M.A. Crisfield
Imperial College of Science,
Technology and Medicine, London, UK

Volume 2: Advanced Topics

Contents

• More continuum mechanics

- Relationships between some strain measures and the structures
- Large strains and the Jaumann rate
- Hyperelasticity
- The Truesdell rate
- Conjugate stress and strain measures with emphasis on isotropic conditions
- Further work on conjugate stress and strain measures

Relationship between ε: and ü
Relationship between the Biot stress, B and the Kirchhoff stress, τ
Relationship between ü, the λ’s and the spin of the Lagrangian
triad, WΝ
Relationship between Ë, the λ’s and the spin, WΝ
Relationship between ε,the λ’s and the spin, WΝ
Relationship between Ë and ε
Specific strain measures
Conjugate stress measures

- Using logeV with isotropy
- Other stress rates and objectivity

• Non-orthogonal coordinates and CO- and contravariant tensor components

- Non-orthogonal coordinates
- Non-orthogonal coordinates
- Transforming the components of a vector (first-order tensor) to a new set of base vectors
- Second-order tensors in non-orthogonal coordinates
- Transforming the components of a second-order tensor to a new set of base vectors
- The metric tensor
- Work terms and the trace operation
- Covariant components, natural coordinates and the Jacobian
- Green’s strain and the deformation gradient

Recovering the standard cartesian expressions

- The second Piola-Kirchhoff stresses and the variation of the Green’s strain
- Transforming the components of the constitutive tensor
- A simple two-dimensional example involving skew coordinates

• More finite element analysis of continua

- A summary of the key equations for the total Lagrangian formulation

The internal force vector
The tangent stiffness matrix

- The internal force vector for the ‘Eulerian formulation’
- The tangent stiffness matrix in relation to the Truesdell rate of Kirchhoff stress

Continuum derivation of the tangent stiffness matrix
Discretised derivation of the tangent stiffness matrix

- The tangent stiffness matrix using the Jaumann rate of Kirchhoff stress

Alternative derivation of the tangent stiffness matrix

- The tangent stiffness matrix using the Jaumann rate of Cauchy stress

Alternative derivation of the tangent stiffness matrix

- Convected coordinates and the total Lagrangian formulation

Element formulation
The tangent stiffness matrix
Extensions to three dimensions

• Large strains, hyperelasticity and rubber

- Introduction to hyperelasticity
- Using the principal stretch ratios
- Splitting the volumetric and deviatoric terms
- Development using second Piola-Kirchhoff stresses and Green’sstrains

Plane strain
Plane stress with incompressibility

- Total Lagrangian finite element formulation

A mixed formulation
A hybrid formulation

- Developments using the Kirchhoff stress
- A ‘Eulerian’ finite element formulation
- Working directly with the principal stretch ratios

The compressible ‘neo-Hookean model’
Using the Green strain relationships in the principal directions
Transforming the tangent constitutive relationships for a ‘Eulerian formulation’

- Examples
A simple example
The compressible neo-Hookean model
- Further work with principal stretch ratios
An energy function using the principal log strains (the Hencky model)
Ogden’s energy function
An example using Hencky’s model

• More plasticity and other material non-linearity-I

- Introduction
- Other isotropic yield criteria
The flow rules
The matrix δa/δσ
- Yield functions with corners
A backward-Euler return with two active yield surfaces
A consistent tangent modular matrix with two active yield surfaces
- Yield functions for shells that use stress resultants
The one-dimensional case
The two-dimensional case
A backward-Euler return with the lllyushin yield function
A backward-Euler return and consistent tangent matrix for the llyushin yield criterion when two yield surfaces are active
- Implementing a form of backward-Euler procedure for the Mohr-Coulomb yield criterion
Implementing a two-vectored return
A return from a corner or to the apex
A consistent tangent modular matrix following a single-vector return
A consistent tangent matrix following a two-vectored return
A consistent tangent modular matrix following a return from a corner or an apex
- Yield criteria for anisotropic plasticity
Hill’s yield criterion
Hardening with Hill’s yield criterion
Hill’s yield criterion for plane stress
- Possible return algorithms and consistent tangent modular matrices
The consistent tangent modular matrix
- Hoffman’s yield criterion
The consistent tangent modular matrix
- The Drucker-Prager yield criterion
- Using an eigenvector expansion for the stresses
An example involving plane-stress plasticity and the von Mises yield criterion
- Cracking, fracturing and softening materials
Mesh dependency and alternative equilibrium states
‘Fixed’ and ‘rotating’ crack models in concrete
Relationship between the ‘rotating crack model’ and a ‘deformation theory’ plasticity approach using the ‘square yield criterion’
Damage mechanics

