influence on the creep of mortar specimens at 7 days (a) and at 28 days (b) For a sustained load of 30%, the strain kinetic evolution is identical at 7 or 28 days for the same type of mortar as we saw previously. On the other hand, for an age of loading given, the kinetics evolution of secondary creep between 1st and the 2nd day under loading represent more than the double in the case of the drying specimens. This can be explained by the fact why drying accelerates the progression of cracking, and thus evolution of the creep strains. In order to evaluate the sealing of the composite and its capacity to reduce drying, creep tests were realized at 28 days on reinforced samples protected or not from the desiccation over the duration of test. An aluminium adhesive was stuck on the four side faces of the various samples. When the mortar sample is reinforced on four faces, its strain due to creep does not seem influenced by drying. Figure 4 shows that the strains curves are almost confused during the 1st day under loading. Beyond that, a variation of about 20% is observed. As the results are comparable for the reinforced or reinforced drying samples protected from the desiccation, it is followed from there that the presence of the composite on the surface prevents the transfers of water of the mortar towards outside, at least throughout our test.

Figure 4 – Drying influence on the creep of reinforced mortar

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Conclusion This paper presents interesting results about creep behavior of externally bonded mortar specimens. Short-term creep tests were carried out on CFRP reinforced specimens and on reference samples at the age of 7 or 28 days. At the young age as at maturity, the strains are clearly reduced in the presence of the composite. The creep kinetic evolution, as well as the progression of cracking, are also decreased. Specimens tested were insensitive to the age of loading, for the studied load. In order to dissociate the strain due to the mechanical load of that due to drying, a series of creep tests were carried out on samples protected from the desiccation during the test. The results obtained on the reinforced and protected specimens are quasi-identical to those obtained on drying similar specimens. The composite plays well the part of a screen reducing drying considerably.

References [1] B. Taljsten, and A. Carolin : Strengthening of a concrete railway bridge in Lulea with carbon fibre reinforced polymers – CFRP: Load bearing capacity before and after strengthening, Technical report :18, Luleâ: Luleâ University of Technology, Structural Engineering, 61p, (1999). [2] A. Carolin : Carbon fibre reinforced polymers for strengthening of structural elements, Doctoral Thesis, Lulea University of Technology, 2003:18, pp.194, (2003). [3] L.C. Hollaway and M.B. Leeming: Strengthening of reinforced concrete structures using externally bonded FRP composites in structural and civil engineering, Woodhead Publishing Limited, Cambridge England, First published 1999, reprinted 2001. [4] P.J. Fanning and O. Kelly : Ultimate response of RC beams strengthened with CFRP plates, ASCE Journal of Composites for construction, May 2001, pp. 122-127, (2001). [5] A. Li, T. Assih and Y. Delmas: Discussion on the failure mechanism of RC beams strengthened in shear by externally bonded CFRP sheets, 1-10, (1999). [6] B. Taljsten: Plate bonding: Strengthening of existing concrete structures with epoxy bonded plates of steel or fibre reinforced plastics, Doctoral thesis. Lulea University, Suede, (1994). [7] V. Sierra Ruiz : Renforcement d’éléments structuraux en béton armé à l’aide de matériaux composites : analyse fine de la zone d’ancrage, XXIèmes Rencontres Universitaires de Génie Civil, pp. 227-236, (2003). [8] Z. Chen and M.L. Wang : A partitioned solution method with moving boundaries for nonlocal creep damage concrete, Creep and shrinkage of concrete proceeding of the fifth international RILEM symposium, Barcelone – London : E & Fn Spon., p. 393 – 398, (1993).

Applied Mechanics and Materials Vol. 61 (2011) pp 71-77 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.71

Contribution of AFM observations to the understanding of Ni3Al yield stress anomaly Joël Bonneville a, Dimitri Charrier b and Christophe Coupeau c Institut P’, Université de Poitiers/CNRS/ENSMA, UPR 3346 SP2MI, Teleport 2, F-86962 Futuroscope, France a

[email protected], [email protected], c [email protected]

Key words: Intermetallics, Stress anomaly, Fine slip line structures, Atomic force microscopy

Abstract. We report in the present paper a practical situation where the use of atomic force microscopy allowed an irrefutable insight in material plasticity for discriminating between different modelling hypotheses concerning the yield stress anomaly of Ni3Al intermetallic compounds with the L12 ordered structure. The contribution of AFM to a better understanding of elementary rate controlling mechanisms as well as collective dislocation motions is highlighted.

Introduction Atomic force microscopy (AFM) is an experimental technique that allows for the investigation of surface structures at the nanometer scale, which is particularly sensitive to off-surface displacements, i.e. to surface relief. It has proved to be an efficient experimental technique for assessing the topography and the detailed structures of stressed surfaces, in particular when surface stress-induced plasticity occurs [1]. In a lot of situations, plastic deformation results from the movement of dislocations that produces steps at the crystal surfaces. The step height depends on the component of the dislocation Burgers vectors perpendicular to the surfaces and on the number of emerging dislocations. Due to dislocation motion, the steps appear in the form of lines at the sample surface. The slip lines are intimately associated with both the crystal structure and plastic deformation. Their tracking gives direct information on dislocation movements concerning the elementary dislocation processes at the origin of the slip trace topologies. For instance, frequent cross-slip events yield wavy slip traces that deviate from the primary slip plane. The distribution, height and length of slip lines reflect the degree of spatial slip heterogeneities, the dislocation source activity and the dislocation mean free path, which are key parameters in the understanding of crystal plasticity. We report in the present paper a practical situation where the use of AFM allowed an irrefutable insight in material plasticity for discriminating between different modelling hypothesis concerning the yield stress anomaly of Ni3Al intermetallic compounds with L12 ordered structure. The contribution of AFM to a better understanding of elementary rate controlling mechanisms as well as collective dislocation motions is highlighted.

