Plates, Laminates and Shells

The following sections are included:

• Preface

• Contents

The following sections are included:

• Introduction

• Function spaces, convex analysis, variational convergence
  • Function spaces: L^p and Sobolev spaces
    • Lebesgue spaces L^p
    • Sobolev spaces and trace operators
  • Elements of convex analysis and duality, minimization theorems, multivalued mappings
    • Convex sets and functions
    • Minimization theorems
    • Normal integrands, integral functionals and Rockafellar's theorem
    • Quasiconvexity and A-quasiconvexity
    • Elements of the duality theory
    • Set-valued maps
  • Variational convergence of sequences of operators and functionals
    • G-convergence
    • H-convergence and the energy method
    • Two-scale convergence
    • Γ-convergence
    • Γ-convergence of sequence of nonconvex functionals convex in highest-order derivatives: non-uniform homogenization
    • Γ-convergence and duality
    • Convergence of sets in Kuratowski's sense
  • Two approximation results
  • An augmented Lagrangian method for problems with unilateral constraints

The following sections are included:

• Introduction

• Three-dimensional analysis and effective models of composite plates
  • Equilibrium problem of a periodic plate
  • Family of problems (P_ɛ)
  • Asymptotic analysis. Effective moduli and local problems
  • Case of transverse symmetry
  • Centrosymmetry of the periodicity cell
  • On computing effective stiffnesses
  • Case of moderately thick periodicity cells
  • Case of thin periodicity cells. Derivation by imposing Kirchhoff's constraints
  • Case of transversely slender periodicity cells of constant thickness
  • Γ-convergence and justification of three models of thin, transversely inhomogeneous and anisotropic plates with constant thickness
    • Basic relations
    • Justification of the effective plate model of Sec. 2.8 by passing to zero: e → 0 and then ɛ → 0
    • Justification of the effective plate model of Section 2.9 by passing to zero: ɛ → 0 and next e → 0
    • Justification of the effective plate model of Section 2.3 by passing to zero: e → 0 and ɛ → 0 simultaneously
  • Effective stiffnesses of longitudinally homogeneous plates

• Thin plates in bending and stretching
  • Kirchhoff type description
  • Asymptotic homogenization. In-plane scaling approach
  • Refined scaling approach
  • Variational formulae for effective stiffnesses
  • Correctors
  • Variational formulae for effective compliances. Dual effective potential
  • Transversely symmetric plates periodic in one direction
  • Ribbed plates. Bending problem
    • Formula of Francfort and Murat for stiffnesses
    • Ribbed plates of higher rank with the stronger phase taken as an envelope
    • Formula of Lurie-Cherkaev-Fedorov for stiffnesses
    • Formula of Francfort-Murat-type for compliances
  • Ribbed plates. Plane elasticity problem
    • Formula of Francfort and Murat for stiffnesses
    • Formula of Francfort and Murat-type for compliances
  • Plates periodic with respect to a curvilinear parametrization. Non-uniform homogenization
  • Effective bending stiffnesses of plates with quadratic inclusions
  • Perforated plates
  • Plates stiffened with rigid inclusions

• Nonlinear behavior of plates
  • Von Kármán equations
  • Homogenization
  • Bifurcation and homogenization of perforated von Kármán plates
    • Homogenization of perforated von Kármán plates
    • Bifurcation of von Kármán plates: basic results
    • Bifurcation points of the homogenized plate and the linearized problem
    • Bifurcating branches of perforated and homogenized plates

• Moderately thick transversely symmetric plates
  • Reissner-Hencky model
  • The in-plane scaling-based asymptotic homogenization
  • The refined scaling analysis
  • Justification of the refined scaling approach
    • Basic relations and auxiliary results
    • Γ-convergence of the sequence {J_ɛ − L_ɛ}_ɛ>0
  • Dual homogenization
  • Orthotropic plates periodic in one direction
    • Effective stiffnesses according to the in-plane scaling approach
    • Effective stiffnesses according to the refined scaling approach
    • Effective torsional stiffness of plates of step-wise varying thickness
    • Formula of Tartar-Francfort-Murat type for effective stiffnesses of ribbed plates. In-plane scaling approach
  • Other linear and nonlinear models of plates with moderate thickness. Homogenization study
    • Reissner's model
    • A refined theory of moderately thick plates undergoing moderately large deflections
    • Homogenization study

