МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ И АНАЛИЗ В КОНТИНУУМЕ

MATHEMATICAL MODELLING AND ANALYSIS IN CONTINUUM MECHANICS OF MICROSTRUCTURED MEDIA

Margaret Woźniak pro memoria

Stan książki/ek: NOWA, papier kredowy

Wydawnictwo Politechniki Śląskiej

Okładka: twarda

Stron: 316

Format: B5

Wydział: Budownictwo

Z okładki:

The book contains an overview of some new trends in continuum theories of microstructured media. The considerations are restricted to solids and structures with a discontinuous, highly oscillating space distribution of local properties, like microperiodic or/and functionally graded composites. The main attention is focused on analytical procedures which make it possible to describe the behaviour ofmicrostructured media ina certain averaged form, i.e. by means of non-oscillating smooth functions. The above procedures are referred to as mathematical modelling of microstructured media. The derived models serve as a tool for an analysis of specific problems.

The book is divided into four parts. In Part I, there are exposed two purely formal new modelling procedures. The first of them is based on the concept of tolerance relation and is referred to as the tolerance modelling. The second procedure represents a certain new asymptotic approach to the modelling problem and is called the semiconsistent asymptotic modelling. The formal hirudina results of Part I are applied in Part II to the formulation and analysis of various engineering theories of microstructured solids and structures. Some alternative modelling procedures are discussed in Part III. The analysis of selected specific problems as well as some experimental results related to the microstructure in soils are discussed in Part IV.

Spis treści:

