كتاب Finite Element Analysis of Composite Materials Using Abaqus
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 كتاب Finite Element Analysis of Composite Materials Using Abaqus

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كتاب Finite Element Analysis of Composite Materials Using Abaqus  Empty
مُساهمةموضوع: كتاب Finite Element Analysis of Composite Materials Using Abaqus    كتاب Finite Element Analysis of Composite Materials Using Abaqus  Emptyالسبت 19 سبتمبر 2020, 11:46 pm

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Finite Element Analysis of Composite Materials Using Abaqus
Ever J. Barbero  

كتاب Finite Element Analysis of Composite Materials Using Abaqus  F_e_a_17
و المحتوى كما يلي :


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
viiviii Finite Element Analysis of Composite Materials
1.13 Engineering Constants 24
1.13.1 Restrictions on Engineering Constants . 27
1.14 From 3D to Plane Stress Equations . 29
1.15 Apparent Laminate Properties . 30
Suggested Problems 32
2 Introduction to Finite Element Analysis 35
2.1 Basic FEM Procedure 35
2.1.1 Discretization . 36
2.1.2 Element Equations 36
2.1.3 Approximation over an Element 37
2.1.4 Interpolation Functions . 38
2.1.5 Element Equations for a Specific Problem . 40
2.1.6 Assembly of Element Equations . 41
2.1.7 Boundary Conditions 42
2.1.8 Solution of the Equations 42
2.1.9 Solution Inside the Elements 42
2.1.10 Derived Results 43
2.2 General Finite Element Procedure . 43
2.3 Solid Modeling, Analysis, and Visualization 46
2.3.1 Model Geometry . 47
2.3.2 Material and Section Properties . 57
2.3.3 Assembly . 61
2.3.4 Solution Steps 63
2.3.5 Loads . 63
2.3.6 Boundary Conditions 65
2.3.7 Meshing and Element Type . 68
2.3.8 Solution Phase 70
2.3.9 Post-processing and Visualization 73
Suggested Problems 89
3 Elasticity and Strength of Laminates 91
3.1 Kinematic of Shells 92
3.1.1 First-Order Shear Deformation Theory . 93
3.1.2 Kirchhoff Theory . 97
3.1.3 Simply Supported Boundary Conditions 99
3.2 Finite Element Analysis of Laminates . 100
3.2.1 Element Types and Naming Convention 101
3.2.2 Thin (Kirchhoff) Shell Elements 104
3.2.3 Thick Shell Elements . 104
3.2.4 General-purpose (FSDT) Shell Elements 104
3.2.5 Continuum Shell Elements 105
3.2.6 Sandwich Shells 106
3.2.7 Nodes and Curvature 106
3.2.8 Drilling Rotation . 106Table of Contents ix
3.2.9 A, B, D, H Input Data for Laminate FEA . 107
3.2.10 Equivalent Orthotropic Input for Laminate FEA . 113
3.2.11 LSS for Multidirectional Laminate FEA 119
3.2.12 FEA of Ply Drop-Off Laminates 129
3.2.13 FEA of Sandwich Shells . 139
3.2.14 Element Coordinate System . 150
3.2.15 Constraints 159
3.3 Failure Criteria 163
3.3.1 2D Failure Criteria 163
3.3.2 3D Failure Criteria 166
3.4 Predefined Fields . 171
Suggested Problems 173
4 Buckling 177
4.1 Eigenvalue Buckling Analysis 177
4.1.1 Imperfection Sensitivity . 183
4.1.2 Asymmetric Bifurcation . 183
4.1.3 Post-critical Path . 184
4.2 Continuation Methods 187
Suggested Problems 192
5 Free Edge Stresses 195
5.1 Poisson’s Mismatch 196
5.1.1 Interlaminar Force 196
5.1.2 Interlaminar Moment 197
5.2 Coefficient of Mutual Influence . 204
5.2.1 Interlaminar Stress due to Mutual Influence 207
Suggested Problems 212
6 Computational Micromechanics 215
6.1 Analytical Homogenization . 216
6.1.1 Reuss Model . 216
6.1.2 Voigt Model 217
6.1.3 Periodic Microstructure Model . 217
6.1.4 Transversely Isotropic Averaging 218
6.2 Numerical Homogenization . 220
6.3 Local-Global Analysis 238
6.4 Laminated RVE 241
Suggested Problems 247
7 Viscoelasticity 249
7.1 Viscoelastic Models 251
7.1.1 Maxwell Model 251
7.1.2 Kelvin Model . 252
7.1.3 Standard Linear Solid 253x 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 ΔT = 0 335
