Mechanics for Materials and Technologies (eBook)

With contributions of W. Müller, Heng Xiao, D.M. Klimov, V.F. Zhuralev, B.D. Annin, V.A. Babeshko, I.G. Goryacheva, V.P. Matveenko, N.F. Morozov, N.N. Bolotnik, A.A. Burenin, R.V. Goldstein, D.A. Indeicev, G.I. Kanel, E.V. Lomakin, A.V. Manzhirov, K.E. Kazankov, V.I. Karev, S.A. Lychev, E.V. Murashkin, D.A. Parshin, Yu.N. Radaev, H. Altenbach, I. Kudish, Yu. Kaplunov, N.K. Gupta, R. Velmurugan, and S. Kapuria

Preface 6
Contents 12
List of Contributors 21
1Multi-Mode Symmetric and Asymmetric Solutions in the Jeffery-Hamel Problem for a Convergent Channel 28
Abstract 28
Key words: 28
1.1 Introduction and Statement of the Problem 29
1.2 Analytical Expressions, Asymptotic Expansions, and Integral 34
1.2 Analytical Expressions, Asymptotic Expansions, and Integral Estimates of the Solutions 34
1.2.1 Perturbation Method for Small Re 34
1.2.2 Perturbation Method for Small Aperture Angles 35
1.2.3 Asymptotic Behavior of the Solution for Large Re 36
1.2.4 Integral Estimates 36
1.3 Numerical-Analytical Accelerated Convergence Method and 38
1.3 Numerical-Analytical Accelerated Convergence Method andContinuation with Respect to a Parameter 38
1.4 Solutions Regularly Depending on the Reynolds Number 40
1.5 Construction of the Velocity Profiles and Analysis of the 44
1.5 Construction of the Velocity Profiles and Analysis of theFluid Flow Modes 44
1.6 Numerical-Analytical Solution of the Problem for theCritical Value of the Channel Angle 48
1.7 New Multi-Mode Asymmetric Solutions that Cannot be 51
1.7 New Multi-Mode Asymmetric Solutions that Cannot beRegularly Continued with Respect to Re 51
1.8 Kinematic and Force Characteristics of Steady Flows 56
1.9 Conclusions 57
Acknowledgements 57
References 57
2Riemann’s Method in Plasticity: a Review 59
Abstract 59
Key words: 59
2.1 Preliminary Remarks 59
2.2 Pressure-Independent Plasticity 61
2.3 Pressure-Dependent Plasticity 66
2.4 Planar Ideal Flows 71
2.5 Conclusions 72
Acknowledgements 72
References 72
3Homogenization of Corrugated Plates Based on the Dimension Reduction for the Periodicity Cell Problem 74
Abstract 74
Key words: 74
3.1 Introduction 75
3.2 Statement of the Problem 77
3.3 Dimension Reduction for the Periodicity Sell Problem 78
3.4 Symmetric Corrugation 83
3.5 Numerical Example 1 - Computation of Effective Stiffness ofthin Corrugated Shells 86
3.6 Computation of the Effective Stiffnesses D2 1212, D2 2121 for Thin Plates 87
3.7 Numerical Example 2 - Corrugated Plates of ArbitraryThickness 89
3.8 Universal Relations Between the Effective Stiffness of Corrugated Plates made of Materials with the same Poisson’s Ratio 95
3.9 Conclusions 96
Acknowledgements 96
References 97
4Consideration of Non-Uniform and Non-Orthogonal Mechanical Loads for Structural Analysis of Photovoltaic Composite Structures 98
Abstract 98
Key words: 98
4.1 Introduction 99
4.1.1 Motivation 99
4.1.2 Objective and Structure 101
4.1.3 Preliminaries and Notation 102
4.2 Mechanical Loads at Photovoltaic Modules 104
4.2.1 Loading at Natural Weathering 104
4.2.1.1 Snow Loads 105
4.2.1.2 Wind Loads 105
4.2.2 Mathematical Description of Mechanical Loads 106
4.2.2.1 Load Vector 106
4.2.2.2 Direction of Loads 106
4.2.2.3 Amplitude and Spatial Distribution of Loads 107
4.3 Solution Approach with eXtended LayerWise Theory 109
4.3.1 Prerequisites 109
4.3.2 Degrees of Freedom 111
4.3.3 Kinematical Measures 111
