كتاب Non-linear Finite Element Analysis of Solids and Structures Volume 1

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Non-linear Finite Element Analysis of Solids and Structures
VOLUME 1: ESSENTIALS
M. A. Crisfield
FEA Professor of Computational Mechanics
Department of Aeronautics
Imperial College of Science, Technology and Medicine
London, UK

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Contents
Preface
Notation
1 General introduction, brief history and introduction to geometric
non-linearity
1 1 General introduction and a brief history
1 1 1 A brief history
1 2 A simple example for geometric non-linearity with one degree of freedom
1 2 1 An incremental solution
1 2 2 An iterative solution (the Newton-Raphson method)
1 2 3 Combined tncremental/iterative solutions (full or modified Newton-Raphson or
the initial-stress method)
1 3 A simple example with two variables
13 1 ‘Exact solutions
1 3 3 An energy basis
List of books on (or related to) non-linear finite elements
References to early work on non-linear finite elements
The use of virtual work
1 4 Special notation
2 A shallow truss element with Fortran computer program
2 1 A shallow truss element
2 2 A set of Fortran subroutines
2 2 1 Subroutine ELEMENT
2 2 2 Subroutine INPUT
2 2 3 Subroutine FORCE
2 2 4 Subroutine ELSTRUC
2 2 5
2 2 6 Subroutine CROUT
2 2 7 Subroutine SOLVCR
2 3 A flowchart and computer program for an incremental (Euler) solution
2 3 1 Program NONLTA
2 4 A flowchart and computer program for an iterative solution using the
Newton-Raphson method
2 4 1 Program NONLTB
2 4 2
A flowchart and computer program for an incrementaViterative solution
procedure using full or modified Newton-Raphson iterations
2 5 1 Program NONLTC
Subroutine BCON and details on displacement control
Flowchart and computer listing for subroutine ITER
Vvi CONTENTS
2 6 Problems for analysis
Single variable with spring
2 6 1 1 Incremental solution using program NONLTA
2 6 1 2 Iterative solution using program NONLTB
2 6 1 3 Incremental/iterative solution using program NONLTC
Perfect buckling with two variables
2 6 4 1 Pure incremental solution using program NONLTA
2 6 4 2 An incremental/\terative solution using program NONLTC with small
increments
2 6 4 3 An incremental/iterative solution using program NONLTC with large
increments
2 6 4 4 An incremental/iterative solution using program NONLTC with displacement
control
2 6 2 Single variable no spring
2 6 4 Imperfect 'buckling with two variades
2 7 Special notation
2 8 References
3 Truss elements and solutions for different strain measures
3.1 A simple example with one degree of freedom
3.1.1 A rotated engineering strain
3.1.2 Green's strain
3.1.3 A rotated log-strain
3.1.5 Comparing the solutions
3.2 Solutions for a bar under uniaxial tension or compression
3.2.1 Almansi's strain
3.3 A truss element based on Green's strain
3.3.3 The tangent stiffness matrix
3.3.4 Using shape functions
3.3.5 Alternative expressions involving updated coordinates
3.3.6 An updated Lagrangian formulation
3.4 An alternative formulation using a rotated engineering strain
3.5 An alternative formulation using a rotated log-strain
3.6 An alternative corotational formulation using engineering strain
3.7 Space truss elements
3.8 Mid-point incremental strain updates
A rotated log-strain formulation allowing for volume change
Geometry and the strain-displacement relationships
Equilibrium and the internal force vector
Fortran subroutines for general truss elements
3.9.1 Subroutine ELEMENT
3.9.2 Subroutine INPUT
3.9.3 Subroutine FORCE
3.10 Problems for analysis
3.10.1 Bar under uniaxial load (large strain)
3.10.2 Rotating bar
3.10.2.1 Deep truss (large-strains) (Example 2.1)
3.10.2.2 Shallow truss (small-strains) (Example 2.2)
3.10.3 Hardening problem with one variable (Example 3)
3.10.4 Bifurcation problem (Example 4)
3.10.5 Limit point with two variables (Example 5)
3.10.6 Hardening solution with two variables (Example 6)
3.10.7 Snap-back (Example 7)
98CONTENTS vii
3.11 Special notation
3.12 References
4 Basic continuum mechanics
Stress and strain
Stress-st rain relationships
4 2 1 Plane strain axial symmetry and plane stress
4 2 2 Decomposition into vo,umetric and deviatoric components
4 2 3 An alternative expression using the Lame constants
Transformations and rotations
4 3 1 Transformations to a new set of axes
4 3 2 A rigid-body rotation
Green’s strain
4 4 1 Virtual work expressions using Green s strain
4 4 2 Work expressions using von Karman s non-linear strain-displacement
relqtionships for a plate
Almansi’s strain
The true or Cauchy stress
Summarising the different stress and strain measures
The polar-decomposition theorem
4 8 1 Ari example
Green and Almansi strains in terms of the principal stretches
4.10 A simple description of the second Piola-Kirchhoff stress
4.11 Corotational stresses and strains
4.12 More on constitutive laws
4.13 Special notation
4.14 References
5 Basic finite element analysis of continua
5 1 Introduction and the total Lagrangian formulation
5 1 1 Element formulation
5 1 2 The tangent stiffness matrix
5 1 3 Extension to three dimensions
5 1 4 An axisymmetric membrane
5 2 Implenientation of the total Lagrangian method
5 2 1 With dn elasto-plastic or hypoelastic material
5 3 The updated Lagrangian formulation
5 4 Implementation of the updated Lagrangian method
Incremental formulation involving updating after convergence
A total formulation for an elastic response
An approximate incremental formulation
