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Roles of plastic spin in shear banding
Pergamon
InternationalJournal of Plasticity,Vol. 12, No. 5, pp. 671~93, 1996 Copyright © 1996 ElsevierScienceLtd Printed in Great Britain.All rights reserved 0749-6419/96 $15.00+ .00 P l h S0749-6419(96)00024-1
ROLES OF PLASTIC SPIN IN SHEAR BANDING Mitsutoshi Kuroda Department of Civil Engineering, Ashikaga Institute of Technology, Omae, Ashikaga, Tochigi 326, Japan (Received in final re visedform 30 January 1996)
Abstract--In this paper, the effects of plastic spin on shear banding and simple shear are examined systematically. Three types of plastic constitutive model with plastic spin are considered: (i) a noncoaxial model in which the direction of the plastic strain rate depends on that of the stress rate; (ii) a strain-softening model based on the J2 flow theory; and (iii) the pressure-sensitive porous plasticity model. AlL1the constitutive models are formulated in viscoplastic forms and in conjunction with nonlocal concepts that have been recently focused and discussed. First, behavior in simple shear is examined by numerical analyses with the aforementioned constitutive models. Moreover, some experimental evidences for stress response to simple shear are shown; that is, several large torsion tests of metal tubes and bars are carried out. Next, finite element simulations of shear banding in plane strain tension are performed. A critical effect of plastic spin on shear banding is observed for the noncoaxial model, while an almost negligible effect is observed for the porous model. The identical effects of plastic spin are observed, whether nonlocality exists or not. Finally, we discuss the relationship between the behavior in simple shear and the shear band formation. It is emphasized that this is a critical issue in predicting shear banding in macroscopic grounds. Copyright © 1996 Elsevier Science Ltd
I. INTRODUCTION M u c h o f the w o r k in finite strain plasticity in the p a s t few decades has focused o n predicting flow l o c a l i z a t i o n in the f o r m o f " s h e a r b a n d s " . W h e n ductile m a t e r i a l s are d e f o r m e d at very high strain rates, the t h e r m a l m a t e r i a l softening due to local h e a t i n g m a y be a n i m p o r t a n t trigger in initiating a n d d e v e l o p i n g the localization. H o w e v e r , the s a m e m a t e r i a l m a y also u n d e r g o the shear l o c a l i z a t i o n at low strain rate, where the t h e r m a l effects are a l m o s t negligible. F o r p r e d i c t i o n s o f the shear l o c a l i z a t i o n in this c i r c u m s t a n c e , one o f the m o s t significant c o n t r i b u t i o n s in p h e n o m e n o l o g i c a l g r o u n d s is the c o n c e p t o f vertex theories in which the d i r e c t i o n o f the plastic strain rate d e p e n d s o n t h a t o f the stress rate (e.g. St6ren & Rice [1975]; Christoffersen & H u t c h i n s o n [1979]; G o t o h [1985]). A n o t h e r i m p o r t a n t c o n t r i b u t i o n is the c o n c e p t o f v o i d d a m a g e theories, such as the G u r s o n m o d e l ( G u r s o n [1977]). T h o u g h these two types o f plasticity t h e o r y h a v e quite different c o n c e p t u a l b a c k g r o u n d s , b o t h o f t h e m are often e m p l o y e d for the same p u r p o s e : t h a t is, p r e d i c t i o n o f a realistic l o c a l i z a t i o n b e h a v i o r in ductile materials. Since N e g t e g a a l a n d de J o n g [1982] have f o u n d an unrealistic o s c i l l a t o r y stress b e h a v i o r in a n analysis o f simple shear, stress b e h a v i o r in shear d e f o r m a t i o n a n d its p r e d i c t i o n have been also actively investigated ( D a f a l i a s [1983]; Lee et al. [1983]; J o h n s o n & B a m m a n n [1984]; Z b i b & A i f a n t i s [1988]; S z a b 6 & Balla [1989]; Y a n g et al. [1992]; K u r o d a [1994]). 671
