Geometrically Non-linear Analysis of Layered Composite Plates and Shells

Geometrically Non-linear Analysis of Layered Composite Plates and Shells
Ireneusz Kerja
Contents
List of Symbol and Abbreviations 9
1. Introduction 13
1.1. Plates and shells – from Nature to space industry . 13
1.2. Laminated composites and sandwich panels . 13
1.3. Scope and objectives of this report 16
2. LITERATURE REVIEW AND MODELING CONSIDERATIONS . 18
2.1. Development of theoretical models for plates and shells 18
2.2. Geometrically non-linear analysis of plates and shells 22
2.3. Theoretical models for layered thin-walled structures 25
2.4. Numerical implementation of plate and shell theories 35
2.5. Modeling considerations resulted from the presented review 51
3. INCREMENTAL FORMULATION OF NONLINEAR SHELL ANALYSIS 52
3.1. Incremental formulation 52
3.2. Shell geometry 53
3.3. Shell deformation . 57
3.4. Strain-displacement relations 59
3.5. Virtual work principle . 72
3.6. Constitutive relations . 77
4. FINITE ELEMENT METHOD IMPLEMENTATION . 82
4.1. Finite element discretization of the problem . 82
4.2. Incremental equilibrium equations of FE model 84
4.3. Solving of incremental equilibrium equations . 86
4.4. Finite elements used in this study 90
4.5. Computer implementation of the proposed FEA algorithm 928 Contents
4.6. Assessment of selected elements in linear analysis . 93
5. LARGE DEFORMATION ANALYSIS EXAMPLES 103
5.1. Instability of clamped-hinged circular arches . 103
5.2. Clamped laminated shallow arch under point load 108
5.3. Hinged laminated cylindrical panels under point load 110
5.4. Glass-epoxy cylinder under internal pressure . 116
5.5. Asymmetric cross-ply simply supported plate strip 117
5.6. Clamped laminated cylindrical panels under point load . 118
5.7. Stretching of an open cylinder . 125
5.8. Pinched hemispherical shell with 18° hole 129
5.9. Axial compression of composite cylindrical panel 133
5.10. Buckling of composite cylindrical panels with square cut-outs . 139
6. CONCLUSIONS AND FUTURE PERSPECTIVES 148
6.1. Concluding Remarks . 148
6.2. Original Contribution . 150
6.3. Recommendations and Future Perspectives 150
7. REFERENCES 151
SUMMARY IN ENGLISH . 177
SUMMARY IN POLISH 178LIST OF SYMBOLS AND ABBREVIATIONS
Symbols
0
aαβ − covariant components of the surface metric tensor in the middle surface 0Ω
0
a
αβ
− contravariant components of the surface metric tensor in the middle surface 0Ω
0
a
α − covariant base vector of the middle surface 0Ω
0bαβ − components of the surface metric tensor of the second order
0
cαβ − components of the surface metric tensor of the third order
0
C – initial configuration, at time 0
1
C – actual configuration, at time t
2
C – searched configuration, at time t+∆t
[C] − constitutive matrix in the 3D constitutive relation
0
gαβ − components of the metric tensor in the shell space
m
d – director, local position vector md = mg3 = ma3
0dV − volume element in the initial configuration 0C
0dΩ − midsurface area element in the initial configuration 0C
m0 E – Green-Lagrange strain tensor
m
E
αβ − components of the Green-Lagrange strain tensor in the configurationm C
E – Young modulus for the isotropic material
E
a – Young modulus in the direction of the material a-axis for the orthotropic material
mF
0
1F
– deformation gradient
0 S – components of the balanced forces vector
2f i − components of external body forces (acting per unit volume element)
0
gi − covariant base vectors in the space of the shell
0
G − metric tensor at any arbitrary point 0P in the shell space
Gab – “in-plane” shear modulus for the orthotropic material
Gbc, Gac – transverse shear moduli for the orthotropic material
h − shell thickness
[H] − constitutive matrix in the 2D constitutive relation
01K (U ) ST − components of the first part of the incremental stiffness matrix
ST
1K (G )
0 − components of the geometrical stiffness matrix
