international journal for numerical methods in engineering volume 26 issue 10 1988 [doi...
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8/9/2019 International Journal for Numerical Methods in Engineering Volume 26 Issue 10 1988 [Doi 10.1002%2Fnme.1620261
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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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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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MULTI-SURFACE PLASTICITY A N D
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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
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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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A N D
VISCOPLASTICITY
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(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
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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
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NON-SMOOTH MULTI-SURFACE PLASTICITY A N D VISCOPLASTICITY
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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)
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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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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$i) onto the boundary of the elastic region
BE
in the metric defined by
G:=[
- ]
i.e. ( ( C l ( ~ : = C : G : C ~ a : C - ' : a + q - D - ' q
D-1
(33)
3.4. Loadinglunloading. Discrete Kuhn-Tucker conditions
exclusively in terms
of
the trial elastic state, as follows.
A
basic result from
a
computational standpoint is that loading/unloading can
be
characterized
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J.
C.
SIMO,
J .
G.
K E N N ED Y A N D
S .
GOVINDJEE
CLOSEST-POINT-PROJECTION
IN THE METRIC DEFINED
I
B Y C
Figure
3.
Geometric illustration of the concept of closest-point-projection
IPE: R 6 4 E
Proposition 3 . 2 . Assuming that f, :R6 R4+R, CL = 1 , 2 ,
. . . ,
m, are convex, one has the
following computational statement of the loading/unloading conditions (28).
fh','i'+G O for
all
tl E (1,2, . . .
,
m)*Elastic step
fi,ii'+> O
for some
BE 1 ,2 , .
.
.
,m)*Plastic step
(34)
Proof:
( i )
I f f e i ' + c0 or all U E
{
1,2 , . . . ,m } then (o 9 is admissible. Thus,
trial
G n + l = o n + l ,
q n + l = q n i
Y : ~ = O
for all CLE 1 , 2 , . . . ,
m}
is a solution to (27H28). Since the solution to (30) is unique, this
constitutes the ac tual solution, and the step is elastic.
(ii) Suppose that there exists at least one B E
{ 1 , 2 , .
.
.
,m} uch thatfl;';', > O . Then, crfy\ is
not admissible; hence, the step is plastic.
0
Remarks 3.1 .
1 . If only
one
yield surface is active (i.e. y +l
> O
for only one
B E
(1,
2, . . .
,m } ) , then the
condition f
F,i;l+
>
0
does imply th at yfl+ >
0;
i.e. the ,+constraint
is
active.
2.
If seoeral
yield co nd itio ns a re ac tive, thenf:,ii'+
> O does not imply
that
y : + > 0
tha t is, one
ma y havefh','i'+ >
0
butf,, ,+ O,
a= 1,2 ,
define a corner mode region r I 2 IM+n
Section
2.4)
in stress space spanned by
{
C
d,f,,,
+
l }
in which
fy : : '+
>
0
and
f:fa,l
>
0. If
oXyi
E
r12,hen on+ is at the intersection (corner) of the two surfaces. O n the other hand , within
regions
rl
nd
T2
on e also hasf'$ +
> 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).
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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
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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
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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 ,
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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
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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
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NON-SMOOTH MULTI-SURFACE PLASTlClTY AND VISCOPLASTICITY
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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)
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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
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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
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MULTI-SURFACE
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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.
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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
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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
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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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H. Matthies, Problems in plasticity and their finite element approximation, Ph.D. Thesis, Department of Mathe-
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J. J. Mo reau , Evolution problem as sociated with a moving convex set in a H ilbert space,
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M. Naghdi, Stress-strain relations in plasticity and thermoplasticity, in
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B. G. Ne al, The effect of she ar and n ormal forces on the fully plastic moment of a beam of rectangular cross section, J .
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M . Ort iz and
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C. S imo, An analysis of a new class of integration algo rithms for elastoplastic constitutive relations,
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