Inlernatlonal Journal of Plasticity, Vol. 3, pp. 391-413, 1987
0749-6.119/87 $3.00 -1- .00
@ 1987 Pergamon Journals Ltd.
Printed in the U . S . A .
A STUDY OF T W O A L T E R N A T E T A N G E N T M O D U L U S
FORMULATIONS AND ATTENDANT IMPLICIT ALGORITHMS
FOR C R E E P AS W E L L AS H I G H - S T R A I N - R A T E PLASTICITY*
S. YOSHIMURA, K.L. CHEN, and S.N. ATLURI
Georgia Institute of Technology
A b s t r a c t - T w o alternate tangent modulus formulations based on the use of material characteristics expressed as ~vp = g(i~.p,#) and 8 = h(i°P,~P), respectively, are presented for the
analysis of material response under conditions such as high temperature creep and high strainrate dynamic plasticity. In each formulation, implicit algorithms of generalized midpoint radial
mapping are presented to compute stress histories at a material point. Several examples to illustrate the stability, accuracy,and convergenceof the presented computational methodologiesare
included. In each instance, the two alternate tangent modulus approaches lead to results of comparable accuracyand compare excellently with each other as well as other available independent solutions.
I. INTRODUCTION
In this article we describe two alternate tangent modulus finite element methods for the
analysis of a variety of problems, ranging from quasistatic problems of material creep
under high temperatures to dynamic response of elastic-plastic solids under very high
strain rates. The material is assumed to harden, exhibiting both strain hardening as well
as strain-rate hardening.
Material constitutive modeling and attendant computational methodologies for rateindependent elastic-plastic behavior are now quite well understood (cf. ArLtrRI [1985]).
Such is not the case, however, for viscoplastic (i.e., rate-sensitive) material behavior.
Early constitutive models for viscoplastic behavior are typified by those of PERZYr~A
[1966], and BODNER & PARXOM [1975]. Constitutive modeling of rate-sensitive inelastic
behavior, using the concept of an internal time, has been discussed recently by W^TArq~E e~ ATLtrRI ([19851, [1986]) based on the original proposals by VAZAmS [1980]. The
information on the strain-rate hardening function, for very high strain rates, that is used
in any constitutive model referred above, is generally not available. Limited experimental results in this regard have been reviewed by CLIFTON [1983], FROST e* ASHBY [1982]
and by CA.'aPBELL& FERGtrSON [1970].
Although the tangent modulus finite element method (wherein the constitutive relation at a material point at the current time depends on an a priori assumption of plastic loading or elastic unloading) generates only a "guess" value for the strain increment,
the appropriate "constitutive-relation-correction" algorithm must be used to correctly
predict the inelastic strain increment and the corresponding stress increment in the considered time increment. An implicit, generalized, midpoint radial mapping algorithm is
*This work was done when S. Yoshimura was a visitor at Georgia Tech from the University of Tokyo,
1985-1986.
391
S. YOSHIMURAet
392
al.
presented in this article, for this purpose, for creep as well as high-strain-rate dynamic
plasticity cases.
The remainder of the article is organized as follows: (1) in Section III we present tangent modulus stress-strain relations in the incremental forms for the two alternate material characterizations of o = h(~t'P,~ vp) and ~'P = g(~VP, O); (2) in Section IV we
present a midpoint radial-return algorithm for computing stress histories for the given
strain histories; (3) in Section V we present several examples pertaining to high-strainrate dynamic plasticity to illustrate the numerical stability, accuracy, and convergence
features of the presented implicit tangent modulus algorithms; and (4) in Section VI we
present some concluding remarks.
I1. ALTERNATE T A N G E N T MODULUS F O R M U L A T I O N S FOR CREEP AND VISCOPLASTICITY
We consider here two types of rate-sensitive phenomena, creep and viscoplasticity.
For metals operating at elevated temperatures, the strain in uniaxial tension is known
to be a function of time, for a constant stress of magnitude even below the conventional
elastic limit. Based on experimental data (for instance, FINNIE a HEtLER [1959]), the
creep strain ~c under constant stress in uniaxial tests, is expressed as:
(1)
~c = A a " t "
where a is the uniaxial stress, and t the time. The creep rate, ~c, may be written as:
~ =f(#,
t) = m A O ' t m-~
(2)
or equivalently,
~c = g ( a , ~ c ) = m A L/m(a)n/"(ec) I - t / "
(3)
The function f ( G t) is often characterized as time hardening, while g(#,~¢) is labeled
as strain hardening. Since eqns (2) and (3) are valid under constant stress, they do not
necessarily give the same results when integrated for variable stress histories. Usually
strain hardening leads to a better agreement with experimental findings for variable
stresses.
Note that, in the case o f creep, although the function g(#, ~c) in eqn (3) may be
determined more naturally, it may be inverted to yield
= h(~C,~ ~)
(4)
.
