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Objective corotational rates and shear oscillation
International Journal of Plasticity, Vol. 8, pp. 643-656, 1992
0749~419/92 $5.00 + .00 Copyright © 1992 Pergamon Press Ltd.
Printed in the U.S.A.
OBJECTIVE COROTATIONAL RATES AND SHEAR OSCILLATION WEI YANG,* LI CHENG,**
and KEH-CHII-I HWANG*
*Tsinghua University and **Beijing Institute of Technology (Communicated by Yannis Dafalias, University of California-Davis) Abstract--Influence of objective corotational rates to shear oscillation is discussed here under
the framework of finite deformation, J2-flow theory. Under an elastic-plastic, mixed hardening constitutive law, the active stress s, as defined in eqn (8) of the text, and back stress b are shown to be governed by two sets of equations formally uncoupled to each other. Formal solutions of complex stresses are obtained for plane strain, isochoric deformation, with the conclusion that the shear oscillation behavior for an elastic-plastic solid can be asymptotically reduced to a rigid-plastic solution and a higher order correction, provided the initial shear strain is small. Consequently, the active stress s appears to be nonoscillatory under a realistic but arbitrary corotational rate, whereas the oscillatory behavior of back stress b is solely determined by the maximum corotation angle, defined in eqns (18) and (41), as well as the variation of material hardening with respect to the corotation angle. A spectrum of back stress responses could be predicted by the incorporation of different objective corotational rates.
I. INTRODUCTION
In finite deformation, elastic-plastic constitutive theory, the objective corotational rate, albeit crucial in the construction of stress rates, can be selected from a variety of candidates ( e . g . FARDSHISHEH & GNAT [1974], LEE, MALLET 8, WERTHEIMER [1983], AGAHTErIRANI et al. [1987]). There have been many works focused on the correct choice of the objective corotational rate, especially since the work of NA~TEOX~a~and DE JONC [1981], who showed an oscillatory stress response for simple shear motion when combining Prager-Ziegler kinematic hardening model with Jaumann rate. Remedies by adopting special corotational rate have been proposed to suppress this shear oscillation, from the early work on finite elastic deformation by Dmr~ES [1979], to the recent investigations on kinematic hardening formulation by LEE et al. [1983], D.'ffALIAS[1985], AG~a-bTEHRAtqIet al. [1987], and by HWANGand CI-Ir.NC [1989]. The above-mentioned researches, however, were confined in the rigid plastic case and subjected to the assignment of a particular corotational rate. The general influence of the objective corotational rate on shear oscillation is still to be explored, especially in connection with the material hardening behavior. The present paper addresses this problem from a finite deformation, mixed hardening J2-flow formulation. A formal solution for plane strain, isochoric deformation is obtained for arbitrary corotational rate, material hardening, and anisotropic hardening parameter c. For the purpose of shear oscillation investigation, we are able to show that the problem is asymptotically reduced to a rigid plastic solution and a higher order correction. Several qualitative and quantitative results about shear oscillation are obtained via rigid plastic model, complemented by examples illustrating some interesting features.
Sponsored by the National Natural Science Foundation of China. 643
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W. YANG et al.
II. OBJECTIVE COROTATIONAL RATES
Corotational rates refer to the rates corotational with a skew-symmetric spin tensor //. If ~ satisfies the frame transformation law during a frame rotation Q carrying frame F to frame F, that is, /~ = Q./-/.Qr + Q . Q r ,
(1)
where the superscript T denotes tensor transpose, then the rate corotational w i t h / / w i l l be objective (LEE, MALLET, R, WERTHEIMER [1983]) A = A-
Q.A + A./'/
(2)
and termed objective corotational rates. Symbol A in eqn (2) denotes any objective symmetric tensor of second rank. It is straightforward to prove that A:B
= .,/i : B
(3)
for any second rank tensor B commutable to A. Various objective corotational rates have been employed in constitutive formulation, such as the Jaumann rate (while Q equals the material spin w), the relative spin r a t e / / = R . R r by DtE~qEs [1979] (with R being the rotation tensor in polar decomposition of the deformation gradient), and the spin rate of material fibre coinciding with the maximum principal direction of back stress (LEE et al. [1983]). The most general form of Q can be deduced axiomatically from the guiding principles of continuum mechanics, such as the requirement of objectivity. This approach was pursued in works by FARDSHISHEHand OSAT [1974] and by AGAH-TEHRANI et al. [1987]. Alternatively, HWANG and CHENG [1989] showed that/1 can be described by a construction approach in terms of the frame spin associated with the principal axes of an arbitrary symmetric objective tensor S of second rank
,~
- -
1
~.~ - , ~
•
s
(S)(~)n~n~,
s
(4)
where A~ (c~ = 1,2,3) are the principal values of S with associated eigenvectors no.S The notation (S)(~0) in eqn (4) describes the components of the material rate of S, namely S, with respect to the principal directions of S, calculable through
(s)~o~ = n~-S.nL
(5)
The parentheses on subscript or/5 emphasize that (S)(~o) is not a tensorial component. Like the frame spin of tensor S, the skew-symmetric tensor t/s can be used to construct the objective corotational rate in eqn (2) so that A is the rate sensed by an observer who travels with the frame formed by triad (n sl, n2, s n3s ). Accordingly, the corotational rate A is formed by the absolute rate ,~ minus that p a r t / / s . A - A-/-/s which the observer cannot feel. The second rank symmetric tensor S is in general arbitrary from the continuum mechanics viewpoint, and there is no consensus to determine it. The frame spin o f S is understood in the sense of Hill's principal axes analysis. Various candidates of
Objective corotational rates and shear oscillation
645
S have been proposed to construct the corotational rates. For example, S can be specified as the back stress b as motivated by the work o f LEE, MAtLET and WERTHEII~tER [1983], or taken as the deformation rate d (or Rivlin-Ericksen tensor o f order n, A tn)) by HwAr~G and O-raNG [1989], or chosen as the left stretch tensor V (or Almansi strain tensor e, etc.). For the last c a s e , / / s becomes the familiar Euler s p i n / / e .
III. M I X E D H A R D E N I N G T H E O R Y
Kirchhoff stress r and deformation rate d are often adopted as the basic work conjugates in the finite deformation theory. If the elastic response of a rate-independent elastic-plastic material is isotropic, one can describe the constitutive law for finite deviatoric deformation as ~' = 2 G ( d ' - d p)
(6)
where r ' and d' represent the deviatoric parts of r and d, respectively, G is the shear modulus, d p is the deviatoric part of plastic deformation rate, and i ' denotes the objective corotational rate as defined in eqn (2) for the stress deviator. The additive decomposition of d in finite deformation, as well as the incompressibility o f plastic deformation, have been used in deriving eqn (6). Under the classical postulate o f Maximum Plastic Work, one can construct d p from a yield surface function. Under a mixed hardening, J2-flow theory, the Mises-type yield surface is formulated by f=
]s:s
- OFz = 0,
(7)
where the active deviatoric stress s is defined as s = r ' -- b,
(8)
with back stress b assumed to be deviatoric. The quantity aF in eqn (7) signifies the radius of the yield surface in r space. It is assumed that or evolves in proportion to the evolution o f the flow stress Y in uniaxial tension, as proposed by KADASCHEVICnand Novoznn~ov [1958] as well as in more recent work of MEAR and HuxcnIr~SON [1985] aF = CYo + (1 -- c)Y.
