international journal for numerical methods in engineering volume 26 issue 10 1988 [doi...

8/9/2019 International Journal for Numerical Methods in Engineering Volume 26 Issue 10 1988 [Doi 10.1002%2Fnme.1620261

1/25

INTERNATIONAL

JOURNAL

FOR NUMERICAL METHODS IN ENGINEERING,

VOL.

26,2161-2185 (1988)

NO N-SM OO TH MULTISURFACE PLASTICITY AN D

VISCOPLASTICITY. LOADING/UNLOADING

CONDITIONS

AND

NUM ERICAL ALGORITHMS

J.

C. SIMO, J . G. KENNEDY

AND S . GOVINDJEEt

Division

of

Applied Mechanics, Department of Mechanical Engineering, Stanford Unioersi ty, Stanford, California

94305,

U.S.A.

SUMMARY

Rate-independent plasticity and viscoplasticity in which the boun dary

of

the elastic dom ain is defined by an

arbitrary number of yield surfaces intersecting in a non-smooth fashion are considered in detail. It is shown

that the standard Kuhn-Tu cker optimality conditions lead to the only computationally useful characteriz-

ation of plastic loading. On the computational side, an unconditionally convergent return mapping

algorithm is developed which places no restrictions (aside from conv exity) on the func tional forms

of

the yield

condition, flow rule and hardening law. The proposed general purpose procedure is amenable to exact

linearization leading to a closed-form expression of the so-called consistent (algorithmic) tangent

moduli.

For

viscoplasticity, a closed-form algorithm is developed based on the rate-indepen dent solution. The methodol-

ogy

is applied to structural elements in which the elastic domain possesses a non-smooth boundary.

Numerical simulations are presented that illustrate the excellent performance

of

the algorithm.

1.

I N T R O D U C T I O N

In recent years, a general methodology for the numerical integration of general elastoplastic

constitutive equations has been developed a s an extension of the classical radial return algorithm

of W i l k i n ~ ~ ~or J,-flow theory . Th e math ema tical analysis of these type of algorithm s goes back

to Moreau, who coined the expression catching

up

algorithms. Related work

is

contained in

N g ~ y e n ~ ~nd Matthies,16 among others. Currently, these algorithms are viewed as product

formulae emanating from a n elastic-plastic op era tor split, a rather useful interpretation in

com putation al imp lementations. Although for single surface plasticity general purpose techniques

a re available ; i.e. Or ti z and P O P O V , ~ ~imo and Taylor,36 Simo and or ti^^^ and Simo and

Hughes,31 with th e excep tion of Maier,13 these ideas have no t been systematically extended to the

case of multiple non-smooth yield surfaces.

This paper is concerned with the formulation a nd numerical implementation of elastoplasticity

and viscoplasticity in the case of an elastic domain defined by multiple convex yield surfaces

intersecting in a non-smooth fashion. This situation is of considerable interest in many

applications, such as soil mechanics (Cam-clay an d ca p models), rock m echanics an d struc tural

mechanics (entailing rods, plates o r shells) where the bou ndary of the elastic doma in is typically

non-smooth when formulated in terms of stress resultants. An essential ingredient of the present

approach concerns the formulation of app ropria te loading/unloading conditions for non-smooth

* Associate Professor of

Applied

Mechanics

Graduate Student

~

0029-598 1/88/102161-25$12.50

0 988

by John Wiley & Sons, Ltd.

Received 4 June 1987

Revised I7 February 1988


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2162

J. C.

SIMO,

1.

G .

KENNEDY A N D

S. GOVINDJEE

multisurface plasticity. We show tha t the sta nd ard Kuhn-Tucker optimality conditions of convex

mathematical programming (see e.g. Luenberger 12) provide the only computationally useful

characterization of plastic loading/unloading . These conditions are essentially equivalent t o the

multisurface co unterpa rt of the cond itions in Ko iter (Reference

11,

page

173).

Within the context

of

strain driven

formulations, which is the standard set-up in computational plasticity, the

Kuhn-Tucker form of the loading/unloading conditions implies the generalization of the

loading/unloading conditions of single surface strain space plasticity as given, for instance, in

Naghdi a nd Trapp. These latter conditions, however, d o not suffice to determine the active

surfaces during plastic loading.

Computationally, a general closest-point-projection algorithm for multi-surface plasticity is

developed, which is unconditionally convergent, an d is capable of accomm odating a n arbitrary

number of yield surfaces intersecting in a non-smooth fashion. In the implementation of this

algorithm, the discrete version of the Kuhn-Tucker cond itions plays a central role. In sh arp

con trast with single surface plasticity, violation of a con strain t (yield con ditio n) by the trial elastic

stress does not insure that the constraint is active. A systematic procedure for determining the

active constrain ts o n the basis of the Kuhn-Tucker cond itions is developed. An addition al

imp ortan t feature

of

the proposed general purpose algorithm is that of being amenable to exact

linearization leading to a closed-form expression of the so-called consistent (algori thmic) tangent

moduli. As shown in Sim o and T aylor,35 hese moduli m ay differ substantially from the classical

continuum elastoplastic tangent moduli. Moreover, use of these moduli is essential in order to

attain qu ad rati c rates of asymp totic convergence in global New ton schemes, and super-linear

rates of convergence in global quasi-Newton meth ods employing periodic re-factorizations, a s in

Matthies and S trang , and H a l lq u i~ t . ~

Fo r viscoplasticity, it is shown t ha t formulatio ns of the P erzyna type2 6 are, in general, not

meaningful when the elastic domain is defined by a number of surfaces intersecting in a non-

smooth fashion. This difficulty is by-passed

by extending the formulation proposed by

Duvaut-Lions3 to accom mod ate hardening variables. Remarkably, for this model, a

closed-form

unconditionally stable

algorithm can be constructed from the trial state and the solution to the

rate-independent problem. This approach is at variance with current computational approaches

to viscoplasticity based on Perzyna-type models, i.e. Zienkiewicz and Cormeau,4O Cormeau,

Hughes and Taylor,6 Pinsky et al.