9.4.2 Boundary Conditions for ΔT = 0 336Table 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.1 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 Tensor 381
B Second-Order Diagonal Damage Models 383
B.1 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.1 Abaqus . 389
C.1.1 Abaqus Programmable Features . 391
C.2 BMI3 . 393
References 395
Index 40
List of Symbols
Symbols Related to Mechanics of Orthotropic Materials
Strain tensor
εij Strain components in tensor notation
α Strain components in contracted notation
e
α Elastic strain


Plastic strain
λ Lame constant
ν Poisson’s ratio
ν12 In-plane Poisson’s ratio
ν23, ν13 Interlaminar Poisson’s ratios
ν
xy Apparent laminate Poisson’s ratio x-y
σ Stress tensor
σij Stress components in tensor notation
σα Stress components in contracted notation
[a] Transformation matrix for vectors
ei Unit vector components in global coordinates
e
i Unit vector components in materials coordinates
fi, fij Tsai-Wu coefficients
k Bulk modulus
l, m, n Direction cosines
u(εij) Strain energy per unit volume
ui Displacement vector components
xi Global directions or axes
x
i Materials directions or axes
C Stiffness tensor
Cijkl Stiffness in index notation
Cα,β Stiffness in contracted notation
E Young’s modulus
E1 Longitudinal modulus
E2 Transverse modulus
E2 Transverse-thickness modulus
Ex Apparent laminate modulus in the global x-direction
G = μ Shear modulus
G12 In-plane shear modulus
G23, G13 Interlaminar shear moduli
xxixxii Finite Element Analysis of Composite Materials
G
xy Apparent laminate shear modulus x-y
Iij Second-order identity tensor
Iijkl Fourth-order identity tensor
Q ij Lamina stiffness components in lamina coordinates
[R] Reuter matrix
S Compliance tensor
Sijkl Compliance in index notation
Sα,β Compliance in contracted notation
[T ] Coordinate transformation matrix for stress
[T ] Coordinate transformation matrix for strain
Symbols Related to Finite Element Analysis
∂ Strain-displacement equations in matrix form
 Six-element array of strain components
θx, θy, θz Rotation angles following the right-hand rule (Figure 2.19)
σ Six-element array of stress components
φx, φy Rotation angles used in plate and shell theory
a Nodal displacement array
ue
j Unknown parameters in the discretization
B Strain-displacement matrix
C Stiffness matrix
K Assembled global stiffness matrix
Ke Element stiffness matrix
N Interpolation function array
N e
j Interpolation functions in the discretization
P e Element force array
P Assembled global force array
Symbols Related to Elasticity and Strength of Laminates
γxy 0 In-plane shear strain
γ4u Ultimate interlaminar shear strain in the 2-3 plane
γ5u Ultimate interlaminar shear strain in the 1-3 plane
γ6u Ultimate in-plane shear strain
0
x, 0 y In-plane strains
1t Ultimate longitudinal tensile strain
2t Ultimate transverse tensile strain
3t Ultimate transverse-thickness tensile strain
1c Ultimate longitudinal compressive strain
2c Ultimate transverse compressive strain
3c Ultimate transverse-thickness compressive strain
κx, κy Bending curvatures
κ
xy Twisting curvatureList of Symbols xxiii
φx, φy Rotations of the middle surface of the shell (Figure 2.19)
c4, c5, c6 Tsai-Wu coupling coefficients
tk Lamina thickness
u0, v0, w0 Displacements of the middle surface of the shell
z Distance from the middle surface of the shell
Aij Components of the extensional stiffness matrix [A]
Bij Components of the bending-extension coupling matrix
Dij Components of the bending stiffness matrix [D]
[E0] Extensional stiffness matrix [A], in ANSYS notation
[E1] Bending-extension matrix [b], in ANSYS notation
[E2] Bending stiffness matrix [D], in ANSYS notation
F1t Longitudinal tensile strength