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Any theory of plasticity is required to predict a realistic stress response to simple shear deformation. One of the most significant contributions on this subject is considered to be the concept of the plastic spin that was originally suggested by Mandel [1971] and developed by Dafalias [1985a, 1985b] and others. The plastic spin concept succeeded not only in reproducing monotonic stress responses to simple shear (Dafalias [1985a, 1985b]), but also in simulating an interesting phenomenon called the Swift's [1947] effect (Zbib [1991 ]; Paulun & Pecherski [ 1992]). However, a possibility of existence of oscillatory stress behavior in simple shear or in fixed-end torsion has been suggested by Montheillet et al. [1984]. In many previous theoretical works, a lack of experimental data documenting large torsion behavior has been pointed out. At present, the large torsion (or simple shear) behavior seems to be still an issue. In these circumstances, some workers have examined the effects of plastic spin on the shear localization behavior. Tvergaard and van der Giessen [1991] investigated the effects of plastic spin on flow localization for porous ductile material, using a simplified plane strain analysis introducing the "band" type initial imperfections. Zbib [1994] studied the influence of plastic spin on the shear band formation in a special rigid plastic softening material during a plane strain compression. Kuroda [1995a] proposed a plastic spin relation associated with a corner theory of plasticity, and investigated its effect on the shear localization in plane strain tension. These studies showed that plastic spin stabilizes the localization phenomena. However, these results cannot be compared directly, because the problems and constitutive models employed to investigate the effects of plastic spin are essentially different. Information on the effects of plastic spin on shear localization has been obtained, as mentioned above. The following important questions, however, still remain: (1) Are the effects of plastic spin equivalent for any constitutive model? If not, how are they different? (2) Can plasticity theories with the plastic spin reproduce both the realistic stress response to simple shear and the localization behavior? (3) What is the realistic stress response to simple shear? The objective of this paper is to answer the foregoing questions. The effects of plastic spin on shear banding and simple shear are systematically examined for several theories of plasticity. The paper is arranged as follows. Section II provides constitutive relations employed in this investigation. Three types of constitutive models with plastic spin are considered: (i) a noncoaxial model in which the direction of the plastic strain rate depends on that of the stress rate (a vertex-like model); (ii) a strain-softening model based on the J2 flow theory; and (iii) the porous plasticity model (Gurson [1977]; Mear & Hutchinson [1985]). All the constitutive models are formulated in viscoplastic forms, and in conjunction with nonlocal concepts that have been recently focused on and discussed, (e.g. Bazant & Pijaudier-Cabot [1988]; Aifantis [1984, 1992]). The first two models are provided as the particular cases of a generalized constitutive model proposed by Kuroda [1995b]. In Section III, the numerical method, namely, finite element formulation, is presented. In Section IV, behavior in simple shear is examined by numerical analyses with the aforementioned three constitutive models. Moreover, some experimental evidences for stress responses to simple shear are shown; i.e. several large torsion tests of metal bar and tube are carried out. In Section V, finite element simulations of shear banding in plane strain tension are performed. Critical or negligible effects of plastic spin on
Roles of plastic spin in shear banding
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shear band formation are observed, depending on the types of constitutive modeling. Moreover, nonlocal effects are examined. Finally, we discuss the relationship between the simple shear behavior and the shear band formation. It is emphasized that this is a critical issue in predicting shear banding on macroscopic grounds.
II. C O N S T I T U T I V E R E L A T I O N S
II.1 General aspects Assuming a small elastic deformation, we can express the results of the kinematics (Dafalias [1985a, 1985b]), L = Le + Lp
(1)
( L ) s = O = D e -'l- O p
(2)
(L)a : W = to q- W p
(3)
where L is the velocity gradient, (L)s and (L)a denote the symmetric and antisymmetric parts of L, and the superscripts e and p denote the elastic and plastic parts, respectively. Usually, D is called the rate of deformation tensor or the strain rate tensor, and W the continuum spin tensor. Within the theoretical framework considered here, to is an instantaneous rigid body spin or spin of the substructure. The elasticity relation is assumed to be ov r = C : De = C :
(D - D p)
(4)
where or is the Cauchy stress, C is the constant elastic moduli defined with the Young's modulus E and the Poisson's ratio v. The superposed V denotes the objective (co-rotational) rate of any second order tensor a with respect to to, i.e. ~7 a :
a -- ~l)a + a o 0 ;
w :
W - W p.