In the case of multiaxial stress states, one m a y define equivalent stress and equivalent
strain rate through the relations
3
0 2 = ~ (,r':o')
2 ~¢:ic
(~c)2 = 3
(5)
where a ' is the deviatoric part of the stress tensor, and i c is the tensor of creep-strain
rate.
Two alteraate tangent modulus formulations
In
393
view of eqn (5), a generalization of eqn (3) to multiaxial states can be written as
(6)
where
O'
N-
(O,.O.,)l/2
~
=
O'
-
(7)
(7
where N is a unit director in the stress space.
Equation (6) can be considered to be the usual flow rule corresponding to a plasticity theory based, in general, on a yon Mises-Hencky-Huber type yield surface that both
expands as well as translates (i.e., isotropic as well as kinematic hardening), of the type
F=
(a' -
t~'):(a'
-
or') -
~2
o2 =
0
(8)
where
~- =- h(~"P, ~"P)
(9)
where or' is the deviator of the back-stress tensor (i.e.,the center of the translating yield
surface) and h is the function identified in eqn (4).
From the normality rule, we have:
• OF
t °p = A ~-~,
(10)
/~ > 0
- 2/~(g' - or') - 2/~J~ NO
(ll)
where
OF
N -=
(o:
\~:~/
o
\2/
Thus, N is now a unit normal to the yield surface, F = 0. Comparing eqn (6) to eqn (11),
the analogy of creep to classical plasticity is complete when A in the flow rule is identified as
/~ = 3 g _= 3 g
48
4h
(13)
394
S . YOSHIMURA
et al.
such that
ivP= ~
~vPN =- ~
gN .
(14)
On the other hand, elastic-plastic materials, under high strain-rate loading, exhibit strain
hardening as well as strain-rate hardening. Thus, in a uniaxial tension test, one normally
finds, under high-rate loading, that the yield stress varies as
O'yield - -
h ( ~ uP, ~ vp) •
(15)
The yield surface in dynamic plasticity will then have the form as in eqn (8), with a
being replaced by cryiela, while the flow rule is identical to eqn (14).
It is worth noting that in the case of creep, it is m o r e natural to specify the function
(g), whereas in dynamic plasticity, it is more natural to think of the function h(~ vp,
~P). The alternate tangent modulus formulations to follow in sections ( I I I . l ) and
(lII.2), respectively, are largely dictated by which of the two functions (i.e., g or h) may
be conveniently employed.
The central themes in both creep and dynamic plasticity are now clear. The stress state
in each case is governed by the yield condition, eqn (8), and the flow rule (14). Furthermore, when loading occurs, the stress state must remain on the expanded and translated
yield surface. Thus, the so-called consistency condition is
dF = 0
(16)
N : ( d o ' - dot') - %~ do = 0 .
(17)
which can easily be seen to be
Moreover, when loading occurs, eqn (10) indicates the development of a viscoplastic strain change. Thus, assuming that the solid is elastically isotropic, we have
do' = 2 # ( d e ' - de ~'p)
(18)
(do:I) = (3A + 2#)(d~:l)
(19)
where A and/~ are Lame constants. We assume linear kinematic hardening, such that
&x' = cde ~'p
(20)
In order to determine the tangent modulus stress-strain relation, de ~p [or dg vp since
de vp is related to d~ vp through eqn (14)] must be expressed in terms of (de). When
eqns (14), (18), and (20) are used in the consistency condition, one obtains a relation
involving dO and d~ vp. This relation can then be used to solve for dg ~p in terms of dE
in several alternate ways, depending on whether it is convenient to work with the function g [as in (~oP = g ( a , EvP)] or with the function h [as in # = h(UP,~P)]. These will
be discussed below.
Two alternate tangent modulus formulations
395
I I. 1. Tangent-modulus, stress-strain relation when h (~"P, ~ "p) is explicitly given
(consistency):
N : ( ` 4 g ' - A ~ ' ) - N/~ AO = 0
(21)
(N:N) = 1
(22a)
z i g ' = 2/~(Ac' - Ae ~p)
(22b)
A ~ ' = ~A ~ ~'p
(23a)
A~ vp = ~
(23b)
A~VPN
when o = h(~"P,~ vp) is expficitly k n o w n ,
Oh
Oh A~W
A e = 0~,,---~ `4~P + 0~----~
(24a)
`4o =- H~Ag,'v
(24b)
+ H2`4~
vp
.
The incremental strain rate is
`4~t'p
~
z t'p
(6)t+A
t --
(~)~'p
(25)
.
We introduce a linear interpolation for `4~"P such that
`4~vp
=
-" t'p
(e)t+j,(O)`4t
+ (~)~'~'(1
O)`4t ,
(26a)
such that
-" vp
(e)t+j, =
A~ t'p -- (~)~'P(1 - - O ) z a t
0.1 t
(26b)
and
`4 ~ ~'P =
Ag w - (i)~'P`4t
0> 0 .