Th e proposed m ethodology is applied to structural elem ents which are typically characterized

by multiple yield surfaces intersecting non-smoo thly. In particular, a n elastoplastic beam m odel

formulated in terms of stress resultants with an elastic domain bounded by two convex yield

surfaces intersecting in a non -smoo th fashion is considered. Th e numerical simulations presented

illustrate the excellent performance of the proposed algorithmic treatment.

2.

RATE - INDEPEND ENT MULTI -SURFACE PLASTICITY. CONT INUU M

F O R M U L A T I O N

Let

L2

c R3 , a boun ded region with smooth boundary

an,

be the reference configuration of the

body of interest, an d let u: R

+

R3 be the displacement field of particles a t points x E R .We denote

by

E

the linearized strain tensor,

= vsu

:

= +[VU + VU)T]

(1)

and we designate by ( z P , q ) the plastic strain tensor and a suitable set of internal variables,

respectively.


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NON-SMOOTH

MULTI-SURFACE PLASTICITY A N D

VISCOPLASTICITY

2163

2.

.

Basic equations

function W ( E

P) so

that

Let a be the stress tensor. The

elastic response

is characterized in terms of a strain energy

a

=

VW(E-EP)

and

C : =

V2W(z-$)

(2)

Typically, one assumes tha t the elasticity tensor C is constant. We consider the case in which the

elastic domain, denoted by

IE

c R6 x

R4,

s defined as

(3)

where

f a ( q

q) are m

2

functions intersecting in a possibly non-smooth fashion. Thus, the

boundary aLE of IE is given by

IE:= {(c,)ER6 x Rq

If.(a,

q) }

d L E : = {(a,q ) E R 6

x

Rq

L(a,

q) = 0, for some

CLE

1,2 , . . . , m } }

(4)

We further assume that the m 2 1 functions f. c, ) are smooth and define

independent (non-

redundant) constraints at an y (a, ) E dlEt and that IE u

d E

s a closed convex set. For simplicity,

the evolution

of zP

is

given by an

associative

flow rule expressed in K oiter's form as (see Koiter,"

Man del" and the review articles of Koiter," and Naghdi")

m

Here j are m

2

I functions, referred to as plastic consistency parameters, which satisfy the

following Kuhn-Tucker complementary conditions for

u

= 1,2 ,

.

. .

,

m:

2 0, d(a,

9) < 0

j f(a,

q)

=

0, and

taf(a,q)

= O f

(6)

Requirement (6)4 is the so-called consistency condition. Conditions ( 6 ) are essentially the

counterpart

in

multisurface plasticity of those in Koiter (Reference

11,

equation

(2.19))

and are

employed by Maier,13 Maier and G r i e g ~ o n , ' ~nd m ore recently by O rtiz and P O P O V ~ ~nd Simo

and co-workers.

The evo lution of the internal variable vector q is specified in term s of a general ha rden ing law of

the form

The associative or potential form of this hardening law may be written asp

where D R qx R4 is a symmetric matrix which, without loss of generality,

is

assumed to be

constant. It can be show n th at the associative form (8) of the harden ing law is the direct result of the

principle of

maximum

plastic dissipation (e.g. see Simo an d Honein3' o r Simo et

~ 1 . ~ ~

'The fact that dim ZE =

6

+

q is finite limits the number of independent surfaces which can intersect at one point (u, ) E d E

in order for the vectors {d,f=(u,q)} (and {d,fh(a,q))) to remain

linearly independent.

For example, if q = O and dim l E = 6

then at most six independent surfaces can intersect at one point

*Thisparameterization is essential to obtain a symmetric from of the discrete return mapping algorithm and algorithmic

elastoplastic tangent moduli in Section 3. a may be interpreted as the thermodynamic force (affinity) conjugate to q

The summation convention on repeated indices is not enforced in this paper


4/25

2164

J.

C.

SIMO, J. G. KENNEDY AND

S.

GOVINDJEE

2.2. Loadinglunloading conditions

denote by Jadmhe set

of mad,

indices associated with these constraints; i.e.

Let

madrn< m be

the number

of

constraints that may

be

active at a given point

(a, ) E dlE,

and

Jadm

:=

D

E

{

1,2,

a ,

m }

I

&(a,

q )

= O }

(9)

Before proceeding further, we make the following additional assumption concerning the

degree

of

allowable softening in the hardening law (8).

