Nonlinear Continuum Mechanics. An Engineering Approach 9783031152061, 9783031152078

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English Pages 358 Year 2023

Nonlinear Continuum Mechanics. An Engineering Approach
9783031152061, 9783031152078

Author / Uploaded
Carlos Agelet de Saracibar

Table of contents :
Preface
Contents
About the Author
1 Tensor Algebra
1.1 Vector Algebra
1.1.1 Linear or Vector Spaces. Real Space
1.1.2 Euclidean Vector Spaces
1.1.3 Lie Algebra
1.2 Vector Algebra Geometry
1.2.1 Inner (or Dot, or Scalar) Product on mathbbR3
1.2.2 Cross (or Vector) Product on mathbbR3
1.2.3 Box (or Scalar Triple) Product on mathbbR3
1.3 Vector Algebra in Components
1.3.1 Standard Basis and Vector Components on mathbbR3
1.3.2 Inner (or Dot, or Scalar) Product on mathbbR3
1.3.3 Cross (or Vector) Product on mathbbR3
1.3.4 Box (or Scalar Triple) Product on mathbbR3
1.4 Tensor Algebra
1.4.1 Vector Space of Linear Transformations mathscrL(mathbbR3,mathbbR3)
1.4.2 Tensor Multiplication on mathscrL(mathbbR3,mathbbR3)
1.4.3 Tensor Product on mathbbR3
1.4.4 The Trace Function
1.4.5 The Determinant Function
1.4.6 Inner Product on mathscrL(mathbbR3,mathbbR3)
1.4.7 Inverse of a Second-Order Tensor
1.4.8 Lie Bracket Operation
1.4.9 Orthogonal Group O(3)
1.4.10 Special Orthogonal Group SO(3)
1.4.11 Spectral Decomposition of a Symmetric Second-Order Tensor
1.5 Tensor Algebra in Components
1.5.1 Standard Basis and Tensor Components
1.5.2 Tensor Multiplication on mathscrL(mathbbR3,mathbbR3)
1.5.3 Tensor Product on mathbbR3
1.5.4 The Trace Function
1.5.5 Inner (or Double Dot) Product on mathscrL(mathbbR3,mathbbR3)
1.5.6 The Determinant Function
1.5.7 Inverse of a Second-Order Tensors
1.5.8 Additive Split of a Second-Order Tensor into Its Symmetric and Skew-Symmetric Parts
1.5.9 Additive Split of a Second-Order Tensor into Its Volumetric and Deviatoric Parts
1.6 Fourth-Order Tensors
1.6.1 Fourth-Order Tensors
1.6.2 Tensor Multiplication on mathscrL(mathbbR3,mathbbR3)timesmathscrL(mathbbR3,mathbbR3)
1.6.3 Tensor Product on mathscrL(mathbbR3,mathbbR3)
1.6.4 Fourth-Order Identity Tensors
1.6.5 Fourth-Order Volumetric and Deviatoric Projection Operators
1.7 Fourth-Order Tensors in Components
1.7.1 Fourth-Order Tensors
1.7.2 Tensor Multiplication on mathscrL(mathbbR3,mathbbR3)timesmathscrL(mathbbR3,mathbbR3)
1.7.3 Tensor Product on mathscrL(mathbbR3,mathbbR3)
1.7.4 Fourth-Order Identity Tensors
1.7.5 Fourth-Order Volumetric and Deviatoric Projection Operators
1.8 Exercises
References
2 Tensor Analysis
2.1 Differential Analysis
2.1.1 The Frechet Derivative
2.1.2 The Directional Derivative
2.1.3 Tangent Spaces and Tangent Maps
2.1.4 Differentiable Manifolds and Charts
2.1.5 Tangent Spaces to a Manifold
2.2 Tangent Space to Lie Groups
2.2.1 Tangent Space to mathbbSmathbbL(3,mathbbR) at the Identity and Lie Algebra mathfraksmathfrakl(3,mathbbR)
2.2.2 Tangent Space to O(3) at the Identity and Lie Algebra mathfraksmathfrako(3)
2.3 Differential Operators in mathbbR3
2.3.1 Differential Operators in mathbbR3
2.3.2 Gradient Vector Differential Operator
2.3.3 Divergence Scalar Differential Operator
2.3.4 Curl Vector Differential Operator
2.3.5 Laplacian Scalar Differential Operator