0_4 t
(27)
Using eqn (27) in eqn (24b), one m a y express/4o in terms of`4~"P and the k n o w n
~ v When the resulting relation between`40 and `4~'P is substituted, along with eqns (22)
and (23) in eqn (21), one obtains (noting that, since N is deviatoric, N:d~' = N.d~, etc.):
.3
A~"u
----
.
(2.+c)
.
.
.
+
.
.
.
H,+~-~
(28)
S. YOSH[MUIL~et al,
396
and
Aet'P = ~
AevPS .
(29)
A a ' = 2 # ( A ¢ ' - A~,'o)
(30)
Thus,
Defining a parameter F such that F = 1 when there is loading and F = 0 when there
is unloading or when the material remains elastic, we have
Ao;j = (2/.@,,,6j,, + A6a6mn -
~f~
r2#2#NijNm.\
-~ )Aem.-F
( H2~fP/O)Nij2#
(31)
where
= (2u+6)+
(32)
-~ H t + OAt]
and N is defined in eqn (12).
In an algorithmic sense, better accuracy may sometimes result if eqn (24b) is replaced
by
A8
=
HIZIg t'p + O(if+Pat) -- O(if p) .
(33)
When eqns (33), (22), and (23) are used in eqn (21), one obtains
[
2/~N:zlc. -. (R~+.¢_t.-- R)
A E t"P
(2~+g)+
N
(34)
~H~
where Rt+a, and R are the radii of the von Mises yield cylinders (R = ~
ov) at times
(t + _4t) and t, respectively, and
Aoij =
2l.£~im~jn+ /~ij~mn-
l"(2#)2#NijNm,,\
p
) A6mn + I ~
(Rt+zt - R)Nq2#
~.
(35a)
where
2
(*=(2.+e)+gH~
.
(35b)
Two alternate tangent modulus formulations
397
iI.2 Tangent modulus stress-strain relation when g(g up, ~) is explicitly given
Here we assume that the relation
~up = g(~Uo,O)
(36)
is explicitly given. As before, to determine A EuP from eqn (23b) one needs to find
A~ uv. As in eqn (26), we first write
A~Up
: r,p
= (~)t+,.(0At)
+ (~)~'P(1 O)at
0<0< 1
(37)
from eqn (36), one has:
Og
Og A~up + ~ 'J O
(i)yP+~,, = (~), + a~---~p
(38)
(i)F + GIAe up + G2ZIO
where Gl -- Og/O~u"; G2 = Og/Og (recall that Hi = Oh/O~UP;H2 = Oh/O~'P). Using
eqn (38) in eqn (37), one has
A~ up
=
(i),~P(At) + OAt(GIAg vp + GzAO)
= Atgt + OAt(GIA~ up + G2AO)
0<0< 1
(39)
where gt is the value of '.t~ ur ,~ evaluated at time , When 0 > 0, we have the socalled "forward-gradient" method (see ATLURI [1980] and REED & ATLURX[1983], CORMEAU [1975], and PmRCE, SHIn ~ NEEDLEMAN [1984], for instance) for determining
A~ up. From the consistency condition, eqn (21), one has
AO = ~
N:CAo' - Aot')
(40)
=- N~ [ 2 # N : A ' ~
(2# + c)A~ up]
wherein eqns (22a), (22b), (23a), and (23b) have been used. Using eqn (40) in eqn (39),
one may obtain
gt + %~ (2#)OG2(N:A~)
A~UP = At[
1+ ~
(41)
G2AtO(2# + ~) - GtOoAt
~
A~Up = NAt I
1
3
gt + 3#G20(N:A~)
1 + ~ G2AtO(2# + ~) - GlOAt
1
(42)
398
S. YOSHIMURAet al.
From which the stresses are derived, as follows:
A a ' = 2#(AE -- A e ~'p)
(43a)
(Ao:|) = (3A + 2 ~ ) ( A e : l ) l .
(43b)
It is perhaps redundant to point out that, in the present approach, if one takes ~ = 0,
eqns (39) and (41) reduce to A~ ~'t' = g t A t ; eqn (43) reduces to
~hen 0 = 0
(44)
while
m
and the consistency condition eqn (40) is merely an identi:y.
A comparison of eqns (28) and eqn (41) reveals tile essential similarity in the expressions for the alternate tangent modulus stress-strain relations. Although the mathematical structures in eqns (28) and (41) are essentially similar, the quantities Ht = 0~/0~ t'p
and H 2 ----- Oa/O~ cp Occur in the former, and the quantities G~ = 0i'P/0~ ~'p and Gz =
0~ vP/0O occur in the latter.
III. GENERALIZED MIDPOINT RADIAL-RETURN ALGORITHM FOR DETERMINING
STRESS HISTORY FOR A GIVEN STRAIN HISTORY
One may use the tangent-stiffness finite element algorithm, based on the weak form
of the m o m e n t u m balance relations (written in terms of compatible velocity fields), to
determine an approximate velocity history and, hence, an approximate strain-rate history. If the particular material point in question (or the Gauss-integration point in a
finite element) is experiencing loading, the tangent modulus constitutive relation at the
point in question may be taken as eqn (31) or eqn (43) [(with (42) substituted in itl. In
both eqns (31) and (43), all the parameters are known at time t. If, on the other hand,
the material simply remains elastic, only the linear elastic constitutive law is employed.