Assumption 2 .1. The hardening law (8) is assumed to obey the following inequality at any

(a, )E am

(10)

gaj (a,q) := Caaf,

:

C

:

, +

aqf,.Daq&l

1

1 a g a s ( a , q ) t s O , for

t a R

U E J a d m BEJadm

For perfect plasticity this assumption follows from the standard requirement that

g:C:e> 0

for

all

gT

=

5.

In addition, note that (10)

does

not

preclude

softening.

For the simplest one-dimensional

linear isotropic hardening model, it can be easily shown (see e.g.

Simo

and Hughes32) hat (10)

reduces to the requirement that the hardening modulus

H >

-E, where

E

>

0

is the Young's

modulus.

Now let aJadm.

y

the chain rule along with ( 5 ) and

(8)

the value o f i is given as

f h

8,

f,:C :

E -

1

[a,f,: C: a, +

dqf,.Daq&]ljs

DEJadm

=

a,f,

:c

:

8 - 1 gap(a,)jJP (11)

If

(a,Q)E

~ L E

nd aEJadm then

i (a ,

q ) < o (12)

PEJadm

where gms(a,q) re defined in (10).A straightforward argument shows that

We have the following.

Proposition

2.1 .

Let

E

be given. The Kuhn-Tucker conditions (6) and assumption

10)

imply the

following (strain space) loading conditions.

If J a d m = 0 hen E P = O and q = O

If

Jadm# 0

hen:

(13)

i) Ifd,f,:C:i


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NON-SMOOTH MULTI-SURFACE PLASTICITY

A N D

VISCOPLASTICITY

2165

(ii) Let

LY eJadm

0 be such that

d,f, :

C :

>

0.Suppose

it

were possible that

Ep = 0

and 4

= 0.

Then,

(1

1) would imply that

fh(a,q)= a , f , : c : t 0

(15)

which is in contradiction with

(12).

Thus (ii) holds.

It should be noted that, if plastic loading takes place at

0,

q)

E

alE and several yield surfaces are

active, then the condition

a,f,

:C

:

> 0 does

not guarantee that

f,will

ultimately

be

actiue.

This

observation is central to our subsequent developments and is illustrated in Section

2.4.

0

BOX

1.

Infinitesimal multi-yield surface plasticity

(i) Elastic stress-strain relations

Q = V W(E- ~ ) , here:

E

:= V

(ii) Associative

f low

rule

m

i p

c

9 &f,(Q,

n)

a =

1

(iii) Hardening law

m

u = 1

4

= 1

aa,f,(Q,

9)

=

- D - 1

(iv) Yield and loading/unloading conditions

f,(Q,q) ,0.

These conditions are a re-statement for multisurface plasticity of classical conditions, see i.e.

(Reference 11, equation (2.19)). Let

matt

be the number of constraints at a given point for which (ii)

holds, and set


6/25

2166

J . C. SIMO. J . G.

EN N ED Y A N D

S.

GOVINDJEE

Then, since j a s non-zero only for CIE

J,,,,

it follows from (1 1) that

fhca-9) =o* c

gap(a,

)?, = a,f, :c :3

(17)

BeJ, I,

for all

CIEJ,,,.

This leads to

a

system of

mact

equations with

m a d m ~ m , , ,

nknowns. Conditions

3,

= 0 if]

(a,

)

< O then provide the remaining madm-

ma=,

quations th at render (17) determinate.

In summary, we have

3,

=

0,

if B

4

J,,,

= 1 8 (a, q)

[ f,

a,

) :

C

: ],

if c1E J,,,

B E J m ,

where gap(a, )= [g,,(a, q)] - By substituting (18) nt o the rate form of the stress-strain relations

(2), we obtain b=C'P:b, where CeP are the elastoplastic tangent mo duli given by the expression

(19)

42

iff

J a c l= O

For convenience, the basic equations governing classical multi-surface rate independent

plasticity are summarized in BOX 1.

2 .4 . Geometric interpretation

to above, that

a constraintf,

may be active; i.e.

For simplicity we consider perfect plasticity. At each

a~i3lE

e have the vector space

We give a geometric interpretation of the loading con ditions (6) and illustrate the fact, alluded

>

0 an d, nevertheless, one may have

&fa :

C bc0.

I M:= span [g, := d,f,. for

a

EJadm

(20)

We equip TM with the inner product induced by

C t

according to

(10);

an d define the dual vectors

(co-vectors)

{ g d } a , J a , m

in the standard fashion, i.e.

g,,

:=

g, :C :g,,

and g = 1 , g,

BE

Jadm

Given 3, conditions (13) define the

accessible elastic region

as the

cone

I M - = g E R6

6

:

C g,

d0)

(22)

whereas the plastic region is IM

-IM-.

he normal cone IM+ s given by (see Figure

1)

A straightforward computation then shows that

[ 3

I n the presence of internal plastic variables, the inner p roduct is induced by


7/25

NON-SMOOTH MULTI-SURFACE PLASTICITY A N D VISCOPLASTICITY

2167

Figure 1 . Illustration

of

the geometry at a singular point

a ~ 8 1 E

ntersection

of

two yield surfaces

( J , , , =

{ l ,

2

Figure 2 Positiveness of the contravariant components y>O does not guarantee positiveness of the covariant

components a& :

C

:k =

y a p

v p

Therefore, for

P

E

IM

+,

3"

and

d,f,

:

C

i

may be interpreted as the

contravariant

and

couariant

components

of

E relative to { g a } , respectively. Th e fact th at

3">0

+ a,f,:c:s>o (25)

is illustrated in Figure

2.

