Comparing Two Algorithms to Add Large Strains to a Small Strain Finite Element Code

Home , Objective stress rate

INTERNATIONAL CENTER FOR NUMERICAL METHODS IN ENGINEERING

A. Rodríguez Ferran B. A. Huerta

Publication CIMNE Nº-91, November 1996

COMPARING TWO ALGORITHMS

TO ADD LARGE STRAINS

TO A SMALL STRAIN

FINITE ELEMENT CODE

Antonio Ro drguezFerran and Antonio Huerta Member ASCE

Research Assistant Departamento de MatematicaAplicada I I I ETS de Ingenieros de

Caminos Universitat Politecnicade Catalunya Campus Nord C E Barcelona Spain

Professor Departamento de MatematicaAplicada I I I ETS de Ingenieros de Caminos

Universitat Politecnicade Catalunya Campus Nord C E Barcelona Spain

y Corresp onding author email huertaetseccpbupces

ABSTRACT

Two algorithms for the stress update ie timeintegration of the constitutive equation

in large strain solid mechanics are discussed with particular emphasis on two issues the in

cremental objectivity and the implementation aspects It is shown that both algorithms are

incrementally objective ie they treat rigid rotations properly and that they can be employed

to add large strain capabilities to a smal l strain nite element code in a simple way Aset

of benchmark tests consisting of simple large deformation paths rigid rotation simple shear

extension extension and compression dilatation extension and rotation have been usedto

test and compare the two algorithms both for elastic and plastic analysis These tests evidence

dierent time integration accuracy for each algorithm However it is demonstrated that the in

general less accurate algorithm gives exact results for shearfree deformation paths

Keywords nonlinear computational mechanics large strain solid mechanics stress up date

algorithms nite elementcodedevelopment large strain b enchmarks

INTRODUCTION

Many problems of interest in physics and engineering are nonlinear Focusing on

solid mechanics two basic typ es of nonlinearity are encountered material and geometric

Material nonlinearity refers to the plastic or more generally inelastic b ehavior shown by

many engineering materials such as metals In some pro cesses moreover the solid undergo es

such large deformations that the variation in shap e may not b e neglected as in a standard

linear computation thus resulting in geometric nonlinearity The two kinds of nonlinearity

invalidate in many cases a classical linear elastic analysis

From the viewp ointofcontinuum mechanics a convenientway to describ e large strains

in a solid is by means of a convected frame whichisattached to the b o dy and deforms with it

Since the convected frame follows the b o dy motion it allows for a simple statementand

handling of the constitutive equations In nonlinear computational mechanics however

the standard approach is to use a xed Cartesian frame like in the linear case This leads

to a simpler description of motion b ecause the frame do es not change but on the other hand

the treatment of constitutivelaws b ecomes more involved b ecause the frame do es not follow

the material

Nonlinear material b ehavior is often describ ed by a rateform constitutive equation re

lating some measure of the rate of deformation to a rate of stress In a largestrain context

the choice of a prop er stress rate is a key p oint b ecause the principle of ob jectivity

should b e resp ected the constitutive equation must b e indep endent of the observer

This is only achieved when ob jectivequantities are employed It can b e shown that the material

derivative of stress is not an ob jective tensor and therefore an alternative ob jective stress rate

is needed The ob jectivity of the constitutive equations should b e resp ected by the numerical

algorithms employed for their timeintegration This requirement is referred to as incremental

ob jectivity

Various stress up date algorithms ie algorithms for the numerical timeintegration of

the constitutive equations can b e found in the literature see for instance In many

cases however employing these algorithms to add largestrain capabilities to a smallstrain FE

co de is a cumbersome task b ecause new quantities not employed for a small strain analysis

must b e computed

This pap er discusses two incrementally ob jective algorithms that allow to transform an

existing smallstrain FE co de into a largestrain co de in a simple way Only the par

ticular case in which the elastic part of the deformation is mo deled byanhyp o elastic lawa

common choice in nonlinear computational mechanics will b e addressed here The rst algo

rithm uses the full Lagrange strain tensor including quadratic terms to account for large strains

The second algorithm presented in employs the same strain tensor as in a smallstrain

analysis but computed in the midstep conguration

Various implementation asp ects for the two algorithms are commented It is shown in

particular that very few additional features must b e added to a co de with smallstrain and

nonlinear material b ehavior to enable its use for largestrain analysis

The two algorithms are tested and compared with the help of a set of b enchmark tests

consisting of simple deformation paths rigid rotation simple shear uniaxial extension ex

tension and compression extension and rotation Moreover it is demonstrated that for some

particular deformation paths the in general less accurate algorithm captures the exact solution

This pap er is organized as follows Some preliminaries including the basic equations of

large strain solid mechanics and the concept of ob jectivity are briey reviewed in Section