(5a, b)
In this investigation, for any constitutive relation for Dp, the plastic spin W p is assumed to arise from the noncoaxiality between the Cauchy stress tr and the plastic rate of deformation D p (Kuroda [1995a]). This can be represented as W p = r/(o'D p -- DPo")
(6)
where r/is a scalar valued function of the stress and structure variables. The objective rate defined in eqn (5a) is reduced to the classical Jaumann objective rate when W p = 0. 11.2 .4 Generalized viscoplastic model involving stress rate dependent noncoaxiality and kinematic tiardening Motivated by the deformation theory (St6ren & Rice [1975]), the following relation for DP is employed as a basis:
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M. Kuroda
D p = iPp -k- ~
(7)
- oe
where ( )' denotes deviatoric quantities, 6- = o- - a, ~ = p : b ' = (3G/#e)(6"' : D - ~ekP), p = 36-'/(2~e), #e = V / ~ ( 6 - ' : 6-,)1/2, o~ is the back stress, G=E/{2(1 +u)}, and ~P is the effective plastic strain rate usually assumed as a function of the effective plastic strain e p : f ~ Pdt and the effective equivalent stress ~e. The modulus H will be specified later by utilizing a strain rate hardening function. In this model, the noncoaxiality between D p and 6- arises from the second term on the right-hand side of eqn (7). By using eqns (4) and (7), the inverted relation for the stress rate ~r can be expressed as =
I) -
(8)
C - --H+3GH C + -~ ~ - --}~ (I ~9 ,)
+ ~2e2to" ® 6-')
P = C : p = _ _ 3G6-, ere
(9) (I0)
where I is the unity tensor. The evolution law for the back stress o~ is taken to be given by a Prager-Ziegler's shift rule ~x' : bp~ p.
(1 I)
The plastic spin relation is derived on the basis of the generalized concept (eqn (6)); that is, the noncoaxiality between D p and o-, W p = v(~D p - DPo") =
( o ' ~ - o'o-) -.]--=-ere
) (o[,o- - o'o[) .
(12)
It should be noted that, in spite of the existence of the two skew tensor generators o-~"- ~ro- and ao- - o'o~, this relation for plastic spin has only one scalar valued function 7. The plastic spin defined in eqn (12) does not vanish even if we assume pure isotropic %r v hardening, because of the existence of the term o-o- - o'o-. Dafalias [1983, 1985a, 1985b] provided the plastic spin relation in the form W p = ~(¢xo"- o-~x) on the basis of the representation theorem (Wang [1970]) for skew-symmetric tensor-valued isotropic functions depending on two tensor variables. Flow theories with kinematic hardening is the case. This plastic spin concept is contained in eqn (12) as the particular case for H ~ c~. It is also noted that the definition of Wp in eqn (6) does not require any particular internal variable. The merits of the present concept were discussed with respect to a corner theory of plasticity (Kuroda [1995a]). Zbib and Aifantis [1988] and Zbib [1991,1993] presented a similar concept for the plastic spin on the basis of somewhat different insights, in the absence of kinematic hardening. Difference between the basic ideas by Zbib [1993] and by Kuroda [1995a] was discussed in Kuroda [1995a]. Considering eqns (3) and (5), the classical Jaumann stress rate 6-is related to the present objective stress rate ~r as follows:
Roles of plastic spin in shear banding
675
0".~ O" -- Wo" + o ' W = ~I -- WPff + o ' W p.
(13)
Using eqns (12) and (13), the constitutive relation (8) can be written finally in the form 6"= [ C + S + Q ]
:D-(P+R)~
p
(14)
with
(s)uk, = 37 ( Iimaj, - ~rimI:, ) ( Imoanp -- amoI, p )-Copkt Q =
3~__A®P, 25eH
A = ( t x t r - trtx)cr - t r ( t x t r - trot)
at/A. R = 25e
(15) (16a, b) (17)
The functions ~ p and H, the kinematic hardening coefficient b, and the plastic spin coefficient r~will be specialized below. The strain rate hardening response is assumed to be represented by the power law including a nonlocal term based on the gradient concept (Aifantis [1984,1992]): O'e ~ B;
\e0/
--Cs~72E p
(18)
and therefore eP = eo~,
(19a, b)
with e: = (1 - r)av + rgtrv.