3.

DISC R ETE FO R M ULA TION. RATE INDE PEN DE NT ELASTOPLASTICITY

The evolution equations

of

multisurface elastoplasticity, as summarized in BOX

1,

define a

unilaterally constrained problem of evolution. By application of an implicit backward Euler

difference scheme, this problem is transformed int o a constrained optimization problem governed

by

discrete Kuhn-Tu cker conditions. We examine the structure of this discrete problem, the

fundam ental role played by the discrete Kuhn-Tuck er conditions an d the geometric interpret-

ation of the solution a s the closest-point-projection in the (com plementary) energy norm of the

trial elastic state onto the elastic domain.

3.1. Strain-driven algorithmic fram ework. Closest-point-projection algorithm

plastic strain fields and the internal variables are known ; tha t is

Let

[0, T ]

R be the time interval of interest. At time t ,

E [0,

T ] we assume that the total an d

{q,,:, a,} given dat a a t t , (26a)


8/25

2168

J. C. SIMO. J.

G.

KENNEDY AND

S.

GOVINDJEE

Note that the

elastic strain

tensor, the

stress

tensor and

q =

-Da are regarded as

dependent

variables which can always be obtained from the basic variables (26a) through the relations

E;:= E , - - E ; ,

a n = V

W(E;) ,

qn=

-Dan (26b)

Let

A u

:

R+R

be the incremental displacement field, which is

assumed to be gioen.

The basic

problem, then, is to update the fields (26) to t , + E [0,

T ]

in a manner consistent with the

elastoplastic constitutive equations summarized in

BOX

1.

By ap plying a n implicit backw ard Euler difference scheme to the evolution equations

(BOX

1)

and making use

of

the initial conditions (26), one is lead to the following discrete

non-linear

coupled system:

E , , + ~= & , + V S ( A u ) (trivial)

Qn

+ 1 = v

W E ,

+ 1

-

: + 1)

&:+I=&::+

c Y i + 1 & L ( Q n + l , q n +l )

m

a = 1

To simplify the no tation in

(27),

we have defined y : + :=

A t

.jpl+ in place of the more appropriate

symbol

A y ; +

1. In additio n, the discrete coun terpart of the Kuhn-Tucker conditions becomes

+ 1 f a a n + 19 q n + 1) = O

for

c t=

1 , 2 , .

.

.

,

m.

As

in the continuum case, the Kuh n-Tucker conditions (28) define the

appropriate notion of loading/unloading.

In order t o interpret geometrically the algorithm (27) an d implement the loading/unloading

conditions (28), one introduces the following

trial elastic state:

e trial

._

+ 1

.-

+ 1 -e

a;?: :=

v w( ;Fy)

trial p

._

trial ._

& , + 1

-- :

an

1- an

qtrial ._

-D

n +

1

.-

jh,::

:=f,( q:

From a physical standpoin t the trial elastic state is obtained

byfreezing plast ic jow

during the time

step. This trial state arises naturally in the context of an

elastic-plastic oper ator split

(see Simo and

or ti^^^

and Ortiz and Simo). Observe that only

function evaluations

are required in definition

(29).

3.2. Geometric interpretation. The notion of closest-point-projection

In terms of the trial sta te (29), he solution to (27) admits a com pelling geometric interpretation

which is crucial to its numerical implementation. We assume that the elasticity tensor


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NON-SMOOTH MULTI-SURFACE PLASTlClTY A N D VISCOPLASTICITY

2169

C:= V2W ( E

E P )

is constant. In addition, we further assume that

D

is positive definite. Then, we

have the following.

Proposition

3 . 1 .

The solution problem (27)- (28) is unique, and is characterized as the argument

of

the minimization problem

I

where

II

qn

-q

II6 :=

[qn -qI

:D- :

Cqn-Q1

Proof:

Assuming that

(an+

qn+

)

is characterized by (30a, b), it is unique. This follows from

standard results in convex analysis (see, e.g. Reference

28,

Section

3)

by noting that (a) ,y :

E-+R

s

strictly convex sinceC nd

D

are positive definite and (b)

ZE

is a closed convex set sincef,

*,

.)

is

convex. To prove the equivalence of

(30)

and

(27)-(28),

consider the Lagrangian

Then, the Kuhn-Tucker optimality conditions for an extremal point t, ,

2')

=(an qn+

y i +

(see Reference

12,

p.

314

or Reference

37,

p.

724)

yield

L = f a ( a n +

1,

qn

+ 1)

0

y :+ 1 2 0

Y:

I f

( o n

+ 1, Q.+

1) = O

which are equivalent to

(27H28).

0

The geometric interpretation of this proposition is shown in Figure 3 for perfect plasticity.

a,+,

=P,

is the closest-point-projection (relative to

C-') of a

onto the admissible

region LEuaE. Here,

P,: R 6 4 E

enotes the

orthogonal projection

(relative to

C-')

onto E.

Since IE is convex,

LP,

at?; is unique for any a R 6 . A similar interpretation is considered in

Reference 16.In the presence

of

internal variables, expression (30b) is consistent with Reference

5 .