Background The yield stress of Ni3Al intermetallic compounds exhibits positive temperature dependence in a specific range of temperature, commonly referred to as ‘YSA’. The YSA has been the subject of extensive experimental studies and theoretical works yielding various interpretations and modelling, which have been critically examined in [2]. Cross-slip models are the more successful in predicting the YSA. It is now considered that the YSA of Ni3Al intermetallic compounds is related to specific

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features of dislocation core configurations and cross-slip mechanisms [3]. The cross-slip models consider a thermally activated cross-slip mechanism of screw dislocations from primary octahedral planes, where they are glissile, onto cube cross-slip planes, where they are sessile. However, the description of dislocation movements may significantly differ. In particular, the cross-slip distance covered by the cross-slipped segments varies drastically in the theoretical models and is often considered as an adjustable parameter. The cross-slip distance can change from less than one lattice spacing, where cross-slip events act as pinning points [4,5], to a complete cross-slip of the whole antiphase boundary (APB) fault [6], called Kear-Wilsdorf (KW) lock, which are considered as very strong lock. Incomplete KW locks have also been proposed as playing key role in YSA of Ni3Al [7]. Importantly, depending on the considered dislocation dynamics, the cross-slip process leads with increasing temperature to a decrease, either in dislocation velocity or in mobile dislocation density [8]. Indeed, the macroscopic plastic strain-rate, εɺ , for plastic flow of crystalline solid is usually expressed in terms of Orowan's equation [9] by εɺ = αρ m bv ,

(1)

where α is a geometric factor, b the Burgers vector magnitude, ρm the density of mobile dislocations and v the mean dislocation velocity of moving dislocations. Eq. 1 shows that, for mechanical test performed under constant strain-rate conditions, the YSA can arise from an anomalous temperature dependence of either v or ρm, or from both. That is, depending on the considered kinetics, a temperature increase is accompanied by a stress increase: • for maintaining v constant, when ρm is constant, • for balancing mobile dislocation exhaustion by a higher multiplication rate, when v has usual positive temperature dependence [8]. Two studies using the double etching technique [10,11] reported that the screw dislocation velocity exhibits negative temperature dependence with increasing temperature, supporting the idea that the YSA results from v-type anomaly. On the other hand, ρm variations were estimated using transient tests for three different Ni3(Al, X) alloys [12,13]. It was shown that, for the three alloys, a drastic exhaustion of ρm takes place in the YSA domain, strongly suggesting that the YSA may be ascribed to ρm-type anomaly. One must notice that for Ni3Al studied by in situ transmission electron microscopy observations, where dislocation movements are directly observed under deformation conditions, plastic deformation proceeds by microscopic instabilities characterised by dislocation bursts [14], which does not allow for establishing whether the YSA arises from either v-type or ρmtype anomaly. Therefore, a direct proof of mobile dislocation exhaustion, which may control the YSA, has not yet been established.

Experimental For our purpose, Ni3(Al,Hf) single crystalline specimens were deformed at room temperature, i.e. in the YSA domain, by compression test along the [ 1 23] crystallographic direction (Fig. 1). The (54 1 ) oriented sample surfaces were examined using a device specially designed for in situ AFM observations of strained samples [1,15]. For convenience, the compression experiment is stopped just prior to scanning the surface, so that each AFM image can be ascribed to one strain (or stress) value only. The AFM images are then recorded while the specimen is under stress relaxation conditions, resulting in a small stress decrease of a few percents of the total applied stress [16]. In this crystallographic configuration, the primary slip system is [10 1 ](111) with an elementary dislocation step height he=0.234 nm, corresponding to the Burgers vector component onto the surface normal.

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AFM investigated surface [54-1]

[-123] Compression axis

[-11-1]

Figure 1: Sample configuration

Results and discussion Figure 2 shows the evolution of the slip traces on a Ni74.8Al21.9Hf3.3, single crystalline sample surface, at various increasing plastic strains. The scan size is about 12 µm × 12 µm. The corresponding plastic strain is indicated for each AFM image. The AFM images have been recorded in signal error contact mode. The scratch feature observed from the left top to the middle bottom of the images can be used as a marker to correlate each AFM image from the others. The sample surface exhibits slip traces that are at nearly 62° away from the compression axis, which corresponds to the crystallographic direction of the primary octahedral slip plane. Note that an unambiguous determination of a slip plane requires the knowledge of slip trace directions at least on two distinct surfaces, but in the present case the identification of the (111) plane as primary slip plane is unambiguous [17]. Two striking features are observable: • First, the slip lines are very rectilinear, without connection between them at the resolution level of AFM. This result clearly demonstrates that cross-slip events are scarce and their extensions on the cross-slip cube plane very small, at least for evolving slip traces. This is in agreement with stability calculations of incomplete and complete KW locks, which indicate that, under the present deformation conditions, both configurations are efficient for definitive dislocation locking [18-20]. • Second, several slip traces do not cross the entire scan area. This specific feature, called ‘ending-lines’ in what follows, is not observed under similar condition observations for materials exhibiting usual yield stress temperature dependence [21]. Here, the AFM scan size used for quantitative analysis is a compromise between (1) a sufficient lateral resolution for resolving parallel slip traces and (2) a scan area large enough for achieving a valuable statistical analysis of slip trace characteristics. It is also observed from Figure 2(a) to Figure 2(g) that after successive plastic strain increments most of existing slip traces does not evolve anymore, while new lines are continuously created. Figure 3 shows the average number of emerging dislocations per slip line, for various plastic strain increments. Since all strain increments are superimposed in Figure 3, it also gathers the cumulative number of slip traces as a function of their emerging dislocations. The graph clearly indicates that the activated sources, which are at the origin of the slip traces, emit a few dislocations only (≈3-4), whatever is the plastic strain. Figure 3 also illustrates the strong increase in number of slip traces with increasing plastic strain.

