large number of increments is needed for plane strain or three-dimensional

39

8.13 APPENDIX: DICTIONARY OF MAIN VARIABLES

problems involving true incompressible materials and displacement control.

8.13 APPENDIX: DICTIONARY OF MAIN VARIABLES

The main variables of FLagSHyP are listed below in alphabetical order:

“ arc-length parameter (=0.0 means no arc

arcln

length)

“ left Cauchy“Green tensor

btens

“ average value of J

barj

“ current loads

cload

“ convergence norm

cnorm

“ fourth-order tensor c ijkl

ctens

“ determinant of F (i.e. J)

detf

“ load component of the displacement

dispf

vector uF

“ Newton“Raphson displacement vector u

displ

“ load parameter increment

dlamb

“ derivatives of geometry with respect to

dxis

isoparametric coordinates

“ element average Cartesian derivatives

elacd

“ boundary elements data

elbdb

“ element Cartesian derivatives at each

elecd

Gauss point

“ element data matrix of dimensions

eledb

(ndime+1,nnode+1,ngaus)

“ external nominal load on each degree

eload

of freedom

“ element type

eltyp

energy “ energy norm for equations (rhs * soln)

“ scaling factor or displacement factor

eta

“ previous value of the parameter eta

eta0

“ Cartesian gradient of shape functions

gradn

“ gravity acceleration vector

gravt

“ nodal boundary codes

icode

“ element number

ie

“ load increment number

incrm

40 COMPUTER IMPLEMENTATION

“ unit number for printed output of

jfile

warning messages

“ first height of each profile column

kprof

then address in stifp of each column

“ boundary elements nodal connectivities

lbnod

“ array containing the degrees of freedom

ldgof

of each node or 0 for fixed nodes. If

the coordinates of these fixed nodes is

to be prescribed, the number stored will

be the prescribed displacement number

“ nodal connectivities of dimensions

lnods

(nnode,nelem)

“ logical input“output unit

lun

“ material number of each element

matno

“ material types

matyp

“ maximum number of boundary pressure

mbpel

elements

“ maximum number of degrees of freedom

mdgof

“ maximum number of elements

melem

“ maximum number of Gauss points per

mgaus

element

“ maximum number of iterations per

miter

increment

“ maximum number of materials

mmats

“ maximum number of nodes per element

mnode

“ maximum number of nodes

mpoin

“ maximum number of off-diagonal terms in

mprof

tangent matrix

“ number of boundary elements with applied

nbpel

pressure

“ nodal connectivities as a linked list of

nconn

dimensions (two, total number of node to

node connections)

“ number of degrees of freedom

ndgof

“ number of dimensions for the given

ndime

element

“ circular queue, storing next nodes™

ndque

degrees of freedom to be numbered

“ number of fixed degrees of freedom

negdf

41

8.13 APPENDIX: DICTIONARY OF MAIN VARIABLES

“ number of elements

nelem

“ number of equations to be solved

neq

“ number of Gauss points per boundary

ngaub

element

“ number of Gauss points per element

ngaus

“ number of load increments

nincr

“ number of iterations

niter

“ number of nodes per boundary element

nnodb

“ number of nodes per element

nnode

“ number of materials

nmats

“ number of mesh nodes

npoin

“ number of entries in the out of diagonal

nprof

part of K

“ number of prescribed displacements

nprs

“ number of stresses per Gauss point

nstrs

“ prescribed displacements of fixed nodes

pdisp

“ external applied pressures

press

“ matrix containing the three principal

princ

column vectors

“ vector of properties. The first is

props

always the initial density, the rest

depend on the material type

“ reactions

react

“ residual forces

resid

“ logical variable: .true. if problem is

rest

restarted .false. if problem started

from scratch

“ residual norm

rnorm

“ current dot product of R by u

rtu

“ initial dot product of R by u

rtu0

“ line search parameter (=0.0 means no

searc

line search)

“ Cauchy stress tensor

sigma

“ principal stresses

sprin

“ diagonal stiffness

stifd

“ profile part of stiffness

stifp

“ element stresses

stres

“ vector containing the stretches

stret

“ program or example title

title

42 COMPUTER IMPLEMENTATION

“ total nominal loads including pressure

tload

“ solution of equations

u

“ determinant of Jacobian J at each Gauss

vinc

point

“ initial element volumes

vol0

“ nodal coordinates

x

“ total displacement over the load

xincr

increment

“ effective kappa value

xkapp

“ current load factor

xlamb

“ maximum load parameter

xlmax

“ effective mu coefficient

xme

“ mu coefficient

xmu

“ initial nodal coordinates

x0

APPENDIX ONE

INTRODUCTION TO LARGE

INELASTIC DEFORMATIONS

A.1 INTRODUCTION

Many materials of practical importance, such as metals, do not behave in

a hyperelastic manner at high levels of stress. This lack of elasticity is

manifested by the fact that when the material is freed from stress it fails

to return to the initial undeformed con¬guration and instead permanent

deformations are observed. Di¬erent constitutive theories or models such

as plasticity, viscoplasticity, and others are commonly used to describe such

permanent e¬ects. Although the mathematics of these material models is

well understood in the small strain case, the same is not necessarily true for

¬nite deformation.

A complete and coherent discussion of these inelastic constitutive models

is well beyond the scope of this text. However, because practical applications

of nonlinear continuum mechanics often include some permanent inelastic

deformations, it is pertinent to give a brief introduction to the basic equa-

tions used in such applications. The aim of this introduction is simply to

familiarize the reader with the fundamental kinematic concepts required to

deal with large strains in inelastic materials. In particular, only the sim-

plest possible case of Von Mises plasticity with isotropic hardening will be

fully considered, although the kinematic equations described and the overall

procedure will be applicable to more general materials.

We will assume that the reader has some familiarity with small strain