Although the tangent modulus relations (31) and (43) involve only the parameters
known at time t, eqn (35a) involves the yet-unknown parameter R t . j t , the radius of the
yield surface at time (t + A t). Finally, note that tangent modulus constitutive relations
of the type of eqn (31), that is, explicitly depending on the known quantities at time t,
were also derived by GHONEIM ~, CriES [1985] and DRYSDAXE e~ ZA~: [1985].
Because it may still be u n k n o w n at time t whether the material is experiencing viscoplasticity or is remaining elastic during the times t and (t + A t ) , it is clear that the
strain-increment Ae evaluated from the above tangent-stiffness finite element method
is still far from correct. The correct solution is obtained through (1) determining the
"correct" A a from the finite-element-computed A~ through the generalized-midpointradial-mapping algorithm (as detailed, for instance, in ATLURI [1976, 1985]), and
through (2) checking the equilibrium of the total internal stress o + n o with the total
Two alternate tangent modulus formulations
399
applied forcesf~+jt (body forces) and Tt+,~t (surface tractions) at time t + At and performing appropriate (Newton-Raphson) iterations to restore equilibrium, if necessary.
We now discuss more details of determining (Aa) from a finite-element-computed A~,
using an implicit algorithm of generalized midpoint radial mapping as follows.
First, consider a viscoplastic process, and assume that at the current time (t), at is
o n F t ( ~ P , ~ p) : O.
1. Let the increment of strain computed from the tangent-stiffness finite element
method be A~, with deviatoric part, dE'.
2. Let the "guess" for increment of deviatoric stress be Ao' = 2#A~'.
3. Let the guesses for total Cauchy stress and back-stress deviators, respectively, be:
got+at
= 0 t -~"
21xA¢'
~a;+,~, = ,,,;
4. Compute ( at+At- been viscoplastic.
g
If
P
Oil
gRt+at< So, the process
): (
g
t
at+A t -- a[ ) ~
gR~+,~tif ~R~+,~, >
So; the process has
has been elastic where
S0Z(~,%0) = ~ - h2(~,%0) .
If the process had been elastic, the "guess" for d o ' had, in fact, been correct.
5. If the process had been viscoplastic, define a parameter 0 </3 < 1, such that A~a =
13A~ and define a director, N3, in the stress space such that
(~r[ + 2 / ~ A E ' )
Na =
-- (0¢; + BAod)
11(o[2~3Ag)- (or; + ~Aa')l[
(46)
6. Now, if the tangent modulus formulation as in eqns (28)-(31) is used, define
(A.VP)3 = N~[2btN~:A. + ~ (H2~P/O)] ~I
(47)
where ~ is defined in eqn (32).
If, on the other hand, the formulation based on eqns (34-35) is used, define
(AeV')~=N~[2t~N~:A'-(gR,+a,--R)]~1
where ~* is defined in eqn (35b).
(48)
400
S. YOSHIMURAet aL
Finally, if the tangent modulus formulation as in eqns (41-43) is used, define
(A¢vP) a = N ~ A t
[ (~J~)
gt + 3#G20(N~:A¢)
] 1
~.--'z
(49)
where
\-~ G2AtO(21s + 6) - GlOAt
7. Compute the "correct" stress increment as:
A o ' = 2/z[AE' -- (A~"~),~]
(50)
d o : l = (3A + 2#)(A~:I)
(51)
8. Compute the correct Rt+.a, a;
Rt+at = R t + A R
(52)
A R = N~3 AO
(53)
If the formulation of eqns (28-31) is used,
A O = (HI + Hz
it~'p ,
\
(54)
0<3,0<1
where Na is given in eqn (46).
If the formulation of eqns (40-45) is used,
(55)
where N~ is defined in eqn (46).
Finally, if the formulation of eqns (33-35) is used,
AO =
Hi
[2p.N-Ae
-
-
(gRt+a,
~.
--
R) )
"-L.p
j + o(et+.at) -- #(~'P)
(56a)
Two alternate tangent modulus formulations
401
where
A~ vp -- (1 - O)~PAt
0.it
6t+.l t
(56b)
9. Compute the correct (x;+j, as
oe;+j, = (x; + ~(/t(~'p)~
where (AEVP)~ is computed from eqns (47), (48), or (49).
The case when f3 = 1 results in a fully implicit algorithm, whereas/3 = 1/2 is found, in
many experiences, to result in an optimal algorithm (the midpoint radial return).
If the process at time t has been elastic, and at time (t + At) becomes viscoplastic,
then divide the strain increment Ae into two parts--the first part a pure elastic one and
the second part a elastic-viscoplastic one. (See, for instance, ATLtrRI [1976], [1985]) for
further details.)
