Generalized Continua as Models for Classical and Advanced Materials (eBook)

Preface 6
Contents 7
Contributors 9
On Strain Rate Tensors and Constitutive Equations of Inelastic Micropolar Materials 13
1 Introduction 13
2 Basic Relations of the Micropolar Mechanics 14
2.1 Kinematics 14
2.2 Motion Equations 15
2.3 Constitutive Equations 15
2.4 Elastic Materials 17
3 Relative Strain Measures 17
4 Relations of Isotropic Materials with Relative Strain Measures 18
5 Rivlin--Ericksen Tensors 19
6 Examples of Constitutive Equations 20
6.1 Linear Viscous Micropolar Fluid 20
6.2 Non-linear Viscous Micropolar Fluid 20
6.3 Viscoelastic Micropolar Fluids 21
6.4 Micropolar Hypo-elasticity 22
6.5 Viscoelastic Materials 22
7 Conclusions and Discussion 23
References 24
On the Modelling of Carbon Nano Tubes as Generalized Continua 26
1 Introduction 26
2 One-Dimensional Beam Model 28
2.1 Strain Measures 29
2.2 Balance 29
2.3 Beam Subjected to an Axial End Displacement 31
3 Graphene 31
4 Energy 33
5 Atomic Model 34
6 1D Continuum Equivalent to the Atomic Model 36
7 Beam with End Displacement: Trivial Solution 38
8 Perturbation Method 39
9 Beam with End Displacement: Bifurcation Analysis 40
10 Beam with End Displacement: Numerical Results 42
11 Conclusions 45
References 45
Isogeometric Analysis of Gradient-Elastic 1D and 2D Problems 47
1 Introduction. Basic Formulae of the Mindlin's Gradient Elasticity Theory 47
2 Weak Form 50
3 Isogeometric Analysis 51
4 Numerical Results 52
4.1 Static Rod in Tension 52
4.2 2D Dynamic Problem 54
References 54
A Fast Fourier Transform-Based Approach for Generalized Disclination Mechanics Within a Couple Stress Theory 56
1 Introduction 57
2 Notations 59
3 Kinematics of Generalized Crystal Defects and Incompatibilities 60
3.1 Linear Theory 60
3.2 Volterra's Crystal Translation and Rotation Line Defects 61
3.3 Generalized Disclination (G-Disclination) Kinematics 62
3.4 Incompatible Field Equations 63
4 Constitutive and Equilibrium Equations 65
4.1 Constitutive Relationships 65
4.2 Equilibrium Equations 67
5 Fourier Method 68
5.1 Solution of Poisson-Type Equations in Fourier Space 68
5.2 Solution of Navier-Type Equation in Fourier Space 69
5.3 Stress and Couple Stress Fields 70
6 Fast Fourier Transform Numerical Implementation 71
6.1 Discrete and Fast Fourier Transforms 71
6.2 Discrete Fourier Transform Differentiation Rules Based on Centered Finite Difference Approximation 71
7 Application to Infinite Straight Wedge Disclinations 74
7.1 Materials and Numerical Data 74
7.2 Two-Dimensional Equations for G-Disclinations 74
7.3 Single Straight Wedge Disclination 75
7.4 Straight Wedge Disclination Dipole 78
8 Concluding Remarks 80
References 82
Some Cases of Unrecognized Transmission of Scientific Knowledge: From Antiquity to Gabrio Piola's Peridynamics and Generalized Continuum Theories 85
1 Introduction 86
2 Some Ancient Examples of Not Recognized Transmission of Knowledge 87
2.1 Galileo and Heron 87
2.2 Galileo and Democritus 91
2.3 The Transmission of the Scientific Explanation of Tides 92
3 Pristine Formulations of the Principle of Virtual Powers (or Work) as a Basic Postulate for Mechanics 93
3.1 The Traité de Dynamique by D'Alembert 94
3.2 The Treatise Méchanique Analytique by Lagrange 96
3.3 Attested Lagrange's Version of the Principle of Virtual Work 103
3.4 Gabrio Piola: An Italian Follower of Lagrange, One of the Founders of Modern Continuum Mechanics 104
3.5 ``Quel Principio Uno, di Dove Emanano Tutte le Equazioni Che Comprendono Innumerabili Verità'' 105