(9)
The constant parameter c varies between zero and one. Extremes c = 0 and c = 1 correspond to isotropic and kinematic hardening, respectively. In eqn (9) Y = Y(gP)
and
Yo = Y(0)
(10)
denote the current and initial flow stresses measured from a uniaxial test, while gP,
~P =
d p dt
dp =
d p : d p,
(11)
is the current effective plastic strain. By means o f the associated flow rule and the uniaxial stress-strain curve, one can write d p as
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W. YANC et al.
dp -
9 ~":s
4h
a2F
dY d~ p'
h -
S
(12)
where h represents the tangential hardening modulus for flow stress with respect to plastic strain in uniaxial tension. The evolution of back stress b is governed by PragerZiegler law. Combining it with the above formula of d p, we obtain 3
~":S
I~ = ~ c ,~o--v-s,
(13)
where the proportional evolution law in eqn (9) has also been utilized. Apparently, the yield surface center will not evolve for the case of isotropic hardening (c = 0). We now rewrite the stress evolution law in eqn (6) by substituting d p in eqn (12)
20(d 9 s) 4h
o~- s .
(14)
The inner product ~' : s in both eqn (13) and eqn (14) can be determined by the consistency condition f = 0 as ~ ' : s = 20F~'.
(15)
To this end, expressions in eqns (3), (7), (8), (9), and (13) have been used. Substituting eqn (15) back in eqn (13) and eqn (12), one obtains 1~ = C - -
~z S,
OF
S ~-~
2haF dp
(16)
317
applicable to both elastic-plastic and rigid plastic cases. For the latter case, d p in eqn (16) is replaced by d, so s can be evaluated directly by the known kinematics. Accordingly, the active deviatoric stress s will not oscillate for rigid plastic solid, irrespective of the objective corotational rate//. For the elastic-plastic case, it is more convenient to replace the last expression in eqn (16) by =2Gd'-
__3 + c
--s.
(17)
OF
We now reduce the mixed hardening theory to the successive solution of uncoupled eqn (17) and then the first equation in (16). IV. P L A N E S T R A I N , I S O C H O R I C D E F O R M A T I O N
Attention is now focused on plane strain deformation. The only nontrivial components of the corotational spin are ~'~12 = --~'~21 = 1 4 .
(18)
in eqn (18) is the corotation angle induced by deviatoric deformation. If we further restrict to isochoric deformation, then from eqns (2), (8), (16), and (17)
Objective corotational rates and shear oscillation
p
d:3 = ri3 = si3 = bi3 = 0
647
i = 1,2,3; (19) t
d~2 = --d~l, r~2 = --rll, $22 = - - S I I , b22 = - b l l , which implies that each of the above four tensors has only two independent components. It is therefore convenient to introduce the following complex, dimensionless quantities D = d~l + ida2
D p = d p + idlP2
(20) S = ~f~ Sll + iSl2
B = xf~ bll + ibl2
Yo
T = x/~ r[l + ir[2
Yo
Yo
They reduce governing eqns (16) and (17) to [~ + i ~ B = rS
r -
qP = 2
S = qPD p
(21)
cy
c + (1 - c ) y c + (1 -
c)y
(22)
4~dp
;~+ (i~b+yP0"0)S= 3'0 2 D
p=
~2 + ~'or,
(23)
where Y
Y = --
Yo
"Yo =
Y0
(24)
4~G
are the dimensionless hardening function and initial yield strain for shear. Integrating (21) from the initial yield instant to, one arrives at
B(t)
=
(25)
r ( ~ ) S ( ~ ) e -it#tt)-#(~)l d~,
where S is given by
S(t) = iexpli[¢(to)
- ¢(t)]
p(t)
-p(to))yo
(26) + -"Y0
I
D(~)exp i[~(~) - ~(t)]
3/0
Equations (25) and (26) provide a formal solution o f the elastic-plastic mixed hardening response in plane strain, isochoric deformation. Their applications, however, are somewhat hindered by the involvement o f d p in the expressions o f p and y.
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W. YANG et al.
V. SIMPLE SHEAR DEFORMATION
The kinematics of simple shear deformation is delineated in Figure 1. The only nontrivial components of deformation rate tensor are d12 = d21 = "Y ~,
(27)
where 7 is the shear amount as defined in Figure 1. For monotonic and rate-independent shear deformation, the corotational speed ~ in eqn (18) can be written explicitly as _
d~(7)
d7
. 7-
(28)
The monotonic behavior of 7 renders it a time-like parameter. Denoting
y(gP(t)) = .v(7),
p(t) = P ( 7 ) ,
(29)
we can specify the general plane strain, isochoric deformation, as invoked from eqns (25) and (26), to the particular case of simple shear, while the integration can be carried out with respect to shear amount from 7o to 7. To facilitate the discussion of shear oscillation, certain restrictions are made to functions ) (7) and • (7), respectively: 1. The material response is nonsoftening, that is, .v'(7) -> 0
v7-
(30)
2. The shear-induced corotational speed is bounded by 0 < ~/v(7) < cz vT.
(31)
The lower bound in eqn (31) implies that the stress rates cannot corotate to a spin that contradicts to the sense of the actual material spin, whereas the upper bound of ~ ' ( 7 ) is motivated by the concept that the stress rate corotation should not be accelerating as 7 increases. If the constant a in eqn (31) is taken as unity, the upper bound on ~ ' ( 7 )
X2
1
......
0/
/
Fig. 1. Simple shear deformation.
X1
Objective corotational rates and shear oscillation
649
would imply that the corotational rates cannot exceed the material spin (namely, Jaumann rate) itself. Two particular cases will be addressed prior to the general discussion on shear oscillation. 1. Nonhardening materials, such that p(3") = 1 for any 3'. In this case, eqns (25) and (26) are reduced to
B(3")
=0
S(3") = iexpli~(3"°) - i~(Y) - "/-3 ¢P(3")13"o (32) + - ,f'ox -
i~(~)
f
3'0 ~ ~o
_
i~(3")
~
_
gP(3") _- gP(~)
'.
3'0
d~. )
Consequently, shear oscillation is effectively suppressed because the back stress vanishes. 2. Noncorotating rate, that is, 4(3') = 0 for any 3". As pointed out by HWAr~G and CnENG [1989], the corotation angle # (3") vanishes for simple shear deformation whenever Q is taken as O a, the frame spin for deformation rate tensor (namely, the spin o f the triad (n a, n2a, na), signifying the principal axes of d), or the frame spin o f Rivlin-Ericksen tensor o f any order. In this case, eqns (25) and (26) reduce to
B(3') =
f
" cp'(~) o c+ (1-c)p(~)
S(~)d~ (33)
--
k
k
--
3"0
3"0
exp
o
d~
,
3'0
and both B and S are purely imaginary. Thus, only the shear components of tensors r', s, and b are nontrivial, and oscillation will never occur, provided condition eqn (30) is satisfied. In the subsequent discussion, we assume .P(3') > O, ~'(3") > 0
v3".