After the "correct" incremental stress is computed, using the above-described elasticviscoplastic stress-integration algorithms, check to see whether the total stress (a + Ao)
is in equilibrium with applied forces. If not, perform equilibrium-correction iterations
(see ATLURI [1976, 1985] for details).
IV. NUMERICAL STABILITY, ACCURACY, AND CONVERGENCE:
SOME ILLUSTRATIVE EXAMPLES
In the analyses of creep problems, the momentum balance equations are of the
quasistatic type (i.e., inertia is ignored). Here, the stability, accuracy, and convergence
of the numerical method will depend on the time increment (At) (dictated by the materials' rate sensitivity), parameter (0) in the forward gradient representation for A~ LR,
and parameter (B) in the radial-return, stress-integration algorithm. In the analysis of
dynamic plasticity problems, material inertia is accounted for. Here, the performance
of the numerical method will depend on At (which is dictated by both wave propagation considerations, as well as the materials' strain-rate dependency), and the parameters 0 and B.
In the following paragraphs, we present several examples to illustrate quantitatively
the features of numerical stability, accuracy, and convergence of the present alternate
algorithms.
IV. 1. Quasistatic creep problems
The problems concern the axisymmetric analysis of creep in a thick-walled cylinder
of an outer to inner radii ratio of 2. A schematic of the problem is shown in Fig. 1; the
problem is solved for prescribed material velocity conditions at the inner surface of the
cylinder. The finite element mesh, consisting of eight-noded quadrilaterals, is also shown
in Fig. 1. Two types of creep behavior are considered: one without strain hardening (i.e.
the function g of eqn (3) or eqn (36) is a function of a alone), and the second with
strain hardening (i.e. g of eqn (36) involves o as well as ~"P).
402
S. YOSHIMURAet al.
Z
Z
o
=100
b
=200
..~ ~9 t9 18 t9
o"
or
/.L
~r
b
•. - - - . - - ~ r
Fig. 1. Schematic of a thick-walled cylinder: problem of creep under prescribed-velocityconditions at the inner
surface.
IV. 1.1. N o n - s t r a i n - h a r d e n i n g
case.
•
Here we assume that
/ a
~-<'P= 7or ~,}
= g(O)
(57a)
ao = constant
(57b)
with E / a o = 500, v = 0.3, eo = 0.002, m = 0.01, and Oo = 40. A similar problem, with
slightly different material parameters, was also solved by PIERCE, S a m , ~, NEr~DLEMAN
[1984]•
The inverse o f g ( e ) in eqn (57) can easily be f o u n d :
#=no
-\ ~o/
-" h ( ~ v p )
•
(58)
Thus, in this case, the function h o f eqn (9) depends on ~P alone. Moreover, since h
o f eqn (9) describes the expansion o f the yield surface, in this example, the initial radius
o f the yield surface is zero. F o r numerical purposes, we take
# = ayo + ao
~
\~o/
=- h ( ~ ~p) •
(59)
We assume that the initial radius o f the yield surface, ayo, is ayo = 0.01. The material
functions h and g are illustrated in Fig. 2 for various values o f m ~ ( l / n ) .
A d r a w b a c k o f the tangent-modulus m e t h o d based on the use o f h(~VP), that is,
Two alternate tangent modulus formulations
403
n=5
.vp= 0 . 0 0 2
5
n-lO
%
4
3
(n • I / m
)
b
Ib
n -100
n -1000
I
o
o
lo-r
I
1o- 5
i
i
*
to-3
to-'
lo
_~
1o 3
*_....
io
~vp( s - I )
Fig. 2. Illustration of the function # (~vp), that is [# = Oyo + ao ( ~ ' / i o ) " ] ,
for various values of n ( = l / m ) .
based on eqns (28-31) or on (33-35), is that when m << 1 in eqn (59), the yield surface
expands rather abruptly in the beginning. Thus, consider the case when the initial radius
is such that ayo = 0.01 in eqn 59. When m = 0.01; ~0 = 0.002; it can be seen from
eqn (59) that (1) ~vp = 0, a = ay0 = 0.01 ; (2) ~vp = 10-9, # = 34.60; (3) for ~vP = 10 -8,
= 35.40. Thus, in the
stages of the present calculations, using the function h,
a fully explicit formulation based on eqn (44), i.e.
initial
zla'= 2#(4E-
NgtAt~)
(60)
is used to compute za a, for the case when m << 1.
Similar drawbacks are also present in the tangent modulus formulation based on the
function g as in eqn (57), when m << 1. For instance, when m = 0.001, difficulties are
encountered in the calculation of
as well as its derivative ( a g / a a ) .