3.6 Truesdell and Toupin in Their Classical Field Theories cite Piola's Works 106
4 The Reconstruction of the Transmission Line of Piola's Ideas and Results 110
5 Non-local Continuum Theories in Piola's Works 114
5.1 Piola's Non-local Internal Interactions 114
5.2 An Explicit Calculation of the Strong Form of the Variational Principle (6) 118
5.3 Modern Perydinamics: A New/Old Model for Deformable Bodies 119
5.4 Piola's Higher Gradient Continua 121
6 Weak and Strong Evolution Equations for Piola Continua 126
7 A Half-Facetious Conclusion: Melittas or the Role of `Ideas Spreader' in the Erasure of Authors and in the Diffusion of Ideas 129
References 130
Computational Analysis of the Size Effects Displayed in Beams with Lattice Microstructures 137
1 Introduction 138
2 Objective of Numerical Modelling 139
3 Micropolar Elasticity 139
3.1 Micropolar Elasticity and Flexure 140
3.2 Constitutive Properties of a Regular Square Lattice 141
4 Computational Representation of Cellular Material 142
5 Results 144
5.1 Uniaxial Tests 144
5.2 Flexural Testing 145
6 Discussion 149
7 Conclusion 150
References 151
Inelastic Interaction and Splitting of Strain Solitons Propagating in a One-Dimensional Granular Medium with Internal Stress 152
1 Introduction 153
2 A Mathematical Model of a One-Dimensional Granular Medium 154
3 The Low-Frequency Approximation 158
4 Soliton Solutions 159
5 Subsonic Solitons 161
6 Supersonic Solitons 164
7 Conclusions 167
References 168
The Eigenmodes in Isotropic Strain Gradient Elasticity 170
1 Introduction 170
2 Isotropic Stiffness Hexadic 171
2.1 An Orthogonal Basis 173
2.2 Eigenvalues and Projectors 174
3 The Eigenmodes and the Harmonic Decomposition 175
3.1 The 7-Dimensional Eigenspace calH3 178
3.2 The 5-Dimensional Eigenspace calH2 179
3.3 The 3-Dimensional Eigenspaces 181
4 Relation to Other Forms of Strain Gradient Elasticity 182
4.1 Mindlin's Second Form of Strain Gradient Elastictiy 182
4.2 Common Strain Gradient Extensions 183
4.3 Gradient Elasticity of Helmholtz Type 183
References 184
Limit Analysis of Lattices Based on the Asymptotic Homogenization Method and Prediction of Size Effects in Bone Plastic Collapse 186
1 Introduction 187
2 Lattice Homogenization Towards 3D Cauchy Continuum 191
2.1 Description of the Lattice Geometry 191
2.2 3D Elementary Unit Cell 192
2.3 Determination of the Plastic Yield and Brittle Fracture Surfaces in Stress Space 194
2.4 3D Plastic Collapse and Brittle Fracture Surface of Trabecular Bone 198
3 Evolution of the Yield Surface with Ongoing Hardening for 2D Extensional Lattices 201
3.1 Homogenized Macroscopic Cauchy Stress and Micro-Stress Relationship 202
3.2 Macroscopic Strain Related to Microscopic Stress 202
3.3 Constitutive Equations at Microscale 203
3.4 Incremental Formulation and Integration Scheme 203
3.5 Applications Accounting for Ongoing Hardening 206
4 Plastic Yield Surface Based on Cosserat Theory---Size Dependent Plastic Yield Criterion 208
5 Conclusions 216
References 217
An Improved Constitutive Model for Short Fibre Reinforced Cementitious Composites (SFRC) Based on the Orientation Tensor 219
1 Introduction 219
2 Phenomenological and Theoretical Considerations 222
2.1 Description of the Fibre Orientation Distribution 222
2.2 Phenomenological: Hyperelastic Material 223
2.3 Material Symmetry, Structural Tensors and Alignment Tensor 223
3 Constitutive Relations for SFRC in the Elastic Range 224
3.1 The Isotropic Matrix 225
3.2 Anisotropic Fibres: Enrichment by the Orientation Tensor 225