4.3.4 Balance Equations and Kinetic Measures 111
4.3.5 Constitutive Equations 114
4.3.6 Boundary Conditions 115
4.3.7 Kinematical Constraints 116
4.3.8 Introduction of Mean and Relative Measures 117
4.3.9 Principle of Virtual Work 118
4.4 Numerical Implementation 120
4.4.1 Basic Procedure in Finite Element Method 120
4.4.2 Shape Functions 120
4.4.3 JACOBI Transformation 121
4.4.4 Discretisation 122
4.4.4.1 Degrees of Freedom 122
4.4.4.2 KinematicalMeasures 123
4.4.5 Constitutive Equations for FEM 124
4.4.6 Element Stiffness Relation 125
4.4.7 Surface Load Vector 126
4.4.8 Assembling 127
4.4.9 Numerical Integration and Artificial Stiffening Effects 128
4.5 Structural Analysis 130
4.5.1 Test Structure 130
4.5.2 Discretisation and Convergence 131
4.5.3 Case Studies 133
4.5.4 Results and Discussion 134
4.5.4.1 Degrees of Freedom 134
4.5.4.2 Kinetic and Kinematic Quantities 137
4.6 Conclusion 140
Acknowledgements 143
4.A Appendix 143
4.A.1 Constitutive Matrices 143
4.A.2 Auxiliary Matrices 144
References 145
5 Block Element Method for the Stamps of the no Classical Form 148
Abstract 148
Key words: 148
5.1 Introduction 149
5.2 Statement of the Problem 150
5.3 Properties of the Integral Equations 151
5.4 The Block Element Method for a System of Integral Equations 153
5.5 Study of the Properties of the Solution of the System of Integral Equations and a Boundary Value Problem 154
5.6 Acknowledgments 156
References 157
6 On the Irreversible Deformations Growth in the Material with Elastic, Viscous, and Plastic Properties and Additional Requirements to Yield Criteria 158
Abstract 158
Key words: 158
6.1 Introduction 158
6.2 Large Deformations Kinematics 159
6.3 Governing Equations 163
6.4 The Flow of Elastic-Viscous-Plastic Solids Inside the Cylindrical Tube 167
6.5 Viscometric Deformation of the Incompressible Cylindrical Layer 171
6.6 Conclusion 175
References 176
7On Nonlocal Surface Elasticity and Propagationof Surface Anti-Plane Waves 177
Abstract 177
Key words: 177
7.1 Introduction 177
7.2 Governing Equations 179
7.3 Anti-Plane Surface Waves in an Elastic Half-Space 181
7.4 Conclusions 184
References 184
8Deformation of Spherical Inclusion in an ElasticBody with Account for Influence of InterfaceConsidered as Infinitesimal Layer withAbnormal Properties 187
Abstract 187
Key words: 187
8.1 Introduction 187
8.2 Model of the Interface Elasticity 188
8.3 Problem of Spherical Inclusion. Various Solutions 190
8.4 Conclusion 192
Acknowledgements 192
References 192
9Analysis of Internal Stresses in a ViscoelasticLayer in Sliding Contact 194
Abstract 194
Key words: 194
9.1 Introduction 194
9.2 Problem Formulation 195
9.3 Method of Solution 196
9.4 Analysis of Internal Stresses 199
9.5 Conclusions 202
Acknowledgements 203
References 203
10On the Problem of Diffusion in Materials UnderVibrations 205
Abstract 205
Key words: 205
10.1 Introduction 205
10.2 The Equation of Impurity Motion 206
10.3 Statement of the Problem: Governing Equations 208
10.4 Localization of Diffusion Process 210
10.5 Structural Transformations of Materials 213
10.6 Conclusion 214
Acknowledgements 214
References 215
11A Study of Objective Time Derivatives inMaterial and Spatial Description 216
Abstract 216
Key words: 217
11.1 Introduction and Outline to the Paper 217
11.2 Frames of Reference – Fundamental Definitions 218
Definition 11.1. 218
Definition 11.2. 219
Definition 11.3. 219
