Thermomechanics of Composite Structures under High Temperatures (eBook)

Preface 6
Contents 8
Notations 17
Introduction 24
1 High-Temperature Environment and Composite Materials 27
1.1 Main Types of High-Temperature Effects on Composite Materials 27
1.1.1 Aerodynamical Heating 28
1.1.2 Gas-Dynamical Heating 36
1.1.3 Heating in Energetic Systems 38
1.1.4 Technological Heating 40
1.1.5 Action of Fire 41
1.2 Ablation Processes in Composites 43
1.2.1 Classification of Ablation Processes 44
1.2.2 Volumetric Ablation 45
1.2.3 Surface Ablation 46
1.3 Phenomena in Composite Materials Under High Temperatures 47
1.4 A Physical Model of Ablative Composite 50
References 51
2 General Equations of Multiphase Continuum Mechanics for Ablative Composites 56
2.1 Conservation Laws 56
2.1.1 Main Concepts of Tensor Analysis 56
2.1.2 System of Conservation Laws for Phases 59
2.1.3 Relationships on a Surface of a Strong Discontinuity 60
2.2 Constitutive Relations for Phases of Ablative Composites 61
2.2.1 The Fourier Law 61
2.2.2 General Thermodynamical Identity 62
2.2.3 Natural Configurations of Phases 63
2.2.4 General Form of Constitutive Relations 65
2.3 Relations at the Phase Interface 68
2.3.1 Main Equations 68
2.3.2 Classification of Phase Interfaces 69
2.3.3 Consequences of General Equations 70
2.3.4 Tensor of Chemical Potential 71
2.4 Equation of Phase Transformation Rate 72
2.5 Infinitesimal Strains of Solid Phases 74
2.5.1 Main Assumptions 74
2.5.2 Constitutive Relations 75
2.5.3 Quasistatic Processes 78
2.5.4 Conservation Equations 78
2.5.5 Conditions on the Phase Interface 79
2.5.6 Rate of the Phase Transformation 81
References 83
3 Mathematical Model of Ablative Composites 85
3.1 Main Assumptions 85
3.2 Method of Asymptotic Averaging 88
3.2.1 Boundary Conditions 88
3.2.2 Initial Conditions 90
3.2.3 Statement of Thermomechanics Problem for Ablative Composites 90
3.2.4 Asymptotic Expansions 91
3.2.5 Zero-Level Local Problem over the Periodicity Cell 91
3.2.6 First-Level Local Problems Over the Periodicity Cell 94
3.3 Averaging of Processes in Ablative Composites 95
3.3.1 Averaged Equations 95
3.3.2 Averaged Constitutive Relations 96
3.3.3 Statement of the Averaged Problem 97
3.4 Analysis of Asymptotic Averaging Method 98
3.5 Statement of Problems for Composites with Ablative Matrix and Fibres 99
3.5.1 Main Equations 99
3.5.2 Motion Equation for the Ablation Surface ? 101
3.5.3 Constitutive Relations 101
3.5.4 Boundary and Initial Conditions 102
3.5.5 Statement of the Problem in Terms of Stresses 104
References 105
4 Behavior of Matrices at High Temperatures 106
4.1 Varying Density of Matrices at High Temperatures 106
4.1.1 Determination of a Volumetric Ablation Rate of Matrices 106
4.1.2 Experimental Data 108
4.1.3 Pore Pressure 111
4.2 Effective Elastic Properties of Ablative Matrices at High Temperatures 112
4.2.1 Solving the Mechanical Local Problem Over a Periodicity Cell 112
4.2.2 Effective Constitutive Relations 116
4.2.3 Experimental Investigation of Changing Elastic Properties of Matrices Under Heating 117
4.3 Heat Expansion/Shrinkage of Matrices at High Temperatures 121
4.4 Strength Properties of Matrices at High Temperatures 123
4.4.1 Microstresses in Phases 123
4.4.2 Failure Criterion for Matrices at High Temperatures 124
4.4.3 Experimental Investigation of Strength Properties of Matrices Under High Temperatures 126
4.5 Heat Conductivity and Heat Capacity of Matrices at High Temperatures 132
4.5.1 Solving the Local Problem of Heat Conduction 132
4.5.2 Experimental Data 133
4.5.3 Heat Capacity 134
4.6 Gas Permeability of Matrices at High Temperatures 134
4.6.1 Solving the Local Problem of Gas Dynamics 134
References 138
5 Reinforcing Fibres Under High Temperatures 140
5.1 Changing Phase Composition of Fibres Under Heating 140
5.2 Heat Conductivity and Heat Capacity of Ablative Fibres 143
5.3 Varying Elastic Properties of Fibres Under Heating 145
5.4 Heat Deformation of Fibres 147
5.5 Strength Properties of Fibres Under High Temperatures 148
5.5.1 Strength of Idealized Fibre 148
5.5.2 Model of a Thread of Fibres with Defects 148
5.6 Short Fibres and Dispersed Particles 155
References 155
6 Unidirectional Composites Under High Temperatures 157
6.1 Structural Model of Unidirectional Composites 157
6.1.1 Peculiarities of Unidirectional Composites Under High Temperatures 157