4 Comparison with Other Models for Short Fibre Reinforced Materials 226
5 Example Distributions and Resulting Elasticity Tensors 228
6 Outlook: Constitutive Relations for Cracked SFRC 230
7 Conclusion 231
References 232
Isogeometric Static Analysis of Gradient-Elastic Plane Strain/Stress Problems 234
1 Equation of Motion of the Generalized Continua 234
2 Numerical Results 235
2.1 Square Domain in the Field of Body Forces 236
2.2 Lamé Problem 237
3 Conclusion 240
References 240
Applications of Higher-Order Continua to Size Effects in Bending: Theory and Recent Experimental Results 241
1 Introduction 241
2 Selected Higher-Order Continuum Mechanical Theories 242
2.1 Strain Gradient Theories 243
2.2 Micromorphic Continuum 249
2.3 Theories of Material Surfaces 251
3 Experimental Analysis 255
3.1 Atomic Force Microscopy 255
3.2 Flexural Vibration Analysis 258
4 Results 260
5 Conclusions 262
References 262
Classification of Gradient Adhesion Theories Across Length Scale 265
1 Introduction 265
2 Theory of Perfect Adhesion of Surface in Classical Elasticity 266
3 Theory of Adhesion Interactions in Gradient Elasticity 270
4 Theory of Gradient Adhesion in Gradient Elasticity 273
5 On the Unified Nature of Cohesive--Adhesive Interactions 275
6 Discussion and Conclusions 278
References 280
Eigenvalue Problems of a Tensor and a Tensor-Block Matrix (TMB) of Any Even Rank with Some Applications in Mechanics 282
1 Introduction 282
2 On Tensors of Module mathbbR2p(?) 283
3 Eigenvalue Problem and Construction of a Complete System of Eigentensor Columns of Symmetric Tensor-Block Matrix 286
3.1 Eigenvalue Problem for mathbbMinmathbbR2p2times2(?) 291
3.2 Construction of the Eigentensor Columns of a TBM 297
3.3 Eigenvalue Problem of a Tensor-Block Diagonal Matrix 301
4 Some Applications to Mechanics 304
4.1 Representations of the Specific Strain Energy and Constitutive Relations in the Linear Micropolar Theory of Elasticity 304
4.2 Presentations of the Specific Strain Energy and the Constitutive Relations Using Eigenvalues and Eigentensor Columns 307
4.3 Construction of the Eigentensor Columns of the TBM of Elastic Modulus Tensor 309
4.4 Micropolar Material with a Center of Symmetry 309
5 Eigenvalue Problem and Construction of the Complete System of Eigentensors for a Symmetric Fourth Rank Tensor 310
6 Classification of the Micropolar Linearly Elastic Anisotropic Materials Without a Center of Symmetry 311
6.1 Symbol of Anisotropy Consisting of One Element 311
6.2 Symbol of Anisotropy Consisting of Two Elements 311
6.3 Symbol of Anisotropy Consisting of Three Elements 312
6.4 Symbol of Anisotropy Consisting of Four Elements 312
6.5 Symbol of Anisotropy Consisting of Five Elements 312
6.6 Symbol of Anisotropy Consisting of Six Elements 313
6.7 Symbol of Anisotropy Consisting of Seven Elements 313
6.8 Symbol of Anisotropy Consisting of Eight Elements 313
6.9 Symbol of Anisotropy Consisting of Nine Elements 313
6.10 Symbol of Anisotropy Consisting of Ten Elements 313
6.11 Symbol of Anisotropy Consisting of Eleven Elements 314
6.12 Symbol of Anisotropy Consisting of Twelve Elements 314
6.13 Symbol of Anisotropy Consisting of Thirteen Elements 314
6.14 Symbol of Anisotropy Consisting of Fourteen Elements 314
6.15 Symbol of Anisotropy Consisting of Fifteen Elements 314
6.16 Symbol of Anisotropy Consisting of Sixteen Elements 315
6.17 Symbol of Anisotropy Consisting of Seventeen Elements 315
6.18 Symbol of Anisotropy Consisting of Eighteen Elements 315
