International Journal o f Plasticity. Vat. t. pp. 63-8"'. 1985
Printed in the U.S.A. All right, reserved.
CONSTITUTIVE
IMPLEMENTATION
0"/49-6419/$5 $3.00 + .00
Copyright © 1985 Pergamon Press Ltd.
MODELING
FOR
AND
FINITE
COMPUTATIONAL
STRAIN
PLASTICITY
KENNETH W. REED* a n d SATYA N. ATLLrRI~
*Southwest Research Institute and tGeorgia Institute of Technology
(Communicated by George J. Dvorak, Rensselaer Polytechnic Institute)
Al~tract-This paper describes a simple alternate approach to the difficult problem of modefing
material behavior. Starting from a general representation for a rate-type constitutive equation,
it is shown by example how sets of test data may be used to derive restrictions on the scalar
functions appearing in the representation. It is not possible to determine.these functions from
experimental data, but the aforementioned restrictions serve as a guide in their eventual definition. The implications are examined for'hypo-elastic, isotropically hardening plastic, and kinematically hardening plastic materials. A simple model for the evolution of the "back.stress,"
in a kinematic-hardepAngplasticity theory, that is entirely analogous to a hypoelastic stress-strain
relation is postulated and examined in detail in modeling a finitely plastic tension-torsion test.
The implementation of rate-type material models in finite element algorithms is also discussed.
!. PRELIMINARIES
We denote the true stress by s, its deviator by s', and its material derivative by s. Invariants p~ o f the tensor are defined as
PI: =s:I
P2: = ( 1 / 2 ) s ' s ' : I P3: = ( 1 / 3 ) s ' s ' s ' : I
where ( A : B ) = tr(AtB) denotes the usual scalar p r o d u c t o f two second order tensors,
A B the tensor product, and A t denotes the transpose o f A. We write e for the stretching, the symmetric part o f the velocity gradient, and J for its trace. We let E denote
the infinitesimal strain tensor. Finally, by an objective stress rate, o f which several exist,
we shall m e a n a stress rate s* which t r a n s f o r m s between frames by the rule
g* = Q s * Q t
when E = Q e Q t and ~ = Q w Q t + QQt, w being a general spin tensor and Q being the
rotation between the frames. W h e n w is chosen as the skew-.symmetric part o f the velocity gradient, we recover "classical" stress t:ates; other definitions result in "non-classical"
stress rates.
It will occasionally be necessary to emphasize the non-invariant character o f certain
stress rates. As a convention, we use direct notation for invariant rates and index notation for non-invariant rates.
!!. INTRODUCTION
T h e deficiencies o f currently used hypo-elastic and kinematically hardening plastic
material laws are briefly discussed below.
63
64
K. ~ ' . REED AND S. N. ATLURI
Since the time of Cauchy, elasticians have used a material law of the form
(I)
s = 2gE + A(I:E)I .
When used in the analysis of stress in infinitesimally deforming bodies, it predicts shear
stresses which vary linearly with the shear strains and no normal stress effects. The stress
exhibits no path-dependence.
In recognition of the Principle of Objectivity of material properties, eqn (1) has
usually been replaced by the hypo-elastic material law
s* = 2~e + A(I :e)l
(2)
in finite element algorithms when finite deformation analyses were anticipated. It seems
to have been presumed that the same sort of idealized material response at finite strain
would be predicted by eqn (2) as was predicted by eqn (1) for infinitesimal strains.
Dm~n~s [1979] was the first to demonstrate in the computational mechanics literature
the fallacy of this presumption. When s ° is identified as the Jaumann stress rate, eqn (2)
predicts shear and normal stresses which oscillate as the shear strain increases monotonically (see Fig. 1), a result hardly to be expected by one familiar with eqn (1). His
work did serve to emphasize the need for a more rigorous method of arriving at ratetype constitutive equations than then in vogue: A mere replacement of the inobjective
material stress rate by an objective rate.
However, Dienes also set a precedent which the present authors regard as unfortunate: he 'blamed' the deficien.cy of eqn (2) on the Jaumann stress rate and showed
that a less objectionable material response could be obtained in the particular problem
of rectilinear shearing by introducing a different stress rate. The results obtained by
Dienes for the so-called "Green Mclnnis" rate are shown in Fig. 2.
o
o
e,i-
::t
LdO.
E
°'~ .
.
.
.
o~
4'oo
o
"
o
o
¥
o
I
Fig. 1. Rectilinear shearing of the material (0*/2t~) -- ~ +
(I:E)I.
O* = J a u m a n n Corotational Rate.
Finite strain plasticity
65
U3,.,3"
w
N°
0
O0
1.00
2.00
3.00
4.00
5.00
6.00
7.00
STRAIN
Fig. 2. Rectilinear shearing o f the material (¢'/2tx) = E +
(l:e)l.
#* = Green-Mclnnis Rate.
Two years later, NAGTEGAAL& DE JONG [1981] found that MELAN'S [1938] kinematic
hardening rule
a;j = 2h@fj,
(3)
which predicts linear back-stress-strain variation and no path dependence, again led to
periodic shear and normal stresses when it is generalized, for finite deformations, to
a hypo-elastic-type rule
a* = 2he p
(4)
and a ° is taken to be the Jaumann rate of the back stress. This result is shown in Fig. 3
(We note that the same result had been discussed nearly ten years earlier by LEHMANN
[1972]). The non-physical material response was at first attributed to the hardening rule,
but eventually "blamed" on the Jaumann stress rate (c.f. LEE et al. [1983]). T o date,
no fewer than four non-classical stress rates have been suggested for use in place o f a*
in Melan's equation (LEE et al. [1981], LEE et aL [1983]), JOHNSON • BAMMANN [1982],
DAFALIAS [1983], and NAGTEGAXL[1983]. The shear and normal stresses arising when
the Green-Mclnnis rate and the rate o f LEE et al. [1983] are used in eqn (4)
are shown in Figs. 4 and 5.
We do not believe the present search for an "ideal" stress rate for these problems to
be well founded. Our dissatisfaction with the models based on the new stress rates stems
from the fact that they cannot be brought into agreement with any realistic idealization o f material behavior, much less with experimental data. None guarantees stable
material behavior in rectilinear shearing. Normal stresses and normal strains predicted
66
K.W.