• More plasticity and other material non-linearity-ll

- Introduction
- Mixed hardening
- Kinematic hardening for plane stress
- Radial return with mixed linear hardening
- Radial return with non-linear hardening
- A general backward-Euler return with mixed linear hardening
- A backward-Euler procedure for plane stress with mixed linear hardening
- A consistent tangent modular tensor following the radial return of Section 15.4
- General form of the consistent tangent modular tensor
- Overlay and other hardening models
Sophisticated overlay model
Relationship with conventional kinematic hardening
Other models
Computer exercises
Viscoplasticity
The consistent tangent matrix
Implementation

• Large rotations

- Non-vectorial large rotations
- A rotation matrix for small (infinitesimal) rotations
- A rotation matrix for large rotations (Rodrigues formula)
- The exponential form for the rotation matrix
- Alternative forms for the rotation matrix
- Approximations for the rotation matrix
- Compound rotations
- Obtaining the pseudo-vector from the rotation matrix, R
- Quaternions and Euler parameters
- Obtaining the normalised quarternion from the rotation matrix
- Additive and non-additive rotation increments
- The derivative of the rotation matrix
- Rotating a triad so that one unit vector moves to a specified unit vector via the ‘smallest rotation’
- Curvature
Expressions for curvature that directly use nodal triads
Curvature without nodal triads

• Three-dimensional formulations for beams and rods

- A co-rotational framework for three-dimensional beam elements
Computing the local ‘displacements’
Computation of the matrix connecting the infinitesimal local and global variables
The tangent stiffness matrix
Numerical implementation of the rotational updates
Overall solution strategy with a non-linear ‘local element’ formulation
Possible simplifications
- An interpretation of an element due to Simo and Vu-Quoc
The finite element variables
Axial and shear strains
Curvature
Virtual work and the internal force vector
The tangent stiffness matrix
An isoparametric formulation
- An isoparametric Timoshenko beam approach using the total Lagrangian formulation
The tangent stiffness matrix
An outline of the relationship with the formulation of Dvorkin et al.
- Symmetry and the use of different ‘rotation variables’
A simple model showing symmetry and non-symmetry
Using additive rotation components
Considering symmetry at equilibrium for the element of Section 17.2
Using additive (in the limit) rotation components with the element of Section 17.2
- Various forms of applied loading including ‘follower levels’
Point loads applied at a node
Concentrated moments applied at a node
Gravity loading with co-rotational elements
Introducing joints

• More on continuum and shell elements

- Introduction
- A co-rotational approach for two-dimensional continua
- A co-rotational approach for three-dimensional continua
- A co-rotational approach for a curved membrane using facet triangles
- A co-rotational approach for a curved membrane using quadrilaterals
- A co-rotational shell formulation with three rotational degrees of freedom per node
- A co-rotational facet shell formulation based on Morley’s triangle
- A co-rotational shell formulation with two rotational degrees of freedom per node
- A co-rotational framework for the semi-loof shells
- An alternative co-rotational framework for three-dimensional beams
Two-dimensional beams
- Incompatible modes, enhanced strains and substitute strains for continuum elements
Incompatible modes
Enhanced strains
Substitute functions
Numerical comparisons

- Introducing extra internal variables into the co-rotational formulation
- Introducing extra internal variables into the Eulerian formulation
- Introducing large elastic strains into the co-rotational formulation
- A simple stability test and alternative enhanced F formulations

• Large strains and plasticity

- Introduction
- The multiplicative FeFp approach
- Using the FeFp approach to arrive at the conventional ‘rate form’
- Using the rate form with an ‘explicit dynamic code’
I- ntegrating the rate equations
- An FeFp update based on the intermediate configuration
- An FeFp update based on the final (current) configuration
The flow rule
- The consistent tangent
The limiting case
- Introducing large elasto-plastic strains into the finite element formulation
- A simple example

• Stability theory

- Introduction
- General theory without ‘higher-order terms’
Limit point
Bifurcation point
- The introduction of higher-order terms
- Classification of singular points
Limit points
Bifurcation points
Symmetric bifurcations
- Computation of higher-order derivatives for truss elements
Amplification of notation
Truss element using Green’s strain
Truss elements using a rotated engineering strain
Computation of the stability coefficients B1 - B3

• Branch switching and further advanced solution procedures

- Indirect computation of singular points
- Simple branch switching
Corrector based on a linearised arc-length method
Corrector using displacement control at a specified variable
Corrector using a ‘cylindrical arc-length method’
- Branch switching using higher-order derivatives
- General predictors using higher-order derivatives
Load control
Displacement control at a specified variable
The ‘cylindrical arc-length method’
- Correctors using higher-order derivatives
- Direct computation of the singular points
- Line-searches with arc-length and similar methods
Line-searches with the Riks/Wempner arc-length method
Line-searches with the cylindrical arc-length method
Uphill or downhill?
- Alternative arc-length methods using relative variables
- An alternative method for choosing the root for the cylindrical arc-length method
- Static/dynamic solution procedures

• Examples from an updated non-linear finite element computer program using truss elements (written in conjunction with Dr Jun Shi)
  