The two stress up date algorithms are presented in Section After some introductory remarks

in Section the notion of incremental ob jectivity is reviewed in Section The two algo

rithms are then shown in Section and their implementation in a smallstrain FE co de is

discussed in Section Section deals with the numerical examples Finally some concluding

remarks are made in Section

PRELIMINARIES

Basic equations

The rst ingredient of continuum mechanics is the equation of motion x xX t

which yields the p osition x of material particles denoted by their material co ordinates X at

time t If the initial spatial co ordinates are employed as material co ordinates the material

displacements can b e dened as ut xt X Once the displacements are dened the kine

matical description continues with strain representation The starting p oint is the deformation

gradient F

Various strain tensors may b e dened by means of F The Lagrange strain tensor for

instance is

E F F I

where T means transp ose and I is the identity Another tensor representing strain is the spatial

gradientofvelo city l This tensor yields relevant tensors if decomp osed into symmetric part

rateofdeformation tensor d and skewsymmetric part spin tensor

l d

Avery common simplication in solid mechanics is that of small deformations If dis

placements rotations and strains are small enough two imp ortantpoints follow i the relation

between displacements and strain is linear and ii the initial conguration of the b o dy can

b e used to solvethegoverning equations Because of this a geometrically linear problem

results

In some other problems on the contrary displacements are large when compared to the

initial dimensions of the b o dy The relation b etween displacements and strains is no longer

linear and moreover the governing equations must b e solved over the current conguration

at time t not over Since the motion that transforms into is precisely the fundamental

0 0

unknown a geometrically nonlinear problem is obtained

The balance laws of continuum mechanics state the conservation of mass momentum and

energy For a wide range of problems in solid mechanics three simplifying assumptions are

common i mechanical and thermal eects are uncoupled ii the density is constantandiii

inertia forces are negligible in comparison to the other forces acting on the b o dy quasistatic

pro cess The mechanical problem is then governed by the momentum balance alone which

b ecomes a static equilibrium equation

is the Cauchy stress tensor and b is the force p er unit volume Equation mo dels where

many problems of practical interest including for instance various forming pro cesses see

Stress tensors

The most common representation of stress is the Cauchy stress tensor dened in the

current conguration and already presented in Eq This tensor has a clear physical

meaning b ecause it involves only forces and surfaces in the current conguration Exp erimen

tal stress measures taken in a lab oratory corresp ond to Cauchy stresses also known as true

stresses

In a large strain context other representations of stress are p ossible and indeed useful

The key idea is that and are dierent congurations so tensors dened in each

conguration cannot b e combined by op erations such as subtraction and addition Let and

b e the Cauchy stress tensors at the initial time t and current time t resp ectively the incre

t 0

ment of stress may not b e dened as b ecause the two tensors are referred to dierent

congurations As stress increments will b e needed to up date stresses a prop er denition is

required

An alternative representation of stress is the second PiolaKirchho tensor S denedas

the pullbackof

1 T

S J F F

where J detF is the Jacobian of the motion which reects the variation of unit volume

asso ciated to the deformation and the inverse of the deformation gradient F is employed

to transform from to see Figure a Equation is called the pullback Piola trans

formation It must b e remarked that S represents the state of stress at time t but referred to

conguration and should not b e confused with the stress at initial time t

0 0

Equation maybereversed and then may b e seen as the pushforward of S

FSF

where F transforms S from to Equation is the socalled pushforward Piola transfor

mation Figure b With the help of Eqs and the stress incrementmay b e represented

either as

t t 1 0 T

J F F a

T 0 1 t 0

b F J F

referred to or resp ectively The Piola transformations are employed to refer the two

tensors to a common conguration where the subtraction can b e prop erly p erformed

Constitutive equations and objectivity

In nonlinear solid mechanics the material b ehavior is often describ ed by a rateform con

stitutive equation relating the stress rate to velocity andor its derivatives and the stress state

and eventually some internal variables The particular case of hypo elastic materials where

the stress rate dep ends linearly on the rateofdeformation tensor d will b e considered here

to presentthetwo stress up date algorithms The two algorithms however can b e extended to

elastoplastic problems by proting from the decomp osition of the rateofdeformation d into

elastic and plastic parts The hypo elastic constitutivelawis

C d

is the material rate of stress and C is the fourthorder mo dulus tensor For isotropic where

materials C can b e written in terms of just two parameters the Lameconstants and just

like in classical elasticity

In fact Eq is only valid for small strains As shown next the material rate of stress

may not b e employed to represent stress variation in a large strain problem b ecause it is not

an ob jective tensor

The principle of ob jectivity is a fundamental requirement regarding the constitutive

equation in large strain solid mechanics if the constitutive equations really describ e

the physical b ehavior of the continuum they must b e indep endent of the observer In other

words they must remain invariant under anychange of reference frame

This requirement is fullled if ob jective quantities app ear in the constitutive equations

A quantity is said to b e ob jective if it transforms in a prop er tensorial manner under a su

p erp osed rigidb o dy motion Let the rigid motion b e represented by an orthogonal rotation

1 T

tensor QQ Q and a translation a The timedep endent relation b etween old and new

new

co ordinates is then x t Qtx at It is p ostulated that the Cauchy stress tensor is

ob jective As it is a secondorder tensor it transforms according to

new T

t Qt tQt

If Eq is derived with resp ect to time it is readily observed that the material derivative

of an ob jective tensor is not ob jective

new T T T T

Q Q Q Q Q Q 6 Q Q

This invalidates the use of as the stress rate in a rateform constitutive equation An

alternative ob jective stress rate is therefore needed In fact Eq whichisvalid for

small strain analysis provides unrealistic stress distributions in very simple large strain tests

as shown in App endix for a rigid rotation test

As for the rateofdeformation tensor d it can b e shown that it is an ob jective tensor

so it may b e employed to represent strains in a constitutive equation Indeed the hypo elastic

constitutive equation is rewritten in a large strain framework as

C d

is not uniquely determined by the ob jectivity principle Some classical The stress rate

options reviewed in are the Jaumann rate

the GreenNaghdi rate

and the Truesdell rate

l l trd c

where is the socalled rateofrotation tensor see and trd denotes the trace of tensor d

