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Finite element analysis of composite materials using Abaqus

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Finite Element Analysis of Composite Materials
Using Abaqus
Ever J. Barbero
CRC Press
Taylor & Francis Croup
Boca Raton London NewYork CRC Press is an imprint of the
Taylor & Francis Group, an informa business

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Contents
Series Preface
xiii
Preface
xv
Acknowledgments
xix
List of Symbols
xxi List of Examples
xxix 1
Mechanics of Orthotropic Materials 1
1.1
Lamina Coordinate System
1 1.2
Displacements
1
1.3
Strain
2
1.4 Stress 3 1.5 Contracted Notation
4
1.5.1 Alternate Contracted Notation 5
1.6
Equilibrium and Virtual Work
6
1.7
Boundary Conditions
8 1.7.1
Traction Boundary Conditions
8
1.7.2
Free Surface Boundary Conditions 8
1.8
Continuity Conditions
8 1.8.1 Traction Continuity 8 1.8.2
Displacement Continuity
9
1.9
Compatibility
9
1.10
Coordinate Transformations
10
1.10.1
Stress Transformation
12
1.10.2 Strain Transformation 14 1.11 Transformation of Constitutive Equations 15 1.12 3D Constitutive Equations 17 1.12.1
Anisotropic Material
18
1.12.2
Monoclinic Material
19
1.12.3
Orthotropic Material
20
1.12.4
Transversely Isotropic Material
21 1.12.5
Isotropic Material
23 vii

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x
Finite Element Analysis of Composite Materials
7.1.4
Maxwell-Kelvin Model
253
7.1.5
Power Law
254 7.1.6
Prony Series
254 7.1.7
Standard Nonlinear Solid
256
7.1.8 Nonlinear Power Law 256 7.2 Boltzmann Superposition 258 7.2.1 Linear Viscoelastic Material 258 7.2.2
Unaging Viscoelastic Material
259 7.3
Correspondence Principle
260 7.4
Frequency Domain
261 7.5
Spectrum Representation
262 7.6
Micromechanics of Viscoelastic Composites
262 7.6.1
One-Dimensional Case
262 7.6.2 Three-Dimensional Case 264 7.7 Macromechanics of Viscoelastic Composites 269 7.7.1 Balanced Symmetric Laminates 269 7.7.2
General Laminates
269 7.8 FEA of Viscoelastic Composites 269
Suggested Problems
280
8
Continuum Damage Mechanics
283 8.1
One-Dimensional Damage Mechanics
284 8.1.1
Damage Variable
284 8.1.2
Damage Threshold and Activation Function
286 8.1.3
Kinetic Equation
287
8.1.4
Statistical Interpretation of the Kinetic Equation 288
8.1.5
One-Dimensional Random-Strength Model
289 8.1.6
Fiber-Direction, Tension Damage
294 8.1.7
Fiber-Direction, Compression Damage
300 8.2 Multidimensional Damage and Effective Spaces 304 8.3
Thermodynamics Formulation
305 8.3.1
First Law 306
8.3.2
Second Law
307
8.4
Kinetic Law in Three-Dimensional Space
313 8.4.1
Return-Mapping Algorithm
316 8.5
Damage and Plasticity
322
Suggested Problems
324 9
Discrete Damage Mechanics
327
9.1
Overview
328 9.2
Approximations
332 9.3 Lamina Constitutive Equation
333
9.4
Displacement Field
334 9.4.1
Boundary Conditions for AT =
0
335
9.4.2
Boundary Conditions for AT ^ 0
336

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Table of Contents xi 9.5
Degraded Laminate Stiffness and CTE
337 9.6
Degraded Lamina Stiffness
338 9.7
Fracture Energy
339 9.8 Solution Algorithm 340 9.8.1 Lamina Iterations 340 9.8.2 Laminate Iterations 340
Suggested Problems
351
10 Delaminations
353
10.1
Cohesive Zone Method 356 10.1.1
Single Mode Cohesive Model
358
10.1.2 Mixed Mode Cohesive Model 361 10.2 Virtual Crack Closure Technique 371
Suggested Problems
375 A Tensor Algebra 377 A.l
Principal Directions of Stress and Strain
377 A.2 Tensor Symmetry 377 A.3 Matrix Representation of a Tensor 378 A.4 Double Contraction 379 A.5 Tensor Inversion 379 A. 6 Tensor Differentiation 380
A.6.1
Derivative of a Tensor with Respect to Itself 380 A.6.2 Derivative of the Inverse of a Tensor with Respect to the Ten¬
sor
381 B
Second-Order Diagonal Damage Models
383 B. l
Effective and Damaged Spaces
383
B.2
Thermodynamic Force Y
384 B.3
Damage Surface
386 B.4
Unrecoverable-Strain Surface
387 C
Software Used
389 C.l
Abaqus
389
C.l.l
Abaqus Programmable Features
391 C.2 BMI3 393
References
395 Index 407

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