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Figure 2: Series at increasing plastic strain, εp, of AFM images showing the fine slip line structure on Ni3(Al,Hf) single crystal surface. The scan size is approximately 12 µm × 12 µm. (a) εp = 0.12% (b) εp = 0.22% (c) εp = 0.32% (d) εp = 0.38% (e) εp = 0.40% (f) εp = 0.52% (g) εp = 0.56%. In addition, topological analysis of AFM observations also indicates that slip line length remains essentially constant with increasing plastic strain. Figure 4 shows the percentage of slip traces that continue to increase their length with increasing number of strain increments. One can deduce from this plot that nearly 75% of the slip traces are frozen after they have been created during a plastic strain increment. Note that in the above analysis no care was taken to identify whether the same or different slip traces were involved, so that 75% represents the minimum percentage of frozen slip traces. Therefore, once a slip line has been created during a plastic strain increment, it does not evolve further, either in height or length. This result suggests a permanent dislocation locking of the leading dislocations creating the slip lines. The process by which dislocations are immobilized is not accessible at the present AFM resolution. Because only single glide is observed, strong interaction with the forest dislocations is not expected [22]. Dislocation locking certainly occurs from a cross-slip mechanism leading to the formation of either incomplete or complete KW locks. One may further remark that Ni3(Al,Hf) single crystal exhibits high strain hardening rate that leads to a strong increase in applied stress, τa, per strain increment [13]. The rise of τa may explain the small number of expanding slip lines with increasing plastic strain increments,

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observed in Figure 4. In this case, further dislocation motion can be attributed to the unlocking of the leading slip line dislocation that was into an incomplete KW lock configuration and which can be a priori stress unlocked [18,19]. Indeed, complete KW locks are considered as very efficient barriers and definitively locked [20]. 10 9

Slip line number

8 0.12%

0.22%

6

0.32%

0.40%

5

0.52%

0.56%

7

4 3 2 1 0 1

2

3

4

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6

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Number of emerging superdislocations per slip line

Figure 3: Statistical AFM analysis of the fine slip line structures in Ni3Al(Hf) single crystals representing the number of emerging superdislocations per slip line for various plastic deformation.

Slip line percentage (%)

100

80

60

40

20

0

∞1

12 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11

Number of strain increments

Figure 4: Percentage of evolving slip traces at increasing number of strain increments. The label (∞) indicates the percentage of slip traces that does not evolve after more than 10 plastic strain increments.

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The limited dislocation number per slip line together with the almost constant length of the slip traces, once there are formed, attest that a strong exhaustion of mobile dislocations takes place. The mobile dislocation exhaustion must be compensated by a continuous activation of new dislocation sources for producing the plastic strain increments, in agreement with the observed drastic increase of number of slip traces. Operation of Frank-Read sources in L12 alloys has been examined in [23]. Briefly summarised, it is argued that in order for a Frank-Read source to operate (1) the dislocation must be 'overstressed' for bowing out up to a critical radius required to overcome the source instability configuration and (2) the stress acting must exceed the necessary stress for dynamic propagation of the screws. Then, plastic deformation proceeds by dislocation bursts of rapidly expanding dislocation loops from sources satisfying these conditions. This description is in agreement with our AFM observations of slip traces which suggest dislocation sources emitting a very few dislocations over 'large' distances on the primary octahedral plane, prior to be definitively locked.

Conclusion To conclude, in situ AFM investigations of the fine slip line structure of Ni3Al intermetallic compounds at increasing plastic strain suggest that (1) the YSA is accompanied by a drastic exhaustion of mobile dislocations and (2) dislocation multiplication occurs by Frank-Read sources. Dislocation exhaustion by cross-slip mechanisms and dislocation multiplication by Franck-Read sources are two ingredients that must be included for modelling the basic features of the YSA of L12 intermetallic alloys. Acknowledgment The authors thank D.P. Pope for kindly providing Ni3(Al,Hf) single crystalline. Thanks are also due to G. Beney at Ecole Polytechnique Fédérale de Lausanne in Switzerland for preparing high quality sample surfaces suitable for AFM observations.

References [1] C. Coupeau, J.-C. Girard, J. Rabier, Dislocations in Solids, in: F.R.N. Nabarro and J.P. Hirth (eds), Vol.12, Elsevier Science, Amsterdam, 67 (2004) p. 275. [2] P. Veyssière, G. Saada, Dislocations in solids, in: F.R.N. Nabarro, M.S. Duesberry (Eds), Vol.10. Elsevier Science, Amsterdam, 53 (1996) p. 253. [3] Y.S. Choi, D.M. Dimiduk, M.D. Uchic, T.A. Parthasarathy, Mat. Science Eng. A 400-401 (2005) p. 256. [4] S. Takeuchi, E. Kuramoto, Acta Met. 21 (1973) p. 415. [5] V. Paidar, D.P. Pope, V. Vitek, Acta Met. 32 (1984) p. 435. [6] B. H. Kear, H. G. F. Wilsdorf Trans. Tms-Aime 224 (1962) p. 382. [7] E. Conforto, G. Molenat, D. Caillard, Phil. Mag. 85 (2005) p. 117. [8] F. Louchet, in: Enclyclopedia of Materials: Science and Technology, Elsevier Science, Amsterdam, (2001) p. 415. [9] E. Orowan, Proc. Phys. Soc. (London) 52 (1940) p. 8.