Figure 3 shows the stability, convergence, and accuracy features of the tangentmodulus method based on g (i.e. the use of eqns (41-43)). In all the calculations shown
in Fig. 3, a fully implicit algorithm (i.e. # = 1 and ~ = 1) is used. Although the material velocity, U, at the inner wall is kept fixed at U = 0.2, the time step in the integration procedure, At, is changed from At = 0.125 tO At = 1, with corresponding changes
g(a)
404
S. YOSHIMURA el al.
in load increments A U. As reported by CORMEAU [1975], the numerical stability of the
fully explicit method (with 0 = 0) demands that
At <
4(1 + v)o0
3EG2
,
Og
G2 = 0o
(61)
For the present case, for the fully explicit method, this implies that At < 1.733 X
All the time increments considered in Fig. 3 exceed this stability limit of
the explicit method by one to two orders of magnitude. Thus, the present results, based
on the use of g, and a fully implicit scheme, are stable for a wide range of zl t, differing by two orders of magnitude. However, the accuracy o f the results, for large time
steps (or equivalently, large load steps in this example), is questionable, especially near
the "knee" of the P-U curve (i.e. for large time steps, the predicted maximum internal
pressure is highly inaccurate). Finally, the results for the predicted internal pressure do
converge for decreasing values of load-increment size (or, in this case, the decreasing
values of time-step size), as seen from Fig. 3.
Figure 4 shows the stability, accuracy, and convergence features of the tangent modulus method based on h, i.e. the use of eqns (28-35). Here also, a fully implicit scheme,
based on 0 = 1, and/3 = l is used. Various values of J t and 4 U, while keeping 0 = conl0 -2 (40/a) 9 .
1.0
(1)
(2)
(3)
(4)
0.8
¢3)
(4)
0.6
0.4
.~
•
EvP=0.002 ( ~ " )
100
E = 20,000
v
0.2
0.0
tl)/kt=l.
,
Au=0.2
( 2 ) A t =0.5
(3)Z~t =0.2,5
,
,
AU=0.1
Z~U=0.05
1 4 ) A t =0.t25
,
AU=0.025
=0.3
=0.2
0
2
4
6
8
10
12
t4
Fig. 3. Convergence o f the tangent modulus method based on the use o f g [as in #~'P= g (~"P, ~ ) ] .
Two alternate tangent modulus formulations
405
stant, are also employed in this case, as in Fig. 3. Here also, all the values of dt
employed far exceed the stability limit for Af imposed by the fully explicit method based
on the use of g. Although the present results based on the use of h are stable for values of At ranging from 0.125-0.5 (which values are already larger than the stability
limit, eqn (61), of the fully explicit method-based g), they are unstable for the largest
value of At employed, that is, At = 1 (correspondingly, A U = 0.2). For the present
example, with M = 0.01 in eqn (58), the material behavior is nearly rate independent.
That the rate-independent classical plasticity solutions (for the present non-strainhardening case) become unstable for very large load increments is not surprising. The
predicted results for pressure, shown in Fig. 4, are far more accurate for all At values
between At = 0.125-0.5, as compared to those shown in Fig. 3. Figure 4 also shows the
superior convergence properties of the results based on the use of the function h as compared to those based on the use of g (Fig. 3).
The converged results (with At = 0.125, A CJ = 0.0125, and 0 = 0.2 in both cases)
using the two alternate formulations based on g and h, respectively, as developed in this
article, and using fully implicit schemes (0 = 0 = 1.0) are shown in Fig. 5. It is seen that
the results from the two alternate formulations are in excellent agreement. Figures 6 and
7 show a comparison of the results of the two alternate formulations, using different
values of At while keeping A U constant, thus resulting in different values of 0 and
hence of i. In Fig. 6, At = 12.5, A U = 0.025, and 0 = 0.002; while in Fig. 7, At =
0.000125, AU = 0.025, and 0 = 200. The strain rate in the problem of Fig. 6 is of the
1.0
0.8
0.6
0.4
= I.
,
Au=o.2
(3)At x0.25
,
Au to.05
(4)At ~0.125
,
Au =0.025
(I)At
0.2
Y = 0.3
0.0
0
2
4
6
8
10
12
14
Fig. 4. Convergence of the tangent modulus method based on the use of h [as in 5 = h (@‘, Fup)1
406
S. YOSlrIIMURAet al.
t.O
0.8
0 o
0.6
@
At=ot25
."
,
Au--oo25
0
O
0.4
~ 'Io0
~'VP=o.o02 1 ~ - 1
E =20,000
0.2
•
•
Tangent modulus method based on
0 the use of ~ v p = g ( ~ v p ~ )
~' =0.3
0.0
i
0
• Tangent modulus method based on
the use of ~ = h l ~ V P , ~vp]
2
4
I
I
I
I
6
8
t0
12
14
Ulae 0
Fig. 5. Comparison of the results for two alternate tangent modulus methods in the non-strain-hardening case
of creep in a thick-walled cylinder ( U = 0.2).
order of (10-5); while that in Fig. 7 is of the order of 1. Figures 6 and 7 show the good
agreement of the results from the two formulations, for both the extreme values of 0
and hence of ~.