(20)
Here cs is a material constant, V 2 denotes the Laplacian operator, ~0 a reference strain rate, m is a strain rate sensitivity parameter, ov is a reference strength, r is an isotropic hardening parameter, and g is the strain hardening function that is defined as g = (1 + eP/eo) n,
(21)
where n and e0 are material constants. It is clear that when Cs = 0 this gradient-dependent formulation corresponds to the conventional local formulation. The characteristic of strain softening is introduced by ~v --
ffYO 1 + (eP/el) 2
(22)
where cry0 is an initial "yield" strength, and el is a softening parameter. If r = 1, pure isotropic hardening is assumed, and pure kinematic hardening corresponds to r = 0.
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M. Kuroda
The coefficient H in (7) is defined to be -O'e H -/3eP '
are =
V/~(O.,
: fit)l/2
(23a, b)
In the original deformation theory,/3 is taken to be unity. In that case, however, an elastic unloading-like behavior cannot be represented. (This has been discussed in Kuroda [1995a].) In the present model,/3 is determined by using • defined in eqn (19) as /3=
1 ~
for • > 1 for~
(24)
By introducing/3, the unloading-like behavior can be represented because/3 ~ 0 when O'e ~( K • If we set [3 = 0 (namely, H ~ ec), the present model is reduced to the viscoplastic version of J2 flow theory. The kinematic hardening coefficient b in (11) is determined as b -- (1 - r) 2(5e + CsV2ep) rd(gav) 3• [ deP
day] depJ'
(25)
according to the condition of coincidence of the uniaxial behavior for isotropic/kinematic combined hardening with the one for pure isotropic hardening at constant plastic strain rate, i.e. =
- & =
3<)
(26)
where ~r is the uniaxial stress for pure isotropic hardening given as O" ---- go'y
--Cs~72~ p.
(27)
Finally, the plastic spin coefficient r/in eqn (12) is assumed to be = a/Se
(28)
where a is taken as a constant (nondimensional) for simplicity. If a = 0, the objective stress rate ~r is reduced to the Jaumann stress rate 6". 11.3 Porous plasticity m o d e l The flow potential for Gurson's model with isotropic/kinematic hardening is expressed in the form (Mear & Hutchinson [1985]) ~)(or, o~, OfF,f) =
5e2
/'1 : ~ + 2ql f c o s h [ w - - - ] - 1 - (q, f ) 2 = 0 \ LO'F /
(29)
where f is the void volume fraction and ql is a material parameter introduced by Tvergaard [1981], and ~rF is the effective matrix flow strength associated with a kinematic
Roles of plasticspin in shear banding
677
hardening characteristic. In this porous plasticity model, o- represents the macroscopic Cauchy stress and tx the corresponding back stress. Although a function accounting for the effects of rapid void coalescence into the flow potential (eqn (29)) was introduced by Tvergaard and Needleman [1984], this function is omitted here. In this investigation, we assume that the increase in void volume fraction arises only from the growth of existing voids for simplicity. The matrix material is taken to be plastically incompressible. In the conventional formulation, the rate of the void volume fraction is given by J'= (1 - f ) I : D p.
(30)
We also consider here a nonlocal damage effect in terms of the integral condition on the rate of increase of the void volume fraction, which is assumed to be given by the following expression (Bazant & Pijaudier-Cabot [1988]; Leblond [1994]; Tvergaard & Needleman [1995]):
et al.