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[10] E. M. Nadgorny, Y. L. Iunin, High Temperature Ordered Intermetallic Alloys VI, in: J. A. Horton, I. Baker, S. Hanada, R. D. Noebe, D. S. Schwartz (Eds), Mat. Res. Soc. Symp., Pittsburgh, (1995) p. 707. [11] C. B. Jiang, S. Patu, Q. Z. Lei, C. X. Shi, Phil. Mag. Letters 78 (1998) p. 1. [12] P. Spätig, PhD thesis, EPF Lausanne, Switzerland (1995). [13] B. Viguier, J.-L. Martin, J. Bonneville, Dislocations in Solids, in: F.R.N. Nabarro, M.S. Duesberry and J.P. Hirth (eds), Vol.11, Elsevier Science, Amsterdam, 62 (2002) p. 489. [14] N. Clément, J. Microsc. Spectrosc. Electron. 11 (1986) p. 195. [15] C. Coupeau, J.-C. Girard, J. Grilhé, J. of Vacuum Sc. and Tech. B 16 (1998) p. 1964. [16] P. Spatig, J. Bonneville, J.-L. Martin, Mat. Sc. and Eng. A 167 (1993) p. 73. [17] A. E. Staton-Bevans, R.D. Rawlings, Phil. Mag. 32 (1975) p. 787. [18] G. Saada, P. Veyssière, Phil. Mag. A 66 (1992) p. 1081. [19] G. Saada, P. Veyssière, Phil. Mag. A 70 (1994) p. 925. [20] D. Caillard, Acta Mater. 44 (1996) p. 2773. [21] C. Coupeau, J. Grilhé, Mat. Sc. and Eng. A 271 (1999) p. 242. [22] Y.S. Choi, D.M. Dimiduk, M.D. Uchic, T.A. Parthasarathy, Phil. Mag. 87 (2007) p. 4759. [23] S.S. Ezz, P.B. Hirsch, Phil. Mag. A 72 (1995) p. 383.

Applied Mechanics and Materials Vol. 61 (2011) pp 79-83 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.79

The non destructive testing methods applied to detect cracks in the hot section of a turbojet Salim Bennoud 1,a, Zergoug Mourad 2 1

Department of aeronautics; Saad Dahlab University, Blida, Algeria. 2 Centre of welding and control CSC, Cherage, Algeria. a [email protected].

Key words: Non Destructive Testing NDT, cracks, hot section, combustion chamber, turbine.

Abstract All aircraft whatever they are; are regularly audited. These controls are mainly visual and external; other controls such as "major inspection" or "general revisions” are more extensive and require the dismantling of certain parts of the aircraft. Some parts of the aircraft remain inaccessible and are therefore more difficult to inspect (compressor, combustion chamber, and turbine). The means of detection must ensure controls either during initial construction, or at the time of exploitation of all the parts. The Non destructive testing (NDT) gathers the most widespread methods for detecting defects of a part or review the integrity of a structure. The aim of this work is to present the different (NDT) techniques and to explore their limits, taking into account the difficulties presented at the level of the hot part of a turbojet, in order to propose one or more effective means, non subjective and less expensive for the detection and the control of cracks in the hot section of a turbojet. To achieve our goal, we followed the following steps: - Acquire technical, scientific and practical basis of magnetic fields, electrical and electromagnetic, related to industrial applications primarily to electromagnetic NDT techniques. - Apply a scientific approach integrating fundamental knowledge of synthetic and pragmatic manner so as to control the implementation of NDT techniques to establish a synthesis in order to comparing between the use of different methods. - To review recent developments concerning the standard techniques and their foreseeable development: eddy current, ultrasonic guided waves ..., and the possibility of the implication of new techniques. Introduction For proper operation of turbojet we must meet various kinds of requirements (mechanical, thermal, aerodynamics, and thermodynamics), among these requirements, among these requirements, we distinguish the one due the increase in the push caused by an increase in the temperature. To meet such a requirement, the very high temperatures at the exit of the combustion chamber are necessary, but these temperatures have drawbacks because of the effects produced on the various components, such as thermal fatigue and corrosion on the one hand, and creep also caused by centrifugal force due to rotation of the elements on the other hand. These problems influence on the thermal and mechanical characteristics of materials and parts of the hot section of a turbojet (compressor, combustion chamber, and turbine). Under these conditions, the elements of this part must be controlled to keep their integrity and achieve maximum lifespan. In the areas of advanced industry (aerospace, nuclear …), Assessing the damage of materials is a key point for controlling the durability and reliability of parts and materials in service. In this perspective, it is necessary to quantify the damage and identify the different mechanisms responsible for this damage. It is therefore essential to characterize materials and identify the most sensitive indicators to the presence of damage to prevent their destruction and use them optimally.