Vl. ASYMPTOTIC SOLUTION FOR ELASTIC-PLASTIC
(34) PROBLEMS
From eqns (25) and (26), it is concluded that shear oscillation can occur only if the corotation angle 4(3,) exceeds a-/2. In combination with eqn (31), shear oscillation would be possible only if 3" reaches the order of unity. On the other hand, the initial yield strain for shear, 3'0, is only about 0.2% for many metals. Thus, it is reasonable to consider an asymptotic solution for 3' >> 3"0
(35)
w. YANGet al.
650
in the study of shear oscillation. Also implied in this statement is the assumption that the hardening modulus h defined in eqn (12) is much less than the elastic shear modulus G. Although less important, shear oscillation can also be caused by elastic response, as discussed by PRAGER [1965], DIENES [1979], and by ATLURI [1984]. The present work is rather focused on the shear oscillation dominated by plastic response. After a straightforward, but lengthy, asymptotic analysis, eqn (26) for active stress can be simplified to
(36) 1-c
3'
3'
where y and h are known functions defined in eqns (10), (12), and (24) and relate to the uniaxial tension data. Accordingly, the leading term in S(3') for the elastic-plastic case is the rigid plastic solution, which precludes the possibility of shear oscillation of the active stress. Substituting eqn (36) into eqn (25), one arrives at an asymptotic expression of back stress.
x/3Yod~Y ~
h
(37) x exp[i,b(~) - i,b(y)] d~ + 0(-/2). The leading term of back stress also corresponds to the rigid plastic solution, and the correction terms due to elastic deformation only slightly increase the possibility of shear oscillation. Consequently, the estimate of eqn (35) would infer that shear oscillation is basically decided by rigid plastic solution. VII. RIGID P L A S T I C S O L U T I O N
For rigid plastic solution, one has gp= !
and
.9(3,)=y{!].
(38)
Introducing the variation rate of material hardening with respect to the corotational angle
d2
R(~) - d~
P'(y)
~'(3') '
(39)
Objective corotational rates and shear oscillation
651
we are able to write the back stress for the rigid plastic case as
(40)
B(3") = iCfo ~t'Y)R(~)e -i~-~) d~. Its oscillatory behavior is solely determined by the function R ( 4 ) and 400 = lim 4(3"),
(41)
termed the m a x i m u m corotation. Conventionally, the oscillatory behavior of back and total stresses is referred to the variation of their shear components
b12(3') = -cYo ~ - fo etv) R ( ~ ) c o s [ 4 - ~1 d~ (42) r12(3") =
c + (1 -
r
c)y
+ Cvo
1
R ( ¢ ) c o s [ 4 - ~] d e .
We introduce the following definitions with respect to the above two stress components: 1. b (or r) decline is referred to the occurrence of a nonpositive derivative of bl2 (or r12) with respect to 3'. Consequently, if there exists a 4 " in (0, 4 = ) such that R(4*) =
ff"
R ( ~ ) s i n ( 4 * - ~) d~,
(43)
then b-decline occurs. Similarly, if eqn (43) is replaced by R(4*) = c
f0 4'*
R(~)sin(4* - ~)d~
(44)
then r-decline occurs. 2. b (or r) return is referred to the occurrence of a nonpositive value of b12 (or 712). Accordingly, if there exists a 4 " in (0, 4o0) such that 0~* R ( ~ ) c o s ( 4 * - ~) d~ = 0,
(45)
then b-return occurs. Similarly, if eqn (45) is replaced by 1+
ff"
then r-return occurs.
R ( ~ ) [1 - c + ccos(l,* - ~)] d~ = 0,
(46)
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W. YANG et al.
3. r-oscillation is referred to the case of shear stress r~2 returning to its initial value of Yo/x/-3, r-oscillation occurs if
£
~* R(~o) [1 - c + c c o s ( ~ * - ,p)] d~o = 0
(47)
for some ~* in (0, ~ ) . Based on the above definitions, we are able to conclude: 1. If ~o0 -< 7r/2; or if ~o~ in [(7r/2), 7r] and R ( ~ ) nondecreasing, b-return will be impossible. 2. If ~oo -> 7r, and R ( ~ ) nonincreasing, b-return should happen at some ~* _< 7r. 3. I f ~= < 1, and R(4i) nondecreasing, b-decline will be impossible. 4. I f ~o0 -> 7r/2, and R ( 4 ' ) nonincreasing, b-decline should occur at some ~* _<_7r/2. 5. For purely kinematic hardening (c = 1), r-oscillation is equivalent to b-return, and r-decline is equivalent to b-decline. 6. r-oscillation is impossible when c ___ ~. 7. If ~= _< 1/c and R ( ~ ) nondecreasing, r-decline will never happen. To demonstrate the above conclusions, let us consider the special case of R ( 4 i) = Ro(1 + e ~ ) ,
(48)
where Ro is positive, and the sign of parameter g determines whether R ( ~ ) is an increasing or a decreasing function of ~. For the latter case, the negative value of ~ must be small enough so that R ( ~ ) would be positive under a realistic • value. Substituting (48) into equations from (43) to (47), we obtain the results as summarized in Table 1. The calculated values of ~/i* in Table 1 agree well with the seven conclusions listed before.
T a b l e 1. O s c i l l a t o r y b e h a v i o r for s i m p l e shear Oscillatory phenomena
C o r o t a t i o n angle 4'* at w h i c h the c o r r e s p o n d i n g p h e n o m e n o n occurs 7r
b-decline
~* = - + tan-~e 2
b-return
~ * = 7r + 2 t a n - ~ e
r-return
(1/c -
l+Ro (l-c)
1)(1 + e ~ * ) + c o s ~ * + e s i n ~ * = 0
~*+~
r-return
1 no s o l u t i o n if c < - or Ro < 1/c 2
(l-c) r-oscillation
( ~~4*, ' + 2 )~
+c[sin,*+~(1-cos~*)] 1 no s o l u t i o n if c -< 2
=0
Objective corotational rates and shear oscillation
653
VIII. EXAMPLES
To illustrate the above results, we assume the material hardening can be described by a power-law response y ( g u ) = Y0 + ~'~P" = Yo(1 + ct3""), ol = 3-tn/2)(ct'/Yo).
(49)
The last equality in eqn (49) for Y(gP) is only valid for rigid plasticity, and o~' and n are the material hardening coefficient and hardening exponent, respectively, with 0 < n < 1. We discuss two classes of the objective corotational rates in the following. VIII. 1. P o w e r - l a w type 0 < m < 1.
~ ( 3 ' ) = !~i03"m
(50)
The particular case of J a u m a n n rate corresponds to ~o = m = 1. Combining eqns (39) and (50), one has R -
an
3",~-m
¢ioo = oo.