I ~ E D AND S. N. A'rtt~'Rl
O
O
c,i-
,u: -CI.00
~ol
1.00
2.00
3.00
4.00
"7"----
STRAIN
5.00
6'.00
7.00
Fig. 3. Rectilinearshearing of the kinematically hardening plasticmaterial (eqn 4).
St2
at2
St=
aH
=-7+1
;
-7='7;
a*=2h~ p ;
a*--JaumannRat¢
;
2h--I
.
LtJ
IX
I--(/3
8
ol-
O
q
30
¢.oo
2'.oo
~'.oo
4'.0o
STRAIN
~.oo
i.oo
i ......
7'.oo
&O0
Fig. 4. Rectilinear shearing of the kinematieaily hardening plastic material (eqn 4).
Stz
atz
T - T +i
;
St t
at t
--~--.--~ ;
a*=2h~a
;
$*--Green-McinnisRate
;
2h--I
.
by the models are generally one or two orders of magnitude larger than have been
reported in the experimental fiterature (c.f. Swn~r [1947], FR~'UD~NTR~ • RONAY
[1966], ROSE~ STUEW~ [1968], Bn~Nm'ON [1977], B~I. • KH+.N [1980]). In Fig. 6, normal strains obtained by using several alternative definitions for a* in eqn (4) are plotted against the average shear strain in the thin-walled tube torsion test.
Finite strain plasticity
67
LJJ
SI2
13£
I-(J3
i
r.O0
1 .OO
2.00
3.00
4.00
STRAIN
5.OO
6.00
i
7.00
Fig. 5. Rectilinear shearing of the kinematically hardening plastic material (¢qn 4),
Siz
a,z
T - T +!
;
Ssl
all
~=-7
;
a*=2he
o
;
a*=Lcc'sRat¢
;
2h=l
.
0
O"
.7
o
Jaumann Rate / /
w~
0
6-
6reen-Mclnnts
Rote
l,l o
I "!.
ZO"
<
o~
iu3
s Rate
O
o-
o"
0
q
~
c :00
0'.33
0.67
1'.00
1.3.3
1.67
G A M M A = R~/L
2.On
2.33
2,a/
Fig. 6. Normal strains in "thin-walled tube torsion test; kinematically hardening plastic material (eqn 4)
( a ' / 2 h ) = e0 ;
T/2h = 207MPa/(.,/3 310 MPa) (of. LEs et al. [1981]).
68
K.W. REED AND S. N. ATLUKI
The goal of this paper is to discuss the relation of certain experimental results to the
theory of finitely deforming kinematically hardening plastic material.
In the present treatment of kinematic-hardening plasticity,we consider the objective
rate of the back stress, i.e. a*, to be a general tensor function of the plastic stretching
e p as well as the current back stress, a. Thus, the present relation represents a generalization of those used by NAGTEGAAL • DE JO~G [1981], LEE et al. [1981,1983] and
AVLUP.I [1983] in which a* is simply related linearly to e p. The firsttwo authors invoke
the rather artificialconstraint that the back stress a must be purely deviatoric and claim,
inadvertently, that this requirement necessitates the exclusion of non-corotational type
stress rates from use in eqn (4). O n the other hand, A ~ V R X [1983] has shown, in essence,
that if a* in eqn (4) is taken to be a convective stress-rateof Truesdell or Cotter-Rivlin,
then non-oscillatory shear-stresses m a y be obtained without recourse to the problematical non-classical stress rates (see Fig. 7). This result of Atluri, and the well-known
classical solutions (as summarized, for instance in ATLURI [1983]) for the hypo-elastic
stress-strain law:
(5)
s* = L(e,s) - linear w.r.t, e ,
0
0
St
•
ll
00.
8
e,i-
5'.00
(/3
S,
22
S,
33
Fig. 7(a). Rectinlinear shearing o f the kinematically hardening plastic material (eqn 4).
Sl2 al2
S~ a,~
T - T + 1 ; 7 -~ (no sum) ; a * = 2 h E p ; a ' = T r u e s d e l l R a t e ; 2 h = l .
Finite strain plasticity
69
suggest that if eqn (4) were to be replaced by:
a* = G(e p, a) - linear w.r.t, e p
(6)
then, non-oscillatory results for a can be obtained for any objective stress rate a* by
properly adjusting the terms in the function G on the right-hand side. Moreover, the
constants in function G m a y be chosen so as to model the test data, as demonstrated
later in this paper. Note that the back-stress a as determined from the evolution eqn (6)
is not necessarily deviatoric; only the deviatoric component of a m a y be used in modeling the classical pressure-insensitive metal plasticity. W e observe that eqn (6) belongs
•to the general class of internal-variable-type evolution equations suggested by O ~ A T
FA~DSmSHEH [1973].
From the one-to-one correspondence of the present kinematic hardening plasticity
model and the classicalhypo-elasticitystems an immediate advantage: from solutions of
eqn (5) can be found solutions of eqn (6) by a mere change of variables.
Our plan is to discuss in the next section the relation of experiment to hypo-elastic
models for a particular class of Ioadings. S o m e specific restrictions on material functions are found. Guided by these restrictions, a new "ideal" hypo-elastic material model
is constructed. W e then propose a new ideal kinematicaIly hardening plastic material
as an analogue to the new ideal hypo-elastic material. The behaviors of the new models
are discussed, as well as the problem of modeling non-ideal material behavior.
The paper is concluded with some brief notes on "objective" numerical integration
schemes which m a y be applied to rate-type constitutive equations.
oi
Sw
33
¢4-
u~g1
?
~
2'.oo
3'.oo
,'.oo
STRAI
N g.00
s'
22
Fig. 7(b). Same as Fig 7(a) except that a" = Cotter-Rivlin Rate.
70
K . W . REED AND S. N. ATLUR/
111.
MODELINGTHE RESPONSE OF THIN-WALLEDHYPO-ELASTICTUBES
IN A TENSION-TORSION-INTERNALPRESSURE TEST
We consider the response of a thin-walled hypo-elastic tube to a certain class of loadings. By hypo-elastic we mean materials whose mechanical response is adequately represented by a constitutive equation o f the form
(7)
s* = L ( e , s ) - l i n e a r w.r.t, e ,
where s* is corotational: s* = s - ws + sw. Since the difference between various acceptable stress rates (e.g. Truesdell, Cotter-Rivlin rates) and corotational rates may be
"absorbed" by the right hand side o f eqn (7), we have lost no generality by restricting
s* in eqn (7) to be corotational. We shall also assume that L is invertible:
e = L-1(s*,s)
.