{6{

It can b e easily checked that the terms in the RHS of Eqs additional to the material

ensure that the dened rates are indeed ob jective Either by means of the spin rate rate

is comp ensated or the gradientofvelo city l the nonob jectivityof the rateofrotation

is obtained and an ob jective

Regarding the Truesdell rate it has b een dened in Eq c in terms of the Eulerian

l and d referred to the current conguration An alternative expression which tensors

provides insightinto its physical meaning and is useful from an algorithmic viewp oint see

F SF

In this equation the Truesdell rate can b e interpreted as the pushforward Piola transformation

of the material derivative of the second PiolaKirchho stress tensor S Thus instead of using

the time derivative of the Cauchy stress tensor which yields the nonob jective material rate

see Eq the Truesdell rate is preferred b ecause it is by construction an ob jective rate In

Eq it is easily observed that the Truesdell rate pro ceeds in three steps i is pulledback

into S ii the material derivativeof S is p erformed and iii the resulting rate is pushedforward

into the current conguration Where the rationale is that the material derivative of a material

tensor ie a tensor referred to the initial conguration yields an ob jective tensor

TWO STRESS UPDATE ALGORITHMS FOR LARGE STRAINS

Intro ductory remarks

If the Finite Element Metho d is employed the partial dierential equation is

transformed into the nonlinear system of equations

r u f u f u

ext

int

where f is the internal force vector f is the external load vector and r are the residual

ext

int

forces which are null if equilibrium is attained Equation is typically solved incrementally

with a displacementbased implicit metho d The fundamental unknowns are then the in

n n

cremental displacements u x x from one known equilibrium conguration at

time t to a new unknown equilibrium conguration at time t t t

n n

Nonlinear systems of equations like Eq maybesolved bya number of iterative tech

niques The twokey ideas are that i a linearized form of Eq is used to predict and

then iteratively correct u and ii the constitutive equation must b e integrated after each

iteration to check equilibrium Indeed each iteration i yields a candidate conguration

n+1

Tocheck whether it is the equilibrium conguration at time t stresses must b e up dated

n+1

from the previous conguration by timeintegrating the rateform constitutive equation By

doing so the internal forces and the residual forces Eq may b e computed

Incremental objectivity

As remarked in stress up date is the central problem in nonlinear solid mechanics and

aects fundamentally the accuracy of the overall algorithm

In the context of the incremental buildup of the solution it is useful to dene the in

n n+1

cremental versions of the tensors presented in Eqs and Let F and F be the

deformation gradients relating and resp ectively to the reference conguration see

n+1 0

Figure The incremental deformation gradient is

n n+1 n 1

F F

which refers conguration to conguration The corresp onding incremental Lagrange

n+1

strain tensor is then

n n Tn

I E

The ob jectivity of the constitutive equation is attained through the denition of

ob jective stress rates Eqs Incremental ob jectivity is a requirement on the algorithm

for the numerical timeintegration of the constitutive equation which is often presented as

the discrete counterpart of the principle of ob jectivity Let the incremental deformation

n n

gradient relating congurations and b e an orthogonal tensor RThenumerical

n+1

algorithm is said to b e incrementally ob jective if it predicts a stress state at t that is

n+1

simply a rotation of the stress state at t

n T n+1 n n

R R

In other words an incrementally ob jective algorithm assumes that the b o dy motion b e

tween t and t is a rigid rotation and rotates stresses in accordance with that assumption

n+1

with no spurious stress variations Eq It must b e remarked however that the incremental

n n

deformation gradient b eing an orthogonal tensor R do es not necessarily imply that the

true unknown b o dy motion b etween t and t is a rigid rotation For this reason incre

n+1

mental ob jectivity is just a reasonable prop ertyofthenumerical algorithm ie a rigid rotation

is assumed when p ossible rather than a physical requirementlike the principle of ob jectivity

Two stress up date algorithms forlarge strains

First stress up date algorithm

It is p ossible to employ the incremental Lagrange strain tensor dened in Eq as the

strain measure in the incrementt The stress incrementisthen

n n

C E

indicates that this tensorial quantity is like E referred to where the sup erscript n in

conguration

n+1

by simply In a largestrain context it is no longer valid to compute the new stresses

n n

b ecause these latter two tensors are in to the old stresses adding the stress increment

n+1

is sought in the conguration It is necessary to trans the conguration and

n+1

form the tensors adequately by means of the pushforward Piola transformation Eq The

numerical algorithm for stress up date is then

n T n T n 1 n n n 1 n n n+1

J J

n n

where the Jacobian J is dened as det and the incremental deformation gradient

n n

into the new conguration and Eq is employed to pushforward b oth

n+1

This algorithm is incrementally ob jective if is an orthogonal tensor Eq yields

anull strain tensor E and Eq reduces to Eq thus predicting a rigid rotation of

stresses with no spurious stress variations Note that the use of the full incremental Lagrange

tensor including quadratic terms is essential for the incremental ob jectivity of the algorithm