• Sandwich plates with soft core
  • Hoff's theory
  • Effective stiffnesses in the periodic case
  • Reissner's approximation and relevant homogenization formulae

• Comments and bibliographical notes

The following sections are included:

• Introduction

• Unilateral cracks in thin plates
  • Cracking modes
  • Periodic layout of cracks. Homogenization
  • Justification: Γ-convergence

• Unilateral cracks in plates with transverse shear deformation
  • Admissible cracking modes
  • Periodic layout of cracks. Homogenization
  • Justification: variational inequality (9.2.1) and the energy method

• Part-through the thickness cracks
  • Two-layer description
  • In-plane scaling and effective model
  • The study of convergence
  • Dual homogenization
  • Passage to classical models of cracked plates
  • Refined scaling and effective model
  • Plates with aligned cracks
  • Cracks of arbitrary position. Three-dimensional local analysis
    • Asymptotic analysis
    • Justification by Γ-convergence

• Stiffness loss of cracked laminates
  • Two-dimensional model of transversely symmetric laminates in stretching and in-plane shearing
  • Modelling the unilateral crack within the internal layer
  • Regular crack systems
  • Moderately thick laminate weakened by transverse cracks of high density. Model (h,l₀)
  • Thin laminate with transverse cracks of high density. Model (h₀,l₀)
  • Thin laminate weakened by transverse cracks of arbitrary density. Model (h₀,l)
  • Justification by Γ-convergence and duality
    • Moderately thick laminate
    • Refined scaling and Γ-convergence
  • Application of the augmented Lagrangian method to solving local problems with unilateral constraints
  • Case of aligned parallel intralaminar cracks. Effective characteristics according to the (h₀, l) approach
  • Degradation of effective stiffnesses of laminates with aligned parallel cracks. The refined scaling approach – model (h₀,l)
  • Degradation of effective stiffnesses of laminates [0°_n/90°_m]_s. Comparison with experimental results and with other analytical predictions
    • Scope of the section
    • [0°/90°₃]_s glass/epoxy laminate tested by Highsmith and Reifsnider (1982)
    • [0°/90°]_s glass/epoxy laminate tested by Ogin et al. (1985)
    • [0°_m/90°_n]_s graphite/epoxy laminates tested by Groves (1986)
    • [0°/90°]_s glass/epoxy laminates tested by Smith and Wood (1990)
    • [0°/90°]_s carbon/epoxy laminate tested by Smith and Wood (1990)
  • Stress distribution around crack tips
  • Crack spacing as a function of the averaged applied stress

• Comments and bibliographical notes

The following sections are included:

• Introduction

• Mathematical complements, homogenization of functionals with linear growth
  • Functional setting: spaces W^1,1, W^2,1, LD, BV, BD, and HB
  • Convex functions and functionals of a measure
  • General homogenization theorems for functional with linear growth
  • Γ-convergence and Dirichlet boundary conditions, relaxation
    • Thin Kirchhoff plates made of Hencky material
    • Moderately thick plates, refined scaling
    • On von Kármán plates made of Hencky material

• Homogenization of plates loaded by forces and moments
  • Thin Kirchhoff plates loaded by boundary forces and moments
  • Three models of thin, transversely inhomogeneous and anisotropic plates with constant thickness
    • Basic relations
    • Derivation of the effective plate model by passing to zero: e → 0 and next ɛ → 0
    • Derivation of the second effective plate model: ɛ → 0 and next e → 0
    • Derivation of the third effective plate model: e → 0 and ɛ → 0 simultaneously