IV. 1.2. Strain-hardening case. Here, we assume that
• / d ~'"
= ,0t
(
,..
ao = 40 1 + 0.--.-~]
)
=f(evP)40
(62)
(63)
such that
(o~t/,~(1 + ~p ~-(0.~/m)
g(a,~°P) = (~o) \ ~ )
b---~-2}
"
(64)
Two alternate tangent m o d u l u s formulations
407
t.0
0.8
AU = 0 . 0 2 5
•:"
oo °
~:~ 0.6
At-
.:"
12.5
o
•
VP=o.o02 " ~ ,t00
0.4
:
•
0.2
o.o o
E
= 20,000
ls
-0.3
• Tangent modulus method based on
the use of ~ =h(~vp ~VPl
0 Tangent modulus met_hod based on
the use of ~.vp, g (~vp, ~ I
t
h
'2
;o
;2
t4
U/a~ 0
Fig. 6. Comparison o f the results for two alternate tangent modulus methods in the non-strain-hardening case
o f creep in a thick-walled cylinder ( U = 0.002).
The function f ( ~ vp) is illustrated in Fig. 8. Inverting this relation, one obtains
a= h(g.vP,~vP)= 40( ~vPlm(l ..k
do/
~vp ~0.1
0.-b-~/
(65)
Figure 9 c o m p a r e s the results o b t a i n e d by the two alternate formulations involving
g and h, respectively, for the strain-hardening case, for the values o f m = 0.01, do =
0.002, E = 2 x 104, v = 0.3, ~1U = 0.0125, At = 0.125, 0 = 0.2, and 0 =/~ = 1.0. The
two formulations are seen to lead to identical results.
IV.2.
Dynamic plasticity analysis
The problem is that o f a clamped beam subject to a Heaviside step-function loading,
as schematically represented in Fig. 10. The material properties are E = 68.9547 G P a ,
v = 0.3, static initial-yield-stress ays = 2.67 x 10 a k g / m 2, p = 2.672 x 103 k g / m 3, the
b e a m g e o m e t r y is given by h = 0.254 m, ( L / 2 ) = 2.54 x 10 -~ m. The viscoplastic
material properties are characterized as follows:
•1.08
S. YOSHIMURA
e t al.
t.0
0.8
b0
n
•:
0.6
~U
•
0.025
Z~ t
- 0.000125
Q
0
Tangent modulus method based on
the use of ~ = h(~'vP ~,vp )
•
•
0.4
:
;
•
~
)100
~vP=0.002 (
E
Tangent modulus m e t h o d b a s e d on
0 the use of ~vp= g ( ~ v p ~ }
= 20,~
=0.3
v
0.2
.0
Ii,
0
I
4
2
I
6
i
8
t0
I
12
t4
Ula~ 0
Fig. 7. Comparison of the results for two alternate tangent modulus methods in the non-strain-hardening case
of creep in a thick-walled cylinder (U = 200).
(
o=oys
/~o /
1 + *~P
-
h(i*'p)
.
(66)
Thus, there is no strain hardening. Inverting eqn (66), one obtains
vp =
•
(-- ]
(1 - -
\
6),s
Oys /
~o -
g(~)
•
(67)
The problem is solved for three values of ~o,~0 = 10, 10 2, and 10 4, respectively. Thus,
as ~0 - ' 00, the material becomes the classical rate-independent plastic material. The
material functions h and g are illustrated in Fig. 11.
The present transient dynamic response problem is analyzed by using N e w m a r k ' s
implicit time integration algorithm. For fully explicit algorithms based on eqn (44), the
time increment At, for stability of numerical solution, as dictated by the materials'
strain-rate sensitivity (from CORMEAU [1975]) is
zl t <
4(1 + v)oys
3E(Og/OO)
(68)
Two alternate tangent modulus formulations
409
n=0.1
t.4
1.3
~
>
lw
)
n
t.2
t.1
1.0
0
'
0.01
'
0.02
O. 3
O.c; 4
~vp
0.05'
'
0.06
Fig. 8. Illustration of the strain-hardening function in the relation: $ = h (~"P, ~"P) ~ h, ($~'P) x f ( ¢ " P ) .
which works out, for the present case, to be A t < (6.94 X 10-3)/g0 that is, At < 6.94 X
10 -4 for ~0 = 10, A t < 6.94 X l0 -5 for ~0 = 102, and A t < 6.94 X 10 -7 for ~0 = 104In the present calculation, fully implicit schemes, with 0 =/3 = 1, are employed where
/3 is the parameter in the radial return algorithm of eqn (46). The time increment zlt,
in Newmark's method, is chosen to be A t = 10 /~s. The parameters in Newmark's
method are taken to be (t5 = 0.55,/3 = 0.276).
Figure 12 compares the time variations of the central deflection of the beam, obtained
by the two alternate tangent-modulus methods, for various values of ~o. In the case of
~0 = 104, the calculated strain rate in the problem is of the order of 10, and thus, from
eqn (46), it follows that the dynamic yield stress ~ is only 1.001 times the static value
eys. Thus, as shown in Fig. 12, the present tangent-modulus methods based on h(~'P),
as in eqns (28-35) for the case of ~0 = l04, agree well with the classical rate-independent plasticity solutions. On the other hand, for the case of ~0 = 104, the tangent modulus method using g ( a ) , as in eqns (41-43) predicts results for deflection that are
slightly higher than the classical rate-independent plasticity solutions. Moreover, the
convergence o f the method using g ( # ) was rather slow for this case of near rateindependent plasticity than that of the method using h(~VP).