(8)
It follows as a special case of WANG'$ [1970] representation theorem that eqn (8) may
be set down as
• -----[Ail/~! -I- A12/~2 + A13(/)3 + 2/3pl,b2)] I
+ [A21~b,+ A22p2 + A2~(.b3 + 2/3p,,b2)ls'
(9)
+ [A3J.bl + A32/b2+A33(A + 2/3.bl.b2)]s's'
+ 2M~s * + M2(s's * + s*s') + M3(s's's * + s*s's') .
The scalars A u and M ~ generally depend on Pi, P2, and ,o3. The states of stress
achieved in the tension-torsion-internal pressure test, when the tube's wall is sufficiently
thin, are o f the form:
s~
s=
z
S,z
s~
0
.
(10)
0
We require t h a t the tension and internal pressure be adjusted so that s ~ = - s ~ z
throughout a test. As a consequence, s may be written as:
s=s'=.~=
[OSo0in20!]
si
0
-cos20
(11)
0
so that PJ = 0, ,o2 = g2, and P3 = 0. Likewise, we write s* as:
"$cos20 + 25(w - 0) sin20
s* =
~sin20
-
25(w - 0)¢os20
0
Ssin20 - 25(w - 0)cos20
- $ c o s 2 0 - 2~(w - 0) sin20
0
il
(12)
Finite strain plasticity
71
The angle between the tube axis and principal axes of stress in the plane of the tube wall
are given by 0 and 0 + 7r/2. Since only the second stress invariant Pz = j2 is non-zero
the A ts and M ~ depend only upon P2 for any load producing a stress system of type
eqn (11).
Substitution of eqns (11) and (12) into (9) gives, after straightforward algebra,
err = 2J,~A12
(13)
(14)
j = 2J~(3A 12 + 2M 2 + 2JzA 32)
[ cos20
-sin20
sin20][(eo-e~z)12f
cos20
L
__[ ,
t.
-2~(w - 0)/~
eo~/.~
2~2A2z
J
(16)
In these equations, the stress is to be regarded as given, the stretching components as
observable, and w as defined (e.g. for the Jaumann rate w: = - e ~ ) . The A is and M t
depend only upon P2 since Pl and P3 vanish for the loadings considered.
The analysis may be continued by considering g ~ 0. Then eqns (13) through (16)
predict
"412 = err~J02 ,
(17)
which determines A 12 for Pl, P3 = 0, and
2M 2 + 2p2A 32 = (e= - err)/,b2 + (e**
-
err)/P2
,
(18)
which restricts M z and A 3z. When w = 0 (at an instant or continuously), eqns (9), (11),
and (12) predict also
2 M I + 2 p 2 M 3 + 2pzA22 = ( e ~ - e=)cos 20/2g + (e~z/g)sin 20
(19)
2 e ¢ ~ l ( e ~ - ez~) = tan 20 = 2s~,zl(s¢, - s ~ ) .
(20)
In the first three equations above, the right hand sides are known as either functions
of the stress, which was controlled in the test, or the deformation which was observed.
Thus, A i~ is determined on the line pi = P3 = 0, and A 22, A 32, and the M t are quantitatively restricted. The fourth equation expresses the precise coaxiality of e with s
necessary when w = 0 (see Appendix B). This condition is fulfilled continuously in a
proportional loading with 0 = 0:
s~, = - s =
,
s+,~= o .
Otherwise (when w ~: 0) eqns (13) through (16) predict (17), (18), and
2M I + 2p2M 3 )
2p2A 22
g/(2~(w-
0))
sin20
-cos201
-sin20
cos20 j
+
0
cos20
o]
sin20
{
e,,b - e~)/2.J'l,
j
(21)
" (22)
72
K . W . REED AND S. N. ATtURt
Entries on the right-hand side of these equations are known, either because they were
prescribed (the stress), observed (the deformation), or defined (the spin). Thus, A z2
may be determined along the line Pt, P3 = 0 and M ~ and M 3 are quantitatively restricted.
Since in eqns (17), (18), (19), (21), and (22) the left-hand sides depend only upon the
invariant 172, it follows that the right-hand sides o f those equations must also depend
only upon P2 for loads of the type~ (11). This comprises a very severe test to be passed
by any genuinely hypo-elastic material.
In general, a series of tests producing stress-systems of the type (11) may be conducted. There result from eqns (17) through (22) restrictions for each test. The material functions may then be defined so as to minimize some invariant measure of the
modeling error. Note that loadings need not be proportional.
IV. E X A M P L E : P U R E TORSION
If we are willing to take it for granted that a particular material is hypo-elastic then
A ~2 and A 22 c a n be found from the results o f a single test. As an illustration, we consider the response to pure torsion, 0 = 7r/4, 0 = 0. We shall assume volume and wallthickness changes to be ignorable in this test, so A 12 M 2, and A 32 may all be set to
zero (under these assumptions the above mentioned material functions can play no role
in the description of the material behavior for this loading). Furthermore, we shall use
the ordinary Jaumann stress rate, so w: = -eoz. Equations (21) and (22) then give
2M I + 2p2M 3)
2p2A 22
= [
(e:: -
e~,~,)/4geo:
(eo:/s) - (e~z - eo~)/4geo:
(23)
(24)
At this juncture, we pause to make the following fascinating observation: in eqns (23)
and (24), which are fully general for the loading considered, the observed shear behavior, (eoz/~), is associated with the material function A 22, not with 2 M *, which is
usually called the shear compliance. Only when t92 = 0 is the statement 2 M I = (eoz/~)
•valid. Generally 2 M j is associated with the phenomenon studied by POYlqTING [1909]
and SWXFT [1947].