02Lmn − components of the effective stress resultants in the configuration mC
0
n − base vector normal to the initial middle surface 0Ω
Nk − isoparametric shape function associated with node k
NNE − number of nodes of the isoparametric element
2
p
i
− components of external surface forces
m
r – position of an arbitrary point mP of the middle surface in the configuration mC
m
r
k
– position vector of the node k at the shell mid-surface in the configuration mC
m
R – position vector of an arbitrary point P at the configuration mC
[R] − rotation matrix
r, s – natural coordinates for the isoparametric element
0
2Smn − components of the second Piola-Kirchhoff stress tensor in the configuration mC10 List of symbols and abbreviations
[T] − transformation matrix between the material axes (a, b, c) and the coordinate
system (θ1, θ2, θ3)
m
V – displacement vector in configuration mC
m
Vi – components of the displacement vector referred to the undeformed shell space
m
υi – components of the displacement vector referred to the undeformed shell mid-surface
αk – ply orientation angle
ε − prescribed tolerance in the convergence criterion
Гmi − Christofel symbol of the second kind (components of the base vector gi derivative)
δ
α
β
− Kronecker delta
δ2eij − variation of components of the infinitesimal strain tensor in the configuration 2 C
δui − covariant components of the virtual displacement vector
2δW
e − external virtual work in the configuration 2C
2δWi − internal virtual work in the configuration 2C
{ } ∆q − vector of displacement increments
m
ϕα − rotation angle, (α = 1, 2)
m
ϕαβ – components of the displacement gradient
κ − transverse shear correction coefficient
0
 −shifter tensor in the initial configuration 0C
βα
0 µ − components of the shifter tensor 0 
0
µ − determinant of the shifter tensor
λ – load parameter
ν – Poisson coefficient
νab – “in-plane” Poisson coefficient for the orthotropic material
νbc, νac – transverse Poisson coefficients for the orthotropic material
2
σij − components of the Cauchy stress tensor in the configuration 2 C
θα − general convected coordinates in the middle surface, (α = 1, 2)
θ3 − thickness coordinate taking values from the interval (-h/2, h/2)
m
« − Rodrigues rotation vector
m
Ω − middle surface of the shell in the configuration mC
[mℜ( m«& − rotation tensor
Abbreviations
2D – two dimensional
3D – three dimensional
ANS – Assumed Natural Strain ( . approach)
CLT – Classical Lamination Theory
DKT – Discrete Kirchhoff Theory
DL – Discrete-Layer ( . theory)
dof(s) – degree(s) of freedom
EAS – Enhanced Assumed Strain ( . method)
ESL – Equivalent Single Layer ( . model)
FE – Finite Element
FEA – Finite Element Analysis
FEM – Finite Element Method
FOSD – First Order Shear Deformation ( . theory)
FRC – Fiber Reinforced Composite
HOSD – Higher Order Shear Deformation ( . theory)List of symbols and abbreviations 11
LRT – Large Rotation Theory
LRT5 – LRT with 5 displacement parameters (inextensible director)
LRT56 – LRT with 5 aggregate and 6 incremental displacement parameters (inextensible
director)
LRT6 – LRT with 6 displacement parameters (extensible director)
MITC – Mixed Interpolation of Tensorial Components
MRT – Moderate Rotation Theory
MRT5 – MRT with 5 displacement parameters (inextensible director)
RT5 – TOSD model of shells with 5 displacement parameters (Başar et al. [37])
RT7 – TOSD model of shells with 7 displacement parameters (Başar et al. [37])
RVK – Refined von Kármán ( . theory)
RVK5 – RVK with 5 displacement parameters (inextensible director)
SLR – Simplified Large Rotation (by Dennis & Palazotto [138, 139])
SOSD – Second Order Shear Deformation ( . theory)
SRI – Selective Reduced Integration
TL – Total Lagrangian ( . formulation)
TOSD – Third Order Shear Deformation ( . theory)
UL – Updated Lagrangian ( . formulation)
URI – Uniformly Reduced Integration
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