For progressively smaller values o f ~o, the rate sensitivity of the material yield stress
increases. For two values o f ~0 (i.e. ~o = 10z and ~0 = 10), the results for the variation
with time, of the central deflection o f the beam, are shown in Fig. 12. It is seen that for
these two values of ~o, the results from the two alternate tangent modulus formulations
agree excellently with each other. For very small values o f ~0, as seen from eqn (66),
the material is extremely rate sensitive, and, in the limit, its behavior approaches that
S. YOSHIMUR~ et al.
410
1.0
a
=
~
-
_
/
0.8
Q
0
0.6
o.
Q.
L~t=O. t25
, AU=O.025
o
0.4
o.
~'VP=o'o02
[ 40(1+ ~VP)o'4J14°°
0.002
0.2
•
E " 20,000
It =O.3
0.0
.
0
I
2
o Tangent modulus method based on
the use of ~.vp= q (~vp ~ )
I
I
4
Tangent modulus method based on
the use of ~ • h(~vP ~.vp)
!
6
8
t0
12
14
Ula~ 0
Fig. 9. Comparisons of the results for two alternate tangent modulus methods in the strain-hardening case
of creep in a thick-walled cylinder ( U = 0.2).
of a linear elastic solid. That this is so is reflected in the results of Fig. 12, wherein the
purely elastic solution for the response of the beam (i.e. the central deflection) is also
plotted. These results in Fig. 12 are also in good agreement with those reported by
THAKKER & STAGG [1980], and BRICKSTAD [1983].
V. CONCLUSION
Two alternate tangent-modulus finite element methods, and attendant implicit algorithms of midpoint radial return for the integration of stress at a material point, are
presented for the analysis of rate-dependent viscoplastic behavior. The presented metholo
ogies are suitable for analyzing a variety of viscoplastic problems ranging from (1) the
quasistatic problem (i.e. with negligible effects of inertia) of creep at high temperatures,
with small strain rates to (2) the dynamic problem (with significant effects of inertia)
of elastic-plastic solids at very large strain rates. In the type of problems considered,
material hardening has two sources: strain hardening as well as strain-rate hardening.
The algorithms presented in this article for quasistatic as well as dynamic-response analyses, are implicit in nature. In all the analyses considered, the two alternate tangent
modulus approaches lead to results of comparable accuracies. The present experiences,
&~"
+
II
il
,-t
%
%
0 t
0 t
%
!
a-i%
M
oq
II
q
0 1 ~...~ .~.
ql
ql
oq
II
qn
-@
-®
I
b
IX)
,.r
~°
o"
0
[
[
[
[
"1"
A
r
"F-\
4--
4--
4--
\\\\\\
\~\\\
A
o~
3:
,--I
=__
0
=.,
0
S. YOSHIMURAet al.
412
Tangent modulus method
base on ~ • h(~VP, ~.vp)
t-
'0
~"
3K
2.0
0
Tangent modulus method
base on ~.vp • g ((~ , ~vp)
I/)
ooo
a3
~o
Rote - independent
plasticity solution
1.5
~0"t04
tO
1.0J-
t Initial yield point
., ~..- 10 2
"10
N
o
t0
U
0.5
'10
~- Elastic solution
0
E
z
0
0
0.2
0.4
0.6
0.8
Normalized
time
1.0
t.2
1.4
t/T 0
( T • 2.0 ms)
Fig. 12. Variation with time of the central deflection of the beam in the problem of Fig. 10.
however, point to the general conclusion that the tangent modulus method based on the
use of the relation ~vp = g(~Vp,#) may be somewhat more convenient to use in the case
of highly rate-dependent material behavior (in the limit of very high strain rates), while
the tangent modulus method based on the use of the relation # = h(~°P,~ op) may be
more convenient to use in the limit o f rate-independent material behavior. There is,
however, no intrinsic reason to prefer one tangent modulus method over another.
Acknowledgements-The results presented herein were obtained during the course of investigations supported
by the U.S. Office of Naval Research. The authors gratefully acknowledge this support as well the encouragement received from Dr. R. Rajapakse. It is a pleasure to thank Ms. Cindi Anderson for her assistance in
the preparation of the final version of this manuscript.
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1959
1966
1970
1975
1975
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413
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PIERCE,D., SHIn, C.F., and NEEDLEMAN, A., "A Tangent Modulus Method for Rate Dependent
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Center for the Advancement of Computational Mechanics
Mail Code 0356
Georgia Institute of Technology
Atlanta, GA 30332
(Received 16 August, 1986; In final revised form 20 January 1987)