Consistent with our earlier assumption that err and .) both vanish, we put coo = -ez~
in eqns (23) and (24). Defining the "tangent shear modulus" 2 # : = (g/eoz) and writing e for I n ( l / L ) , we get
2 M t + 2/72M3 --- 2#t (de/dP2)
(25)
A 22 = [(1/2#,) -- 2tzt(d~ldp2)]12p2
(26)
A value for A 22 in the limit 172---*0 (if one exists) can be found by application of
l'H6pital's rule. Thus, we have shown that the material's behavior for this loading
depends upon only two observable functions of P2, namely 2 # t ( P 2 ) a n d ~(Pz)In order to continue the example further, we shall assume forms for 2/zt and ~. It
should be emphasized that we are not claiming that these functional forms have special physically significant attributes; they are merely the simplest forms which do not
lead us to a trivial result, and were chosen for that reason only. We assume for 2#t
and ~:
Finite strain plasticity
73
2P.t = 2~to (a constant with the dimensions of stress)
e = ln[l + 2cp2/(2po)2] l/2c (c is a dimensionless constant) .
(27)
(28)
According to eqn (27), the shear stress rate should be linearly related to the shear strain
rate for this loading, as in Cauchy elasticity. It is apparent from eqn (28) that the normal strain can be made arbitrarily small at any given value of P2 simply by choosing
a large enough positive value for c. This is an important point, for the parameter c gives
us direct control over the magnitude of the normal strains in the torsion test, and, as
shall be seen, indirect control over the magnitude o f the normal stresses in rectilinear
shearing. This is not a feature offered by any other model known to the authors. We
can show that for p 2 / ( 2 p o ) 2 << 1 eqn (28) reduces to POYNTING'S [1909] result.
Putting eqns (27) and (28) into (25) and (26) gives
2 M l + 2p2M3 = (1/2/~o)[1 + 2cP2/(21Xo)2] -I
(29)
A 22 = c(1/2/~0)3[1 + 2cpz/(21Zo)21 - t ,
(30)
or, choosing M 3 = 0 and recalling eqn (9), we get
e = [ 1 + 2Cp2/(2p0) 21-I [I -I- C(S'/2p0 ) (~ (S'/2p0) ] :($*/2/~0)
+ [terms associated with A t~ and A t3 ,
I = 1,2,3] .
(31)
The constitutive eqn (31) predicts (27) and (28) exactly. Had we chosen different forms
for 2pt and e [instead of those given in eqns (27) and (28)], they too would have been
predicted exactly by the constitutive equation resulting from application of eqns (25),
(26), and (9). The material functions A w not restricted by this single test may be used
to fit predictions o f eqn (9) to data from other sorts o f tests. If, for example, the bulk
behavior of the material is known to follow a law of the form
J = (1/3k)/~ ,
(32)
in which k may depend upon p~, then, by choosing A ~t as
Air= (1/3){(1/3k)
- (1/2/~o) [1 + 2cp2/(2tto)2] -1}
(33)
and setting.the other undetermined terms to zero, eqn (3t) becomes
e = [1 + 2cp2/(21Zo)21 -~ I! - ( 1/3)I ® I + c(s'/2/~0) ® (s'/2/z0)l :(s*/2#o)
+ (2/zo/9k) (I ® I ) :(s*/2/z0) .
(34)
Equation (34) satisfies each o f (27), (28), and (32) exactly.
In Figs. 8, 9, and 10 typical stress-strain curyes produced by the model 34 in pure torsion, rectilinear shearing, and uniaxial extension, respectively, are shown. In Fig. 8, the
shear stress is plotted against the so-called "average" shear strain 7 = R D / L which is
usually used in experiment (c.f. BELL & KUA~ [1980]). As "~ -> 2e,~ for a tube which
extends during torsion, there is a slight negative curvature apparent in the soz-'t curve.
As the parameter c increases, the extension o f the tube for any given 7 is reduced, and
74
K, W. REz~ At,rD S. N. ATLUI~Z
o
c = 50
0
0
c=
I0
c=2
0
ci"
GAMMA = ~ ~ / L
I.
,
1
I
I
.
_
_
_
.5o
I
~
10
,_o
~-c
= 2
.
Fig. 8. Thin.walled tube torsion of hypo-elastic material (35). Shear stresses and normal strains w.r.t, variations of the parameter c.
o
i
sii
8
!
,
~
0.00
-
i .......
1.00
i .....
2.00
•....
~
3.00
:
,
4.00
.....
:r
5.00
w
'
6.00
7.00
STRAIN
Fig. 9(a). Rectilinear shearing of the hypo-elastic material (35) with c == I0.
. . . .
!
-
B.00
Finite strain plasticity
75
to
o"
.
,,
.00
i
i
0.20
.
.
'
0.40
0.60
'
Sll
.
i
0.80
STRAIN
'
1.00
1.20
I'.40
1'.60
Fig. 9(b). Same as Fig 9(a) except that c = 100.
uiO'5
o
~ oo
~
~-
u = 0,2
9k
0.3
2 % + 6k
0~
O
O
~:-
e,io
°c].oo
,'.oo
2'.00
Loo
+'.oo
5',o0
o'.oo
~'.oo
LogorIthmlc Stroln
Fig. I0. Uniaxial extension of a hypo-clastic material (35) (independent of parameter c).
!
o.on
76
K . W . RI~ED AND S. N. ATLURI
hence, the soz-7 curve approaches a straight line. In Fig. 9, typical stresses accompanying rectilinear sheafing are shown. It is apparent that as the parameter c grows, the ideal
Cauchy-type response is approached more and more closely for this load. It can be
shown that the magnitude o f the normal stress s,t/2/Zo approaches the value I / c
asymptotically. In Fig. 10 typical stresses accompanying uniaxial extension are shown,
plotted against the 'natural' strain e : = I n ( l / L ) .
For future reference, we give the inverse of the constitutive eqn (34) in the following uncoupled form:
(s'*/2/~ o) = [( 1 +
2cp2/(21,to)2)[
-
c(s'/2/Zo) ® (s'/2/~o) ]:e'
Pl = 3k(I:e) .
(35)
(36)
When rectilinear shearing motions are studied, eqn (35) reduces to
SII = ~S22
(st t/2/Zo)/et2 = 2(st2/2#o) (1 - c(sti/2po))
(37)
(st2/2/Zo)/et2 = 1 -- 2(s,i/2po) (1 - c(s,t/2/.,.o)) •
If we stipulate that ( s 1 2 / 2 ~ o ) / e l 2 be unconditionally positive, a "stability" condition
other workers have sought to fulfill by introduction of new stress rates, we are led by
eqn (37) immediately to the restriction:
C > 1/2 .
