Zienkiewicz O.C., Taylor R.L. The Finite Element Method for Solid and Structural Mechanics

6th ed. — Elsevier, 2005. — 631 p.General problems in solid mechanics and non-linearitySmall deformation solid mechanics problems
Variational forms for non-linear elasticity
Weak forms of governing equations
Concluding remarksGalerkin method of approximation – irreducible and mixed formsFinite 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 remarksSolution of non-linear algebraic equationsIterative techniques
General remarks – incremental and rate methodsInelastic and non-linear materialsViscoelasticity – history dependence of deformation
Classical time-independent plasticity theory
Computation of stress increments
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 remarksGeometrically non-linear problems – finite deformationGoverning 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 remarksMaterial constitution for finite deformationIsotropic elasticity
Isotropic viscoelasticity
Plasticity models
Incremental formulations
Rate constitutive models
Numerical examples
Concluding remarksTreatment of constraints – contact and tied interfacesNode–node contact: Hertzian contact
Tied interfaces
Node–surface contact
Surface–surface contact
Numerical examples
Concluding remarksPseudo-rigid and rigid–flexible bodiesPseudo-rigid motions
Rigid motions
Connecting a rigid body to a flexible body
Multibody coupling by joints
Numerical examplesDiscrete element methodsEarly 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 remarksStructural mechanics problems in one dimension – rodsGoverning 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 remarksPlate bending approximation: thin (Kirchhoff) plates and C1 continuity requirementsThe plate problem: thick and thin formulations
Rectangular element with corner nodes (12 degrees of freedom)
Quadrilateral and parallelogram elements
Triangular element with corner nodes (9 degrees of freedom)
Triangular element of the simplest form (6 degrees of freedom)
The patch test – an analytical requirement
Numerical examples
General remarks
Singular shape functions for the simple triangular element
An 18 degree-of-freedom triangular element with conforming shape functions
Compatible quadrilateral elements
Quasi-conforming elements
Hermitian rectangle shape function
The 21 and 18 degree-of-freedom triangle
Mixed formulations – general remarks Hybrid plate elements
Discrete Kirchhoff constraints
Rotation-free elements
Inelastic material behaviour
Concluding remarks – which elements?Thick’ Reissner–Mindlin plates – irreducible and mixed formulationsThe 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 refinementShells as an assembly of flat elementsStiffness of a plane element in local coordinates
Transformation to global coordinates and assembly of elements
Local direction cosines
‘Drilling’ rotational stiffness – 6 degree-of-freedom assembly
Elements with mid-side slope connections only
Choice of element
Practical examplesCurved rods and axisymmetric shellsStraight element
Curved elements
Independent slope–displacement interpolation with penalty functions (thick or thin shell formulations)Shells as a special case of three-dimensional analysis – Reissner–Mindlin assumptionsShell element with displacement and rotation parameters
Special case of axisymmetric, curved, thick shells
Special case of thick plates
Convergence
Inelastic behaviour
Some shell examples
Concluding remarksSemi-analytical finite element processes – use of orthogonal functions and ‘finite strip’ methodsPrismatic bar
Thin membrane box structures
Plates and boxes with flexure
Axisymmetric solids with non-symmetrical load
Axisymmetric shells with non-symmetrical load
Concluding remarksNon-linear structural problems – large displacement and instabilityLarge 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 remarksMultiscale modellingAsymptotic 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
Numerically developed global self-consistent elastic–plastic constitutive law 5
Global solution and stress-recovery procedure
Concluding remarksComputer procedures for finite element analysisSolution of non-linear problems
Eigensolutions
Restart option
Concluding remarksAppendix A Isoparametric finite element approximations
Appendix B Invariants of second-order tensors
Author index
Subject index