(51)
~o m Substitution o f eqn (51) into eqn (40) leads to otn [~ = C -~0 (n/m),
(52)
m
where the first and the second terms in the brackets are complete and incomplete g a m m a functions. The oscillation o f back stress only relies on the exponent ratio n / m . For large ~, eqn (52) can be expanded as
n(3") =~ il"_~
ei[(nTr/2m)--~] q- ~(n/m)--ll~=O ~i~li.,k ~ --
•
(53)
When n < m, the back stress B approaches a limit cycle in the complex plane as 3' increases. This limit cycle, as characterized by the first term in eqn (53), is centered at the origin and of radius f31-'(n/m). When n > m, the back stress (or trajectory o f the yield surface center) wriggles away toward bxl direction as 3' increases. For the coincidental case of n = m, interestingly enough, a periodical solution tangent to the origin emerges B(3") = ~(1 - e -i~)
m = n
~ = ca/~o,
(54)
which means the solution has a trifurcation with respect to parameter m / n in the neighb o r h o o d o f unity. None of the three cases discussed above can be considered as satisf a c t o r y for the simple shear response. To d e m o n s t r a t e that, Figure 2 shows the trajectories of back stress in the cast of n / m = ~,~1,2. When n / m equals El-or 2, eqn (52) can be reduced to
654
W. YANGet al.
B ( . / ) = i3e it¢~/4)-¢l 4 ~ b ( x / ~ ) ,
n m
1 2
(55) B('y) = ~ [ ¢ i + i ( 1 - e - i ~ ) ] ,
n
=2,
m
where cI,(.) represents the probability integral as defined in standard mathematical handbooks (e.g., see GRADSnTEYN & RvzniI¢ [1980]). VIII.2. Trigonometric type ~(3') = ~ o t a n - l 3'
k'
~oo = ~07r/2.
(56)
The above expression covers several known corotation rates proposed in the literature, such as the rates used by DIENES [1979] (when #o = 2, k -- 2) and by LEE et aL [1983] (when #0 = 2, k = l) as the present authors derived from their works. Furthermore, we are able to show that the case o f 4~o = l, k = 2 corresponds to the objective rate corotating with the Euler spin. The value of #¢o is reduced by half by using #o = 1, indicating a better behavior on shear oscillation. Combining eqns (39), (49), and (56), we arrive at otn k2).yn_l - ~ n k " R = ~ (,~2..1._ ~0
[sin(~/i/~o)]"-I [cos(~/¢~o)] "+1"
(57)
Then the back stress becomes B ( ~ ) = ikncc~ tan" ~
;
for ~o = 2
(58) B(~I) = i k ' c ~ n e - i ¢ { I , ( ~ )
+ iI,+l(~)}
- 2n + l 2#) I , ( # ) = c o s ~ tan"-l~b EEl ( 1/2,1;---:---;sin n-1
for #0 = 1,
where 2El is the Gaussian hypergeometric function. When # tends to its limit value #oo = #o7r/2, eqn (58) is reduced to B ( ~ ) -- ica~"
(59)
which implies the desirable feature that the leading term of B(q~) radiates toward the
positive bl2 direction for large 3, under all values of n. This result also agrees qualitatively with the result by taking f/d frame spin as the corotational rate (HwAN~ & CHEN~ [1989]). From eqn (58), the value of k does not influence the oscillatory behavior of B, so the schemes of DIENES [1979] and LEE et al. et al. [1983] would rather produce iden-
Objective corotational rates and shear oscillation
655
tical oscillatory behavior o f the back stress. It is interesting to check the value o f B evaluated at 7 = k (where ~ equals ~0 x / 4 , half way between zero and ~oo), especially the ratio between b]2(k) and hi1 (k)
bl2(k)
for ~o
2,
=
(60)
-
where ~b denotes psi function. This ratio decreases as n decreases f r o m one to zero, and takes the values o f 0.146, 0.290, 0.787, and 1.214 for n = 0.1, 0.2, 0.5, and 1, respectively. Thus, even if we use the corotational rates o f Dienes or Lee et ai., the response o f back stress still deviates substantially f r o m a shear d o m i n a n t response when the material hardening is relatively weak. F o r the case o f linear hardening (n = 1), eqn (58) is reduced to
~/3b12/~Yo
x/3b12/kc~tYo
"
Euler spin fie
t
Lee,Mallet
/ & Wertheimer [1983]
Dienes [19791
/
Hwang & Cheng [1989]
/
n=2m
11i"
1
L i m i t circle m = 2n
for
-2
! t -1
I
I
~
0
~,
')/)
I
/
~/3611/Yo~ ,
~bll/kc~Yo
m=2n
-1 J
-2
Fig. 2. Trajectories of yield surface center under different corotational rates, where two sets of coordinates correspond to the cases of eqn (50) and eqn (56), respectively.
656
W. YANG et al.
B(7) = ikcote i ~ - ~ ) [tan ~ -
+ 2iln(cos ~ ) 1
for 40 = 2 (61)
B ( 7 ) = ikcoLe-i*lln(tan • + sec 4) + i(sec # -- 1)}
for #o = 1.
Their trajectories are also plotted in Figure 2. Obviously, the deviation from the shear stress response is largely reduced by the engagement of Euler spin. The trajectories of yield surface center shown collectively in Figure 2 exhibit a spectrum of the back stress responses. It shows that enormous uncertainty can be brought about for the plastic response at very large deformation by different choice of the objective corotational rates, and the latter cannot be determined purely from the continuum mechanics principles. This result would render a study on corotational rates from the physical viewpoint necessary. Nevertheless, the authors consider the choice of Eulet spin (by which the stress response is traced from the current configuration) to be appropriate from the continuum mechanics viewpoint, and our calculations seem to support this selection. The merit of Euler spin, however, is still subjected to further investigation. REFERENCES 1958
1965 1974 1979 1980 1981
1983 1984 1985 1985 1987 1989
KADASCHEVICH,Y.N., and NovozmLov, B.B. "Theory of Plasticity With Consideration of Residual Micro-Stresses," Appl. Math. and Mech., 22 (1), 78 (in Russian). PRAGER,W., Introduction to Continuum Mechanics. FARDSmSnEH,E, and Or~AT, E.T., Proc. Syrup. on Foundation of Plasticity, Warsaw, Noordhoff, 89. DroNES,J.K., "On the Analysis of Rotation and Stress Rate in Deformation Bodies," Acta Mech., 32, 217. Ga~,DSaTEYN,I.S., and RYZItIK, I.M., Table of Integrals, Series and Products, Academic Press, New York, 930. NAGTEGAAL,J.C., and DE JONG, J.E., "Some Aspects of Non-lsotropic Hardening in Finite Strain Plasticity," in LEE, E.H., and MALLET, R.L. (eds.), Proc. of Workshop on Plast. of Metals at Finite Strain, Div. of Appl. Mech., Stanford Univ., Stanford, p. 65. LEE, E.H., MALLET, R.L., and WERTHEIMER,T.B., "Stress Analysis for Kinematic Hardening of Finite Deformation Plasticity," J. Appl. Mech., 50, 554. ATLURI,S.N., Comput Meth. Appl. Mech. Engng., 43, 137. DAFALIAS,Y.F., "The Plastic Spin," J. Appl. Mech., 52, 865. MEAR,M.E., and HUTCmr~SON, J.W., "Influence of Yield Surface Curvature on Flow Localization in Dilatant Plasticity," Mech. of Mater., 4, 395. AGAH-TErmANI,A., LEE, E.H., MALLET, R.L., and ONAT, E.T., "The Theory of Elastic-Plastic Deformation at Finite Strain With Induced Anisotropy Modeled as Combined Isotropic-Kinematic Hardening," J. Mech. Phys. Solids, 35, 519. HWANG,K.C., and CHENG, L., "Several Fundamental Problems in the Elastic-Plastic Constitutive Theory at Finite Deformation," Acta Mech. Sinica, Special Issue on Plasticity, 3 (in Chinese).