The final state (a,,+

,

qn+

)

is now the closest-point-projection in

Lspace, Z:=

(a, ), of

(a;?;,

qt$i) onto the boundary of the elastic region

BE

in the metric defined by

G:=[

- ]

i.e. ( ( C l ( ~ : = C : G : C ~ a : C - ' : a + q - D - ' q

D-1

(33)

3.4. Loadinglunloading. Discrete Kuhn-Tucker conditions

exclusively in terms

of

the trial elastic state, as follows.

A

basic result from

a

computational standpoint is that loading/unloading can

be

characterized


10/25

2170

J.

C.

SIMO,

J .

G.

K E N N ED Y A N D

S .

GOVINDJEE

CLOSEST-POINT-PROJECTION

IN THE METRIC DEFINED

I

B Y C

Figure

3.

Geometric illustration of the concept of closest-point-projection

IPE: R 6 4 E

Proposition 3 . 2 . Assuming that f, :R6 R4+R, CL = 1 , 2 ,

. . . ,

m, are convex, one has the

following computational statement of the loading/unloading conditions (28).

fh','i'+G O for

all

tl E (1,2, . . .

,

m)*Elastic step

fi,ii'+> O

for some

BE 1 ,2 , .

.

.

,m)*Plastic step

(34)

Proof:

( i )

I f f e i ' + c0 or all U E

{

1,2 , . . . ,m } then (o 9 is admissible. Thus,

trial

G n + l = o n + l ,

q n + l = q n i

Y : ~ = O

for all CLE 1 , 2 , . . . ,

m}

is a solution to (27H28). Since the solution to (30) is unique, this

constitutes the ac tual solution, and the step is elastic.

(ii) Suppose that there exists at least one B E

{ 1 , 2 , .

.

.

,m} uch thatfl;';', > O . Then, crfy\ is

not admissible; hence, the step is plastic.

0

Remarks 3.1 .

1 . If only

one

yield surface is active (i.e. y +l

> O

for only one

B E

(1,

2, . . .

,m } ) , then the

condition f

F,i;l+

>

0

does imply th at yfl+ >

0;

i.e. the ,+constraint

is

active.

2.

If seoeral

yield co nd itio ns a re ac tive, thenf:,ii'+

> O does not imply

that

y : + > 0

tha t is, one

ma y havefh','i'+ >

0

butf,, ,+ O,

a= 1,2 ,

define a corner mode region r I 2 IM+n

Section

2.4)

in stress space spanned by

{

C

d,f,,,

+

l }

in which

fy : : '+

>

0

and

f:fa,l

>

0. If

oXyi

E

r12,hen on+ is at the intersection (corner) of the two surfaces. O n the other hand , within

regions

rl

nd

T2

on e also hasf'$ +

> O

andf;:t+

>0,

but y l + c in region

r l

and y j + < O n

region

Tz.

A systematic procedure for determining the active yield surfaces will be discussed in

Section

3.5.2.

C

3.5.

General multisurface closest-point-projection solution algorithm

By way of motivation, we consider first the case of single surface plasticity.

In w hat follows, we present a ge neral algorithm for the numerical solutio n of problem

(27H28).


11/25

NON-SMOOTH

MULTI-SURFACE

PLASTICITY A N D VISCOPLASTICITY

2171

110 f,=O

\

f,=o

(c)

Figure

4.

Ccom etric illustration

of

the geometry at a comer point a ~ a l Entersection of two yield surfaces .I,,,=1,.2}):

(a) definition of regions rl .rZrnd

rL2;

b) region

rl

s characterized by v, +

I

>O, v:+, O , Y , 2 + 1 > 0

3.5.

. Motivation. Convex programming.

Without loss of generality we shall restrict our

attention to perfect plasticity. With reference to the characterization in

Proposition

3.

I,

we

consider the L agrangian

(35)

(t, I.

:=

x t ) +

Observe that the derivatives of

L(r, .

are given by

a6 (t,

4

V f ( 4

We then consider the following Newton algorithm.

1 .

Define the residual at iteration k)by

2. Check whether convergence is attained.

I F IIVL k)llO, is

in the metric ddfined

by

the elasticities Che closest point of that level set to the previous iterate


13/25

NON-SMOOTH MULTI-SURFACE PLASTICITY A N D VlSCOPLASTlClTY

2173

Conceptually, the extension of the preceding algorithm to the case of several constraints in the

presence of internal variables is straightforward and is constructed based on the following

Lagrangian:

where~ t ,

)

is defined in (30) and Jac,

{ 1,2,

. . .

,m }

is the set of indices associated with the active

constraints at the unknown solution point (en+

q,,,

that is,

Jact

:=

{a

192, * * m }

If,(am+ 1, q n 1)=0}

(39)

The difficulty associated with multisurface plasticity, however, is that

J,,,

is not known in advance,

since, as noted in Remarks 3.1,fc: + O does not guarantee thatf,,,,,, = O .

3.5.2. Determination

of

the active constraints. A yield surfacef,,

,,+

, s termed active if y :+ >O.

A systematic enforcement of the discrete Kuhn-Tucker condition (28),which relies on the solution

of

the plastic return mapping equations (27), then serves as the basis for determining the active

constraints. The starting point in enforcing (28) is to define the trial set

J:::':= { a ~ { 1 , 2 , ..

m } I f ~ : ' + l > o }

(40)

where

J,,,

c

: In

order to determine the final set Jat two procedures may be adopted.