2.3.6 Hessian Second-Order Tensor Differential Operator
2.4 Integral Theorems
2.4.1 Gradient or Green Theorem
2.4.2 Divergence or Gauss Theorem
2.4.3 Curl or Stokes Theorem
2.5 Exercises
References
3 Kinematics
3.1 Continuum Body, Deformation Map, and Motion
3.1.1 Continuum Body
3.1.2 Deformation Map and Motion
3.1.3 Material and Spatial Functions and Maps
3.1.4 Material and Spatial Time Derivatives
3.2 Displacement, Velocity, and Acceleration
3.2.1 Material and Spatial Displacement Fields
3.2.2 Material and Spatial Velocity Fields
3.2.3 Material and Spatial Acceleration Fields
3.3 Fundamental Geometric Maps
3.3.1 Deformation Gradient. Tangent Map
3.3.2 Volume Deformation. Jacobi Map
3.3.3 Area Deformation. Normal Map
3.4 Deformation and Strain Tensors
3.4.1 Stretch Vector, Stretch, and Deformation Tensors
3.4.2 Variation of Angles
3.4.3 Strain Tensors
3.4.4 Polar Decomposition of the Deformation Gradient
3.4.5 Spectral Decomposition of Deformation and Strain Tensors
3.5 Velocity Gradients
3.5.1 Material and Spatial Velocity Gradients
3.6 Deformation Rate and Rotation Rate Tensors
3.6.1 Deformation Rate and Rotation Rate Tensors
3.7 Material Time Derivative of Kinematic Fields
3.7.1 Material Time Derivative of the Inverse of the Deformation Gradient
3.7.2 Material Time Derivative of the Strain Tensors
3.7.3 Material Time Derivative of the Differential of Volume
3.8 Push-Forward and Pull-Back Operations
3.8.1 Push-Forward and Pull-Back Operations
3.9 Lie Derivative of Spatial Fields
3.9.1 Lie Derivative of Spatial Fields
3.10 Material Frame Indifference
3.10.1 Deformation Map and Motion
3.10.2 Area Deformation
3.10.3 Deformation and Strain Tensors
3.10.4 Material and Spatial Velocity Gradients, Deformation Rate, and Rotation Rate (or Spin) Tensors
3.11 Linear Kinematics
3.11.1 Displacement Field
3.11.2 Displacement Gradient Field
3.11.3 Infinitesimal Strain Tensor and Infinitesimal Rotation Tensor
3.11.4 Linearization of Kinematic Fields
3.12 Exercises
References
4 Stresses
4.1 Body Forces and Surface Forces
4.1.1 Body Forces
4.1.2 Surface Forces
4.1.3 Resultant of Forces
4.2 Stressess
4.2.1 Cauchy Stress Tensor
4.2.2 First Piola–Kirchhoff Stress Tensor
4.2.3 Piola Transformation and Piola Identity
4.2.4 Kirchhoff Stress Tensor
4.2.5 Second Piola–Kirchhoff Stress Tensor
4.3 Material Frame Indifference
4.3.1 Objective Stress Tensors
4.3.2 Objective Stress Rate Tensors
4.4 Exercises
References
5 Balance Principles
5.1 Conservation of Mass
5.1.1 Global Material Form
5.1.2 Local Material Form
5.1.3 Global Spatial Form
5.1.4 Local Spatial Form
5.2 Convective Flux
5.3 Reynolds Transport Theorem
5.4 Balance of Linear Momentum
5.4.1 Global Material Form
5.4.2 Local Material Form
5.4.3 Global Spatial Form
5.4.4 Local Spatial Form
5.5 Balance of Angular Momentum
5.5.1 Global Material Form
5.5.2 Local Material Form
5.5.3 Global Spatial Form
5.5.4 Local Spatial Form
5.6 Internal Mechanical Power
5.6.1 Global Spatial Form
5.6.2 Global Material Form
5.7 Mechanical Energy Balance
5.7.1 Global Spatial Form
5.7.2 Global Material Form
5.7.3 Quasistatic Problem
5.7.4 Free Vibration Problem
5.7.5 Conservative Mechanical System
5.8 External Thermal Power
5.8.1 Global Spatial Form
5.8.2 Global Material Form