Second stress up date algorithm

An alternative more accurate numerical algorithm will b e shown next Following

the hypo elastic constitutive equation is written in terms of the Truesdell ob jective stress rate

Eq c

C d

A basic ingredient of this algorithm is that d is evaluated in the midstep conguration

dened through linear interpolation b etween and The midstep spatial co or

dinates are

n n n n

x x x x u

the asso ciated deformation gradientis

n n n

F F F

and similarly to Eq the incremental deformation gradient relating the midstep and nal

congurations is

1 1

n n n

2 2

F F

The dierent deformation gradient tensors are summarized in Figure Using a midp oint

rule algorithm to integrate Eq the stress up date b ecomes

1 1 1

n n n n T n n T n n

2 2 2

J J t C d j

1 1

n n

2 2

J dened as det As in Eq tensors referred to the initial with the Jacobian

and midstep congurations are pushedforward into the nal one by means of the appropriate

incremental deformation gradients and

Recalling the denition of d Eq the approximation to d needed in Eq will

9 8

# " # "

= <

u u

1 1

; n n :

t t

2 2

x x {10{

The stress increment is then

9 8

# " # "

= <

u u

C t C d j

1 1

; : n+ n+

2 2

x x

It must b e noted that in Eq the strain increment is represented bythe

symmetrized gradient of the incremental displacements like in a small strain analysis No ad

ditional quadratic terms are needed in Eq b ecause large strains are prop erly mo deled by

employing the midstep conguration to compute the gradient of displacements As discussed in

this algorithm is also incrementally ob jective Moreover the numerical tests of next Sec

tion show that its numerical p erformance in terms of accuracy is sup erior to that of the rst

algorithm An accuracy analysis of the two algorithms showing that the rst one is rstorder

accurate in time and the second one is secondorder accurate can b e found in

The two stress up date algorithms can also b e employed in elastoplasticity The basic idea

is to mo del the elastic part of the deformation with an hypo elastic law and use any of the two

algorithms to compute the elastic trial stress After that a plastic corrector a radial

return algorithm for instance is required to account for material nonlinearity

Implementation asp ects

It is shown in this Section that any of the two stress up date algorithms can b e employed

to add largestrain capabilities to a smallstrain FE co de in a simple way

The basic idea is that the incremental deformation gradients required in Eqs and

can b e computed in a straightforward manner by using quantities that are available in a

small strain co de Consider for instance the incremental deformation gradient relating

to Eq Recalling the denition of F in Eq and the expression of incremental

n+1

displacements it can b e easily checked that can b e put as

n+1

x u

n n

x x

If an Up dated Lagrangian formulation is used the conguration is taken as a

reference to compute the incremental displacements In such a context can b e computed {11{

from Eq with the aid of standard no dal shap e functions by expressing u in terms of

the no dal values of incremental displacements Since the derivatives of shap e functions are

available in a standard FE co de no new quantities must b e computed to obtain

n n

Recalling the expression of the incremental Lagrange strain tensor E in terms of

Eq it can written as

( )

T T

@ u @ u @ u @ u

E :

n n n n

@ x @ x @ x @ x

As for combining Eqs and renders

n+1

@ x @ u

I ;

1 1

n+ n+

2 2

@ @ x x

so can also b e directly computed with the aid of the shap e functions once the congu

ration of the mesh has b een up dated from to

As a result the only two additional features that are required to handle large strains

are 1. the up dating of mesh conguration and 2. the computation of incremental gradient

gradients Eqs and This can b e seen by comparing the schematic algorithm for a

smallstrain analysis with nonlinear material b ehavior shown in Box with the largestrain

versions depicted in Box a rst stress up date algorithm and Box b second stress up date

algorithm In Boxes a and b large strains the mo dications with resp ect to Box small

strains are highlighted with b oldface and the symbol is employed to designate additional

steps {12{

FOR EVERY TIMEINCREMENT t t

FOR EVERY ITERATION k WITHIN THE TIMEINCREMENT

Compute the incremental displacements u by solving a linearized form of

as the symmetrized gradient of displace Compute the incremental strains

ments

8 9

" # " #

< =

k k

u u

: ;

X X

via the elastic mo dulus Compute the elastic trial incremental stresses

trial

tensor

k k

trial

Compute the elastic trial stresses at t

k n k

trial trial

Compute the nal stresses at t by p erforming the plastic correction

k k

byintegrating the stresses Compute the internal forces f

int

Check convergence If it is not attained go backtostep

Box Smallstrain analysis with nonlinear material b ehavior

FOR EVERY TIMEINCREMENT t t

FOR EVERY ITERATION k WITHIN THE TIMEINCREMENT

Compute the incremental displacements u by solving a linearized form of

Compute the incremental strains accounting for quadratic terms

9 8

" # " # # " # "

T T

= <

k k k k

u u u u

n n n n

; :

x x x x

via the elastic mo dulus Compute the elastic trial incremental stresses

trial

tensor

k k

trial

Compute the incremental deformation gradient Eq and its

determinant J

and Compute the elastic trial stresses at t pushing forward b oth

from conguration to Eq

trial

n T n T n n k k n n n

J J

trial trial

Up date the conguration from to by using the incremental

displacements of the current step

at t by p erforming the plastic correction Compute the nal stresses

k k

Compute the internal forces f byintegrating the stresses

int

Check convergence If it is not attained recover conguration and go

backtostep

Box a First stress up date algorithm

FOR EVERY TIMEINCREMENT t t

FOR EVERY ITERATION k WITHIN THE TIMEINCREMENT

Compute the incremental displacements u by solving a linearized form of

Up date the conguration from to Eq

Compute the incremental strains as the symmetrized gradient of displace

ments

9 8

# " # "

= <

k k

u u

1 1

n n ; :