j'(x) = A(x)
[I -f(y)]l : DP(y)qo(x - y)dV(y)
(31)
with
A(x)=/v~(X-y)dV(y),
qo(x - y) = exp -
(32a, b)
where V is the volume under consideration and I is a material characteristic length. If we employ the gradient concept (Aifantis [1992]), the simplest form of the rate of the void volume fraction may be written, for example, a s j ' = (1 - f ) I : DP + cV2f. However, some preliminary calculations showed that the weight function-type expression in eqn (31) yields more appropriate results in the case of this porous model. Using the flow potential (eqn (29)) and the flow rule, the macroscopic plastic rate of deformation Dp takes the form
D p = h ~-~,Oq'
A - (1 --f)O'F~ pO~ ~-:--
(33)
0o" The matrix plastic strain rate k P is assumed to be
~P=ko(~rr~ '/m, ~G=(1--r)avo+rgayo
(34a, b)
\~GJ
where the ,;train hardening function g is the same as eqn (21). The evolution equation for aF is obtained from the consistency condition ~ = 0. Substituting eqn (33) into eqn (4) gives the following stress rate-strain rate relation tr=C:D-AC:0t
r
(35)
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M. Kuroda
Following Mear and Hutchinson [1985], the evolution equation for the back stress is taken to be given by
o o(o
"dg
~x= ( l - r )
0ti)-~ -1
6- : ~-~j
_ :
2
A q av0(1 - g) 0~
(1 - f)CrF#~G
(36)
Based on eqn (6), the plastic spin associated with this porous theory is given by Wp
:
r/(tyOp _ Opo. ) = a_~ (txo" -- O'IX) = r/(txO p -- DPct).
(37)
o" F
where the scalar valued function rl is assumed to be r / = a / c r F. Consequently, this relation is equivalent to that provided by Dafalias [1983, 1985a, 1985b] and employed by Tvergaard and van der Giessen [1991]. Considering eqns (13) and (37), the constitutive relation (35) can be written in terms of the Jaumann rate ~- as b=C:D-A
OO 3r# \ C:~--~+~-~FA ) .
(38)
IlL NUMERICAL M E T H O D
An updated Lagrangian finite element method is used. The FE equation is derived on the basis of the rate-type principle of virtual work (Hill [1959]) with neglect of the body force effect
f
s : 6LdV=/p.
6vdS,
i.e. fj,6LodV=/p,6v,dS
(39)
where V and S are the current volume and surface, s is the nominal stress rate, p is the nominal rate of the force per unit current surface, v is the velocity, and 6v and 6L denote virtual velocity and virtual velocity gradient, respectively. The relation between the nominal stress rate and the Jaumann rate of the Cauchy stress ~ is given by = ~- + Wcr - o'W - Lo- + (I : L)o'•
(40)
The finite element equation is derived by substituting eqn (40) into eqn (39) and by assuming the distribution of the velocity v in a finite element. All the calculations performed in this paper employ the constant strain triangular element. The FE analysis may be performed with the constitutive equation (14) or (38). However, in such a case, a very small time increment is usually required due to numerical unstable problems• In order to perform stable numerical calculations, a powerful rate tangent modulus method (Peirce e t al. [1984]) is employed in the present
Roles of plastic spin in shear banding
679
numerical c,alculations shown below. The procedures for deriving the rate tangent modulus are essentially identical to those in Peirce et al. [1984], Becker and Needleman [1986], Tvergaard and Needleman [1995], and Kuroda [1995b]. Therefore, they are not repeated here. IV. BEHAVIOR IN SIMPLE SHEAR
Simple shear is one of the simplest homogeneous deformation modes. This problem is expressed by xl = X1 + 7X2,
x2 = )(2,
x3 = )(3
(41)
where -~ is the shear strain measure, x is the position vector of the particle in the deformed (current) configuration and X the one in the undeformed (initial) configuration. Generally, constitutive relations aimed at large strain analysis are required to provide a reasonable stress behavior in the simple shear deformation. In fact, validity of various constitutive relations has been actively discussed on the basis of stress responses to simple shear. However, the lack of experimental evidence has been pointed out by many workers. Reliable experimental data for simple shear is essential to evaluate the effectiveness of constitutive relations for large strain analysis. In this section, stress behavior in simple shear is examined by both numerical analyses and