Department of Engineering Mechanics Tsinghua University Beijing 100084 P.R. China Department of Applied Mechanics Beijing Institute of Technology Beijing 100081 P.R. China (Received 22 November 1989; in final revised forrn 10 October 1991 )
0749~419/92 $5.00 + .00 Copyright © 1992 Pergamon Press Ltd.
Printed in the U.S.A.
OBJECTIVE COROTATIONAL RATES AND SHEAR OSCILLATION WEI YANG,* LI CHENG,**
and KEH-CHII-I HWANG*
*Tsinghua University and **Beijing Institute of Technology (Communicated by Yannis Dafalias, University of California-Davis) Abstract--Influence of objective corotational rates to shear oscillation is discussed here under
the framework of finite deformation, J2-flow theory. Under an elastic-plastic, mixed hardening constitutive law, the active stress s, as defined in eqn (8) of the text, and back stress b are shown to be governed by two sets of equations formally uncoupled to each other. Formal solutions of complex stresses are obtained for plane strain, isochoric deformation, with the conclusion that the shear oscillation behavior for an elastic-plastic solid can be asymptotically reduced to a rigid-plastic solution and a higher order correction, provided the initial shear strain is small. Consequently, the active stress s appears to be nonoscillatory under a realistic but arbitrary corotational rate, whereas the oscillatory behavior of back stress b is solely determined by the maximum corotation angle, defined in eqns (18) and (41), as well as the variation of material hardening with respect to the corotation angle. A spectrum of back stress responses could be predicted by the incorporation of different objective corotational rates.
I. INTRODUCTION
In finite deformation, elastic-plastic constitutive theory, the objective corotational rate, albeit crucial in the construction of stress rates, can be selected from a variety of candidates ( e . g . FARDSHISHEH & GNAT [1974], LEE, MALLET 8, WERTHEIMER [1983], AGAHTErIRANI et al. [1987]). There have been many works focused on the correct choice of the objective corotational rate, especially since the work of NA~TEOX~a~and DE JONC [1981], who showed an oscillatory stress response for simple shear motion when combining Prager-Ziegler kinematic hardening model with Jaumann rate. Remedies by adopting special corotational rate have been proposed to suppress this shear oscillation, from the early work on finite elastic deformation by Dmr~ES [1979], to the recent investigations on kinematic hardening formulation by LEE et al. [1983], D.'ffALIAS[1985], AG~a-bTEHRAtqIet al. [1987], and by HWANGand CI-Ir.NC [1989]. The above-mentioned researches, however, were confined in the rigid plastic case and subjected to the assignment of a particular corotational rate. The general influence of the objective corotational rate on shear oscillation is still to be explored, especially in connection with the material hardening behavior. The present paper addresses this problem from a finite deformation, mixed hardening J2-flow formulation. A formal solution for plane strain, isochoric deformation is obtained for arbitrary corotational rate, material hardening, and anisotropic hardening parameter c. For the purpose of shear oscillation investigation, we are able to show that the problem is asymptotically reduced to a rigid plastic solution and a higher order correction. Several qualitative and quantitative results about shear oscillation are obtained via rigid plastic model, complemented by examples illustrating some interesting features.
Sponsored by the National Natural Science Foundation of China. 643
644
W. YANG et al.
II. OBJECTIVE COROTATIONAL RATES
Corotational rates refer to the rates corotational with a skew-symmetric spin tensor //. If ~ satisfies the frame transformation law during a frame rotation Q carrying frame F to frame F, that is, /~ = Q./-/.Qr + Q . Q r ,
(1)
where the superscript T denotes tensor transpose, then the rate corotational w i t h / / w i l l be objective (LEE, MALLET, R, WERTHEIMER [1983]) A = A-
Q.A + A./'/
(2)
and termed objective corotational rates. Symbol A in eqn (2) denotes any objective symmetric tensor of second rank. It is straightforward to prove that A:B
= .,/i : B
(3)
for any second rank tensor B commutable to A. Various objective corotational rates have been employed in constitutive formulation, such as the Jaumann rate (while Q equals the material spin w), the relative spin r a t e / / = R . R r by DtE~qEs [1979] (with R being the rotation tensor in polar decomposition of the deformation gradient), and the spin rate of material fibre coinciding with the maximum principal direction of back stress (LEE et al. [1983]). The most general form of Q can be deduced axiomatically from the guiding principles of continuum mechanics, such as the requirement of objectivity. This approach was pursued in works by FARDSHISHEHand OSAT [1974] and by AGAH-TEHRANI et al. [1987]. Alternatively, HWANG and CHENG [1989] showed that/1 can be described by a construction approach in terms of the frame spin associated with the principal axes of an arbitrary symmetric objective tensor S of second rank
,~
- -
1
~.~ - , ~
•
s
(S)(~)n~n~,
s
(4)
where A~ (c~ = 1,2,3) are the principal values of S with associated eigenvectors no.S The notation (S)(~0) in eqn (4) describes the components of the material rate of S, namely S, with respect to the principal directions of S, calculable through
(s)~o~ = n~-S.nL
(5)
The parentheses on subscript or/5 emphasize that (S)(~o) is not a tensorial component. Like the frame spin of tensor S, the skew-symmetric tensor t/s can be used to construct the objective corotational rate in eqn (2) so that A is the rate sensed by an observer who travels with the frame formed by triad (n sl, n2, s n3s ). Accordingly, the corotational rate A is formed by the absolute rate ,~ minus that p a r t / / s . A - A-/-/s which the observer cannot feel. The second rank symmetric tensor S is in general arbitrary from the continuum mechanics viewpoint, and there is no consensus to determine it. The frame spin o f S is understood in the sense of Hill's principal axes analysis. Various candidates of
Objective corotational rates and shear oscillation
645
S have been proposed to construct the corotational rates. For example, S can be specified as the back stress b as motivated by the work o f LEE, MAtLET and WERTHEII~tER [1983], or taken as the deformation rate d (or Rivlin-Ericksen tensor o f order n, A tn)) by HwAr~G and O-raNG [1989], or chosen as the left stretch tensor V (or Almansi strain tensor e, etc.). For the last c a s e , / / s becomes the familiar Euler s p i n / / e .
III. M I X E D H A R D E N I N G T H E O R Y
Kirchhoff stress r and deformation rate d are often adopted as the basic work conjugates in the finite deformation theory. If the elastic response of a rate-independent elastic-plastic material is isotropic, one can describe the constitutive law for finite deviatoric deformation as ~' = 2 G ( d ' - d p)
(6)
where r ' and d' represent the deviatoric parts of r and d, respectively, G is the shear modulus, d p is the deviatoric part of plastic deformation rate, and i ' denotes the objective corotational rate as defined in eqn (2) for the stress deviator. The additive decomposition of d in finite deformation, as well as the incompressibility o f plastic deformation, have been used in deriving eqn (6). Under the classical postulate o f Maximum Plastic Work, one can construct d p from a yield surface function. Under a mixed hardening, J2-flow theory, the Mises-type yield surface is formulated by f=
]s:s
- OFz = 0,
(7)
where the active deviatoric stress s is defined as s = r ' -- b,
(8)
with back stress b assumed to be deviatoric. The quantity aF in eqn (7) signifies the radius of the yield surface in r space. It is assumed that or evolves in proportion to the evolution o f the flow stress Y in uniaxial tension, as proposed by KADASCHEVICnand Novoznn~ov [1958] as well as in more recent work of MEAR and HuxcnIr~SON [1985] aF = CYo + (1 -- c)Y.