(i)

Procedure

1

(conceptual). One proceeds as follows (see Figure

4).

(il) Solve the closest-point-projection iteration with Jnct=J

(i2) Check the sign of

?;+

to obtain (a,,+, En+,,q,,, 1),

If

Ti+

< O ,

for some B E F;:', drop the p-constraint from .It:: and

along

with ?:+ 1,

aEJFL,B'.

go to (il). Otherwise exit.

BOX

2a. Elastic predictor

1.

Compute elastic predictor state

a

= v W(&,,+l

&,P)

jz;l

:=fa(etrial

a +

1, qn),

for (1,2,. . . m }

2. Check for plastic process

IF

f ~ ~ ' + , ~ O f o r a Z l a ~ { 1 , 2 ,

. . ,

m}THEN:

Set (*),,+

,

( *) ? and EXIT

Ji:i:=

{ ~ 1 ~ { 1 , 2 , ..

, m } I f ~ i ~ : l > O }

&,Py)1=

,P

y;y\ =0

0

ELSE

a ,?

=

a,,

GO TO

BOX

2b

I

ENDIF


14/25

2174

J. C. SIMO, J .

G. E N N E D Y

A N D S .

GOVINDJEE

(ii) Procedure 2. In Procedure

1

the working set J remains unchanged during the iteration

process, and the admissibility req uirement tha t

y i + > O

is tested only w hen a converged solu tion is

obtained. In this procedure, on the othe r hand, the set

J :

is allowed to chan ge during the iteration

process, as follows.

(iil) Let

J i t \

be the w orking set at the (k)th iteration. com pute increments

AYE($)^,

C X E J ~ ~ ) , .

(ii2) Up date a nd check the sign of

yiki

If y t y ,


15/25

NON-SMOOTH

MULTI-SURFACE

PLASTICITY A N D

VlSCOPLASTlClTY

2175

N ote tha t the structure of (47) is entirely analo gous to expression

(19).To

obtain the algorithmic

tangent m oduli all th at is needed is to replace the elastic moduli C,+ in the expression for the

continuum

elastoplastic mod uli by the

algorithmic

moduli

En+

defined by (43). Th e cou nterp art of

(47) for single surface plasticity is derived in Simo [1986].

BOX 2b. Gen eral multisurface closest-point-projection iteration

3. Evaluate

flow

rule/hardening law residuals

c i k i l = VW ( & n + l - & ~ y ) l ) ; qikl1= D a ( k )+ 1

4. Check convergence

6. Obtain increment to consistency parameter and check active constraints

A

I J ~

c

ccO pxki c f - aUjP,aPfPi A

:~ i i ~ i

P E

J

it\

- a ( k

+ 1)=

a ( k )

Y

+

1

IF: -2t(:T1)0)

GO T O 4.

ELSE

? a ( k + l ) - - a ( k + l )

J it;

1) = J

( k )

n + l - Y n + l 7

ac t

G O T O B O X 2 c 0

E N D I F


16/25

21 76

J. C. SIMO, . G. K E N N E D Y A N D S.G O V I N D J E E

4.

EXTENSION TO

VISCOPLASTICITY

Here we consider an extension to viscoplasticity based on the model proposed by Duvaut and

Lions.3 Th e stand ard formulation of viscoplasticity as suggested by P erzyna Z6 s not well suited

for the multisurface context, as the following discussion shows.

BOX

2c. Cont. Closest-point-projectioniteration

7. Obtain incremental plastic strains and internal variables

4 . 1 . Motivation. Perzyna-type models

could be obtained by postulating a flow rule of the form

It would ap pe ar tha t a straightforward extension of inviscid plasticity t o the rate depen dent case

where

yl,

E 0, co) s a fluidity parameter, and ) s the ram p function defined as ( x ) = (x + Ixl)/2.

Unfortunately, as

q + O ,

this model would not reduce

t o

the rate-independent ormulation in

BOX

1,

as the following example illustrates.

Example 4.1. Con sider the case in which two convex functionsf,(o) andf,(a) intersect in a non-

smo oth fashion, as shown in Figure 6.

In

the limit as q + O , since bothf, >0 andf, >0, (48)would

predict the return path and plastic strain rate to be a s shown in Figure 6(a) and a+& The actual

solution for the inviscid case, however, corresponds to the solution shown in Figure 6(b) in which

o n l y f ,

= O

is active. Hence, as in the inviscid case, we have

fa(a)

>

0

+

a-constraint is active

Th e model discussed below precludes this difficulty and properly reduces to the inviscid limit.

0

4.2. Extension

of

the Duvaut-Lions model

Lions3 proposed the following constitutive model:

Fo r the case of a single loading surface characterized by perfect viscoplasticity, D uv au t and


17/25

NON-SMOOTH MULTI-SURFACE PLASTlClTY AND VISCOPLASTICITY

2177

12'0

f,=O

Figure 6. (a) Inviscid limit return path

for

Perzyn a-type multisurface mod els. (b) Actual inviscid return path

where

E,:= {ueR6 f(u)


18/25

2178

J . C.

SIMO. J.

G. KENNEDY AND S .

GOVINDJEE

may then b e integrated in closed form t o o btain

Using the approximations

an d proceeding in th e same m anner with equation (51)2, one obtains the algorithm for

viscoplasticity sum marized in

BOX

3. Note that , with q =

-Da,

51), takes the form

Remarks

4.2.