F2t Transverse tensile strength
F3t Transverse-thickness tensile strength
F1c Longitudinal compressive strength
F2c Transverse compressive strength
F3c Transverse-thickness compressive strength
F4 Interlaminar shear strength in the 2-3 plane
F5 Interlaminar shear strength in the 1-3 plane
F6 In-plane shear strength
Hij Components of the interlaminar shear matrix [H]
IF Failure index
Mx, My, Mxy Moments per unit length (Figure 3.3)
Mn Applied bending moment per unit length
Nx, Ny, Nxy In-plane forces per unit length (Figure 3.3)
Nn
Applied in-plane force per unit length, normal to the edge
Nns Applied in-plane shear force per unit length, tangential
Qijk Lamina stiffness components in laminate coordinates, layer k
Vx, Vy Shear forces per unit length (Figure 3.3)
Symbols Related to Buckling
λ, λi Eigenvalues
s Perturbation parameter
Λ Load multiplier
Λ(cr) Bifurcation multiplier or critical load multiplier
Λ(1) Slope of the post-critical path
Λ(2) Curvature of the post-critical path
v Eigenvectors (buckling modes)
[K] Stiffness matrix
[Ks] Stress stiffness matrix
PCR Critical loadxxiv Finite Element Analysis of Composite Materials
Symbols Related to Free Edge Stresses
ηxy,x, ηxy,y Coefficients of mutual influence
ηx,xy, ηy,xy Alternate coefficients of mutual influence
F
yz Interlaminar shear force y-z
Fxz Interlaminar shear force x-z
Mz Interlaminar moment
Symbols Related to Micromechanics
α Average engineering strain components
εij Average tensor strain components
0
α, ε0 ij Far-field applied strain components
σα Average stress components
Ai Strain concentration tensor, i-th phase, contracted notation
2a1, 2a2, 2a3 Dimensions of the RVE
Aijkl Components of the strain concentration tensor
Bi Stress concentration tensor, i-th phase, contracted notation
Bijkl Components of the stress concentration tensor
I 6 × 6 identity matrix
Pijkl Eshelby tensor
Vf Fiber volume fraction
Vm Matrix volume fraction
Symbols Related to Viscoelasticity
ε˙ Stress rate
η Viscosity
θ Age or aging time
σ˙ Stress rate
τ Time constant of the material or system
Γ Gamma function
s Laplace variable
t Time
Cα,β(t) Stiffness tensor in the time domain
Cα,β(s) Stiffness tensor in the Laplace domain
Cα,β(s) Stiffness tensor in the Carson domain
D(t) Compliance
D0, (Di)0 Initial compliance values
Dc(t) Creep component of the total compliance D(t)
D, D Storage and loss compliances
E0, (Ei)0 Initial moduli
E∞ Equilibrium modulus
E, E0, E1, E2 Parameters in the viscoelastic models (Figure 7.1)
E(t) RelaxationList of Symbols xxv
E, E Storage and loss moduli
F [] Fourier transform
(Gij)0 Initial shear moduli
H(t − t0) Heaviside step function
H(θ) Relaxation spectrum
L[] Laplace transform
L[]−1 Inverse Laplace transform
Symbols Related to Damage
α Laminate CTE
α(k) CTE of lamina k
αcr Critical misalignment angle at longitudinal compression failure
ασ Standard deviation of fiber misalignment
γ(δ) Damage hardening function
γ0 Damage threshold
δij Kronecker delta
δ Damage hardening variable
ε Effective strain
ε Undamaged strain
εp Plastic strain
γ˙ Heat dissipation rate per unit volume
γ˙ s Internal entropy production rate
λ Crack density
λlim Saturation crack density
λ, ˙ λ˙ d Damage multiplier
λ˙ p Yield multiplier
ρ Density
σ Effective stress
σ Undamaged stress
τ13, τ23 Intralaminar shear stress components
ϕ, ϕ∗ Strain energy density, and complementary SED
χ Gibbs energy density
ψ Helmholtz free energy density
ΔT Change in temperature
Ω = Ωij Integrity tensor
2a0 Representative crack size
di Eigenvalues of the damage tensor
f d Damage flow surface
f p Yield flow surface
f(x), F (x) Probability density, and its cumulative probability
g Damage activation function
gd Damage surface