experiments. IV.1 Numerical analysis Fixed material constants used in the calculations are assumed to be E = 200 GPa, v=0.333, av0 = 400 MPa, n = 0.1, e0=0.002, E0 = 0.0002/s and m=0.002. The problem specified by (41) is solved by the finite element analysis with two triangular elements. And the applied strain rate is specified as "~ = 10x/-3~0. We consider here the following three types of models: • A stress rate dependent noncoaxial model with strain hardening (eqn (14) with eqns (23) and (24), and el ~ c~ in eqn (22); this model is henceforth referred to simply as the "noncoaxial model"). • A strain-softening model based on J2 flow theory (eqn (14) with H --* oo, and el = 1.5 in eqn (22); this model is referred to as the "softening model"). • A porous model (eqn (38) with ql = 1.5). For the noncoaxial model, both pure isotropic hardening and pure kinematic hardening are considered, since even if isotropic hardening is assumed the plastic spin still exists due to the stress rate dependent term trot-~rtr (as seen in eqn (12)). For the softening model and the porous model, pure kinematic hardening is considered, since the plastic spin arises only from the back stress. As the purpose of the present investigation is to study the effects of plastic spin, such extreme cases are chosen and examined. Though the principal motivation for using a kinematic hardening model is to represent deformation-induced anisotropy, the kinematic hardening model in large strain plasticity analysis has succeeded in reproducing axial effects in torsion or in shear deformation (Paulun & Pecherski [1992]; van der Giessen et al. [1992]). The
680
M. Kuroda a-3
2.0 1.5 (",1
1.0 0.5
-0
a-3
~'- -0. 5 I II
~ -1.0 a-1
-1.5 Fig. 1. Stress responses to simple shear predicted by the noncoaxial model with isotropic hardening. The stress components are normalized by avo/vr3. axial effects have been widely recognized as an i m p o r t a n t and interesting p h e n o m e n o n at large strain since Swift [1947]. Figure 1 shows the c o m p u t e d curves o f the nonzero stress c o m p o n e n t s versus shear strain 7 for the noncoaxial model with isotropic hardening (r = 1). The curves o f n o r m a l stress cr22 =-0.11 and shear stress o12 corresponding to the plastic spin p a r a m e t e r a = 0, 1 and 3 are illustrated. Strong decrease in 0.12 and significant development in 0.22 are observed when no plastic spin is assumed (a = 0). This is a c o m m o n predictive feature o f "vertex-type" constitutive models. The calculation was terminated when 0.12 reached zero, since the situation of 0.12 < 0 seems to be physically meaningless. In the case of a = 3, the
a-3
2.0
a~l
1.5 c'q
1.0
0.5 a-3
~-0.5 I
a~l
n -1.0 -1.5
a~0
Fig. 2. Stress responses to simple shear predicted by the noncoaxial model with kinematic hardening. The stress components are normalized by av0/v~.
Roles of plastic spin in shear banding
681
2.0 1.5 a-3 ~
1.0 0.5 NN~ "0 i
o
~3
i
~' i
I
'
1:1=-0.5 fl ~e~-l. 0 -1.5 Fig. 3. Stress responses to simpleshear predicted by the softeningmodel with kinematichardening. The stress components are normalizedby av0/v~. shear stress a12 increases monotonically. Figure 2 shows the results for kinematic hardening case, and almost the same effect of plastic spin is observed. Figure 3 ,.shows the results for the softening model with kinematic hardening (r = 0). The shear and normal stresses decline even for an increased amount of plastic spin, because the material is assumed to soften essentially. Figure 4 shows the results for the porous model with kinematic hardening. The initial void volume fraction J~ is assumed to be 0.01. Although a12 for a = 0 does not become negative, the oscillatory behavior is observed as well-known since Nagtegaal and de Jong [1982]. The voids do not grow from the initial state because no hydrostatic pressure exists in the simple shear deformation. Therefore, no progress in damage occurs during the simple shear process. Corresponding to this fact, a monotonic stress response is observed for the large amount of plastic spin, a = 3.
2.0
a-3
1.5
a-1
t"q
1.o
0.5
~=-0. 5 i II
~ -1.0
a-O
--
-1.5 Fig. 4. Stress responses to simple shear predicted by the porous model with kinematichardening. The stress components are normalizedby avo/v~.