1. The elastic and inuiscid cases are recovered from th e preceding algo rithm in the following

limiting situations:

(l a ) Let At/q+O. It follows that exp[-At/q]+l an d (1 -exp[-At/q])/(At/q)+l. Hence,

(lb) Let:

At/q+co.

It

follows th at exp[-At/q]+O, an d

(1

-exp[-At/q])/(At/q)+O. Hence,

n,, -+C

:

A

E,+

+

n,,,and qn+ +qn. Therefore, one ob tains th e elastic case.

cn++ifn +

a n d q n + +qn+ an d o ne recovers the inviscid plastic case.

BOX 3. Closed-form algorithm for viscoplasticity

1. Compute the closest-point projection

2. O bt ain the viscoplastic solution by the formulae

q n + J by BOX 2a-2c

On+

1 =exP(-A t /V)an

+

1 -exp (- A t/?)I*,+ 1

1- xp(

-

A t / q )

A t/V

+ C : A E n + l

2. A lternatively, from (51), by application of an im plicit backw ard E uler algorithm we obtain

the first order accurate formulae


19/25


20/25

2180

J. C.

SIMO,

J.

G. KENNEDY AND

S.

GOVINDJEE

-N

Figure 7. Plot of yield criterion (63) in stress resultant space

Figure 8. Built-in beam. Finite element mesh

Example 1

A double built-in cantilever beam of length 10 is subjected to a linearly increasing point

load 3/4 of the way down its span. The material properties of the beam are: EA=2.55e+06,

G A = 1.25e

+06,

E l = 1.3e +06, kappa = 0.83333,

N o

=403 V o= 23-4, M , = 25.3 (see Figure 8).

First, to assess the accuracy of the algorithm, the computed solution for the problem is

compared w ith the exact solution obtained by assuming moment dominated yielding. Co mp uta-

tionally, the mom ent yield dom inated so lution is obtaine d by a penalty procedure in which N o / M o

and

V o / M ,

are large

(>

lo). Figure 9 shows the load versus displacement curve at the lo ad point

which was generated by the program. Th e break p oints in the graph correspo nd to the formation

of

plastic hinges, first at the wall closest to the load point, then at the load p oint and finally at the

wall furthest from the load point. Exact agreement

is

found between the computed and exact

solutions. Figure 9 is generated with a mesh of 16 elements, using 18 time steps of

h=

1.0,

80

times step of

h = 0 * 1

and

100

time time steps of

h=0*01.

Also shown in Figure 9 (dashed line) is the load versus displacement curve at the load point

when N o and V o are set to their actual values. The s ha rp reduction in the load levels at w hich th e


21/25

NON-SMOOTH

MULTI-SURFACE

PLASTICITY

AN D VISCOPLASTICITY

2181

U

124

30

I I I I

0 2 4 6 0

10

x - position

Figure 10. Built-in beam. Mom ent diagrams for progressive plastic states-ombined moment/shear model

plastic hinges

form

is the effect of combined moment, shear and axial yielding. Figure 10contains

the moment diagram for the beam at load

160

(just before the first plastic hinge forms), at load

17.5

when there is one plastic hinge and at load 21-91just before the collapse load.

To assess the robustness of the algorithm, the problem is solved with substantially larger time

steps. One load step of value

h

= 16.0 and three steps of h= 2.0 are used. A summary of the

residual norms, energy norms and states for each time step can be found in Table I. As expected,

within the radius of convergence of the global problem, a quadratic rate of asymptotic

convergence is obtained. In addition, attention is drawn to the importance of the Kuhn-Tucker

conditions. During the solution process

13

'pseudo corner region situations' were encountered.


22/25

2182

J. C.

SIMO. J . G .

K E N N E D Y A N D S . GOVINDJEE

Table

1.

Built-in beam. Convergence

of

global residual norm

Iteratio n Load 16 18 20 22

State e e e e

1

R

norm

0.1

6e

$02

0.20e

+

0

1

0.20e

+

01

0920e

+

0

1

E norm

0.9le -03

0.14e- 4

0.14e

-

4 048e

-

4

2 R norm

0.39e -03

0%0e

+

00 0.30e+01

0.18e +01

E norm

0.75e- 3

0-20e- 5 0.27e - 4

0.15e

-

4

3 R norm 0.7 le - 8

0 2 5 e

+00 0.1

6e

+

01 066e +00

E

norm

0.37e- 4

0.17e- 7 0.16e- 5

0.28e

-

6

State

e

P

P P**

State

e

P

P*

P* *

State

4

R norm

E norm

State

5

R

norm

E norm

P P* P**

P

P* P**

0.25e

-

2 0.64e

-

0.56e

-

2

0.97e

1

1 0.1

5e- 7 0.46e

-

9

0.25e

-

5

0.1

2e

-

2

0.8

1

e

-

5

0.22e

- 7

0.10e- 10 047e - 16

State

P

P*

P**

6 R norm

0.26e- 8 0.32e -07

0.34e- 8

E norm

0.41

e

-

4

0.1

6e

- 0

O l l e - 2 3

*Recovery from a pseudo corner region, defined by

c

Er,

r

r2 n Figure 4

**Recovery from two pseudo corner regions

f

igure 11. Three

beam

frame. Finite element mesh

Without a proper enforcement of the loading/unloading conditions, the iteration process either

fails to converge to the correct values, or diverges.

