Higher Gradient Materials and Related Generalized Continua (eBook)

Preface 6
Contents 8
List of Contributors 14
1 A Computational Approach for Determination of Parameters in Generalized Mechanics 18
1.1 Introduction 18
1.2 Homogenization Between Micro- and Macroscales 21
1.3 Determination of Parameters 23
1.4 Computation of One Specific Case 25
1.5 Algorithm for All Deformation Cases 26
1.6 Examples 27
1.7 Conclusion 30
References 31
2 Extensible Beam Models in Large Deformation Under Distributed Loading: a Numerical Study on Multiplicity of Solutions 36
2.1 Introduction 37
2.2 The Model 38
2.2.1 Kinematics and Deformation Energy 38
2.2.2 Lagrange Multipliers Method 39
2.3 Numerical Simulations 40
2.3.1 Numerical Methods 40
2.3.2 The Number of Equilibrium Configurations when the Load Increases 42
2.3.3 Equilibrium Configurations 43
2.3.4 Parametric Study on the Extensional Stiffness 46
2.4 Conclusions 49
Appendix 53
References 54
3 On the Characterization of the Nonlinear Reduced Micromorphic Continuum with the Local Material Symmetry Group 59
3.1 Introduction 59
3.2 Micromorphic Continua 60
3.3 Local Material Symmetry Group 62
3.4 Relaxed Micromorphic Medium as a Micromorphic Subfluid 63
3.5 Conclusions 68
References 68
4 Structural Modeling of Nonlinear Localized Strain Waves in Generalized Continua 71
4.1 Introduction 71
4.2 Principles of Structural Modeling 72
4.3 One-dimensional Model of a Nonlinear Gradient-elastic Continuum. 76
4.4 Nonlinear Strain Waves 78
4.5 Conclusions 82
References 83
5 A Diffusion Model for Stimulus Propagation in Remodeling Bone Tissues 85
5.1 Introduction 86
5.2 Accepted Assumptions and Main Variables 88
5.3 Poromechanical Formulation 91
5.4 Evolutionary Equations for Bone Remodeling 93
5.5 Stimulus ModelingWithout Time Delay and Diffusion Phenomena 95
5.6 An Improved Version of Stimulus Modeling 96
5.7 Numerical Simulations 98
5.8 Conclusions 103
References 105
6 A C1 Incompatible Mode Element Formulation for Strain Gradient Elasticity 111
6.1 Introduction 111
6.2 From Local Balance of Momentum to Minimization of the Elastic Potential 113
6.3 Ciarlets Elastic Energy 115
6.3.1 Strain Gradient Extension 115
6.3.2 Stress-strain Relations 116
6.3.3 Material parameters 117
6.4 Element Formulation 117
6.4.1 Overview 117
6.4.2 Incompatible Mode Element Formulation 118
6.4.3 Numerical Integration 121
6.4.4 Testing of the Implementation 122
6.5 Single Force Indentation Simulations 123
6.5.1 The Boundary Conditions 123
6.5.2 Meshing 124
6.5.3 Results 124
6.5.3.1 Transition Behavior as ? = 0. . .? 124
6.5.3.2 Singularity 124
6.5.4 A Comment on Pseudorigid Bodies 126
6.6 Sharp Corner Simulations 127
6.6.1 The Boundary Conditions 127
6.6.2 Meshing 128
6.6.3 Results 128
6.6.3.1 Transition Behavior as ??? 128
6.6.3.2 Convergence on Mesh Refinement 130
6.7 Improvement of the Element Formulation 132
6.8 Convergence Study 132
6.9 Conclusion 134
References 135
7 A Comparison of Boundary Element Method and Finite Element Method Dynamic Solutions for Poroelastic Column 137
7.1 Introduction 137
7.2 Mathematical Model 139
7.2.1 usi–p-formulation in Laplace Domain 140
7.2.2 usi–p-formulation in Time Domain 141
7.3 Boundary Integral Equation and Boundary Element Methodology 141
7.4 Laplace Transform Inversion 143
7.5 Numerical Example 143
7.6 Conclusion 146
Appendix 147
References 149
8 From Generalized Theories of Media with Fields of Defects to Closed Variational Models of the Coupled Gradient Thermoelasticity and Thermal Conductivity 151