Definition 11.4. 220
Definition 11.5. 220
11.3 Changing Frames of Reference 223
11.3.1 Kinematic Quantities and Their Images 224
11.3.2 Rotation of one Reference Frame with Respect to Another 228
11.3.3 Motion of FoRs with Respect to Each Other 231
11.4 Frame Indifference of Operators 233
11.4.1 Transformation Properties of Spatial Gradients 233
11.4.2 Transformation Properties of the Total and Material Time Derivatives 240
11.5 Conclusions and Outlook 245
9.A Appendix 246
9.A.1 Rotational Tensors and Angular Velocity Vectors 246
References 249
12On Electronically Restoring an ImperfectVibratory Gyroscope to an Ideal State 251
Abstract 251
Key words: 251
12.1 Introduction 252
12.2 Notation 254
12.3 Kinetic Energy, Prestress and Potential Energy 256
12.4 Tangentially Anisotropic Damping 257
12.5 Electrical Energy 258
12.6 Eliminating Frequency Split 262
12.7 Parametric Excitation 265
12.8 Principal and Quadrature Vibration 266
12.9 Numerical Experiment 266
12.10 Averaging 270
12.11 Graphical Comparisons and Quantitative Analysis of theExact and Averaged ODE 271
12.12 Isotropic Damping and the Meander Electrodes 274
12.13 Conclusion 274
Acknowledgements 274
References 275
13ShockWave Rise Time and the Viscosity ofLiquids and Solids 277
Abstract 277
Key words: 277
13.1 Introduction 277
13.2 Experiments and Their Results 278
13.3 Conclusions 282
Acknowledgements 282
References 283
14Lowest Vibration Modes of StronglyInhomogeneous Elastic Structures 284
Abstract 284
Key words: 284
14.1 Introduction 284
14.2 Antiplane Shear Motion 285
14.2.1 Stiffer Outer Domain 287
14.2.2 Stiffer Inner Domain 289
14.3 Model Examples 290
14.3.1 Two-Layered Circular Cylinder 290
14.3.2 Square Cylinder with a Circular Annular Inclusion 292
14.4 Concluding Remarks 294
12.A Appendix 294
References 295
15Geometrical Inverse Thermoelastic Problem forMultiple Inhomogeneities 297
Abstract 297
Key words: 297
15.1 Introduction 297
15.2 Mathematical Formulation of the Direct Problem 298
15.3 Reciprocity Principle and Reciprocity Gap Functional 300
15.4 Statement of the Inverse Problem and a Method of itsSolving 302
15.5 Numerical Procedure and Numerical Examples 306
15.6 Conclusions 312
Acknowledgements 312
References 312
16Indentation of the Regular System of Punchesinto the Foundation with Routh Coating 314
Abstract 314
Key words: 314
16.1 Statement of the Problem 315
16.2 Dimensionless Form and Operator Representation 317
16.3 Transformation of Main Equation and Special Basis 318
16.4 Solving the Problem 320
16.5 Main Results and Conclusions 325
Acknowledgements 325
References 325
17Physical Modeling of Rock Deformation andFracture in the Vicinity of Well for DeepHorizons 326
Abstract 326
Key words: 326
17.1 Introduction 326
17.2 Experimental Facility and Loading Programs for Specimens 327
17.3 Rock Specimens Test Results 330
17.4 Conclusion 333
Acknowledgements 333
References 334
18Full Axially Symmetric Contact of a RigidPunch with a Rough Elastic Half-Space 335
Abstract 335
Key words: 335
18.1 Introduction 335
18.2 Problem Formulation 336
18.3 Some Generalizations 340
References 342
19Geometric Aspects of the Theory ofIncompatible Deformations in Growing Solids 343
Abstract 343
Key words: 343
19.1 Introduction 344
19.2 Naive Geometric Motivation 345
19.3 Material Manifold 347
19.4 Growing Solids 350
19.5 Mappings Between Manifolds 351
19.6 Deformations 353
19.7 Material Connection 355
