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nique has not been implemented in FLagSHyP, which implies that a very
large number of increments is needed for plane strain or three-dimensional
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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
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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

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