April 04, 2008
M. A. Crisfield
FEA Professor of Computational Mechanics
Department of Aeronautics
Imperial College of Science, Technology and Medicine
London, UK
Contents
- General introduction, brief history and introduction to geometric non-linearity
- General introduction and a brief history
A brief history
- A simple example for geometric non-linearity with one degree of freedom
An incremental solution
An iterative solution (the Newton-Raphson method)
Combined tncremental/iterative solutions (full or modified Newton-Raphson or the initial-stress method)
A simple example with two variables
‘Exact solutions
The use of virtual work
An energy basis
- Special notation
List of books on (or related to) non-linear finite elements
References to early work on non-linear finite elements
- A shallow truss element with Fortran computer program
- A shallow truss element
- A set of Fortran subroutines
Subroutine ELEMENT
Subroutine INPUT
Subroutine FORCE
Subroutine ELSTRUC
Subroutine BCON and details on displacement control
Subroutine CROUT
Subroutine SOLVCR
- A flowchart and computer program for an incremental (Euler) solution
Program NONLTA
- A flowchart and computer program for an iterative solution using the Newton-Raphson method
Program NONLTB
Flowchart and computer listing for subroutine ITER
- A flowchart and computer program for an incrementaViterative solution procedure using full or modified Newton-Raphson iterations
Program NONLTC
- Problems for analysis
Single variable with spring
Incremental solution using program NONLTA
Iterative solution using program NONLTB
Incremental/iterative solution using program NONLTC
Single variable no spring
Perfect buckling with two variables
Imperfect ‘buckling’ with two variades
Pure incremental solution using program NONLTA
An incremental/\terative solution using program NONLTC with small increments
An incremental/iterative solution using program NONLTC with large increments
An incremental/iterative solution using program NONLTC with displacement
control
- Special notation
- References
- Truss elements and solutions for different strain measures
- A simple example with one degree of freedom
A rotated engineering strain
Green’s strain
A rotated log-strain
A rotated log-strain formulation allowing for volume change
Comparing the solutions
- Solutions for a bar under uniaxial tension or compression
Almansi’s strain
- A truss element based on Green’s strain
Geometry and the strain-displacement relationships
The tangent stiffness matrix
Using shape functions
Alternative expressions involving updated coordinates
An updated Lagrangian formulation
- An alternative formulation using a rotated engineering strain
- An alternative formulation using a rotated log-strain
- An alternative corotational formulation using engineering strain
- Space truss elements
- Mid-point incremental strain updates
- Fortran subroutines for general truss elements
Subroutine ELEMENT
Subroutine INPUT
Subroutine FORCE
- Problems for analysis
Bar under uniaxial load (large strain)
Rotating bar
Deep truss (large-strains) (Example 2.1)
Shallow truss (small-strains) (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 variables (Example 6)
Snap-back (Example 7)
- Special notation
- References
- Basic continuum mechanics
- Stress and strain
- St ress-strain relationships
Plane strain axial symmetry and plane stress
Decomposition into vo,umetric and deviatoric components
An alternative expression using the Lame constants
- Transformations and rotations
Transformations to a new set of axes
A rigid-body rotation
- Green’s strain
Virtual work expressions using Green s strain
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
Ari example
- Green and Almansi strains in terms of the principal stretches
- A simple description of the second Piola-Kirchhoff stress
- Corotational stresses and strains
- More on constitutive laws
- Special notation
- References
- Basic finite element analysis of continua
- Introduction and the total Lagrangian formulation
Element formulation
The tangent stiffness matrix
Extension to three dimensions
An axisymmetric membrane
- Implementation of the total Lagrangian method
With an elasto-plastic or hypoelastic material
- The updated Lagrangian formulation
- Implementation of the updated Lagrangian method
Incremental formulation involving updating after convergence
A total formulation for an elastic response
An approximate incremental formulation
- Special notation
- References
- Basic plasticity
- Introduction
- Stress updating incremental or iterative strains?