(38)
Thus, when the condition (38) is fulfilled, there will be no unstable behavior in rectilinear shearing for any initial stress o f the type (11) + a superposed pressure.
We close this section by remarking that a test for path independence in hypo-elastic
materials has been devised by BERr~STEIr~ [1960].
V. A MODEL OP AN IDEAL KINEMATICALLY HARDENING RIGID-PLASTIC MATERIAL
In view of the formal correspondence between hypo-elasticity and the present kinematically hardening plasticity, we adopt eqn (35) as a model for an ideal kinematically
hardening rigid-plastic material. Accordingly, we replace the tangent shear modulus
2/~o by Melan's hardening modulus 2h, the stress s by the back-stress a, the second
invariant of the stress P2 by the second invariant of the back-stress q2, and finally the
stretching e' by the plastic stretching eP:
(a'*/2h) = [(1 + 2cq2/(2h)~ - c(a'/2h) ® (a'/2h)] :e a .
(39)
The path-independence shown by this model may be investigated by application of
BER~STEIN'S [1960] result. Inasmuch as eqn (35) gave results very close to the ideal of
Cauchy elasticity, we expect eqn (39) to give results approaching the ideal Melan
plasticity.
So that we may work a few examples illustrating the behavior of this model, we adopt
a Mises-type description o f the loading surface and associated flow rule:
f ( s ' , a ' ) : = (s' - a'):(s' - a ' ) / 2 - T 2 = 0 ,
(40)
Finite strain plasticity
77
e u = Adf/ds
(41)
in which T is the radius of the surface. In Fig. 11 are shown the normal strains predicted
by the model in pure torsion and the data of Swift. As can be seen, the agreement is
excellent for monotonic loading. Agreement is less satisfactory once there has been a
reversal, but nevertheless we continue to obtain results of the proper order of magnitude, in contrast to other models which overestimate normal strains by one to two
orders of magnitude. Moreover, the magnitude of the Swift effect exhibited by the
model can be adjusted independently through the parameter c, without affecting the rate
of hardening. In Fig. 12, we show the prediction of the rigid-plastic model for the shear
stress-average shear strain curve against the data of Swift. Within the obvious limitations of the rigid-plastic idealization, the agreement is excellent. In Fig. 13, the response
of the model in rectilinear shearing is shown. We have not found data to which such
data as this could be compared, but the normal stresses are small enough to be regarded
as the second-order effect, as they are suspected to be. The shear stress does not oscillate for any. initial value of a of the type (I I) so long as c > I / 2 . In Fig. 14 is shown
the uniaxial stress-strain curve predicted by the model. Finally, in Fig. 15 is shown the
response o f the model to several reversed loadings in torsion.
VI. REMARKS ON MODELING ELASTIC-KINEMATICALLY HARDENING
ELASTIC MATERIAL BEHAVIOR
There are both practical and theoretical reasons for not making the rigid-plastic idealization, even when there can be no doubt that plastic strains are much the larger. First
of all, rigid-plastic models provide no information about the stresses in the non-plastic
o.
O"
T-- 0
z
<
n~
I-=~.
ttlo
o"
0
o.
I
-o.oo
I
~.oo
2.00
I
~.oo
GAI"~A = R~/L
Fig. 11. Thin-walled tube torsion test, kinematically hardening plastic material (39). T = IS tons/in.2;
2h = 7.5 t o n s / i n ? ; c = 3.7.
78
K . W . RE~D A.~;DS. N. ATLURI
0
q
B
cq.i,
U3
I
03
U3
u)
0
0
(.)
- swift
[]947]
0
0
~oo
1
o'.so
.
.
~.oo
.
.
.
,'.so
i'
2'.o0
~..so
6,.%qRA = Ra/L
Fig. 12. Same as Fig. 11; Shear-stress-average strain curve.
o
B
e-
p-
s
Et
A'
0
22.50
2 ~.5o
GN4MA= R~IL
II
B', C'
~.50
20 s~
oo.
I"
0
0
I
Fig. 13. Rectilinear shearing ofa kinematicatly hardening plastic material (39) with load reversal. Same material
as Fig, 11. St2 follows A-B.C-D-E; Sit follows A'-B'.C'-D'-E'.
Finite strain plasticity
79
~q'l"
SIt
e,i"
2o
00
u
~_
oo
0'.23
'
0.67
1'.00
{
1.33
1'.67
21.00
2.33
2.67
Looarlthmlc Strain
C
q
7-
Fig. 14. Uniaxial extension of a kinematically hardening material (39) (independent of parameter c); 2h = 1.
regions of a deforming body. Secondly, it is known that elastic-plastic structures may
become unstable when, under the same loads, the corresponding rigid-plastic structure
remains stable (c.f. RIcE & Rtn~NXC[I [1980]). The more useful material model incorporates elastic and combined isotropic-kinematic hardening plastic behavior. We discuss very briefly the basis for describing material behavior by such a "combined" model.
Without making any constitutive assumption at all we can express the true stress as
a sum of two stress-like tensors:
s = (s - a) + a .
(42)
When s and a are coaxial, it is possible to represent them as ordinary vectors in the
principal-stress-space. The loading surface (40) becomes a circular cylinder with orientation (1,1,1) and diameter 2T in that space.
The back-stress a' lies in the ,-plane and serves to locate the center of the loading
surface. If only T changes as the material deforms then the hardening is described as
"isotropic." If only a' changes as the material deforms then the hardening is described
as "kinematic." When both change during deformation, the hardening is described as
"combined isotropic-kinematic." It is clear that the isotropic part of the hardening (the
growth of the vector ( s - a ) ) must be tied to the growth of T, but it is not necessarily
related to the growth of a. Thus, the simplest possible model incorporating both
isotropic and kinematic hardening leaves the two "uncoupled." Mathematically:
(s - a) ° = L(e, (s - a) )
(43)
80
K.W.