Example

2

A

double built-in

3

bar frame is subjected to a linearly increasing off-centre point load on its

horizontal member. See Figure

11

for dimensions and material properties.

As

an accuracy

assessment the exact solution is obtained assuming moment dominated yielding and compared


23/25

N O N - S MO O T H

MULTI-SURFACE PLASTICITY

A N D

VISCOPLASTICITY

2183

with the computed solution. Figure 12 shows the load versus displacement curve under the

applied load. The break points correspond t o the formatio n of plastic hinges, first under the load,

second at the corner nearest the load and lastly at the corner furthest away from the load. The

exact solution is matched exactly using a mesh of

30

elements in conjunction with 3 time steps of

h=4.0, 2 time steps of h = O 5 and 1 1 time steps of h=0.1.

T o assess the robustness of the algorithm a nd th e impo rtance of the proper co rner conditions,

the problem is again solved with the actu al values of the material constants sho wn in Figure 1 I

and employing substantially larger time steps. Th e results are sum marized in Tab le 11. Attention is

again draw n t o the q ua dra tic rate of asymptotic convergence exhibited by the iteration process,

and the role of the proper statement of the loading/unloading at corner regions. A total of 21

'pseudo corner regions' were encountered in the course of the solution, and successfully handled.

7.

C L O S U R E

A

systematic numerical treatment

of

elastoplasticity, for the case in which the b ound ary of the

elastic domain is defined by an arbitrary num ber of yield surfaces intersecting in a non -smoo th

fashion, depends crucially on the formulation of the loading/unloading conditions in Kuhn

Tuck er form. Within the context of strain driven problems, these conditions imply the strain space

loadin g conditio ns unde r mild assum ptions o n the degree of allowable softening. Algorithmically,

the determination

af

the set

of

active constraints in o u r closest-point-projection algorithm

involves merely a systematic iterutiue enforcement of the Kuhn-Tucker conditions. We have

shown that the resulting procedure is amenable to exact linearization and places no

restrictions

(excepting convexity) on the functional forms of the yield condition, flow rule or hardening law.

Since

IE

is conuex, it follows that the proposed closest-point-projection algorithm is un-

cond itionally convergent-wh en comb ined with a line search technique.

Fo r viscoplasticity, the Duvaut-Lions formulation h as been extended to accomm odate

hardening variables, and a closed-form algorithm based on the inviscid solution has been

developed. Formulations

of

the Perzyna-type are, in general, not meaningful when the elastic

dom ain is defined by several surfaces intersecting in a non-sm ooth fashion.

" 0 0.0006

0.0012

0.0018

Displacement at Load

Figure

12.

Three beam

frame. Load4isplaceme nt

curve


24/25

2184

J. C. SIMO.

J.

G. KENNEDY AND S. GOVINDJEE

Table TI Three beam frame. Convergence of global residual norm

Iteration Load 1 1 12 13 13.1 13.15

State

1 R

n o r m

E n o r m

State

2 R

n o r m

E norm

State

3 R

n o r m

E norm

State

4 R n o r m

E

n o r m

State

5

R n o r m

E n o r m

State

6

R

n o r m

E n o r m

State

7 R n o r m

E n o r m

e

O.llef02

038e-02

e

0.91e

+00

0-lle-70

e

0.29e- 6

0-51e- 1

e

O . l O e + O l

032e-04

P

0.3Oe +01

0.24e

- 4

022e+01

015e -05

P

0 13e+00

0.2Oe- 7

P*

P*

055e

-

2

O.lle-11

0.72e- 5

0.16e- 7

031e-06

0 13e

-

1

P*

P*

e

0.10e

+

01

0.32e- 4

P

0 4 e

+

01

0.83e- 4

0.29e+01

0.42e

-

5

0.22e

+ 00

053e-

7

0.1

3e

-

1

0.12e- 10

0.45e

-

4

0.12e-

15

0.26e- 5

0.41e-18

P*

P*

P*

P*

P*

e

0.12e- 5

P

0~4Oe+O1

013e-09

P*

0.9Oe-01

0.8Oe- 4

043e- 4

P*

033e -06

0 18e- 1

e

0.10e

+01

032e-06

P

0.18e+01

014e-04

017e+01

019e

-05

0.75e

-01

063e- 7

052e

-02

0.32e- 1

066e

-

6

041e-21

P**

P**

P**

P**

*Recovery from a pseudo corner region, defined by

a ~ r ,r

r2 n Figure 4

**Recovery from two pseudo com er regions

In the past, the rate-independent case has often been obtained as the inviscid limit of

viscoplastic formulations; a procedu re essentially equivalent to a penalty method . In the present

approach, o n the other hand, the rate-dependent case is obtained from the rate-indepen dent limit

by a closed-form algorithm. Among other things, this avoids well-documented ill-conditioning

problems.

ACKNOWLEDGEMENTS

We thank T. J. R. Hughes and R. L. Taylor for helpful comments. Support for this research was

provided by a g rant from the National Science Foundation. J. G. Kennedy was supported

by

a

Fellowship from the Shell Development Company. This support is gratefully acknowledged.

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international journal for numerical methods in engineering volume 26 issue 10 1988 [doi...

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