19.8 Example 357
References 361
20Free Vibrations of a Transversely Isotropic Platewith Application to a Multilayer Nano-Plate 364
Abstract 364
Key words: 364
20.1 Introduction 364
20.2 Equations of Motion and Their Transformation 366
20.3 Principal Natural Frequency in the Dependence of Boundary Conditions 368
20.4 Numerical Results and Their Discussion 370
20.5 The Generalized Kirchhoff–Love (GKL) Model for a Multilayer Plate 372
20.6 Continuum Model of a Multilayer Graphene Sheet (MLGS) Vibrations 373
20.7 Identification of Graphite and Graphene Parameters and some Numerical Results 374
20.8 Numerical Results and Their Discussion 375
Acknowledgements 376
References 376
21On Thermodynamics of Wave Processes of HeatTransport 378
Abstract 378
Key words: 379
21.1 Preliminary Remarks 379
21.2 Thermodynamic Orthogonality and Constitutive Equationsof the Perfect Plasticity 380
21.3 Internal Entropy Production for a Heat Transport Processin Thermoelastic Continua 384
21.4 Constitutive Equations for Type-III Thermoelasticity byVirtue of Thermodynamic Orthogonality 387
21.5 Conclusions 389
References 390
22The Technological Stresses in a VaultedStructure Built Up on a Falsework 392
Abstract 392
Key words: 392
22.1 Introduction 392
22.2 Statement of the Problem 393
22.3 Boundary Value Problem for the Built-up Structure 395
22.4 Analytical Solution of the Problem. Determining theStresses in the Vault Supported by the Falsework 398
22.5 Residual Stresses in the Finished Structure 399
Acknowledgements 400
References 400
23Reversible Plasticity Shape-Memory Effect inEpoxy Nanocomposites: Experiments, Modelingand Predictions 402
Abstract 402
Key words: 403
23.1 Introduction 403
23.2 Experimental Methods 405
23.2.1 Material Selection and Sample Preparation 405
23.2.2 Material Characterization 405
23.2.3 RPSM Characterization 406
23.3 Mechanism 407
23.4 Model Description 407
23.4.1 Kinematics 407
23.4.2 Structural Relaxation and Thermal Deformation 409
23.4.3 Constitutive Equations for Stress 409
23.4.4 Flow Rule 411
23.5 Results and Discussions 411
23.5.1 Mechanical Properties 411
23.5.2 Thermal Properties 413
23.5.3 Morphological Properties 414
23.5.4 RPSM Properties 415
23.6 Conclusion 423
18.A Appendix 423
18.A.1 Parameter Determination and Effect of MWCNT on theMaterial Parameters 423
18.A.2 Determination of ?r, k, G and l 424
18.A.3 Determination of C1, C2, tos, ag and ar 425
18.A.4 Determination of hg, ss, Q and h 427
References 427
24The Dynamics of an Accreting Vibrating Rod 431
Abstract 431
Key words: 432
24.1 Introduction 432
24.2 Equations of Motion and Their Transformations 433
24.3 Theoretical Treatment: Solution of Mixed Problem to 435
24.4 Numerical Simulations and Discussions 438
24.5 Conclusion 443
Acknowledgements 443
References 444
25A New, Direct Approach Toward ModelingRate-Dependent Fatigue Failure of Metals 446
Abstract 446
Key words: 446
25.1 Introduction 446
25.2 New Rate-Dependent Elastoplasticity Model 448
25.3 Failure Under Monotone and Cyclic Loadings 451
25.3.1 Governing Equations in the Uniaxial Case 451
25.3.2 Parameter Identification with Monotone Strain Data 452
25.3.3 Predictions for Fatigue Failure Under Cyclic Loadings 453
25.4 Numerical Results 454
25.4.1 Failure Under Monotone Strain 454
25.4.2 Predictions for Fatigue Failure Under Cyclic Loadings 455
25.5 Concluding Remarks 458
Acknowledgements 458
References 459