6.1.2 Multilevel Internal Structure of Unidirectional Composite 158
6.2 Model of Microcomposite 160
6.2.1 Elastic Properties 160
6.2.2 Heat Deformations and Phase Interactions 163
6.2.3 Microstresses 164
6.2.4 Heat Conductivity 165
6.3 Thermo-Elastic Characteristics and Heat Conductivity of Unidirectional Composites 166
6.3.1 Theoretical Relations 166
6.3.2 Experimental Data 166
6.4 Strength Properties of Unidirectional Composite Under High Temperatures 169
6.4.1 Thermal Strength of Unidirectional Composite in Ension Along Reinforcing Direction 170
6.4.2 Experimental Data 175
6.4.3 Thermal Strength of Unidirectional Composite in Compression Along Reinforcing Direction 178
6.4.4 Thermal Strength of Unidirectional Composite in Transverse Tension/Compression and Shear 180
6.4.5 Thermal Microstresses and Microcracking 181
6.4.6 Thermal Strength of Unidirectional Composite in Longitudinal Shear 183
6.4.7 Multiaxial Loading of Unidirectional Composite 185
6.5 Heat Expansion/Shrinkage 186
References 187
7 Textile Ablative Composite Materials 188
7.1 Model of a Structure of Ablative Textile Composite Material 188
7.2 Model of a Layer with Curved Threads 190
7.2.1 Elastic Properties 190
7.2.2 Heat Deformations and Coefficients of Phase Interaction 193
7.2.3 Microstresses 193
7.2.4 Heat Conductivity of Layers with Curved Threads 195
7.3 Constitutive Relations for Ablative Textile Composites 196
7.4 Thermo-Elastic Moduli and Heat Conductivity Coefficients of Textile Composites 197
7.4.1 Theoretical Results 197
7.4.2 Experimental Data 199
7.5 Heat Deformations 202
7.5.1 Theoretical Relations 202
7.5.2 Experimental Data 203
7.6 Coefficients of Phase Interaction 205
7.7 Thermal Strength 206
7.7.1 Destruction by Types (A?) and (B?) 208
7.7.2 Destruction by the Types (C) and (D) 213
7.7.3 Experimental Data 214
7.8 Thermal Properties of Textile Composites 215
7.8.1 Heat Conductivity 215
7.8.2 Density 217
7.9 Gas Permeability 220
References 221
8 Composites Reinforced by Dispersed Particles 222
8.1 Model of the Composite 222
8.2 Thermo-Elastic Characteristics 223
8.2.1 One-Dimensional Model 223
8.2.2 Three-Dimensional Relations 226
8.3 Strength 227
8.3.1 Strength in Tension 227
8.3.2 Strength in Compression 228
8.4 Thermal Properties 229
8.4.1 Heat Conductivity 229
8.4.2 Density and Heat Capacity 230
8.4.3 Gas Permeability 230
References 232
9 Phenomena in Composite Materials Caused by Gradient Heating 233
9.1 Internal Heat-Mass-Transfer and Stresses in Ablative Composites Under Gradient Heating 234
9.1.1 Problem Statement and Solution 234
9.1.2 Computed Results 235
9.2 Plane Problems of Thermomechanics for Composites Under High Temperatures 239
9.3 Heat Deformations, Stresses and Load-Bearing Capacity of a Composite Plate Under Gradient Heating 243
9.3.1 Problem Statement 243
9.3.2 Other Cases of Loading a Plate 247
9.3.3 Computed Results 248
References 253
10 Linear Ablation of Composites 254
10.1 Main Types of Linear Ablation of Composites 255
10.2 Combustion Rate 255
10.2.1 General Equations 255
10.2.2 Combustion Rate of a Composite in Air Flow 258
10.2.3 Computed Results 259
10.3 Sublimation Rate 260
10.4 Thermomechanical Erosion Rate 262
10.4.1 General Relationships 262
10.4.2 Isotropic Composites 264
10.4.3 Transversally Isotropic Composites 269
10.4.4 Textile Composites 272
10.4.5 Computed Results 273
10.5 Melting Rate 274
10.6 Comparison of Theoretical and Experimental Results 275
10.6.1 Effect of a Matrix Type on the Rate of Linear Ablation of Composites 276
10.6.2 Effect of a Fibre Type on the Rate of Linear Ablation of Composites 277
10.6.3 Effect of a Pressure Head on the Rate of Linear Ablation of Composites 278
10.6.4 Particle-Reinforced Composites 280
10.7 Heat Balance on Ablative Surface 281
10.8 Criteria of Efficiency of Composites 282
References 285
11 Thermal Stresses in Composite Structures Under High Temperatures 287
11.1 Axisymmetric Problems of Thermomechanics ƒ 287
11.1.1 Basic Equations 287
11.1.2 Constitutive Relations 288
11.1.3 Functions of Stresses 290
11.1.4 Boundary Conditions 290
11.1.5 Statement of the Axisymmetric Problem in Terms of Stresses 291
11.1.6 The Problem Statement in Terms of Displacements 291
11.2 Thermal Stresses in Composite Structures of Heat-Energetic Systems 293
11.2.1 External Shell of the Inlet of STJE 293