REED AND S. N. ATLURI
0
o
0
o.
e",- o
0=6_
,41--'~
~'J.
i 00
,,.00
2'.00
i
i
3..no
GAMHA
I,-- o
4.~o
=
5'.00
R~/L
0
I0
0
e,I
I-
8
c5
I
Fig. 15. Thin-walled tube torsion test; kinematically hardening plastic material (39). Three complete cycles
between 7 = 0 and 3, = 2.4. Curves are coincident. Accumulated axial strain E. - 19%. c= 0.55; T = 19;
2hffi 1.
a* = G(eP, a) .
(44)
This is different from the usually encountered model because of the appearance of a*
on the left-hand side of eqn (43). However its omission would spoil the uncoupled character of the model. Technically, eqn (43) would be styled after ordinary isotropically
hardening plasticity and eqn (44) after a model of the type discussed in the previous sections.
VII. NUMERICAL INTEGRATION: STABILITY AND OBJECTIVITY
Stability
A stress.dependent critical strain increment E: = ieH[ can be found for the constitutive eqn (5) and hence eqn (6), as
E < 2/Ids*/dsU
(45)
in which H is the time step, and Uds*/dsl signifies the max over e of [~ {d(L:e)/ds}lJ/
UeJ]. The critical strain increment decreases in magnitude as ~s~ grows. Critical strain
increments for eqns (35) and (39) are found in Appendix C.
Finite strain plasticity
81
Objectivity
In order to be called objective, a numerical approximation for a physical entity must
transform between frames of reference by the same rule as the entity itself. The true
stress s, for example, transforms between frames according to the rule
= QsQ t
(46)
(where Q is an orthogonal tensor), and we thus require any objective approximation
for s to transform by the same rule (46). The advantage of this definition of numerical objectivity is that it may be applied directly to any invariant entity; it is not intrinsically bound to any particular stress tensor, constitutive equation, or numerical
technique. This is an important point because the potential exists for confusion between
the presently discussed "preservation of invariance" and the Principal of Objectivity
of material properties (c.f. HUOHES • L I t r [1981]).
Unfortunately, we can only achieve degrees of objectiv!ty in numerical analysis, and
thus the absolute errors present in an approximate solution depend, to put it crudely,
on the choice of coordinates (REED ~ Arr.trgx [1983]). This point has not been brought
out by other discussors (HuGrms a, WIr~OET [t980], PtNSKY et al. [19831, RtrBE~rSTEI~r
& ATLURI [1983]).
The key to the construction of objective algorithms is the recognition of the fact that,
just as in time-differentiation of tensors, in their time-integration it is necessary to specify the "group" of reference frames in which the integration is to be carried out, whether
that integration is numerical or not. The property which distinguishes an acceptable
group of frames is that the relative rotation between any two members of the group is
constant w.r.t, time. This in itself does not suffice to determine any unique group;
examples of groups of frames possessing this property are (i) the group of all inertial
frames; (ii) the group of all Lagrangian frames; and (iii) groups of corotational frames.
Groups of the latter two types are regarded as the significant ones with regard to constitutive equations.
If we specify that integration be carried out in a corotational frame then we may
apply the " J a u m a n n integral" of GODD~D & MfLLER [1966]:
s(t) = Q t ( t ) s ( r ) Q ( t ) +
fTIQ t ( t ) Q ( x ) s * ( x ) Q t ( x ) O ( t )
dx .
(47)
In eqn (47) Q is the orthogonal tensor relating quantities in the working frame to their
counterparts in the corotational frame. We have previously shown (REED ~, ATLURI
[1983]) that Q is defined through the initial value problem
Q(t) = - Q ( t ) w ( t ) ,
Q(T) = I ,
(48)
in which w(t) is the ordinary spin tensor. Any of the standard formulae for approximation of definite integrals may be applied to eqn (47) and the result is an "objective"
integration rule. When Q(t) can be found exactly within the interval of interest then
the result is precisely objective; that is eqn (47) will yield the same result no matter what
frame it is applied in.
However, in practice it is generally only possible to approximate the solution of eqn
(48). The "degree" to which our numerical result may be called objective is then directly
82
K.W. l ~ : o AND S. N. ArtuRx
related to the degree of accuracy with which Q has been found. The initial value problem (48) may easily be reduced to a scalar problem by introduction of "angles of
rotation":
(dQ/dOi)Oi=
-Q(Oi)w(t)
,
(49)
Oi(r) = 0 ,
"3
¢02C03
C0t$03 "1"S0tS02C03
SOTS03 -- C01S02C03 I
l
Q(01,02,03)
=
--C02,S03
sO2
(50)
cOtcO 3 -- sOts02sO 3 $OtcO 3 + cOts02sO 3 [
-sOl c02
col c02
J
in which the abbreviations sO~ = sin0~ and c0~ = cos0/have been used. The angles are
absolute scalars and may thus be integrated objectively by standard numerical schemes.
The error of such schemes is usually expressed in the form
le(tk)
- 0k[< CHN+'o(tv+t)(tk)
(51)
where 0 eN+t) is the ( N + 1)th time derivative of 0 and H is the time step size. When
eqn (47) is used in conjunction with a Q of Nth-order accuracy, eqn (46) will be exactly
satisfied only for those Q whose angles are Nth-order polynomials w.r.t, time. Hence,
numerical objectivity cannot be absolute.
As an example, we derive an objective midpoint rule by approximation of eqn (47).
By inspection, we can write:
sN+I = Q'(tN+l )sNQ(tN+t) + HQt(t~c+t)Q(t~c÷x)s~c+rQ,(tN+r)Q(tN+t)
•
(52)
When s* is eliminated by use of the constitutive equation and K set to 1/2 then eqn (52)
reduces to the formulate of KEY et al. [1981] ~, Rtrm~NST~ ~ ATLUm [1983]. The essential point to be made is that through eqn (47) much more general objective schemes may
be easily and accurately constructed (see I ~ D & ATttrSa [1983]).
Acknowledgements-The results presented herein were obtained during the course of investigations supported
by NASA-Lewis Research Center under a grant, NAG-3-346 to Georgia Institute of Technology. This support as well as the encouragement of Drs. C. Chamis and L. Berke are gratefully acknowledged. Appreciation is also expressed to Ms. Joyce Webb and Ms. Tanya Jackson for their assistance in the preparation of
the manuscript.
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1909
1938
1947
1960
1966
1966
1968
1970
1972
POYNTtNC,J.H., "On Pressure Perpendicular to the Shear Planes in Finite Pure Shears and on the
Lengthening of Loaded Wires when Twisted," Proc. Roy. Soc. (Lon.) A82, 546.