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To address these problems; the (NDT) aims to verify the integrity of a part or a material without damaging it. The need to control complex parts leads to design increasingly advanced appropriate methods. Hence the need to develop innovative methods for complex controls. The aim of this work is to present the different (NDT) techniques and to study their limits, taking into account the difficulties presented at the hot section of a turbojet. Defects in the industry and the importance of NDT In industry, several defects can be met, we distinguish: manufacturing defects or degradation, such as: corrosion, peeling, variations of thicknesses or gauges, damages, cracks, material or geometrical anomalies of fabrication, weld defects, oxidation, and others which are caused by: mechanical constraints, chemical actions, thermal actions,… (NDT) is a set of methods for characterizing the state of integrity of industrial structures without degrading them,, either during production (the parts that come out of the foundries are never free from defects), or during use (appearance of defect), so, the purpose of (NDT) is to seek by a nonintrusive process, the presence of defects in materials and its major interest is to highlight the consideration of everything has a consequence that the inspected part is not different from the nominal one. The NDTfalls under the general concept of quality assurance, it plays a key role in all the applications claiming a maximum of safety and reliability (nuclear, aeronautics, automobile…). Constraints A turbojet must meet the design criteria that are dictated by three constraints: creep, thermal fatigue, oxidation and corrosion. The three constraints are dependent on the temperature or the temperature difference, although other factors such as part geometry and mechanical constraints are equally important. These constraints are causing development of cracks and fissures that can cause ruptures and destructions. Cracks The crack designates an anomaly (a defect) which occurs as a fracture whose appearance at the microscopic level is due to a rupture on compression with an important deformation of cells accompanied by numerous transverse cracks in the walls. The result is the appearance of irregular wrinkles on the surface of the material in the perpendicular direction to the fibers; it creates in most cases a very low tensile strength of longitudinal traction. The compression cracks indicate a final break of the material caused by excessive load or shock in the fibers of the wall.

Figure 1: Crack and its deformation Inspection procedure A summary of the recent repair and maintenance actions of engines is established. Visual inspections are realized to reveal the presence of a large crack at the weld on the front side of the outer casing of the combustion chamber and others parts.

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Samples of fuel and oil were removed, and no obvious anomalies were detected, the chip was less than clean, and the oil contained no obvious metallic particles. After dismantling engines, several parts of components have been sent to the laboratory to further review. Tests realized In this section, we include some realized tests (about 27 tests) on different parts located in the hot part of a turbojet (compressor blades, the casing of the combustion chamber, turbine blades ...). Visual method The wheel of the first stage of the turbine has revealed numerous cracks (about three cracks). Damage by friction has been observed on the end of several rotor blades. In addition, damage by friction observed on the interior joint of this stage (see Figure 2). The casing of the combustion chamber has an open crack to 10cm along the weld. This fracture has characteristics corresponding to a recently reported fatigue which is the result of engine vibration.

Figure 2: Crack on the interior joint Optical method An optical test of the wheel of the turbine showed that the initial cracking of tiredness followed a radial direction towards its axis. The metallurgical analysis of compressor components did not reveal anything of abnormal on the level of manufacture or the materials which could have contributed to the appearance of tiredness.

Figure 3: An optical sight of a crack Other tests were performed. Penetrating fluid method: This technique is mainly used to: the motor housing, the blades of the turbine and compressor, it allows the detection of thermal fatigue cracking, and corrosion on the parts. Eddy current method: This technique is used on piece of conductive material, it provides: the detection of fatigue crack on surface, sorting the conductivity of materials, and boundaries of thermal-affected zone. Magnetic method: This technique is used on ferromagnetic steel piece, whatever the shape of the piece; it allows the detection of cracks.

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Ultrasonic method: This technique is used to detect internal cracking in the material. Results and analysis A crack is not characterized by a cracking of the material but by a deformation comparable with a pleating. It is important to mention that the morphological analysis of the ruptures reveals many modes of ruin, which are functions of the stress type, of the presence of defects or plans of weakness of material. It is particularly mentioned that the development of the ruin does not depend only on the zone of principal weakness but also on the history of the loads applied to the part. (The way these loads have been applied is a major factor). Table 1 Advantages and disadvantages of realized methods Method Visual

Application Detection of defects in structures of all materials.

Advantages Simple to use, possibility of involving optical means of assistance. Simple to use, fast, easy to interpret.

Penetrating fluid

Detection of defects on structures of all materials.

Magnetic

Detection of discontinuities in ferromagnetic materials

Simple, easy, fast, portable apparatus.

Eddy Current

Detection of cracks, corrosion, measures the conductivity.

Best technique for detecting cracks

Ultrasound

Crack detection, detection of internal defects.

Easy, immediate, portable apparatus.

Disadvantages Depends on the experience of the user.

The defect must be accessible and open; the surface must be cleaned before and after testing. Magnetic field must be normal to the plane of failure; it is difficult to apply for large parts. Specific recommendations for each application, Difficult for complex shapes, Requires a power source, and a qualified operator. The orientation of crack plane must be known to select the wavelengths used

To valorize the different techniques, we give the following summary: The visual inspection allows subjective interpretation of the observations. The method of penetrating fluid can detect surface defects like open cracks. The optical methods are non contact methods for measuring displacements and deformations on the surface. The ultrasonic method allows in-depth inspection with direct access. The X-Ray and Gamma-Ray allow inspection depth without direct access. The Electromagnetic Method: this method applies to all conducting materials. The Magnetic Method: This technique applies only to ferromagnetic materials and alloys.