(9)
The constant parameter c varies between zero and one. Extremes c = 0 and c = 1 correspond to isotropic and kinematic hardening, respectively. In eqn (9) Y = Y(gP)
and
Yo = Y(0)
(10)
denote the current and initial flow stresses measured from a uniaxial test, while gP,
~P =
d p dt
dp =
d p : d p,
(11)
is the current effective plastic strain. By means o f the associated flow rule and the uniaxial stress-strain curve, one can write d p as
646
W. YANC et al.
dp -
9 ~":s
4h
a2F
dY d~ p'
h -
S
(12)
where h represents the tangential hardening modulus for flow stress with respect to plastic strain in uniaxial tension. The evolution of back stress b is governed by PragerZiegler law. Combining it with the above formula of d p, we obtain 3
~":S
I~ = ~ c ,~o--v-s,
(13)
where the proportional evolution law in eqn (9) has also been utilized. Apparently, the yield surface center will not evolve for the case of isotropic hardening (c = 0). We now rewrite the stress evolution law in eqn (6) by substituting d p in eqn (12)
20(d 9 s) 4h
o~- s .
(14)
The inner product ~' : s in both eqn (13) and eqn (14) can be determined by the consistency condition f = 0 as ~ ' : s = 20F~'.
(15)
To this end, expressions in eqns (3), (7), (8), (9), and (13) have been used. Substituting eqn (15) back in eqn (13) and eqn (12), one obtains 1~ = C - -
~z S,
OF
S ~-~
2haF dp
(16)
317
applicable to both elastic-plastic and rigid plastic cases. For the latter case, d p in eqn (16) is replaced by d, so s can be evaluated directly by the known kinematics. Accordingly, the active deviatoric stress s will not oscillate for rigid plastic solid, irrespective of the objective corotational rate//. For the elastic-plastic case, it is more convenient to replace the last expression in eqn (16) by =2Gd'-
__3 + c
--s.
(17)
OF
We now reduce the mixed hardening theory to the successive solution of uncoupled eqn (17) and then the first equation in (16). IV. P L A N E S T R A I N , I S O C H O R I C D E F O R M A T I O N
Attention is now focused on plane strain deformation. The only nontrivial components of the corotational spin are ~'~12 = --~'~21 = 1 4 .
(18)
in eqn (18) is the corotation angle induced by deviatoric deformation. If we further restrict to isochoric deformation, then from eqns (2), (8), (16), and (17)
Objective corotational rates and shear oscillation
p
d:3 = ri3 = si3 = bi3 = 0
647
i = 1,2,3; (19) t
d~2 = --d~l, r~2 = --rll, $22 = - - S I I , b22 = - b l l , which implies that each of the above four tensors has only two independent components. It is therefore convenient to introduce the following complex, dimensionless quantities D = d~l + ida2
D p = d p + idlP2
(20) S = ~f~ Sll + iSl2
B = xf~ bll + ibl2
Yo
T = x/~ r[l + ir[2
Yo
Yo
They reduce governing eqns (16) and (17) to [~ + i ~ B = rS
r -
qP = 2
S = qPD p
(21)
cy
c + (1 - c ) y c + (1 -
c)y
(22)
4~dp
;~+ (i~b+yP0"0)S= 3'0 2 D
p=
~2 + ~'or,
(23)
where Y
Y = --
Yo
"Yo =
Y0
(24)
4~G
are the dimensionless hardening function and initial yield strain for shear. Integrating (21) from the initial yield instant to, one arrives at
B(t)
=
(25)
r ( ~ ) S ( ~ ) e -it#tt)-#(~)l d~,
where S is given by
S(t) = iexpli[¢(to)
- ¢(t)]
p(t)
-p(to))yo
(26) + -"Y0
I
D(~)exp i[~(~) - ~(t)]
3/0
Equations (25) and (26) provide a formal solution o f the elastic-plastic mixed hardening response in plane strain, isochoric deformation. Their applications, however, are somewhat hindered by the involvement o f d p in the expressions o f p and y.
648
W. YANG et al.
V. SIMPLE SHEAR DEFORMATION
The kinematics of simple shear deformation is delineated in Figure 1. The only nontrivial components of deformation rate tensor are d12 = d21 = "Y ~,
(27)
where 7 is the shear amount as defined in Figure 1. For monotonic and rate-independent shear deformation, the corotational speed ~ in eqn (18) can be written explicitly as _
d~(7)
d7
. 7-
(28)
The monotonic behavior of 7 renders it a time-like parameter. Denoting
y(gP(t)) = .v(7),
p(t) = P ( 7 ) ,
(29)
we can specify the general plane strain, isochoric deformation, as invoked from eqns (25) and (26), to the particular case of simple shear, while the integration can be carried out with respect to shear amount from 7o to 7. To facilitate the discussion of shear oscillation, certain restrictions are made to functions ) (7) and • (7), respectively: 1. The material response is nonsoftening, that is, .v'(7) -> 0
v7-
(30)
2. The shear-induced corotational speed is bounded by 0 < ~/v(7) < cz vT.
(31)
The lower bound in eqn (31) implies that the stress rates cannot corotate to a spin that contradicts to the sense of the actual material spin, whereas the upper bound of ~ ' ( 7 ) is motivated by the concept that the stress rate corotation should not be accelerating as 7 increases. If the constant a in eqn (31) is taken as unity, the upper bound on ~ ' ( 7 )
X2
1
......
0/
/
Fig. 1. Simple shear deformation.
X1
Objective corotational rates and shear oscillation
649
would imply that the corotational rates cannot exceed the material spin (namely, Jaumann rate) itself. Two particular cases will be addressed prior to the general discussion on shear oscillation. 1. Nonhardening materials, such that p(3") = 1 for any 3'. In this case, eqns (25) and (26) are reduced to
B(3")
=0
S(3") = iexpli~(3"°) - i~(Y) - "/-3 ¢P(3")13"o (32) + - ,f'ox -
i~(~)
f
3'0 ~ ~o
_
i~(3")
~
_
gP(3") _- gP(~)
'.
3'0
d~. )
Consequently, shear oscillation is effectively suppressed because the back stress vanishes. 2. Noncorotating rate, that is, 4(3') = 0 for any 3". As pointed out by HWAr~G and CnENG [1989], the corotation angle # (3") vanishes for simple shear deformation whenever Q is taken as O a, the frame spin for deformation rate tensor (namely, the spin o f the triad (n a, n2a, na), signifying the principal axes of d), or the frame spin o f Rivlin-Ericksen tensor o f any order. In this case, eqns (25) and (26) reduce to
B(3') =
f
" cp'(~) o c+ (1-c)p(~)
S(~)d~ (33)
--
k
k
--
3"0
3"0
exp
o
d~
,
3'0
and both B and S are purely imaginary. Thus, only the shear components of tensors r', s, and b are nontrivial, and oscillation will never occur, provided condition eqn (30) is satisfied. In the subsequent discussion, we assume .P(3') > O, ~'(3") > 0
v3".
Vl. ASYMPTOTIC SOLUTION FOR ELASTIC-PLASTIC
(34) PROBLEMS
From eqns (25) and (26), it is concluded that shear oscillation can occur only if the corotation angle 4(3,) exceeds a-/2. In combination with eqn (31), shear oscillation would be possible only if 3" reaches the order of unity. On the other hand, the initial yield strain for shear, 3'0, is only about 0.2% for many metals. Thus, it is reasonable to consider an asymptotic solution for 3' >> 3"0
(35)
w. YANGet al.