5 5 Special notation
5 6 References
6 Basic plasticity
6 1 Introduction
6 2 Stress updating incremental or iterative strains?
6 3 The standard elasto-plastic modular matrix for an elastic/perfectly plastic
von Mises material under plane stress
6 3 1 Non-associative plasticity
6 4 Introducing hardening
159viii CONTENTS
6 4 3 Kinematic hardening
Von Mises plasticity in three dimensions
6 5 1 Splitting the update into volumetric and deviatoric parts
6 5 2 Using tensor notation
6 6 Integrating the rate equations
6 6 1 Crossing the yield surface
6 6 2 Two alternative predictors
6 6 3 Returning to the yield surface
6 6 4 Sub-incrementation
6 6 5 Generalised trapezoidal or mid-point algorithms
6 6 6 A backward-Euler return
6 6 7 The radial return algorithm a special form of backward-Euler procedure
The consistent tangent modular matrix
6 7 1 Splitting the deviatoric from the volumetric components
6 7 2 A combined formulation
6 8 Special two-dimensional situations
6 8 1 Plane strain and axial symmetry
6 8 2 Plane stress
6 8 2 1 A consistent tangent modular matrix for plane stress
Isotropic strain hardening
Isotropic work hardening
6 9 Numerical examples
6 9 1 Intersection point
6 9 2
6 9 3 Sub-increments
6 9 4
6 9 5 Backward-Euler return
A forward-Euler integration
Correction or return to the yield surface
6 9 5 1 General method
6 9 5 2 Specific plane-stress method
6 9 6 Consistent and inconsistent tangents
6 9 6 1 Solution using the general method
6 9 6 2 Solution using the specific plane-stress method
6 10 Plasticity and mathematical programming
6 10 1
6 1 1 Special notation
6 12 References
A backward-Euler or implicit formulation
7 Two-dimensional formulations for beams and rods
7 1 A shallow-arch formulation
A simple corotational element using Kirchhoff theory
7 2 1 Stretching 'stresses and 'strains
7 2 8 Some observations
7 3 A simple corotational element using Timoshenko beam theory
74 An alternative element using Reissner's beam theory
The tangent stiffness matrix
Introduction of material non-linearity or eccentricity
Numerical integration and specific shape functions
Introducing shear deformation
Specific shape fur,ctions, order of integration and shear-locking
Bending 'stresses' and 'strains
The virtual local displacements
The virtual work
The tangent stiffness matrix
llsing shape functions
Including higher-order axial terms
221CONTENTS ix
7.5 An isoparametric degenerate-continuum approach using the total Lagrangian
formulation
7.6 Special notation
7.7 References
The introduction of shape functions and extension to a general
isoparametric element
8 Shells
8 1 A range of shallow shells
8 1 1 Strain-displacement relationships
8 1 2 Stress-strain relatiomhips
8 1 3 Shape functions
8 1 4 Virtual work and the internal force vector
8 1 5 The tangent stiffness matrix
8 1 6 Numerical integration matching shape functions and 'locking
8 1 7 Extensions to the shallow-shell formulation
8 2 A degenerate-continuum element using a total Lagrangian formulation
8 2 1 The tangent stiffness matrix
8 3 Special notation
8 4 References
9 More advanced solution procedures
9 2 Line searches
The total potential energy
9 2 1 Theory
Flowchart and Fortran subroutine to find the new step length
9 2 2 1 Fortran subroutine SEARCH
Implementation within a finite element computer program
9 2 3 1 Input
9 2 3 2 Changes to the iterative subroutine ITER
9 2 3 3 Flowchart for Iine-search loop at the structural level
The need for arc-length or similar techniques and examples of their use
Various forms of generalised displacement control
93 2 1 The 'spherical arc-length method
9 3 2 2 Linearised arc-length methods
9 3 2 3 Generalised displacement control at a specific variable
Flowchart and Fortran subroutines for the application of the arc-length constraint
9 4 1 1 Fortran subroutines ARCLl and QSOLV
Flowchart and Fortran subroutine for the main structural iterative loop (ITER)
9 4 2 1 Fortran subroutine ITER
9 3 The arc-length and related methods
9 4 Detailed formulation for ttre 'cylindrical arc-length' method
9 4 3 The predictor solution
Automatic increments, non-proportional loading and convegence criteria
9 5 3 Non-proportional loading
9 5 4 Convergence criteria
Automatic increment cutting
The current stiffness parameter and automatic switching to the arc-length method
Restart facilities and the computation of the lowest eigenmode of K,
rcjrtran subroutine LSLOOP
Input for incremental/iterative control
9 6 2 1 Subroutine INPUT2
flowchart and Fortran subroutine for the main program module NONLTD
9 6 3 1 Fortran for main program module NONLTD
9 6 The updated computer prcgram
299X CONTENTS
Flowchart and Fortran subroutine. for routine SCALUP
9 6 4 1 Fortran for routine SCALUP
Flowchart and Fortran for subroutine NEXINC
9 6 5 1 Fortran for subroutine NEXINC
9 7 Quasi-Newton methods
9 8 Secant-related acceleration tecriniques
9 8 1 Cut-outS
9 8 2 Flowchart and Fortran for subroutine ACCEL
9 8 2 1 Fortran for subroutine ACCEL
9 9 Problems for analysts
9 9 1 The problems
Small-strain limit-point cxample with one variable (Example 2 2)
Hardening problem with one variable (Example 3)
Bifurcation problem (Example 4)
Limit point with two variables (Example 5)
Hardening solution with two variable (Example 6)
Snap-back (Example 7)
9 10 Further work on solution procedures
9 11 Special notation
9 12 References
Appendix Lobatto rules for numerical integration
Subject index
Author index

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