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So the methods can be regrouped in three classes - Strongly widespread methods with the requirements of control of the hot part of a turbojet: acoustic methods, electromagnetic method, magnetic method. - Fairly widespread methods: optical methods (image analysis), - Complementary methods: visual test, method of penetrant. The combination of several of these methods (using methods coupled) may increase the detection capacity of the implemented control protocol. Conclusion The work aimed to use the techniques of nondestructive testing to detect and monitor damage evolution of materials in the hot part of a turbojet. The study was conducted in several phases: - Collect existing data. - Define the problem of nondestructive testing and development of a typology of cracks. - Analyze specialized bibliographical articles relating to the cracks. - Censure control methods applicable to the problem of crack detection. Initially we showed the potential of each control method for the detection and characterization of the damage. (We came to value limits for each method). Our work has enabled us to implement the technical, scientific and practical basis for magnetic fields, electrical and electromagnetic industrial applications primarily related to the magnetic NDT techniques, we have also given a survey on recent developments and involvement of new technology: video-assisted camera control, the magnetization without contact by a rotating magnetic field; techniques and competitive developments predictable: eddy current, ultrasonic guided, magneto-optics… The satisfactory results obtained by the NDT open very interesting prospects for the developments and the implication for the new techniques for the parts which work at excessive temperatures. References [1]-S.Bennoud: Numerical Simulation of the Electromagnetic Interaction with Structures of Various Materials Having Complex Geometries, Aes-atema' 2009 Fourth International Conference (Hamburg, Germany: September 01 - 04, 2009). [2]- Airbus compagny :Non Destructive Testing Manual A330, (Octobre 2007). [3]- Airbus compagny : TTM A300-200 ATA 29, (2005). [4]- Airbus company: Computer based training Airbus A300-200 maintenance (ATA29), (2005). [5]- T.Clauzon, F.Thollon, A. Nicolas. : Flaws Characterization with Pulsed Eddy Currents N.D.T. ,IEEE TRANSACTIONS ON MAGNETICS, VOL. 35, NO. 3, (MAY 1999). [6]- G.Orland : in French : Contribution au développement et à la modélisation d’un traducteur ultrasonore multiéléments pour l’inspection au contact de composants à géométrie complexe, Paris university 6. [7]-S.Bennoud, M.Zergoug : Les méthodes de contrôle non destructif appliquées pour la détection des criques dans la partie chaude d'un turboréacteur ,International Symposium On Aircraft Materials -ACMA 2010-Marrakech, Maroc, ISBN :9782953480412 /P104.

Applied Mechanics and Materials Vol. 61 (2011) pp 85-93 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.85

Simulation of thermo mechanical behavior of structures by the numerical resolution of direct problem B.Aouragh1,a, J.Chaoufi1,b, H.Fatmaoui1,c,J.-C. Dupré2,d, C. Vallée2,e and K.Atchonouglo3,f. 1

LESPPM Faculty of Sciences BP 8106 Agadir Morocco.

2

LP' Boulevard Marie and Pierre Curie (Teleport 2 - BP 30179) 86962 Futuroscope CEDEX France. 3

University of Lomé - Department of Physics, BP 1515, Lomé – Togo.

a

[email protected] [email protected] [email protected] d [email protected] [email protected] f [email protected]

Key words: direct problem, numerical resolution, the finite element method, discretization, heat flux, infrared thermography, thermal conductivity, volumetric heat.

Abstract The aim of our study is to develop an approach to both experimental and numerical modeling to the thermal behavior of a material by identifying these thermal parameters. The theoretical part is based on the finite element method which is a starting point to solve a two-dimensional inverse heat. The experimental measurements are performed by infrared thermography. All these experimental and numerical techniques give this method properties valued in the industrial world as the non-intrusive measurements and real-time calculations. For this, we have supported a system of equations and the temperature field, so before starting the inverse problem, we addressed the direct problem by finite element method that has been compared to measures experimental infrared thermography well to check the validity of equations, so it’s the purpose of this work. 1.

Introduction

The precise determination of thermo physical properties of materials calls growing interest to scientists. Indeed, in the field of mechanical and design knowledge, the thermals properties of materials are necessary for a more realistic modeling. It is also crucial for the design of photovoltaic cells. Identification of thermo-physical parameters has been addressed in a first-time, using a heating wire, the technique is to modify "manual" settings. In this case, precise knowledge sources imposed is not necessary [1-3]. In a previous work by the identification procedure has been discussed by K. Atchonouglo [4] initially treating the direct problem in one-dimensional by finite element discretization. Then, we can simplify the writing of the heat equation in the form linear differential equation. Our study concerns a two-dimensional approach for to generalize the description to some anisotropic materials such as composites. The thermo-physical characterization of materials is a prerequisite for the establishment of comprehensive energy balances during mechanical tests. The goal is to develop simple protocol, quick, reliable and nondestructive.

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Position of the problem

This part is devoted to solving a problem of heat transfer [5]. Consider a rectangular homogeneous solid plate with thickness e, the length l and width h. The temperature distribution in the plate is evaluated in 3-D with the temperature function T (x, y, z, t). Assuming that the plate is homogeneous with short thickness, it can validly consider a two-dimensional conduction where the temperature distribution in the plate is given by the function T (x, y, t). Suppose then that the plate occupies the interval [0, l] of the axis Ox, [0, h] of the axis Oy and at time t = 0, the temperature distribution is known at all M (x, y) of the field, and equal to: T0 ( x, y ) Suppose also be placed in the area bounded by x=0 a constant heat flux .

ϕs

The surfaces

Γ 2 , Γ3 and Γ 4

are assumed isolated (Fig 1).