2 2

x x

via the elastic mo dulus Compute the elastic trial incremental stresses

trial

tensor

k k

trial

Compute the incremental deformation gradients and

Eqs and and its determinants J and J

k n

and Compute the elastic trial stresses at t pushing forward

trial

to Eq

1 1 1

n T n k n n T k n n n

2 2 2

J J

trial trial

Up date the conguration from to

at t by p erforming the plastic correction Compute the nal stresses

k k

Compute the internal forces f byintegrating the stresses

int

Check convergence If it is not attained recover conguration and go

backtostep

Box b Second stress up date algorithm

NUMERICAL TESTS

Intro ductory remarks

The two stress up date algorithms presented in Section are compared with the help

of various simple largestrain deformation paths rigid rotation simple shear uniaxial exten

sion extension and compression extension and rotation Both elastic and elastoplastic cases

are considered

Elastic b ehavior will b e represented byYoungs mo dulus E and Poissons co ecient or

equivalentlyby the Lame constants and All the tests have b een p erformed with

resulting in and E For the plastic cases the constitutivelaw is assumed to b e

bilinear with a plastic mo dulus of E E and a yield stress of E

p y

The rigid rotation test demonstrates that as predicted bytheory b oth algorithms are

incrementally ob jective Dierences b etween the two algorithms arise for the other nonrigid

test cases The rst algorithm is rstorder accurate in time while the second one is second

order accurate see the pro of in As a consequence the general b ehavior is that results

dep end heavily on the numb er of timeincrements if the rst algorithm is employed and only

slightly with the second one However the rst algorithm shows a sup erior p erformance for

some stress comp onents in various tests as shown in Section and App endix For com

parison purp oses a smallstrain analysis is also p erformed The results show that neglecting

the large strains lead to qualitatively dierent resp onses The need for a largestrain analysis

is clearly demonstrated

Rigid rotation

A rectangle AB C D is initially sub jected to a uniaxial stress eld

;

and rotated around vertex A see App endix Both algorithms are capable of rotating {16{

the stress eld accordingly

;

even if only one timestep is employed This result illustrates the incremental ob jectivityofthe

two algorithms

Simple shear

The problem statement for the simple shear test is shown in Figure where the initial

t and nal t congurations are presented The initial stress eld is zero The

equations of motion are

xt X Yt

;

y t Y

and the deformation gradientis

" #

If the Truesdell ob jective rate is employed Eq yields the system of ordinary dierential

equations

xx xy

xy yy

;

which can b e complementedwithnull initial conditions and solved to provide the analytical

solution

> 2

t t

;

yy {17{

Both algorithms have b een employed with dierentvalues of the timestep

increments to integrate the constitutive equation for the given deformation path

Eq from t to t

Figure presents the results for the elastic case increments It can b e seen

that the second algorithm gets for the three comp onents of stress the exact analytical values

whereas the rst one grossly overestimates stresses when not enough increments are employed

and demands a small timestep to get close to the analytical solution The output of a small

strain analysis increment is also presented in Figure the null and linear are cor

yy xy

rectly predicted but this linear analysis is not able to repro duce the quadratic Eq

Since the analytical solution is known Eq the error in the nal stress t can b e

computed and plotted versus the number of timeincrements In a loglog scale an straightline

with a slop e equal to the order of the algorithm is exp ected that is for the rst algorithm

and for the second see An average slop e can b e computed by tting a straightline

via a linear regression Figure depicts the results for the shear test Only the rst algorithm

is shown b ecause the second one provides the exact analytical solution except for rounding

errors for anynumb er of timesteps It can b e seen that for the three comp onents of stress

Figures a b and c the observed slop es are very close to the exp ected value of for

for and for The rstorder accuracy of the rst algorithm is thus

xx xy yy

corrob orated by this simple test

The shear test has also b een carried out in the elastoplastic case The results are pre

sented in Figure increments As in the elastic problem the rst algorithm grossly

overestimates the nal stress if only one increment is employed while the second one predicts

much more accurate values With a higher numb er of timesteps b oth algorithms converge

to the same resp onse It may b e observed that of course when plastication starts around

t the curves dier from their elastic counterparts of Figure

A small strain analysis this time with increments to account for material nonlinearity

is again not satisfactory As commented ab ove is incorrectly kept at its zero initial value

throughout the rst steps of the computation with elastic b ehavior As is also zero the

yy {18{

only nonzero stress is and the elastic stress tensor is

" #

that is a fully deviatoric tensor As the plastic correction once plastication b egins is carried

precisely along the deviatoric part of the stress tensor is the only comp onent of stress

aected by plastication while and keep their elastic b ehavior The need for a large

xx yy

strain analysis is again demonstrated

Uniaxial extension

A unit square is sub jected to uniaxial extension in the x direction see Figure The

equations of motion are

xt X t

;

y t Y

and the deformation gradientis

" #

The analytical solution of Eq is this case

t Et

;