7 Materials with Negative Poisson's Ratio 315
8 Orthotropic Micropolar Material with a Center of Symmetry 316
9 Conclusions 318
References 318
Analytical Solutions in the Theory of Thin Bodies 321
1 Parametrization of Thin Domain with One Small Size Under an Arbitrary Base Surface 322
2 Representation of the Second Rank Isotropic Tensor and Its Components 328
3 Representations of Gradient, Divergence, Repeated Gradient and Laplacian 330
4 Equations of Motion and Constitutive Relations in the Micropolar Theory of Thin Bodies 331
5 Classical Theory of Elasticity in Displacements 333
5.1 Equations of the Classical Theory of Elasticity in Displacements 333
5.2 On Boundary Conditions in the Linear Theory of Elasticity. Stress Tensor Operator 335
5.3 Quasi-Static Problems of the Classical Theory of Elasticity in Displacements 336
5.4 Quasi-Static Problems of the Theory of Prismatic Bodies 336
5.5 Equations of Quasi-Static Problems of the Theory of Prismatic Bodies 337
6 Micropolar Theory of Elasticity in Displacements and Rotations 343
6.1 Equations of Motion of 3D Micropolar Theory of Elasticity in Displacements and Rotation Vectors 343
6.2 On Stress Boundary Conditions. Tensor Operators of Stress and Couple Stress 347
6.3 Quasi-Static Problems of Micropolar Elasticity Theory in Displacements and Rotations 351
6.4 Prismatic Bodies with Constant Thickness in Displacements and Rotations and in Moments of Displacement and Rotation Vectors 352
6.5 Multilayer Prismatic Bodies 357
6.6 Prismatic Bodies with Two Small Sizes 359
References 362
Method for Calculating the Characteristics of Elastic State Media with Internal Degrees of Freedom 364
1 Introduction 364
2 Theoretical Statements 365
3 Numerical Results 371
4 Conclusion 376
References 376
Variational Theories of Two-Phase Continuum Poroelastic Mixtures: A Short Survey 378
1 Introduction 379
2 Variational Theories of the 70's and the 80's 382
2.1 Cowin's Theories Including Porosity 383
2.2 Mindlin's Variational Single-Phase Theory of Materials with Microstructure 384
2.3 The Variational Theory of Immiscible and Structured Mixtures by Bedford and Drumheller 385
3 Most Recent Theories 386
3.1 Variational Theories by Lopatnikov and Co-workers 387
3.2 Variational Higher Gradient Theories by dell'Isola and Co-workers 387
3.3 The VMTPM Framework and the Extrinsic/Intrinsic Treatment 389
4 Conclusions 391
References 392
Buckling of Sandwich Tube with Foam Core Under Combined Loading 396
1 Introduction 397
2 Initial Strain State of Sandwich Tube 397
3 Perturbed State of Sandwich Tube 401
4 Numerical Results 403
5 Conclusion 407
References 412
Frequency-Dependent Attenuation and Phase Velocity Dispersion of an Acoustic Wave Propagating in the Media with Damages 414
1 Introduction 414
2 Self-Consistent Model for Damage Description 415
3 Dispersion Relations 418
4 Dispersion Analysis 420
5 Conclusions 423
References 423
A Statistically-Based Homogenization Approach for Particle Random Composites as Micropolar Continua 425
1 Introduction 426
2 Micropolar Homogenization 427
3 Statistical Homogenization Convergence 432
3.1 Computational Multiscale Procedure 432
3.2 Finite Size Scaling of Elastic Moduli. Numerical Simulations 433
4 Final Remarks 439
References 440
Paradoxical Size Effects in Composite Laminates and Other Heterogeneous Materials 442
1 Introduction 443
2 Size Effects in a Two Phase Laminated Beam 444
3 Size Effects in a Two Dimensional Material with Periodic Heterogeneity 449
4 Discussion and Conclusions 454
References 455