Fig 1: Diagram of the problem of heat transfer. 3.

Experimental Part

The goal is to create a temperature field, we will have to heat the sample studied to identify variations in temperature. A heater is confined between two plates, and the electrical power is supplied by a generator. Under these conditions, it is assumed that the imposed flow is divided equally between the two plates. The heater is controlled by a voltmeter and an ammeter. It requires very low current intensities. To access the temperature fields of the material, we use an infrared thermography. Finally, a video monitor connected to it can follow the evolution of the thermal mapping (Fig. 2).

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video monitor

infra-red camera

Specimen

87

electric generator

Fig. 2: Mounting experimental. The parameters used are: Plate dimensions: l = 145 mm and h = 45 mm Step: ∆x = l / n = 24.16 mm and ∆y = h / n = 7.5 mm avec n = 6 Thermal conductivity: k = 0.45 W / m / °C

Initial temperature:

T0 = 20 °C

6 3 Volumetric heat: ρ c = 1.74 10 J / m / °C Constant heating power: ϕ s = 397 w / m ² Time step: ∆t = 1 s

So, from this experimental study we determine the temperature field which we will compare with numerical results by solving the direct problem of conduction (Fig. 4, 5, 6, 7, 8 and 9).

4.

Problem formulation 4.1.

Strong formulation

If ρ , c and k representing respectively the volume density, the specific heat and thermal conductivity, for determination of the temperature distribution of the plate at the point M of coordinates (x, y) and at time t, we solve the following system of equations [5]: Search T such that:

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T : Ω → ℜ and :

ρc

∂T = div (k. grad T ) ∂t

lim it conditions : k . grad T .n = k

∂T   = ϕs ∂ y  x =0

for Γ1

k . grad T .n = k

∂T   =0 ∂ y  x= L

for Γ3

k . grad T .n = k

∂T   =0 ∂ x  y =0

for Γ2

k . grad T .n = k

∂T   =0 ∂ x  y =h

for Γ4

(1)

Initial condition : T ( x, y, 0) = T0 ( x, y )

In an experimental condition, T0 ( x, y ) is the temperature of the plate in the ambient air. This is the strong formulation of the problem considered. 4.2.

Weak formulation

We consider the boundary problem (1) with the initial temperature reflecting strong language. As part of solving the problem, we can proceed by multiplying the equation of heat conduction through a regular test function T* and transforming it into an integral form. This calculation is mainly formal in the sense that we assume the existence and regularity of the solution T (x, y, t) so that all calculations are lawful. Then multiply each member of the heat equation by T *. dS, where dS a surface element, and sum over the whole domain
, so we get the following equivalent problem:

∫ρ

c T *(

Ω

∂T ) dS = ∂t

∫ Γ1

T * .ϕ s dl −

∫

grad

T * . k . grad

T dS

(2)

Ω

And that whatever the temperature weighting T * (x, y, t) with initial condition: T ( x, y, 0) = T0 ( x, y ) 4.3.

Resolution of the direct problem by finite element method

If we want to solve the system resulting in the direct problem, there are several numerical techniques: finite differences, finite volumes and finite elements. The finite element method [6], to which we are concerned, is the numerical method gives more information on the temperature field T (x, y, t). We will show how the problem is reduced to ordinary differential systems in t. To do this, proceed to the discretization of domain
into Nt subdomains called elements. The geometry of these elements is triangular with three-node. The total number of nodes is N. The discretization is shown in (Fig. 3) where the length and width are each divided into n segments, for example we get for n=6 :

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28

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9

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1

2

3 4 5 Fig. 3: Discretization of domain
.

6

7

4.3.1 Local representation In this study, the approximate solution is the temperature function T(x,y,t) with the form: 3

T e ( x, y , t ) =

∑T

i

e

(t ) N ie ( x, y )

i =1

 1 if i = j As interpolation functions: N ie ( x j , y j ) = δ i j =   0 else T e (t ) The coefficients i are interpreted as approximate numerical value of T (x, y, t) at the nodes, and similarly for the test function (or weighting) T *. After simplification of the calculations, we get: e e T e ∂T  e e e T T* (c  fe ∀ T *e + k T ) = T *  ∂t  With:

{ } [ ]

[ ]{ } { } [ ]

[c ] = ρ c ∫ [N ] [N ] dS e

e

T

e

{ }

e

Ωe

[k ] = ∫ [B ] . k. [B ] .dS {f } = ∫ [N ] ϕ .dl e T

e

e

e

Ωe

e

e T

s

Γ1 ∩∂Ω e

4.3.2 Global representation and assembly If we want to treat the problem by finite element, we discuss, firstly, the study at the element, and at a second time, we treat whole structure. Let us now to pass in the element Ωe to the overall structure
.