Mm.AN,E., "Zur Plastizitit des riumlichen Kontinuums," Ing.-Arch., 9, 116.
SwiFr, H.W., "Length Changes in Meufls Under Torsional Overstrain," Engineering, 163, 253.
BE~S'rEI~, B., "Hypo-Elasticity and Elasticity," Arch. Rat. Mech. Anal., 6, 89.
FL-n~3E~n'rt~a.,A.M. and RONAY, M., "Second Order Effects in Dissipative Media," Proe. Roy. Soc.
(Lon.), A292, 14.
GODDAttD,J.D. and Mll.tut, C., "An Inverse for the Jaumann Derivative and Some Applications
to the Rheology of Viscoelastic Fluids," Rheolosica Acta, $, 177.
R a t . W. and S'riJw~, H.-P., "Der Einfluss der Textur auf die l.~nfeiinderung im Torsionversuch,"
Z. Metallkde, 59, 396.
WANG,C.-C., "A New Representation Theorem for lsotropic Functions," Arch. Rat. Mech. Anal.,
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Lm-B~ANN,Trt., "Einige Bemerkungen zu einer allgemeinen Klasse yon Stoffgesetzen ftir grosse elastoplastische Formhnderungen," lng.-Arch., 41, 297.
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1981
1981
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1982
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1983
1983
1983
1983
1983
1983
83
ON^T,E.T. and FS.gDSmSh~, F., "Representation of Creep, Rate Sensitivity, and Plasticity," SIAM
J. Appl. Math., 25, 522.
BU.L~6"rON,E.W., "Non-linear Mechanical Response of Various Metals [, If, Ill," J. Phys. D: Appl.
Phys., 10, 519.
HszL,coY, A., "Three Dimensional Analysis of the Elastico-viscous Lubrication in Short Journal Bearings," Rheologica Acta, 16, 51.
Dmt,r~s, J.K., "On the Analysis of Rotation and Stress Rate in Deforming Bodies," Acta Mech.,
32, 217.
ASTU¢,J and JoNEs, R.S., "A Note on Harnoy's Equations of State," J. Non-Newtonian Fluid
Mech. 4, 359
BELl.,J.F. and KP,~H, A.S., "Finite Plastic Strain in Annealed Copper During Non-proportional
Loading," Int. J. Sol. Struct., 16, 683.
RicE, J.R. and Rtmmcg,, J.W., "A Note on Some Features of the Theory of Localization of Deformation," Int. J. Sol. Structu., 16, 597.
HUGHES,T.J.R. and Wtr~OET, J., "Finite Rotation Effects in Numerical Integration of Rate Constitutive Equations Arising in Large-deformation Analysis," Int. J. Numer. Meth. Eng., 15, 1862.
NAGT~G~L, J.C. and DE JoNc, J.E., "Some Aspects of Non-isotropic Workhardening in Finite
Strain Plasticity," Proceedings of the Workshop on Plasticity of Metals at Finite Strain: Theory,
Experiment and Computation, eds. LE~, E.H. and MALLETT,R.L., Stanford University, 65.
LE~, E.H., MXLL~Tr, R.L., and WEaTaEtMEX, T.B., "Stress Analysis for Kinematic Hardening of
Finite Deformation Plasticity," Scientific Rept. to the Office of Naval Research, Dept. of Navy.
HuoR~s, T.J.R. and L~u, W.K., "Nonlinear Finite Element Analysis of Shells: Part I. Threedimensional Shells," Comput. Meth. Appl. Mech. Eng., 26, 331.
KEY, S.W., STONE, C.M., and KaEm, R.D., "Dynamic Relaxation Applied to the Quasistatic Large
Deformation Inelastic Response of Axisymmetric Solids," in Nonlinear Finite Element Analysis in
Structures, Springer, Berlin, 585.
JomcsoH, G.C. and B^m,ts~N, D.J., "A Discussion of Stress Rates in Finite Deformation Problems,"
Rept. No. SAND-82-8821, Sandia National Laboratories, Alburquerque, New Mexico.
LEE, E.H., MALLE~r, R.L., and WEaTHEIMEa, T.B., "Stress Analysis for Anisotropic Hardening
in Finite-Deformation Plasticity," J. Appl. Mech., 50, 554.
D^vsJ.h~s,Y.F., "Corotational Rates for Kinematic Hardening at Large Plastic Deformations," J.
Appl. Mech., 50, 561.
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Meth. Appl. Mech. & Eng. (In press).
RSED,K.W. and ATLUR[, S.N., "Analysis of Large Quasistatic Deformations on Inelastic Bodies
by a New Hybrid-Stress Finite Element Algorithm," Comput. Meth. Appl. Mech. Eng., 39, 254.
NAGTI:GAAL,J.C., "A Note on the Construction of Spin Tensors," as discussed by T.J.R. Hughes
at the Workshop on the Theoretical Foundations for Large-Scale Computations of Nonlinear Material
Behavior, Northwestern Univ., Evanston, Illinois, Oct. 24-26.
PlHsg'~, P.M., OaTLZ, M., and PISTEa, K.S., "Rate Constitutive Equations in Finite Deformation
Analysis: Theoretical Aspects and Numerical Integration," Comput. Meth. Appl. Mech. Eng., 39.
RUaENSTEIN,R. and ATLURI,S.N., "Objectivity of Incremental Constitutive Equations Over Finite
Time Steps in Computational Finite Deformation Analysis," Computer Meth. Appl. Mech. Eng.,
36, 277.
APPENDIX A
ON THE SOLUTION OF LEE ET AL. 119S31 TO THE RECTILINEAR SHEARING PROBLEM
We m a y represent the back-stress a in this problem as in eqn (I I) in the text above.
When Lee's stress rate is used w = 2et2sin20 for this problem, which, together with
Melan's rule (4) yields
(~'/h) = s i n 2 0
20'=
[(1 + ( a / h ) ) / ( a / h ) ] c o s 2 0 - 1 .