• Comments and bibliographical notes

The following sections are included:

• Introduction

• Linear and nonlinear models of elastic shells
  • Theory of shells with transverse shear deformation
  • Koiter's version of thin shell model
  • Budiansky-Sanders-Koiter version of a thin shell model
  • Thin shell model with moderately large rotations around tangents
  • The models of Mushtari-Donnell-Vlasov and Mushtari-Marguerre

• Homogenization of stiffnesses of thin periodic elastic shells. Linear approach
  • Koiter's shell. Asymptotic analysis and the convergence theorem
    • Asymptotic analysis
    • Justification by the Γ-convergence method
  • Dual homogenization
  • Effective stiffnesses of ribbed orthotropic cylindrical shells
    • Geometrical and material characteristics of a cylindrical shell
    • Strong formulations of the local problems
    • Case of stiffeners along the generating lines
    • Case of circumferential stiffening
  • Shallow shells of periodic structure: effective properties

• Homogenized properties of thin periodic elastic shells undergoing moderately large rotations around tangents

• Perfectly plastic shells

• Comments and bibliographical notes

The following sections are included:

• Introduction

• Mathematical complements
  • Alternative representation of second and fourth order tensors
  • Y-transformation
  • Fourier representation of Y-periodic functions
  • Examples of quasiconvex and quasiaffine functions
    • A quasiaffine function of the strain tensor κ
    • A quasiaffine function of two strain tensors
    • An aggregate form of the previous results
    • A quasiaffine function of the stress tensor m
    • A quasiaffine function of the stress tensor n
    • A quasiaffine function of two stress tensors
    • An aggregate form of the two previous results
  • Harmonic mean as a lower bound for effective energy
  • Elements of the theory of Young measures

• Two-phase plate in bending. Hashin-Shtrikman bounds
  • Lower bound for the Kelvin modulus
  • Upper bound for the Kelvin modulus
  • Lower bound for the Kirchhoff modulus
  • Upper bound for the Kirchhoff modulus
  • Attainability of Hashin-Shtrikman bounds. The Francfort – Murat construction

• Two-phase plate. Hashin-Shtrikman bounds for the in-plane problem
  • The upper and lower bounds for the Kelvin modulus
  • The upper and lower bounds for the Kirchhoff modulus
  • Attainability of Hashin-Shtrikman bounds
  • Summary of the main results

• Explicit formulae for effective bending stiffnesses and compliances of ribbed plates
  • First rank ribbed structure
  • Second rank ribbed structure with soft phase taken as a core
  • Second rank ribbed structure with the strong phase taken as a core

• Explicit formulae for effective membrane stiffnesses and compliances of ribbed plates
  • First rank ribbed plates
  • Second rank ribbed plates

• Thin bending two-phase plates of minimum compliance
  • Ill-posedness of the initial formulation
  • Relaxation
  • Bounding the potential W* by the translation method of Cherkaev-Gibianskii
  • Attainability of the translation bound
    • Regime (2): ζ_M ∊ [ζ₂, ζ₁]
    • Regime (3): ζ_M ≤ ζ₂
    • Regime (1): ζ_M ≥ ζ₁
  • Physical interpretation of the relaxed problem
  • Primal formulation of the relaxed problem
  • On the shape design
  • Square clamped plates of minimum compliance
  • Optimal perforated plates of small volume

• Minimum compliance problem for thin plates of varying thickness: application of Young measures

• Thin shells of minimum compliance
  • Setting of the problem
  • Relaxation
  • Primal formulation of the relaxed problem
  • On the in-plane minimum compliance problem of two-phase plates

• Truss-like Michell continua
  • Structures of minimum weight. Discrete versus continuum formulations
  • Dual formulation
  • A symmetric cantilever problem

• Comments and bibliographical notes

The following sections are included:

• References

• Index

• Series on Advances in Mathematics for Applied Sciences