- The standard elasto-plastic modular matrix for an elastic/perfectly plastic von Mises material under plane stress
Non-associative plasticity
- Introducing hardening
Isotropic strain hardening
Isotropic work hardening
Kinematic hardening
- Von Mises plasticity in three dimensions
Splitting the update into volumetric and deviatoric parts
Using tensor notation
- Integrating the rate equations
Crossing the yield surface
Two alternative predictors
Returning to the yield surface
Sub-incrementation
Generalised trapezoidal or mid-point algorithms
The radial return algorithm a special form of backward-Euler procedure
- The consistent tangent modular matrix
Splitting the deviatoric from the volumetric components
A combined formulation
- Special two-dimensional situations
Plane strain and axial symmetry
Plane stress
A consistent tangent modular matrix for plane stress
- Numerical examples
Intersection point
A forward-Euler integration
Sub-increments
Correction or return to the yield surface
Backward-Euler return
General method
Specific plane-stress method
Consistent and inconsistent tangents
Solution using the general method
Solution using the specific plane-stress method
- Plasticity and mathematical programming
A backward-Euler or implicit formulation
- Two-dimensional formulations for beams and rods
- A shallow-arch formulation
The tangent stiffness matrix
Introduction of material non-linearity or eccentricity
Numerical integration and specific shape functions
Introducing shear deformation
Specific shape functions, order of integration and shear-locking
- A simple corotational element using Kirchhoff theory
Stretching ’stresses and ’strains
Bending ’stresses’ and ’strains
The virtual local displacements
The virtual work
The tangent stiffness matrix
Using shape functions
Including higher-order axial terms
Some observations
- A simple corotational element using Timoshenko beam theory
- An alternative element using Reissner’s beam theory
The introduction of shape functions and extension to a general isoparametric element
- An isoparametric degenerate-continuum approach using the total Lagrangian formulation
- Shells
- A range of shallow shells
Strain-displacement relationships
Stress-strain relationships
Shape functions
Virtual work and the internal force vector
The tangent stiffness matrix
Numerical integration matching shape functions and ‘locking’
Extensions to the shallow-shell formulation
- A degenerate-continuum element using a total Lagrangian formulation
The tangent stiffness matrix
- More advanced solution procedures
- The total potential energy
- Line searches
Theory
Flowchart and Fortran subroutine to find the new step length
Fortran subroutine SEARCH
Implementation within a finite element computer program
Input
Changes to the iterative subroutine ITER
Flowchart for Iine-search loop at the structural level
- The arc-length and related methods
The need for arc-length or similar techniques and examples of their use
Various forms of generalised displacement control
The ’spherical arc-length’ method
Linearised arc-length methods
Generalised displacement control at a specific variable
- Detailed formulation for ttre ‘cylindrical arc-length’ method
Flowchart and Fortran subroutines for the application of the arc-length constraint
Fortran subroutines ARCLl and QSOLV
Flowchart and Fortran subroutine for the main structural iterative loop (ITER)
Fortran subroutine ITER
The predictor solution
- Automatic increments, non-proportional loading and convegence criteria
Automatic increment cutting
The current stiffness parameter and automatic switching to the arc-length method
Non-proportional loading
Convergence criteria
Restart facilities and the computation of the lowest eigenmode of K1
- The updated computer prcgram
Fortran subroutine LSLOOP
Input for incremental/iterative control
Subroutine INPUT2
Flowchart and Fortran subroutine for the main program module NONLTD
Fortran for main program module NONLTD
Flowchart and Fortran subroutine. for routine SCALUP
Fortran for routine SCALUP
Flowchart and Fortran for subroutine NEXINC
Fortran for subroutine NEXINC
- Quasi-Newton methods
- Secant-related acceleration tecriniques
Cut-outs
Flowchart and Fortran for subroutine ACCEL
Fortran for subroutine ACCEL
- Problems for analysts
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)
- Further work on solution procedures
Preface
This book was originally intended as a sequal to my book Finite Elements and Solution
Procedures for Structural Analysis, Vol 1 - Linear Analysis, Pineridge Press, Swansea, 1986. However, as the writing progressed, it became clear that the range of contents was becoming much wider and that it would be more appropriate to start a totally new book. Indeed, in the later stages of writing, it became clear that this book should itself be divided into two volumes; the present one on ‘essentials’ and a future one on ‘advanced topics’. The latter is now largely drafted so there should be no further changes in plan!
Some years back, I discussed the idea of writing a book on non-linear finite elements with a colleague who was much better qualified than I to write such a book. He argued that it was too formidable a task and asked relevant but esoteric questions such as ‘What framework would one use for non-conservative systems?’ Perhaps foolishly, I ignored his warnings, but 1 am, nonetheless, very aware of the daunting task of writing a ‘definitive work’ on non-linear analysis and have not even attempted such a project.