TABLE OF CONTENTS

FROM THE EDITORS. XIII

PHOTO ..XV

MEMORIES OF MARGARET WOZNIAK .XVII

MAIN PUBLICATIONS OF MARGARET WOZNIAK. XXI

Aim and scope of the book XXV

Overview of the recent results .XXVII

PART I TOLERANCE VERSUS ASYMPTOTIC MODELLING

Summary of notions .. 3

1. BASIC CONCEPTS . 7

1.1. Tolerance parameter 7

1.2. Cell distribution . 8

1.3. Tolerance - periodic (TP) functions 9

2. AVERAGING OF INTEGRAL FUNCTIONALS .. 13

2.1. Averaging of TP functions 13

2.2. Micro - macro decomposition . 15

2.3. Tolerance averaging . 16

2.4. Consistent asymptotic averaging 17

2.5. Semiconsistent asymptotic averaging .. 20

3. MODELLING OF DIFFERENTIAL EQUATIONS. 23

3.1. Extended stationary action principle. 23

3.2. Tolerance modelling . 24

3.3. Asymptotic modelling . 26

3.4. Combined modelling 28

3.5. Illustrative examples. 31

PART II APPLICATIONS

1. DYNAMIC BEHAVIOUR OF A BEAM RESTING ON PERIODICALLY SPACED VISCOELASTIC SUPPORTS . 43

1.1. Preliminaries.. 43

1.2. Fundamental equations. 45

1.3. Tolerance model ..45

1.4. Asymptotic model .. 48

1.5. Combined model . 48

1.6. Generalizations. 50

2. ELASTIC RESPONSE OF THE PERIODIC ROD UNDER FUZZY RANDOM LOADING . 53

2.1. General solution.. 54

2.2. Tolerance modelling method 57

2.3. Probabilistic characteristic of random dynamic influence function .. 58

3. WAVE PROPAGATION IN THE PERIODICALLY RIBBED ELASTIC PLATES. 61

3.1. Propagation of the long waves. 63

3.2. Propagation of the certain short waves 64

3.3. Comparison of results.. 66

4. ELASTIC RESPONSE OF A SEMI-SPACE UNDER HIGHLY OSCILLATING BOUNDARY EXCITATIONS. 69

4.1. Model equations . 69

4.2. Time oscillating boundary conditions.. 72

4.3. Time decaying boundary conditions. 73

4.4. Remark 73

5. ELASTODYNAMIC BEHAVIOUR OF A LAMINATED LAYER IN THE UNIAXIAL STRAIN. 75

5.1. Preliminaries. 75

5.2. Asymptotic modelling . 77

5.3. Superimposed tolerance modelling 79

5.4. Generalizations 82

6. MODELLING OF THE QUASI - LINEAR HEAT CONDUCTION PROBLEMS FOR FUNCTIONALLY GRADED MATERIALS. 85

6.1. Formulation of the problem .. 85

6.2. Modelling procedure. 86

6.3. Tolerance model equations 88

6.4. Asymptotic model equations 89

7. QUASI - LINEAR HEAT CONDUCTION IN THE PERIODICALLY -LAYERED MEDIUM.. 91

7.1. Model equations . 91

7.2. Heat conduction across laminae . 93

7.3. Heat conduction along laminae 95

7.4. Conclusions .. 98

8. FREE VIBRATIONS OF A THIN FUNCTIONALLY GRADED PLATE BAND .. 99

8.1. Formulation of the problem .. 99

8.2. Modelling procedure.. 100

8.3. Example 102

8.4. Results .. 104

8.5. Remarks 106

9. DYNAMIC BEHAVIOUR OF A CERTAIN LONGITUDINALLY GRADED PLATE 107

9.1. Preliminaries 107

9.2. Model equations 109

9.3. Example of applications 111

9.4. Conclusions . 115

10.ON THE MICRO-DYNAMICS OF REINFORCED CYLINDRICAL SHELLS 117

10.1. Formulation of the problem 117

10.2. Tolerance model .. 121

10.3. Asymptotic model .. 125

10.4. Micro-vibrations of the shell . 127

10.5. The wave propagation problem 129

10.6. Final remarks . 130

11. ASYMPTOTIC MODELLING IN ELASTODYNAMICS OF FGM 133

11.1. Introductory concepts 133

11.2. Consistent asymptotic modelling. 134

11.3. Semiconsistent asymptotic modelling 137

11.4. Combined asymptotic modelling . 139

PART III ALTERNATIVE MODELLING TECHNIQUES

1. COMBINED APPROACH TO THE FGM - MODELLING. 145

1.1. Introduction 145

1.2. Overall strategy of the joint, FE and ANN computations ANN 146

1.3. ANN as a tool of numerical hirudina approximation 147

1.4. ANN as a numerical representation of a constitutive relationship in a global BVP for FGM 150

1.5. Modelling of composite materials by FE virtual testing .. 155

1.6. Overall algorithm of the joint application of ANN and FEM 157

1.7. Example illustrating the proposed, combined procedure of FGM -modelling. 158

1.8. Example of application. 162

1.9. Conclusions 165

2. MACROSCOPIC MODELLING OF POROUS MATERIALS.. 167

2.1. Introduction 167

2.2. Kinematics and balance equations in Lagrangian description 171

2.3. Second law of thermodynamics 180

2.4. Simplified constitutive relations .. 184

2.5. Thermodynamical results. 186

2.6. Biot's model .. 189

2.7. Unsaturated porous media .. 193

3. DISCRETIZED HOMOGENIZATION BY MICROLOCAL PARAMETERS. 197

3.1. Introductory concepts 198

3.2. Foundations of modelling 199

3.3. Application to elastodynamics.. 200

3.4. Alternative form of the model equations . 201

3.5. Effective moduli .. 201

4. HOMOGENIZED ELASTICITY VIA MICROLOCAL PARAMETERS. 203

4.1. Governing relations 203

4.2. The weak formulation of the problem 204

4.3. Existence result. 206

4.4. Existence of He^ and its properties . 210

4.5. Final remarks. 211

PART IV SELECTED PROBLEMS

1. BOUSSINESQ PROBLEM FOR A MICROLAYERED ELASTIC MEDIUM .. 215

1.1. Formulation and solution of the problem. 215

1.2. Final remarks . 227

2. THE RIGID INCLUSION PROBLEM FOR A MICROLAYERED MEDIUM .. 229

2.1. Problem formulation and governing equations . 229

2.2. Solution of the boundary-value anticrack problem . 234

2.3. Analysis of the stress field near the rigid inclusion edge . 239

3. FRICTIONAL HEATING IN A MICROLAYERED SOLID UNDER BRAKING 243

3.1. Preliminaries.. 243

3.2. Formulation of the problem 244

3.3. Temperature 246

3.4. Wear .. 247

3.5. Numerical analysis . 249

3.6. Conclusions 254

4. MICROSTRUCTURE IN SOILS .. 255

4.1. Introduction .,. 255

4.2. Characteristics of the soil structural elements 256

4.3. Microstructure study using the scanning electron microscope (SEM).. 257

4.4. Classification of microstructures .261

4.5. Polish soil microstructures.. 264

4.6. Soil behaviour and properties 278

4.7. Recapitulation and conclusions 282

Bibliography . 285

Subject index. 309

Authors affiliations 313