Both algorithms correctly predict null values for and Dierences are found on

xy yy

the contrary for while the rst algorithm grossly overestimates it the second one slightly

underestimates it see Figure It can b e seen in Figure that for this particular test a small

strain analysis with one timestep pro duces the exact solution This is due to the fact that the

two sources of error neglecting quadratic terms in the strain tensor and the timediscretization

error comp ensate each other Of course this is not the situation in a general case as shown

for the other tests {19{

The error in the nal value of versus the numb er of timesteps is presented in Figure

The observed slop es for the two algorithms are in this case for the rst algorithm and

for the second one in very go o d agreement with exp ected values of and resp ectively

The plastic resp onse is presented in Figure When plastication b egins at t the

plastic correction is p erformed along the deviatoric part of the stress tensor thus resulting in

an increase of at the exp ense of Again the second algorithm b ehaves much b etter

yy xx

than the rst one if a small number of timeincrements is employed esp ecially for see

Figure a With a large number of timesteps the two algorithms provide very similar

results dierent from the small strain analysis see Figure b

Extension and compression

A unit square undergo es extension in the x direction and compression in the y direction

with no change in volume see Figure The equations of motion are

xt X t

;

y t Y t

the deformation gradientis

" #

and the analytical solution of Eq is

t E t

" #

;

Again b oth algorithms are capable of predicting null for the elastic test but dier

ences app ear for and see Figure The observed orders are in this case rst

xx yy

algorithm and second algorithm for see Figure a and rst algorithm and

second algorithm for see Figure b

yy {20{

As for the plastic test results are shown in Figure Once again convergence to the

reference solution with timeincrements is faster with the second algorithm than with

the rst one while the small strain analysis yields a qualitatively dierent resp onse

Dilatation

A unit square undergo es biaxial extension see Figure The equations of motion are

xt X t

;

y t Y t

resulting in the deformation gradient

" #

The analytical solution of Eq is

t E ln t

;

t E ln t

As in the two previous tests b oth algorithms provide qualitatively correct results in the

sense that is zero and equals for anynumber of timesteps and in b oth elastic

xy xx yy

and plastic mo des There are sharp dierences however concerning convergence b ehavior

For the elastic case for instance the second algorithm provides a b etter prediction with one

timeincrement than the rst one with ve see Figure It can b e seen in Figure that

once again the algorithms b ehave as exp ected The computed slop es are in this case for

the rst algorithm and for the second one

A similar comparison is valid in the plastic test where one increment with the second

algorithm gets closer to the reference solution than ten steps of the rst algorithm see Figure

{21{

Extension and rotation

In this last deformation path a unit square undergo es a uniaxial extension and a sup er

p osed rigid rotation see Figure The equations of motion are

xt X tcost Y sint

;

y t X tsint Y cos t

the deformation gradientis

" #

tcost sint

tsint cos t

and the analytical solution of Eq is

t Et cos t

t Et sin t

;

t Et sintcost

which is as exp ected a rotation of the solution to the uniaxial extension test Eq

The output of the elastic analysis with and timesteps can b e seen in Figure

If only one increment is employed the rotation part of the motion is not captured and the

predicted stress is identical to that of the uniaxial extension test With a higher number of

steps the comparative p erformance of the two algorithms is dierent from that of the previous

tests For the stress comp onents and for instance the rst algorithm is the one which

xy yy

predicts correct null values for anynumb er of timeincrements In fact this result illustrates

a more general b ehavior A unied treatment of shearfree deformation paths can b e found in

App endix It is shown that the rst algorithm correctly predicts null shear stresses for any

number of timesteps while the second one do es not

As for the stress comp onent the rst algorithm p erforms b etter if a reduced number

of timesteps is employed and a larger numb er is required for the second algorithm to pro

duce more accurate results This b ehavior is illustrated by Figure a where the error curves {22{

for of the two algorithms intersect each other Again the observed order of b oth schemes

for the rst one and for the second one is in accordance with the exp ected values

Figures b and c show the convergence b ehavior of the second algorithm for the other two

stress comp onents It can b e seen that the convergence to the exact analytical value is very

fast esp ecially for Figure c

The outcome of the plastic analysis is depicted in Figure Once again a smallstrain

analysis with nonlinear material b ehavior turns out to b e completely unsatisfactory providing

a solution which is qualitatively dierent from that of a largestrain analysis

CONCLUDING REMARKS

Twonumerical stress up date algorithms for large strains have b een discussed The rst

one uses the full incremental Lagrange strain tensor including quadratic terms as the strain

measure The second one works in the midstep conguration where the symmetrized gradient

of the displacement increment can b e employed as the strain measure as in a small strain

analysis

Any of the two algorithms may b e employed to enhance a small strain nite elementcode

into a large strain co de in a simple way In particular it has b een shown that if the co de already

handles material nonlinearity adding large strains only involves two additional features the

up dating of the mesh conguration and the computation of incremental deformation gradients

which are readily available from standard shap e functions

These two algorithms have b een compared with the aid of a set of simple deformation

paths rigid rotation simple shear uniaxial extension extension and compression extension

and rotation b oth in elasticity and elastoplasticity The rigid rotation test shows that the

twoschemes are incrementally ob jective that is they rotate the stresses accordingly and yield

no spurious stress variations For the other nonrigid tests the results show that the second

algorithm is generally more accurate that the rst one the main reason b eing the use of the

midstep conguration as a reference In fact it can b e shown that the rst algorithm is rst

order accurate and the second one is secondorder accurate Accuracyhowever is not the only

relevant p oint for shearfree deformation paths there is no error in the shear comp onentof

the stress tensor computed with the rst algorithm In the numerical tests the two algorithms {23{

b ehaveinvery go o d agreement with their exp ected b ehavior Analytical solutions are provided

for the elastic analyses which can b e employed to verify the implementation of these or any

other stress up date algorithms

ACKNOWLEDGEMENTS

The authors thank the graduate student A Vila for carrying out the various numerical

tests The authors also want to express their appreciation to Prof A Millard for fruitful dis

cussions on the topic of large strains The nancial supp ort from DGICYT PB and the