(A1)
(A2)
W e a s s u m e h to be a c o n s t a n t , as in t h e p u b l i s h e d examples~ W h e n 0 lies in t h e r a n g e
0 < 0 < a-/2 then
(a'/h) > 0, so we m a y divide eqn (A2) by (A1) to obtain
d20/d~ = [ ( 1 + ~)cos20 - ~ ] / ~ sin20 ,
(A3)
84
K.W. REED A N D S. N. ATLUR!
in which ¢~: = ( a / h ) . Equation (A3) may be integrated for t~ ~e 0 as
(A4)
cos20 = (a - l ) / a - [ao(1 - cos20o) - l l ( l / a ) e x p { - ( a - ao)}
The stress-strain curves published by LEE et al. [1983] correspond to a singular solution o f eqn (A3). Although Lee's numerical results agree reasonably well with our own
for a range o f strain, the present result indicates that 0 approaches an asymptote o f 0
degrees for large strains. This is in conflict with the numerical result of Lm~ et al. [1983],
which approached 15 ° asymptotically.
The stability condition which motivated Lee to introduce a new stress rate may be
written as
(AS)
(s{2/h) = (tisin20)' = a'sin20 + ~(20')cos20 > 0 ,
or, after using eqns (AI) and (A2) to eliminate ~' and 20',
(A6)
I - acos20(l - cos20) > 0 .
In Fig. AI we have plotted eqn (A6) as well as (A4) for various initial ~ and 20. As can
be seen there is a zone in the ti - 20 plane inside o f which the stability condition (A6)
8
12
t~
x
<0
St
~.
12
8
0
0
1'.00
2.00
i/
~.oo
4'.oo
s'.oo
^
o
Fig. AI. Solutions of the eqn (A3).
i
e.oo
i
7.oo
~.oo
Finite strain plasticity
85
is not satisfied. In Fig. A2 we show a typical shear stress-shear strain curve corresponding to a solution (A4) which passes through this zone. We note that D,~,aa.tAs [1983]
has shown that the Green-McInnis rate leads to analogous stress-strain curves for nonvanishing initial d.
We close by remarking that our solution to this problem is in no way pathological.
It also indicates, and we have found numerically, that when the hardening modulus h
decreases with strain, as would be required to describe uniaxial strain softening, the use
of Lee's rate can then lead to unstable behavior even in infinitesimal shearing.
APPENDIX B
HARNOY'S STRESS RATE
HARNOY [1977] a LEE et al. [1981] have proposed that a corotational stress rate be
based on the spin of the eigenvector triad of the stress and on the spin of the material
elements instantaneously coincident with the eigenvector associated with the maximum
eigenvalue of the back-stress, respectively. As-rr~ * J o ~ s [1979] have observed correctly
that Harnoy's rate must always be coaxial with the stress itself. This is also made evident by defining w: = ~) in eqn (1,2). We now recall a constitutive postulate which
encompasses a very wide class of rate-type materials, including hypo-elasticity, ordinary
• plasticity, and some engineering theories of creep and viscoelasticity:
e
= F(s*,s) ,
(B1)
in which F is not necessarily linear w.r.t, s*. The coaxiality of s and s* in any material o f class (B1) ~lcarly implies that s and e are also necessarily coaxial. All solutions
8
,..:,
C'q
~uO
i o(/3
O3
n,,"
I.-- o
o-
o
O
O
%'.00
i%3
2%7
4'.00
5'.s3
8'.87
8'.o0
9'.~3
STRAIN
Fig. A 2 . Stress-strain curve ['or LEE'S [1983] rate; initial values ~ = 8, 20 = ~r/2.
i
10.67
86
K . W . REED Am) S. N, ATLURI
of eqn (B1) must have this character, leading us to conclude that this corotational rate
is unsuitable for use in this class of constitutive equations.
APPENDIX C
CRITICAL STRAIN INCREMENTS
Evaluation of the matrix norm ] d s ' / d s ] in the stability condition (45) is equivalent
to finding the maximum over • of ][d(L:e)/ds][/]e], i.e., an eigenvaiue problem. For
the constitutive eqn (35) d(L'e)/ds is given by
d(L:e)/ds =
-cw(1)
-
c(s' ® e') + 2c(e' ® s')
(co
in which the stress has been normalized by 2#o and w: = s':e'. This is an unsymmetric fourth-order tensor, whose least eigenvalue for [e[ = 1 and fixed s' will determine
the critical strain increment E. [Material stability in the ordinary sense requires that all
of these ¢igenvalues be less than or equal to zero; it is not shown here that this condition is fulfilled by eqn (35).] So that we do not have to work with an unsymmetric tensor, we pre-multiply eqn (CI) by its transpose, obtaining the positive-indefinite
fourth-order tensor:
c 2 [ 4 ( e ' : e ' ) ( s ' ® s ') - 3 w ( s ' ® e ' + e ' ® s )
+ ( s ' : s ' ) ( e ' ® e ' ) + w2(I)] .
(C2)
The trace D of this tensor is the sum of its eigenvalues, and since all of these are necessarily non-negative, D must be greater than or equal to the greatest eigenvalue R.
Hence
R -< c2[5(s':s')(e':e ') + 3w 2] ,
(c3)
which we must maximize for ie'] = 1. It is clear that this is accomplished by choosing
e' parallel to s'; then (C3) yields
R ~: 8c2(s':s ') .
(C4)
It follows that the least eigenvalue r of the original matrix (C1), which is necessarily
negative, is bounded from below by
r_> -2x/2 c~s'[ ,
(C5)
so the critical strain increment E must be (in terms of un-normalized stress)
E<
(2p.o)/[~ c~s'~] .
(C6)
It is noteworthy that this restriction becomes more acute as c and ~s'~ grow.
In view of the correspondence between the ideal hypo-elastic and kinematically
hardening plastic models, we may also establish the following critical plastic strain increment E p for the latter (39) as
E p = (2h)/[x/2 c~a'~l .
(C7)
Finite strain plasticity
87
We suspect that these bounds are not sharp because no use has been made of the fact
that e' and s' are traceless. Nevertheless, they are of the proper order of magnitude:
when the ideal hypo-elastic model (35) is specialized to the case of rectilinear shearing
a critical strain increment may be found independently from the above analysis as
E ~ 2~0/csl2, which is only about twice as large as that given by eqn (C6).
Southwest Research Institute
San Antonio, TX 78284, USA
Georgia Instiuteof Technology
Atlanta, G A 30332, U S A
(Received 15 June 1984)