5.9 First Law of Thermodynamics. Balance of Energy
5.9.1 First Law of Thermodynamics
5.9.2 Global Spatial Form
5.9.3 Local Spatial Form
5.9.4 Global Material Form
5.9.5 Local Material Form
5.10 Second Law of Thermodynamics
5.10.1 Second Law of Thermodynamics
5.10.2 Global Spatial Form
5.10.3 Local Spatial Form
5.10.4 Global Material Form
5.10.5 Local Material Form
5.11 Isothermal, Isentropic, and Adiabatic Processes
5.11.1 Isothermal Processes
5.11.2 Isentropic Processes
5.11.3 Adiabatic Processes
5.12 Governing Equations
5.12.1 Local Spatial Form
5.12.2 Local Material Form
5.13 Exercises
References
6 Finite Deformation Hyperelasticity
6.1 Constitutive Equations for Finite Hyperelasticity
6.1.1 Free Energy
6.1.2 Material Frame Indifference. Free Energy of the Form overlineΨ( C )
6.1.3 Material Isotropy. Free Energy of the Form tildeψ( b )
6.1.4 Spatial Covariance. Free Energy of the Form overlineψ( g,F ) = overlineΨ( C )
6.2 Constitutive Equations for Isotropic Finite Hyperelasticity
6.2.1 Free Energy in Terms of the Principal Invariants of the Right (or Left) Cauchy–Green Deformation Tensor Ψ( IC ,IIC ,IIIC ) or ψ( Ib ,IIb ,IIIb )
6.2.2 Free Energy in Terms of the Principal Stretches Ψ( λ1 ,λ2 ,λ3 )
6.3 Constitutive Equations for Incompressible Finite Hyperelasticity
6.3.1 Free Energy of the Form Ψ( F )
6.3.2 Material Frame Indifference. Free Energy of the Form overlineΨ( C )
6.3.3 Material Isotropy. Free Energy of the Form tildeψ ( b )
6.4 Constitutive Equations for Incompressible Isotropic Finite Hyperelasticity
6.4.1 Free Energy in Terms of the Principal Invariants of the Right (or Left) Cauchy–Green Deformation Tensor Ψ( IC ,IIC ,IIIC ) or ψ( Ib ,IIb ,IIIb )
6.4.2 Free Energy in Terms of the Principal Stretches Ψ( λ1 ,λ2 ,λ3 )
6.5 Constitutive Equations for Compressible Finite Hyperelasticity
6.5.1 Free Energy of the Form overlineΨ( C ) = U( J ) + overlineΨiso ( overlineC )
6.5.2 Free Energy of the Form widetildeψ( b ) = U( J ) + widetildeψiso ( overlineb )
6.5.3 Free Energy of the Form overlineψ( g,F ) = U( J ) + overlineψiso ( overlineg,F )
6.6 Constitutive Material Models for Isotropic Finite Hyperelasticity
6.6.1 Neo-Hookean Material Model
6.6.2 Mooney–Rivlin Material Model
6.6.3 Kirchhoff–Saint Venant Material Model
6.6.4 Compressible Neo-Hookean Material Model
6.7 Exercises
References
7 Finite Deformation Plasticity
7.1 Finite Deformation Plasticity Based on the Multiplicative Decomposition: Intermediate Configuration Formulation
7.1.1 Free Energy, Elastic Constitutive Response and Reduced Plastic Dissipation
7.1.2 Yield Function, Associative Plastic Flow Rule and Isotropic Hardening Rule
7.2 Finite Deformation Plasticity Based on the Multiplicative Decomposition: Spatial Formulation
7.2.1 Free Energy, Elastic Constitutive Response and Reduced Plastic Dissipation
7.2.2 Yield Function, Associative Plastic Flow Rule and Isotropic Hardening Rule
7.3 Finite Deformation Plasticity with Uncoupled Volumetric/Isochoric Response: Spatial Formulation
7.3.1 Multiplicative Volumetric/Isochoric Split for Isochoric Plasticity
7.3.2 Uncoupled Volumetric/Isochoric Free Energy
7.3.3 Yield Function, Associative Plastic Flow Rule and Isotropic Hardening Rule