Human Capital and Mobility ERBPL is gratefully acknowledged

APPENDIX MATERIAL STRESS RATE VERSUS OBJECTIVE STRESS RATE

It has b een commented in Section that the material stress rate cannot b e employed

in rateform constitutive equations for large strains b ecause it is not ob jective This fact is

illustrated here with a simple test

Consider the rigid b o dy rotation problem depicted in Figure A The rectangle AB C D is

and rotates around vertex A with a constant angular initially sub jected to a uniaxial stress

velocity The velocity eld is

yx A v v

x y

and the spatial gradientofvelo city l is a skewsymmetric tensor thus coincident with the spin

rate

" #

l A

As exp ected the rateofdeformation d is a null tensor in this rigidb o dy strainfree

motion If is selected as stress rate Eq then

d ) C d A

{24{

the stress distribution is constantly equal to the initial one and do es not Since

follow the rotation After a turn for example the stress of Figure A is predicted where

the uniaxial stress is now through the short dimension of the rectangle AB C D This result is

fully unsatisfactory a simple rotation of the initial stress should b e obtained see Figure A

This correct answer can b e achieved by using an ob jective stress rate as commented in Section

APPENDIX STRESS UPDATE FOR SHEARFREE DEFORMATION PATHS

Here a particular b ehavior of the rst stress up date algorithm is demonstrated This al

gorithm correctly predicts null shear stresses for any shearfree deformation path The second

algorithm on the contraryonlyprovides similar results for some particular deformation paths

of this typ e

The general expression for shearfree deformation paths is

xtXa tcos t Ya tsin t

x y

;

y tXa tsin t Ya tcos t

x y

where a tand a t represent the axial deformations in the x and y directions resp ectively

x y

and t t accounts for the rigid rotation Taking a tt and a t yields the

x y

extensionrotation test of Section An extensioncompressionrotation test with no change

in volume can b e obtained with a t t and choosing a t a t t

x x y

a t

renders a dilatationrotation test

The deformation gradient of motion A is

" #

a tcos t a tsin t

x y

F A

a tsin t a tcos t

x y

and the Jacobian is J a ta t

x y

To assess the b ehavior of the two stress up date algorithms it is enough to study the stress

predicted after one timestep and to check if the shear stress comp onent in rotated axes to

account for the rigid rotation is null or not {25{

First stress up date algorithm

For the rst timestep t the incremental Lagrange strain tensor is Eq

" #

E A

with A a t and A a t Recalling the expression of the rst algorithm Eq

x x y y

the stress after one timestep is

" #

2 2

B cos B sin B sincos B sincos

x y x y

2 2

A A

x y

B sincos B sincos B sin B cos

x y x y

2 2 2 2

with t B A A and B A A

x y

x x y y

0 T

R R whereR is a orthogonal If the stress is transformed into the rotated stress

tensor that accounts for the rigid rotation the result is

# "

Equation A shows that null shear stresses are predicted indep endently of the

0 0

x y

axial deformations A and A Itcanbeeasilychecked that the analysis just p erformed for the

x y

rst timestep can b e extended to the successive increments In conclusion the rst algorithm

predicts null shear stresses in the rotated axes for anynumb er of timesteps and for any

0 0

x y

choice of a tand a t

x y

Second stress up date algorithm

A similar analysis has b een p erformed for the second algorithm Since the midstep con

guration is employed the required computations are rather cumbersome in this case The

incremental displacements for the rst timestep can b e computed from Eq A After that

the midstep co ordinates are obtained following Eq Then the incremental displacements

are written in terms of the midstep co ordinates so the strain increment can b e written as

A A A A cos

x y x y

" #

A A A A cos A A sin

x y x y x y

A A sin A A A A cos

x y x y x y {26{

is computed by employing the incre Once the strain incrementisknown the stress

mental deformation gradient Eqs and and then transformed into the rotated stress

The nal result is that the shear stress is prop ortional to a factor that dep ends on

0 0

x y

axial deformations A and A

x y

3 2 2 3 2 2

A A A A A A A A A

0 0

y x

x y x y x y

x y

It can b e seen from Eq A that is not null for anychoice of a tand a t

0 0

x y

asshown for the Taking a tt and a t for instance yields a nonnull

0 0

x y

extensionrotation test of Section On the contraryfor a ta t dilatationrotation

x y

test and for a ta t extensioncompressionrotation test with no volume change null

x y

are predicted In conclusion the second stress up date algorithm predicts cor values for

0 0

x y

rect null shear stresses in the rotated axes for some particular deformation paths of the form