11.2.2 A Shell of a Central Body of STJE Inlet 301
11.3 Thermal Stresses in Thermoprotective Structures Under Gas-Dynamical Heating 305
11.3.1 The Problem Statement 305
11.3.2 Numerical Analysis of the Problem 308
11.4 Thermal Stresses in Thermoprotective Structures Under Aerodynamical Heating 312
11.4.1 The Problem Statement 312
11.4.2 Computed Results 314
11.5 Thermal Stresses in Composites Under Local Technological Heating 319
11.5.1 Statement of the Problem 320
11.5.2 Computed Results 320
References 324
12 Mechanics of Composite Thin-Walled Shells Under High Temperatures 326
12.1 General Equations for Thin-Walled Ablative Shells Under High Temperatures 327
12.1.1 Model of a Multilayer Composite Shell 327
12.1.2 Constitutive Relations of Ablative Composites 329
12.1.3 Boundary and Initial Conditions 332
12.2 Main Assumptions for Thin-Walled Ablative Shells 334
12.3 Peculiarities of the Theory of Composite Shells Under High Temperatures 335
12.3.1 Mechanical Equations for Thin-Walled Ablative Shells 335
12.3.2 Strains and Stresses in a Multilayer Ablative Shell 336
12.3.3 Constitutive Relations for a Multilayer Ablative Shell 338
12.3.4 The Problem Statement on Heat-Mass-Transfer and Deforming for a Multilayer Ablative Shell 339
12.4 Cylindrical Composite Shells Under High Temperatures 341
12.4.1 Basic Equations 341
12.4.2 Computed Results 342
12.5 Failure of Composite Structures Under High Temperatures 347
12.5.1 Conditions of the Appearance of Failure 347
12.5.2 Behavior of a Composite Shell After the Appearance of Failure 349
12.5.3 Computed Results 351
12.5.4 Experimental Results 353
References 353
13 Finite-Element Method for Modeling of Thermomechanical Phenomena in Composite Shells Under High Temperatures 355
13.1 Variational Statements of Problems for Composite Shells ƒ 355
13.1.1 A Variational Statement of a Space Problem for Ablative Shell Mechanics 355
13.1.2 The Hellinger--Reissner Variational Principle for a Space Problem 360
13.1.3 The Hellinger--Reissner Variational Principle for Ablative Shells 362
13.2 Finite-Element Method for Ablative Composite Shells 366
13.3 Computational Methods for Modeling of Internal Heat-Mass-Transfer ƒ 370
13.3.1 The General Method Algorithm 370
13.3.2 The Dimensionless Form of Heat-Mass-Transfer Equation System 371
13.3.3 The Numerical Solving Algorithm for the Local Problem 373
13.3.4 The Asymptotic Method of Solving the Heat-Mass-Transfer Problem in Domain V2 376
13.3.5 The Numerical Method for Solving the Heat-Mass-Transfer Equations in Domain V2 378
13.4 Modeling of Cylindrical Composite Shells Under Local ƒ 380
13.4.1 Initial Data and Loading Parameters 380
13.4.2 Analysis of Modeling Results of Internal Heat-Mass-Transfer in a Shell 381
13.4.3 Analysis of Temperature Deformations 384
13.4.4 Analysis of Results for Displacement U1 387
13.4.5 Analysis of Results for Displacement U2 388
13.4.6 Analysis of Results for Flexure W of the Shell 392
13.4.7 Analysis of Results for Rotation Angle ?1 of the Normal 397
13.4.8 Analysis of Results for Rotation Angle ?2 of the Normal 399
13.4.9 Analysis of Results for Stress ?1 402
13.4.10 Analysis of Results for Stress ?2 402
13.4.11 Analysis of Results for Stress ?12 406
13.4.12 Analysis of Results for Stress ?13 406
13.5 Modeling of Axisymmetric Composite Shells Under Local ƒ 406
13.5.1 Initial Data and Loading Parameters 406
13.5.2 Analysis of Results for Displacements U1 and U2 408
13.5.3 Analysis of Results for Flexure W 409
13.6 Modeling of Composite Plates Under Local High-Temperature Heating 419
13.6.1 Initial Data and Loading Parameters 419
13.6.2 Analysis of Results for Displacement U1 419
13.6.3 Analysis of Results for Flexure W 423
13.6.4 Analysis of Results for Rotation Angle ?2 of the Normal 423
References 430
14 Methods of Experimental Investigation of High-Temperature Properties of Composite Materials 432
14.1 Determination of Density Under Heating 432
14.2 Determination of Thermal Characteristics Under Heating 433
14.2.1 Experimental Device 433
14.2.2 Determination of Thermoconductivity 434
14.2.3 Determination of Heat Conductivity 436
14.3 Determination of Gas Permeability 437
14.4 Determination of Heat Deformations Under Heating 438
14.5 Determination of Strength and Elastic Modulus ƒ 439
14.6 Gas-Dynamical Testing of Composites 442
References 444
Index 445