650
in the study of shear oscillation. Also implied in this statement is the assumption that the hardening modulus h defined in eqn (12) is much less than the elastic shear modulus G. Although less important, shear oscillation can also be caused by elastic response, as discussed by PRAGER [1965], DIENES [1979], and by ATLURI [1984]. The present work is rather focused on the shear oscillation dominated by plastic response. After a straightforward, but lengthy, asymptotic analysis, eqn (26) for active stress can be simplified to
(36) 1-c
3'
3'
where y and h are known functions defined in eqns (10), (12), and (24) and relate to the uniaxial tension data. Accordingly, the leading term in S(3') for the elastic-plastic case is the rigid plastic solution, which precludes the possibility of shear oscillation of the active stress. Substituting eqn (36) into eqn (25), one arrives at an asymptotic expression of back stress.
x/3Yod~Y ~
h
(37) x exp[i,b(~) - i,b(y)] d~ + 0(-/2). The leading term of back stress also corresponds to the rigid plastic solution, and the correction terms due to elastic deformation only slightly increase the possibility of shear oscillation. Consequently, the estimate of eqn (35) would infer that shear oscillation is basically decided by rigid plastic solution. VII. RIGID P L A S T I C S O L U T I O N
For rigid plastic solution, one has gp= !
and
.9(3,)=y{!].
(38)
Introducing the variation rate of material hardening with respect to the corotational angle
d2
R(~) - d~
P'(y)
~'(3') '
(39)
Objective corotational rates and shear oscillation
651
we are able to write the back stress for the rigid plastic case as
(40)
B(3") = iCfo ~t'Y)R(~)e -i~-~) d~. Its oscillatory behavior is solely determined by the function R ( 4 ) and 400 = lim 4(3"),
(41)
termed the m a x i m u m corotation. Conventionally, the oscillatory behavior of back and total stresses is referred to the variation of their shear components
b12(3') = -cYo ~ - fo etv) R ( ~ ) c o s [ 4 - ~1 d~ (42) r12(3") =
c + (1 -
r
c)y
+ Cvo
1
R ( ¢ ) c o s [ 4 - ~] d e .
We introduce the following definitions with respect to the above two stress components: 1. b (or r) decline is referred to the occurrence of a nonpositive derivative of bl2 (or r12) with respect to 3'. Consequently, if there exists a 4 " in (0, 4 = ) such that R(4*) =
ff"
R ( ~ ) s i n ( 4 * - ~) d~,
(43)
then b-decline occurs. Similarly, if eqn (43) is replaced by R(4*) = c
f0 4'*
R(~)sin(4* - ~)d~
(44)
then r-decline occurs. 2. b (or r) return is referred to the occurrence of a nonpositive value of b12 (or 712). Accordingly, if there exists a 4 " in (0, 4o0) such that 0~* R ( ~ ) c o s ( 4 * - ~) d~ = 0,
(45)
then b-return occurs. Similarly, if eqn (45) is replaced by 1+
ff"
then r-return occurs.
R ( ~ ) [1 - c + ccos(l,* - ~)] d~ = 0,
(46)
652
W. YANG et al.
3. r-oscillation is referred to the case of shear stress r~2 returning to its initial value of Yo/x/-3, r-oscillation occurs if
£
~* R(~o) [1 - c + c c o s ( ~ * - ,p)] d~o = 0
(47)
for some ~* in (0, ~ ) . Based on the above definitions, we are able to conclude: 1. If ~o0 -< 7r/2; or if ~o~ in [(7r/2), 7r] and R ( ~ ) nondecreasing, b-return will be impossible. 2. If ~oo -> 7r, and R ( ~ ) nonincreasing, b-return should happen at some ~* _< 7r. 3. I f ~= < 1, and R(4i) nondecreasing, b-decline will be impossible. 4. I f ~o0 -> 7r/2, and R ( 4 ' ) nonincreasing, b-decline should occur at some ~* _<_7r/2. 5. For purely kinematic hardening (c = 1), r-oscillation is equivalent to b-return, and r-decline is equivalent to b-decline. 6. r-oscillation is impossible when c ___ ~. 7. If ~= _< 1/c and R ( ~ ) nondecreasing, r-decline will never happen. To demonstrate the above conclusions, let us consider the special case of R ( 4 i) = Ro(1 + e ~ ) ,
(48)
where Ro is positive, and the sign of parameter g determines whether R ( ~ ) is an increasing or a decreasing function of ~. For the latter case, the negative value of ~ must be small enough so that R ( ~ ) would be positive under a realistic • value. Substituting (48) into equations from (43) to (47), we obtain the results as summarized in Table 1. The calculated values of ~/i* in Table 1 agree well with the seven conclusions listed before.
T a b l e 1. O s c i l l a t o r y b e h a v i o r for s i m p l e shear Oscillatory phenomena
C o r o t a t i o n angle 4'* at w h i c h the c o r r e s p o n d i n g p h e n o m e n o n occurs 7r
b-decline
~* = - + tan-~e 2
b-return
~ * = 7r + 2 t a n - ~ e
r-return
(1/c -
l+Ro (l-c)
1)(1 + e ~ * ) + c o s ~ * + e s i n ~ * = 0
~*+~
r-return
1 no s o l u t i o n if c < - or Ro < 1/c 2
(l-c) r-oscillation
( ~~4*, ' + 2 )~
+c[sin,*+~(1-cos~*)] 1 no s o l u t i o n if c -< 2
=0
Objective corotational rates and shear oscillation
653
VIII. EXAMPLES
To illustrate the above results, we assume the material hardening can be described by a power-law response y ( g u ) = Y0 + ~'~P" = Yo(1 + ct3""), ol = 3-tn/2)(ct'/Yo).
(49)
The last equality in eqn (49) for Y(gP) is only valid for rigid plasticity, and o~' and n are the material hardening coefficient and hardening exponent, respectively, with 0 < n < 1. We discuss two classes of the objective corotational rates in the following. VIII. 1. P o w e r - l a w type 0 < m < 1.
~ ( 3 ' ) = !~i03"m
(50)
The particular case of J a u m a n n rate corresponds to ~o = m = 1. Combining eqns (39) and (50), one has R -
an
3",~-m
¢ioo = oo.