gp Yield surface
h Laminate thicknessxxvi Finite Element Analysis of Composite Materials
hk Thickness of lamina k
m Weibull modulus
p Yield hardening variable
p Thickness average of quantity p
p Virgin value of quantity p
p Volume average of quantity p
q Hear flow vector per unit area
r Radiation heat per unit mass
s Specific entropy
u(εij) Internal energy density
A Crack area
[A] Laminate in-plane stiffness matrix
Aijkl Tension-compression damage constitutive tensor
Bijkl Shear damage constitutive tensor
Ba Dimensionless number (8.57)
Cα,β Stiffness matrix in the undamaged configuration
Ced Tangent stiffness tensor
Dij Damage tensor
Dcr
1t Critical damage at longitudinal tensile failure
Dcr
1c Critical damage at longitudinal compression failure
Dcr
2t Critical damage at transverse tensile failure
D2, D6 Damage variables
E(D) Effective modulus
E Undamaged (virgin) modulus
Gc = 2γc Surface energy
GIc, GIIc Critical energy release rate in modes I and II
Jijkl Normal damage constitutive tensor
Mijkl Damage effect tensor
N Number of laminas in the laminate
{N} Membrane stress resultant array
Q Degraded 3x3 stiffness matrix of the laminate
R(p) Yield hardening function
R0 Yield threshold
S Entropy or Laminate complinace matrix, depending on context
T Temperature
U Strain energy
V Volume of the RVE
Yij Thermodynamic force tensor
Symbols Related to Delaminations
α Mixed mode crack propagation exponent
βδ, βG Mixed mode ratios
δ CZM separation of the interface
δm Mixed mode separationList of Symbols xxvii
δ0
m Mixed mode separation at damage onset
δ0
m Mixed mode separation at fracture
σ0 CZM critical separation at damage onset
 Delamination length for 2D delaminations
σ0 CZM strength of the interface
ψxi, ψyi Rotation of normals to the middle surface of the plate
Ω Volume of the body
ΩD Delaminated region
Πe Potential energy, elastic
Πr Potential energy, total
˙ Γ Dissipation rate
Λ Interface strain energy density per unit area
∂Ω Boundary of the body
d One-dimensional damage state variable
k
xy, kz Displacement continuity parameters
[Ai], [Bi], [Di] Laminate stiffness sub-matrices
DI, DII, DIII Damage variables for modes I, II, and III of CZM
G() Energy release rate (ERR), total, in 2D
G Energy release rate (ERR), total, in 3D
GI, GII, GIII Energy release rate (ERR) of modes I, II, and III
Gc Critical energy release rate (ERR), total, in 3D
Gc
I Critical energy release rate mode I
[Hi] Laminate interlaminar shear stiffness matrix
K Penalty stiffness
K˜ Virgin penalty stiffness
KI, KII, KIII Stress intensity factors (SIF) of modes I, II, and III
Ni, Mi, Ti Stress resultants
U Internal energy
W Work done by the body on its surroundings
Wclosure Crack closure work
List of Examples
Example 1.1, 7
Example 1.2, 11
Example 1.3, 11
Example 1.4, 28
Example 1.5, 30
Example 1.6, 31
Example 2.1, 46
Example 2.2, 47
Example 2.3, 54
Example 2.4, 75
Example 2.5, 77
Example 2.6, 79
Example 3.1, 108
Example 3.2, 113
Example 3.3, 116
Example 3.4, 121
Example 3.5, 130
Example 3.6, 136
Example 3.7, 139
Example 3.8, 151
Example 3.9, 153
Example 3.10, 157
Example 3.11, 159
Example 3.12, 167
Example 3.13, 168
Example 3.14, 171
Example 4.1, 179
Example 4.2, 185
Example 4.3, 188
Example 4.4, 191
Example 5.1, 199
Example 5.2, 199
Example 5.3, 207
Example 5.4, 209
Example 6.1, 219
Example 6.2, 226
Example 6.3, 236
Example 6.4, 240
Example 6.5, 243
Example 7.1, 257
Example 7.2, 259
Example 7.3, 263
Example 7.4, 267
Example 7.5, 269
Example 7.6, 271
Example 7.7, 272
Example 7.8, 276
Example 8.1, 288
Example 8.2, 292
Example 8.3, 296
Example 8.4, 311
Example 8.5, 317
Example 9.1, 341
Example 10.1, 364
Example 10.2, 368
Example 10.3, 373  

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