A but not for every shearfree path

APPENDIX REFERENCES

Bathe KJ Finite Element Pro cedures in Engineering Analysis Prentice Hall New

JerseyUSA

Criseld MA Nonlinear Finite Element Analysis of Solids and Structures John

Wiley Sons Ltd England

Healy BE and Do dds RH A Large Strain Plasticity Mo del for Implicit Finite

Element Analyses Computational MechanicsVol pp

Hughes TJR The Finite Element Metho d Prentice Hall International Stanford

USA

Hughes TJR Numerical Implementation of Constitutive Mo dels RateIndep endent

Deviatoric Plasticity Chapter of Theoretical Foundation for Largescale Computations for

Nonlinear Material Behavior Eds S NematNasser RJ Asaro and GA Hegemier Mar

tinus Nijho Publishers Dordrecht

Hughes TJR and Winget J Finite Rotation Eects in Numerical Integration of

Rate Constitutive Equations Arising in LargeDeformation Analysis Int J Num Meth

EngrgVol pp {27{

Key SW and Krieg RD On the Numerical Implementation of Inelastic Time

Dep endent and Time Indep endent Finite Strain Constitutive Equations in Structural Me

chanics Comp Meths Appl Mech EngrgVol pp

Khan AK and Huang S Continuum Theory of Plasticity John Wiley Sons Inc

New York

Malvern LW Introduction to the Mechanics of a Continuous Medium PrenticeHall

Series in Engineering of the Physical Sciences Englewoo d Clis New Jersey

Marsden JE and Hughes TJR Mathematical Foundations of Elasticity Prentice

Hall USA

Pegon P and Guelin P Finite Strain PlasticityinConvected Frames Int J

Num Meth EngrgVol pp

PinskyPM Ortiz M and Pister KS Numerical Integration of Rate Constitutive

Equations in Finite Deformation Analysis Comp Meths Appl Mech EngrgVol pp

Ro drguezFerran A Pegon P and Huerta A Accuracy Analysis of Two Stress

Up date Algorithms for Large Strain Solid Mechanics Research Rep ort n Int Centre

for Numerical Metho ds in Engng CIMNE Barcelona

Ro drguezFerran A and Huerta A A Comparison of Two Ob jective Stress Rates

in Ob jectOriented Co des Monograph CIMNE No ISBN Barcelona

Simulation of Materials Pro cessing Theory Metho ds and Applications Pro c of the Fifth

Int Conf on Numer Meths in Industrial Forming Pro cesses NUMIFORM Ithaca New

York

Zienkiewicz OC and Taylor RL The Finite Element Metho d Vols and Mc

Graw Hill London {28{

FIGURE CAPTIONS

Figure Piola transformations a Pullback Piola transformation b Pushforward Pi

ola transformation

Figure Deformation gradients in an incremental analysis

Figure Simple shear test Initial and nal congurations

Figure Simple shear test elastic analysis Stress vs time curves computed with the

two algorithms and timesteps and smallstrain analysis timestep

a b c

xx xy yy

Figure Simple shear test elastic analysis Error in the nal value of stress t vs

number of timesteps a b c

xx xy yy

Figure Simple shear test elastoplastic analysis Stress vs time curves computed with

the two algorithms and timesteps and smallstrain analysis time

steps a b c

xx xy yy

Figure Uniaxial extension test Initial and nal congurations

Figure Uniaxial extension test elastic analysis Horizontal normal stress vs time

curves computed with the two algorithms and timesteps and small

strain analysis timestep

Figure Uniaxial extension test elastic analysis Error in the nal value of horizontal

normal stress t vs number of timesteps

Figure Uniaxial extension test elastoplastic analysis Stress vs time curves computed

with the two algorithms and timesteps and smallstrain analysis

timesteps a b

xx yy

Figure Extension and compression test Initial and nal congurations

Figure Extension and compression test elastic analysis Stress vs time curves com

puted with the two algorithms and timesteps and smallstrain analysis

timestep a b

xx yy

Figure Extension and compression test elastic analysis Error in the nal value of

stress t vs numb er of timesteps a b

xx yy

Figure Extension and compression test elastoplastic analysis Stress vs time curves

computed with the two algorithms and timesteps and smallstrain

analysis timesteps a b

xx yy

Figure Dilatation test Initial and nal congurations

Figure Dilatation test elastic analysis Normal stress vs time curves computed with

the two algorithms and timesteps and smallstrain analysis time

step

Figure Dilatation test elastic analysis Error in the nal value of normal stress t

vs number of timesteps

Figure Dilatation test elastoplastic analysis Normal stress vs time curves computed

with the two algorithms and timesteps and smallstrain analysis

timesteps

Figure Extension and rotation test Problem statement

Figure Extension and rotation test elastic analysis Stress vs time curves computed

with the two algorithms and timesteps and smallstrain analysis

timestep a b c

xx xy yy

Figure Extension and rotation test elastic analysis Error in the nal value of stress

t vs numb er of timesteps a b c

xx xy yy

Figure Extension and rotation test elastoplastic analysis Stress vs time curves com

puted with the two algorithms and timesteps and smallstrain analysis

timesteps a b c

xx xy yy

Figure A Rigid rotation test Initial stress state

Figure A Rigid rotation test Stress state predicted with material stress rate

Figure A Rigid rotation test Stress state predicted with ob jective stress rate