- A two-bar truss with an asymmetric bifurcation
Bracketing
Branch switching
- The von Mises truss
Bracketing
Branch switching
- A three-dimensional dome
Bracketing
Branch switching
The higher-order predictor
The higher-order correctors
Line searches
- A three-dimensional arch truss
- A two-dimensional circular arch

• Contact with friction

Introduction
A two-dimensional frictionless contact formulation using a penalty approach
Some modifications
The ‘contact patch test’
Introducing ‘sticking friction’ in two dimensions
Introducing Coulomb ‘sliding friction’ in two dimensions
Using Lagrangian multipliers instead of the penalty approach
The augmented Lagrangian methods
An augmented Lagrangian technique with Coulomb ’sliding friction’
A symmetrised version
A three-dimensional frictionless contact formulation using a penalty approach
The consistent tangent matrix
Adding ‘sticking friction’ in three dimensions
The consistent tangent matrix
Coulomb ‘sliding friction’ in three dimensions
A penalty/barrier method for contact
Amendments to the solution procedures

• Non-linear dynamics

- Introduction
- Newmark’s method
- The ‘average acceleration method’ or ‘trapezoidal rule’
- The ‘implicit solution procedure’
- The ‘predictor step’
- The ‘corrector’
- An explicit solution procedure
- A staggered, central difference, explicit solution procedure
- Stability
- The Hilber-Hughes-Taylor α method
- More on the dynamic equilibrium equations
- An energy conserving total Lagrangian formulation
The ‘predictor step’
The ‘corrector’
- A co-rotational energy-conserving procedure for two-dimensional beams
Sophistications
Numerical solution
- An alternative energy-conserving procedure for two-dimensional beams
- Automatic time-stepping
- Dynamic equilibrium with rotations
- An ‘explicit co-rotational procedure’ for beams
- Updating the rotational velocities and accelerations
- A simple implicit co-rotational procedure using rotations
- An isoparametric formulation for three-dimensional beams
- An alternative implicit co-rotational formulation
- (Approximately) energy-conserving co-rotational procedures
- Energy-conserving isoparametric formulations

Preface

In the preface to Volume 1, I expressed my trepidation at starting to write a book on non-linear finite elements and the associated mechanics. These doubts grew as
I worked on Volume 2, which attempts to cover ‘advanced topics’. These topics include many areas which are still the subject of considerable controversy. None the less, I have finally completed this second volume, although in so doing, I have almost certainly made mistakes. In persevering, I have received much encouragement from a number of readers of Volume 1 who have urged me not to abandon the second volume and who have made me believe that there is some need for a book of this kind.
As with the subject-matter of the first book, there are many specialist texts which
cover the background mechanics. My aim has not been to replace such books and,
indeed, I have attempted to reference these books with a view to encouraging wider
reading. Instead, my aim has been to emphasise the numerical implementation. As with the earlier volume, an engineering approach is adopted in contrast to a strict mathematical development.
At theend of the Preface of Volume 1, I indicated the subjects that I intended to cover in this second volume. These topics have all been included, but so have a number of other topics that I did not originally envisage including. In particular Chapter 23 covers ‘Contact and friction’ and Chapter 24 covers ‘Nonlinear dynamics’ (both ‘implicit’ and ‘explicit’). These important subjects are included because I have now conducted some research in these areas. This is true of most of the topics in the book. However, while I have often given the background to some of my own research, I have also attempted to cover important developments by others. Often, in so doing, I have reinterpreted these works in relation to my own ‘viewpoint’. Often, this will not coincide with that of the originator. The reader should, of course, read the originals as well!
The previous paragraph gives the impression that the book is related to research.
This is only partially true in that any book, attempting to cover advanced topics, must be concerned with the recent research in the field. However, in addition to these research-related topics, there are many other topics in which the ground work is fairly well established. In these areas, the book is closer to a traditional ‘textbook’.
I referred earlier to ‘my own research’. Of course, I should have referred to ‘the work of my research group’. In particular, I must thank the following (in alphabetical order) for their important contributions: Mohammed Asghar, Michael Dracopoulos,
Zhiliang Fan, Ugo Galvanetto, Hans-Bernd Hellweg, Gordan Jelenic, Ahad Kolahi,
Yaoming Mi, Gray Moita, Xiaohong Peng, Jun Shi and Hai-Guang Zhong.
Indeed, I wrote Chapter 22 on ‘Examples from an up-dated non-linear finite element computer program using truss elements’ in conjunction with Dr Shi. This chapter describes a finite element computer program that can be considered as the extension of the simple computer programs described in Volume 1. As with the latter programs, the new program is available via anonymous FTP (ftp: //ftp.cc.ic.ac.uk/pub/depts/aero1nonlin2). The aim of the new program is purely didactic and it is intended to illustrate some of the ‘path-following’ and ‘branch-switching techniques’ described in Chapter 21.