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At the element e, the ordinary differential system obtained written: ∂T e  T e e e T T *e ( c e  fe + k T ) = T *  ∂t 

{ } [ ]

[ ]{ } { } [ ]

[ ]

We consider Ae the matrix positioning. Local and global numberings are shown in (Fig. 3). We have on one hand:

{T } = [A ]{T }

{ }

[ ]

T

, T *e T = {T *} T A e On the other hand, as the whole domain
can be partitioned into disjoint subdomains Ωe , that is to say: e

e

e= N t

Ω=

∪Ω

e

e =1

Then the integral over the domain
is identical to the sum of integrals over the subdomains, so the summation is over the total number of elements Nt (Nt = 72 in this case). Finally, through the assembly elementary quantities, we can write a matrix system, obtained from the discretization as follows: • [C ]. T  + [K ]{T } = {F } (3)   With : • e= Nt T  T   : The vector derivative of the vector temporary temperature. [C] = Ae c e Ae : The

∑ [ ] [ ][ ] e =1

matrix capacity. e= Nt

[K ] = ∑ [A e ]T

[k ][A ] e

e

: The conductance matrix.

e =1

And e= Nt

{F } = ∑ [Ae ]T

[f ] e

: The vector resulting flux boundary conditions.

e =1

Solving a problem of conduction leads us to solve a differential system of first order with time t (Eq. 3) with initial condition follows which we must add (Eq. 4):  T1 (0)   T ( 0)  {T0 } =  2  = {20} ( 4)   TN (0) Determining the temperature field in the material back to look for temperature values at the nodes over time, and the numerical solution of the previous system can determine the temperature evolution in the material for thermo-physical parameters known. For determination of the matrices C, K and the vector F, the integrals involved are calculated without any approximation.

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Problem solving

There are several methors to integrate this system. We chose the generalized trapezoidal method •

which assumes that over time, the temperature vector T(t) is known and that the vector derivative T of temperature is constant over each time interval [t , t + ∆t [ . After the simplification of calculations we obtain the following system:

 (C +ν .∆t K )T (t + ∆t ) = ∆t.F + C.T (t ) − ∆t.(1 − ν ).K .T (t )   T0 (t + ∆t ) = T0

(5)

Finally, we start with an initial condition at t=0, then look for the solution at each time t by successive increment ∆t . The resulting solution is an approximate solution in space and time [7]. 5.

Comparison of theoretical and experimental results

First step: fixing the time t and the ordinate y, and varying the abscissa x, we obtain the following result:

Fig. 4: Numerical and experimental temperatures depending on the abscissa x for ordinate y = 7.5mm at the time t = 100s. We note that the numerical temperatures are comparable with the experimental temperatures.

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Second step: fixing the ordinate y and the abscissa x, and varying the time t, we get:

Fig. 5: Numerical and experimental temperatures depending on the time t for abscissa x=0 and ordinate y = 7.5mm. We also note that the numerical temperatures are comparable with the experimental temperatures. Step Three: fixing the abscissa x and the time t, and varying the ordinate y, we get:

Temperature=f(ordinate) for abscissa x=0 at time t=100s 30

Temperature (C°)

25 20 Numerical temperature

15

Experimental temperature

10 5 0 0

7,5

15

22,5

30

37,5

45

Ordinate y(mm)

Fig. 6: Numerical and experimental temperatures depending on the time t for abscissa x=0 at the time t=100s. We note that the numerical temperatures are relatively comparable with the experimental temperatures.

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Explanation The shape of the plate and wire in the middle do not give experimentally a temperature field quite symmetrical. This is probably due to the behavior of the material or to the flows imposed experimentally. These points remain to be worked out, but does not affect the approach used and prepare the description optimization problems. Conclusion The finite element method has enabled us to adapt the mesh to geometric area studied. The weak formulation has transformed the conduction equation in a differential equation of first order. These circumstances explain our enthusiasm expressed for the application of the finite element method to solve the equation of conduction of heat in no-stationary regime. With this approach, the theoretical model is connected to the physical observable: temperature. The experimental measurements have been useful to ascertain the validity of our system of equations of our resolution. Prospects concern the treating the inverse problem, also the extension of this resolution to the case of no-homogeneous materials and no-isotropic. References [1] Wilson Nunes dos Santos, Paul Mummeryb and Andrew Wallwork: Thermal diffusivity of polymers by laser flash technique, Polymer Testing 24 (2005) 628-634. [2] A. Germaneau and JC Dupré: Termam exchanges and termomechanical couplings in amorphous polymers, Polymers & Polymer Composites, Vol. 16, No. 1, pp. 9-17, 2008 [3] K. Atchonouglo, M. Banna, C. Vallée and J.C Dupré, Inverse Transient Heat Conduction Problems and Application to the Estimation of Heat Transfer Coefficients, Heat and Mass Transfer, Vol. 45,Number 1, pp. 23-29, November 2008 [4] K. Atchonouglo, Identification des Paramètre Caractéristiques d’un Phénomène Mécanique ou Thermique Régi par une équation différentielle ou aux dérivées Partielles, Thèse de Doctorat, Université de Poitiers, 2007. [6] E. Barkanov, Introduction to the Finite Element Method, Institute of Materials and Structures, Faculty of Civil Engineering, Riga Technical University. [7] P. Corde et A. Fouilloux, Cours Langage Fortran, 2008.

Applied Mechanics and Materials Vol. 61 (2011) pp 95-99 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.61.95

Low temperature sintering and characterization of MgTiO3 M. ALIOUAT 1,2,a, B. ITAALIT 2,b, N. AMAOUZ 2,c, A. CHAOUCHI 2,d 1

Département de chimie Faculté des sciences, Université M.B. Boumerdes 35000 Algérie 2 Laboratoire L.C.A.& G.C Université M. M. Tizi-Ouzou 15000 Algérie. a [email protected], b [email protected] , c [email protected] d [email protected]

Key words:, Dielectric properties, MgTiO3, glass phases, low temperature sintering

Abstract. The development of consumer electronics has meant that the component market is subjected of major economic issues. The economic battle goes through a search of lower manufacturing costs of electronic components together with an improvement in their performance, especially multilayer ceramic capacitors. The dielectric properties of the compound MgTiO3 (εr ≈ 17, τε =100 ppm/°C, Tan (δ)