Instead, the books are attempts to bring together some concepts behind the various strands of work on non-linear finite elements with which I have been involved. This involvement has been on both the engineering and research sides with an emphasis on the production of practical solutions. Consequently, the book has an engineering rather than a mathematical bias and the developments are closely wedded to computer applications. Indeed, many of the ideas are illustrated with a simple non-linear finite element computer program for which Fortran listings, data and solutions are included (floppy disks with the Fortran source and data files are obtainable from the publisher by use of the enclosed card). Because some readers will not wish to get actively involved in computer programming, these computer programs and subroutines are also represented by flowcharts so that the logic can be followed without the finer detail.
Before describing the contents of the books, one should ask ‘Why further books on non-linear finite elements and for whom are they aimed?’ An answer to the first question is that, although there are many good books on linear finite elements, there are relatively few which concentrate on non-linear analysis (other books are discussed in Section 1. I). A further reason is provided by the rapidly increasing computer power and increasingly user-friendly computer packages that have brought the potential advantages of non-linear analysis to many engineers. One such advantage is the ability to make important savings in comparison with linear elastic analysis by allowing, for example, for plastic redistribution. Another is the ability to directly simulate the collapse behaviour of a structure, thereby reducing (but not eliminating) the heavy cost of physical experiments.
While these advantages are there for the taking, in comparison with linear analysis, there is an even greater danger of the ‘black-box syndrome’. To avoid the potential dangers, an engineer using, for example, a non-linear finite element computer program to compute the collapse strength of a thin-plated steel structure should be aware of the main subject areas associated with the response. These include structural mechanics, plasticity and stability theory. In addition, he should be aware of how such topics are handled in a computer program and what are the potential limitations.
Textbooks are, of course, available on most of these topics and the potential user of a non-linear finite element computer program should study such books. However, specialist texts do not often cover their topics with a specific view to their potential use in a numerical computer program. It is this emphasis that the present books hope to bring to areas such as plasticity and stability theory.
Potential users of non-linear finite element programs can be found in the aircraft, automobile, offshore and power industries as well as in general manufacturing, and it is hoped that engineers in such industries will be interested in these books. In addition, it should be relevant to engineering research workers and software developers. The present volume is aimed to cover the area between work appropriate to final-year undergraduates, and more advanced work, involving some of the latest research. The second volume will concentrate further on the latter.
It has already been indicated that the intention is to adopt an engineering approach and, to this end, the book starts with three chapters on truss elements. This might seem excessive! However, these simple elements can be used, as in Chapter 1, to introduce the main ideas of geometric non-linearity and, as in Chapter 2, to provide a framework for a non-linear finite element computer program that displays most of the main features of more sophisticated programs. In Chapter 3, these same truss elements have been used to introduce the idea of ‘different strain measures’ and also concepts such as ‘total Lagrangian’, ‘up-dated Lagrangian’ and ‘corotational’ procedures. Chapters 4 and 5 extend these ideas to continua, which Chapter 4 being devoted to ‘continuum mechanics’ and Chapter 5 to the finite element discretisation.
I originally intended to avoid all use of tensor notation but, as work progressed, realised that this was almost impossible. Hence from Chapter 4 onwards some use is made of tensor notation but often in conjunction with an alternative ‘matrix and vector’ form.
Chapter 6 is devoted to ‘plasticity’ with an emphasis on J,, metal plasticity (von Mises) and ‘isotropic hardening’. New concepts such as the ‘consistent tangent’ are fully covered. Chapter 7 is concerned with beams and rods in a two-dimensional framework. It starts with a shallow-arch formulation and leads on to ‘deepformulations’ using a number of different methods including a degenerate-continuum approach with the total Lagrangian procedure and various ‘corotational’ formulations. Chapter 8 extends some of these ideas (the shallow and degeneratecontinuum, total Lagrangian formulations) to shells.
Finally, Chapter 9 discusses some of the more advanced solution procedures for non-linear analysis such as ‘line searches’, quasi-Newton and acceleration techniques, arc-length methods, automatic increments and re-starts. These techniques are introduce into the simple computer program developed in Chapters 2 and 3 and are then applied to a range of problems using truss elements to illustrate such responses as limit points, bifurcations, ‘snap-throughs’ and ‘snap-backs’.
It is intended that Volume 2 should continue straight on from Volume 1 with, for example, Chapter 10 being devoted to ‘more continuum mechanics’. Among the subjects to be covered in this volume are the following: hyper-elasticity, rubber, large strains with and without plasticity, kinematic hardening, yield criteria with volume effects, large rotations, three-dimensional beams and rods, more on shells, stability theory and more on solution procedures.