7.4 Finite Deformation Plasticity Based on the Maximum Plastic Dissipation and the Multiplicative Decomposition: Material Convected Formulation
7.4.1 Multiplicative Decomposition
7.4.2 Material and Spatial Tensors. Push-Forward/Pull-Back Operations
7.4.3 Lie Derivatives of Spatial Tensors
7.4.4 Hyperelastic Response: Free Energy. Spatial Covariance
7.4.5 Hyperelastic Response: Hyperelastic Constitutive Equations
7.4.6 Plastic Response: Yield Function. Material Convected Strain-Space Formulation
7.4.7 Plastic Response: Hardening Rule
7.4.8 Plastic Response: Plastic Dissipation
7.4.9 Plastic Response: Maximum Plastic Dissipation Principle and Associative Plastic Flow Rule
7.4.10 Plastic Response: Elastoplastic Tangent Moduli
7.5 Finite Deformation Elastoplasticity with Uncoupled Volumetric/Isochoric Response: Material Convected Formulation
7.5.1 Multiplicative Volumetric/Isochoric Split
7.5.2 Uncoupled Volumetric/Isochoric Free Energy and Hyperelastic Response
7.5.3 Associative Isochoric Plastic Response
7.6 Exercises
References
8 Finite Deformation Coupled Thermoplasticity
8.1 Constitutive Equations for Finite Deformation Coupled Thermoplasticity: Intermediate Configuration Formulation
8.1.1 Constitutive Equations in Terms of the Internal Energy
8.1.2 Constitutive Equations in Terms of the Free Energy
8.1.3 Evolution of the Total Entropy
8.2 Evolution Equations for Thermoplastic Flow
8.3 Exercises
References
9 Finite Deformation Contact Mechanics
9.1 Contact Surfaces Parametrization
9.1.1 Parametrization of the Contact Surfaces
9.2 Contact Normal Constraints
9.3 Frictional Contact Constraints
9.3.1 Convected Basis, Metrics, and Curvature Tensors at the Closest-Point-Projection
9.3.2 The Convected Relative Velocity
9.3.3 Frictional Traction
9.3.4 Frictional Constraint Evolution Problem
9.4 Penalty Regularized Frictional Contact Problem
9.4.1 Regularization of the Contact Normal Constraints
9.4.2 Regularization of the Frictional Slip Rule
9.4.3 Fractional Step Method
9.5 Exercises
References
10 Variational Methods
10.1 Variational Principle. Euler–Lagrange Equations
10.1.1 Functionals, Gateaux Derivative, and Frechet Derivative
10.1.2 Variational Principle. Euler–Lagrange Equations
10.2 Variational Form of the IBVP in Solid Mechanics
10.2.1 Admissible Function Spaces
10.2.2 Local (or Strong) and Variational (or Weak) Spatial Forms of the IBVP
10.2.3 Local (or Strong) and Variational (or Weak) Material Forms of the IBVP
10.3 Mixed Deformation/Pressure Variational Form of the Nearly Incompressible IBVP in Solid Mechanics
10.3.1 Admissible Function Spaces
10.3.2 Mixed Deformation/Pressure Local (or Strong) and Variational (or Weak) Material Form of the Nearly Incompressible IBVP in Solid Mechanics
10.4 Variational Form of the Coupled Thermo-mechanical IBVP in Solid Mechanics
10.4.1 Admissible Function Spaces
10.4.2 Local (or Strong) and Variational (or Weak) Forms of the Coupled Thermo-mechanical IBVP
10.5 Variational Form of the Frictional Contact IBVP in Solid Mechanics
10.5.1 Admissible Configuration Spaces
10.5.2 Variational Formulation of the Frictional Contact Problem IBVP in Solid Mechanics
10.5.3 Directional Derivatives of Frictional Contact Kinematics Quantities
10.5.4 Frictional Contact Virtual Work
10.6 Exercises
References