(51)
~o m Substitution o f eqn (51) into eqn (40) leads to otn [~ = C -~0 (n/m),
(52)
m
where the first and the second terms in the brackets are complete and incomplete g a m m a functions. The oscillation o f back stress only relies on the exponent ratio n / m . For large ~, eqn (52) can be expanded as
n(3") =~ il"_~
ei[(nTr/2m)--~] q- ~(n/m)--ll~=O ~i~li.,k ~ --
•
(53)
When n < m, the back stress B approaches a limit cycle in the complex plane as 3' increases. This limit cycle, as characterized by the first term in eqn (53), is centered at the origin and of radius f31-'(n/m). When n > m, the back stress (or trajectory o f the yield surface center) wriggles away toward bxl direction as 3' increases. For the coincidental case of n = m, interestingly enough, a periodical solution tangent to the origin emerges B(3") = ~(1 - e -i~)
m = n
~ = ca/~o,
(54)
which means the solution has a trifurcation with respect to parameter m / n in the neighb o r h o o d o f unity. None of the three cases discussed above can be considered as satisf a c t o r y for the simple shear response. To d e m o n s t r a t e that, Figure 2 shows the trajectories of back stress in the cast of n / m = ~,~1,2. When n / m equals El-or 2, eqn (52) can be reduced to
654
W. YANGet al.
B ( . / ) = i3e it¢~/4)-¢l 4 ~ b ( x / ~ ) ,
n m
1 2
(55) B('y) = ~ [ ¢ i + i ( 1 - e - i ~ ) ] ,
n
=2,
m
where cI,(.) represents the probability integral as defined in standard mathematical handbooks (e.g., see GRADSnTEYN & RvzniI¢ [1980]). VIII.2. Trigonometric type ~(3') = ~ o t a n - l 3'
k'
~oo = ~07r/2.
(56)
The above expression covers several known corotation rates proposed in the literature, such as the rates used by DIENES [1979] (when #o = 2, k -- 2) and by LEE et aL [1983] (when #0 = 2, k = l) as the present authors derived from their works. Furthermore, we are able to show that the case o f 4~o = l, k = 2 corresponds to the objective rate corotating with the Euler spin. The value of #¢o is reduced by half by using #o = 1, indicating a better behavior on shear oscillation. Combining eqns (39), (49), and (56), we arrive at otn k2).yn_l - ~ n k " R = ~ (,~2..1._ ~0
[sin(~/i/~o)]"-I [cos(~/¢~o)] "+1"
(57)
Then the back stress becomes B ( ~ ) = ikncc~ tan" ~
;
for ~o = 2
(58) B(~I) = i k ' c ~ n e - i ¢ { I , ( ~ )
+ iI,+l(~)}
- 2n + l 2#) I , ( # ) = c o s ~ tan"-l~b EEl ( 1/2,1;---:---;sin n-1
for #0 = 1,
where 2El is the Gaussian hypergeometric function. When # tends to its limit value #oo = #o7r/2, eqn (58) is reduced to B ( ~ ) -- ica~"
(59)
which implies the desirable feature that the leading term of B(q~) radiates toward the
positive bl2 direction for large 3, under all values of n. This result also agrees qualitatively with the result by taking f/d frame spin as the corotational rate (HwAN~ & CHEN~ [1989]). From eqn (58), the value of k does not influence the oscillatory behavior of B, so the schemes of DIENES [1979] and LEE et al. et al. [1983] would rather produce iden-
Objective corotational rates and shear oscillation
655
tical oscillatory behavior o f the back stress. It is interesting to check the value o f B evaluated at 7 = k (where ~ equals ~0 x / 4 , half way between zero and ~oo), especially the ratio between b]2(k) and hi1 (k)
bl2(k)
for ~o
2,
=
(60)
-
where ~b denotes psi function. This ratio decreases as n decreases f r o m one to zero, and takes the values o f 0.146, 0.290, 0.787, and 1.214 for n = 0.1, 0.2, 0.5, and 1, respectively. Thus, even if we use the corotational rates o f Dienes or Lee et ai., the response o f back stress still deviates substantially f r o m a shear d o m i n a n t response when the material hardening is relatively weak. F o r the case o f linear hardening (n = 1), eqn (58) is reduced to
~/3b12/~Yo
x/3b12/kc~tYo
"
Euler spin fie
t
Lee,Mallet
/ & Wertheimer [1983]
Dienes [19791
/
Hwang & Cheng [1989]
/
n=2m
11i"
1
L i m i t circle m = 2n
for
-2
! t -1
I
I
~
0
~,
')/)
I
/
~/3611/Yo~ ,
~bll/kc~Yo
m=2n
-1 J
-2
Fig. 2. Trajectories of yield surface center under different corotational rates, where two sets of coordinates correspond to the cases of eqn (50) and eqn (56), respectively.
656
W. YANG et al.
B(7) = ikcote i ~ - ~ ) [tan ~ -
+ 2iln(cos ~ ) 1
for 40 = 2 (61)
B ( 7 ) = ikcoLe-i*lln(tan • + sec 4) + i(sec # -- 1)}
for #o = 1.
Their trajectories are also plotted in Figure 2. Obviously, the deviation from the shear stress response is largely reduced by the engagement of Euler spin. The trajectories of yield surface center shown collectively in Figure 2 exhibit a spectrum of the back stress responses. It shows that enormous uncertainty can be brought about for the plastic response at very large deformation by different choice of the objective corotational rates, and the latter cannot be determined purely from the continuum mechanics principles. This result would render a study on corotational rates from the physical viewpoint necessary. Nevertheless, the authors consider the choice of Eulet spin (by which the stress response is traced from the current configuration) to be appropriate from the continuum mechanics viewpoint, and our calculations seem to support this selection. The merit of Euler spin, however, is still subjected to further investigation. REFERENCES 1958
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1983 1984 1985 1985 1987 1989
KADASCHEVICH,Y.N., and NovozmLov, B.B. "Theory of Plasticity With Consideration of Residual Micro-Stresses," Appl. Math. and Mech., 22 (1), 78 (in Russian). PRAGER,W., Introduction to Continuum Mechanics. FARDSmSnEH,E, and Or~AT, E.T., Proc. Syrup. on Foundation of Plasticity, Warsaw, Noordhoff, 89. DroNES,J.K., "On the Analysis of Rotation and Stress Rate in Deformation Bodies," Acta Mech., 32, 217. Ga~,DSaTEYN,I.S., and RYZItIK, I.M., Table of Integrals, Series and Products, Academic Press, New York, 930. NAGTEGAAL,J.C., and DE JONG, J.E., "Some Aspects of Non-lsotropic Hardening in Finite Strain Plasticity," in LEE, E.H., and MALLET, R.L. (eds.), Proc. of Workshop on Plast. of Metals at Finite Strain, Div. of Appl. Mech., Stanford Univ., Stanford, p. 65. LEE, E.H., MALLET, R.L., and WERTHEIMER,T.B., "Stress Analysis for Kinematic Hardening of Finite Deformation Plasticity," J. Appl. Mech., 50, 554. ATLURI,S.N., Comput Meth. Appl. Mech. Engng., 43, 137. DAFALIAS,Y.F., "The Plastic Spin," J. Appl. Mech., 52, 865. MEAR,M.E., and HUTCmr~SON, J.W., "Influence of Yield Surface Curvature on Flow Localization in Dilatant Plasticity," Mech. of Mater., 4, 395. AGAH-TErmANI,A., LEE, E.H., MALLET, R.L., and ONAT, E.T., "The Theory of Elastic-Plastic Deformation at Finite Strain With Induced Anisotropy Modeled as Combined Isotropic-Kinematic Hardening," J. Mech. Phys. Solids, 35, 519. HWANG,K.C., and CHENG, L., "Several Fundamental Problems in the Elastic-Plastic Constitutive Theory at Finite Deformation," Acta Mech. Sinica, Special Issue on Plasticity, 3 (in Chinese).
Department of Engineering Mechanics Tsinghua University Beijing 100084 P.R. China Department of Applied Mechanics Beijing Institute of Technology Beijing 100081 P.R. China (Received 22 November 1989; in final revised forrn 10 October 1991 )