كتاب The Finite Element Method for Solid and Structural Mechanics

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The Finite Element Method for Solid and Structural Mechanics
Zienkiewicz & Taylor

ويتناول الموضوعات الأتية :

General problems in solid mechanics and non-linearity
Introduction
Small deformation solid mechanics problems
Variational forms for non-linear elasticity
Weak forms of governing equations
Concluding remarks
References
Galerkin method of approximation – irreducible and mixed forms
Introduction
Finite element approximation – Galerkin method
Numerical integration – quadrature
Non-linear transient and steady-state problems
Boundary conditions: non-linear problems
Mixed or irreducible forms
Non-linear quasi-harmonic field problems
Typical examples of transient non-linear calculations
Concluding remarks
References
Solution of non-linear algebraic equations
Introduction
Iterative techniques
General remarks – incremental and rate methods
References
Inelastic and non-linear materials
Introduction
Viscoelasticity – history dependence of deformation
Classical time-independent plasticity theory
Computation of stress increments
viii Contents
Isotropic plasticity models
Generalized plasticity
Some examples of plastic computation
Basic formulation of creep problems
Viscoplasticity – a generalization
Some special problems of brittle materials
Non-uniqueness and localization in elasto-plastic deformations
Non-linear quasi-harmonic field problems
Concluding remarks
References
Geometrically non-linear problems – finite deformation
Introduction
Governing equations
Variational description for finite deformation
Two-dimensional forms
A three-field, mixed finite deformation formulation
A mixed–enhanced finite deformation formulation
Forces dependent on deformation – pressure loads
Concluding remarks
References
Material constitution for finite deformation
Introduction
Isotropic elasticity
Isotropic viscoelasticity
Plasticity models
Incremental formulations
Rate constitutive models
Numerical examples
Concluding remarks
References
Treatment of constraints – contact and tied interfaces
Introduction
Node–node contact: Hertzian contact
Tied interfaces
Node–surface contact
Surface–surface contact
Numerical examples
Concluding remarks
References
Pseudo-rigid and rigid–flexible bodies
Introduction
Pseudo-rigid motions
Rigid motions
Contents ix
Connecting a rigid body to a flexible body
Multibody coupling by joints
Numerical examples
References
Discrete element methods
Introduction
Early DEM formulations
Contact detection
Contact constraints and boundary conditions
Block deformability
Time integration for discrete element methods
Associated discontinuous modelling methodologies
Unifying aspects of discrete element methods
Concluding remarks
References
Structural mechanics problems in one dimension – rods
Introduction
Governing equations
Weak (Galerkin) forms for rods
Finite element solution: Euler–Bernoulli rods
Finite element solution: Timoshenko rods
Forms without rotation parameters
Moment resisting frames
Concluding remarks
References
Plate bending approximation: thin (Kirchhoff) plates and C continuity
requirements
Introduction
The plate problem: thick and thin formulations
Rectangular element with corner nodes ( degrees of freedom)
Quadrilateral and parallelogram elements
Triangular element with corner nodes ( degrees of freedom)
Triangular element of the simplest form ( degrees of freedom)
The patch test – an analytical requirement
Numerical examples
General remarks
Singular shape functions for the simple triangular element
An degree-of-freedom triangular element with conforming
shape functions
Compatible quadrilateral elements
Quasi-conforming elements
Hermitian rectangle shape function
The and degree-of-freedom triangle
Mixed formulations – general remarks
x Contents
Hybrid plate elements
Discrete Kirchhoff constraints
Rotation-free elements
Inelastic material behaviour
Concluding remarks – which elements?
References
‘Thick’ Reissner–Mindlin plates – irreducible and mixed formulations
Introduction
The irreducible formulation – reduced integration
Mixed formulation for thick plates
The patch test for plate bending elements
Elements with discrete collocation constraints
Elements with rotational bubble or enhanced modes
Linked interpolation – an improvement of accuracy
Discrete ‘exact’ thin plate limit
Performance of various ‘thick’ plate elements – limitations of thin
plate theory
Inelastic material behaviour
Concluding remarks – adaptive refinement
References
Shells as an assembly of flat elements
Introduction
Stiffness of a plane element in local coordinates
Transformation to global coordinates and assembly of elements
Local direction cosines
‘Drilling’ rotational stiffness – degree-of-freedom assembly
Elements with mid-side slope connections only
Choice of element
Practical examples
References
Curved rods and axisymmetric shells
Introduction
Straight element
Curved elements
Independent slope–displacement interpolation with penalty functions
(thick or thin shell formulations)
References
Shells as a special case of three-dimensional analysis – Reissner–Mindlin
assumptions
Introduction
Shell element with displacement and rotation parameters
Special case of axisymmetric, curved, thick shells
Special case of thick plates
Contents xi
Convergence
Inelastic behaviour
Some shell examples
Concluding remarks
References
Semi-analytical finite element processes – use of orthogonal functions
and ‘finite strip’ methods
Introduction
Prismatic bar
Thin membrane box structures
Plates and boxes with flexure
Axisymmetric solids with non-symmetrical load
Axisymmetric shells with non-symmetrical load
Concluding remarks
References
Non-linear structural problems – large displacement and instability
Introduction
Large displacement theory of beams
Elastic stability – energy interpretation
Large displacement theory of thick plates
Large displacement theory of thin plates
Solution of large deflection problems
Shells
Concluding remarks
References
Multiscale modelling
Introduction
Asymptotic analysis
Statement of the problem and assumptions
Formalism of the homogenization procedure
Global solution
Local approximation of the stress vector
Finite element analysis applied to the local problem
The non-linear case and bridging over several scales
Asymptotic homogenization at three levels: micro, meso
and macro
Recovery of the micro description of the variables of the problem
Material characteristics and homogenization results
Multilevel procedures which use homogenization as an ingredient
General first-order and second-order procedures
Discrete-to-continuum linkage
Local analysis of a unit cell
Homogenization procedure – definition of successive yield surfaces
xii Contents
Numerically developed global self-consistent elastic–plastic
constitutive law
Global solution and stress-recovery procedure
Concluding remarks
References
Computer procedures for finite element analysis
Introduction
Solution of non-linear problems
Eigensolutions
Restart option
Concluding remarks
References
Appendix A Isoparametric finite element approximations
Appendix B Invariants of second-order tensors
Author index
Subject index

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