8.1 Introduction 151
8.2 Kinematics of Gradient Continuous Media and Gradient Media with Fields of Defects 154
8.3 Variational Statement of Generalized Gradient Media with Fields of Defects 156
8.4 Mathematical Statement for Generalized Gradient Dilatation Model 158
8.5 Identification of Generalized Stress Factors of the Model 159
8.6 Particular Model: Gradient Thermoelasticity 163
8.7 Particular Model: Gradient Thermal Conductivity Model 164
8.8 Conclusion 167
References 168
9 Mathematical Modeling of Elastic Thin Bodies with one Small Size 171
9.1 Introduction 172
9.2 On Parametrization of a Thin Body Domain With one Small Size with an Arbitrary Base Surface 174
9.3 Presentations of the Equations of Motion, Heat Influx and Constitutive Relations of Micropolar Theory 184
9.3.1 Presentations of the Equations of Micropolar Theory 184
9.3.2 Representation of the Equation of Heat Influx in Micropolar Mechanics of a Deformable Thin Solids 185
9.3.3 Representations of Hooke’s Law and Fourier’s Heat Conduction Law 185
9.4 Some Recurrence Relations of the System of Legendre Polynomials on the Segment [?1, 1] 187
9.4.1 Main Recurrence Relations 187
9.4.2 Additional Recurrence Relations 188
9.5 Moments of Some Expressions Regarding the Legendre Polynomial System 188
9.5.1 Moments of Some Expressions Regarding the Legendre Polynomial System 189
9.6 Different Representations of the System of Motion Equations in Moments 192
9.6.1 Presentations of the System of Motion Equations in Moments with Respect to Systems of Legendre Polynomials 193
9.7 Representations of Constitutive Relations in Moments 197
9.8 On Boundary and Initial Conditions in Micropolar Mechanics of a Deformable Thin Body 199
9.8.1 The Boundary Conditions on the Front Surface 199
9.8.2 Boundary Conditions in Moments in the Theory of Thin Bodies 201
9.8.3 Kinematic Boundary Conditions in Moments 202
9.8.4 Physical Boundary Conditions in Moments 203
9.8.5 Boundary Conditions of Heat Content in Moments 205
9.8.5.1 Boundary Conditions of the First Kind in Moments 205
9.8.5.2 Boundary Conditions of the Second Kind in Moments 206
9.8.5.3 Boundary Conditions of the Third Kind in Moments 206
9.8.6 Initial Conditions in Moments 206
9.9 Problem Statements in Moments of Micropolar Thermomechanics of a Deformable Thin Body 207
9.9.1 Statement of the Coupled Dynamic Problem in Moments of (r,N) Approximation 207
9.9.2 Statement of a Non-stationary Temperature Problem in Moments of (r,N) Approximation 208
9.9.3 Statement of the Uncoupled Dynamic Problem in Moments of the (r,N) Approximation 209
References 210
10 Application of Eigenvalue Problems Under the Study of Wave Velocity in Some Media 216
10.1 Kinematic and Dynamic Conditions on the Strong Discontinuity Surface in Micropolar Mechanics 216
10.2 Equations for Determining the Wave Velocities in an Infinite Micropolar Solid 219
10.3 Classical Materials with Anisotropy Symbols {1,5} and {5,1} 223
10.4 Classical Material with an Anisotropy Symbol {1, 2, 3} (Cubic Symmetry) 228
10.5 Classical Material with an Anisotropy Symbol {1,1,2,2} (Transversal Isotropy) 229
10.6 Micropolar Material with a Center of Symmetry and the Anisotropy Symbol {1,5,3} 232
10.7 Conclusion 233
References 234
11 Theoretical Estimation of the Strength of Thin-film Coatings 236
11.1 Introduction 237
11.2 Theoretical Position 238
11.3 Basic Assumptions 241
11.4 Comparison of Calculation Results with Known Data 244
11.5 Conclusion 244
References 245