682
M. Kuroda
IV.2 Experimental evidence The simple shear deformation can be reproduced approximately by a large strain torsion test of thin-walled tubes or solid circular bars with fixed ends. In particular, the deformation in the tube specimen, is almost identical to simple shear deformation. The close relationship between the behaviors in simple shear and in solid bar torsion was studied in detail by Neale and Shrivastava [1990] and by van der Giessen et al. [1992]. In the fixed end torsion tests, we observe usually the development of axial force (normal stress) in addition to the torque (shear stress). Torsion tests were carried out for tube specimens with fixed ends at room temperature and static loading condition. The testing machine was newly designed and assembled. Tube specimens of aluminum-magnesium alloy with outside diameter 8 mm, inside diameter 6 mm, and gage length 15 mm were prepared. They were machined from commercial bars (JIS A 5056B) of diameter 10 mm, and were not annealed. The sample surfaces were sufficiently polished. An example of the test results is shown in Fig. 5. In the figure, -~represents the shear strain at the mid-thickness of the wall of the tube specimen; this is defined by "~= 2NTrRo/16, where/~0 denotes the average value of outside and inside radii, N is the number of revolutions, and lc is the gage length. The reference strength ~rR is taken to be 118 MPa, which is the nominal yield strength for the commercial A1-Mg alloy. The torque T is macroscopically monotonic, and the axial force F shows a saturation. But, a buckling was observed clearly at ~ 1.4. The subsequent torque and axial force have been denoted by the broken lines. For comparison purposes, solid bar specimens of AI-Mg alloy and copper (the diameter 6 mm and the gage length 40 mm) were also tested. Some of them were annealed (3 h at 400°C and 3 h at 700°C for AI-Mg and Cu, respectively). Though the results for the solid bars are omitted here, it was confined that their torsion behavior was very similar to the results for the tube specimens. In the case of the solid bars, the monotonic torque and the saturation of axial force were observed up to failure at 7 -~ 5 (any buckling did not occur). Thus, we see no "softening" behavior during the torsion processes.
2.01.5-
~II~
1.0.5"
Not annealed
0
Fig. 5. Example of fixed-endtorsion test results: torque T and axial force F versus shear strain -~ for a thinwalled tube specimen of AI-Mg alloy. (--Ro = average value of outside and inside radii of the tube, to= initial thickness of the tube.)
Roles of plastic spin in shear banding
683
Although we do not carry out here any parameter-fitting study that corresponds to the above experimental results, the global stress behavior observed in the experiments has been captured semi-quantitatively by the noncoaxial model (Figs 1 and 2) and the porous model (Fig. 4) with the large amount of plastic spin. Anand and Kalidindi [1994] showed a torsion test result for a thin-walled tubular specimen of copper that was annealed at 800°C (1 h). This torsion test was a reverse test in which a direction of twist was reversed at -~= 1.5. The shear stress response up to -~= 1.5 in Anand and Kalidindi [1994] is very similar to the one in Fig. 5. The monotonic increase in shear stress in a fixed end torsion was also shown for a mild steel in the earlier work of Swift [194711. However, in the Swift report, the corresponding response in axial force was not shown. Montheillet et al. [1984] observed that, in solid bar torsion tests with aluminum and copper specimens at room temperature, the axial forces decreased notably after the attainment of a clear peak value and then they increased considerably again. But, the correspondiLng responses in torque at room temperature were not shown in their work. This behavior of axial forces seemed to be "oscillatory" and suggested a possibility of occurrence of stress oscillations in simple shear deformation. In fact, Zbib and Aifantis [1988] tried to simulate this type of oscillatory stress behavior by using a complex back stress model. However, in the present experiments, such "oscillatory" behavior of axial force has n~ot been observed. Meanwhile, at very high temperatures, a clear decline in torque was observed in the experiments by Montheillet et al. [1984], which may correspond to the results for the softening model shown in Fig. 3.
Fig. 6. Plane strain tensile specimen and its finite element discretization.
684
M. Kuroda V. EFFECT OF PLASTIC SPIN ON SHEAR BANDING
V. 1 Problem formulation The plane strain tension problem to be analyzed here is illustrated in Fig. 6. The initial configuration of the plane strain specimen is specified by the initial length 2L0 and the initial width 2 W0 = 2L0/3, where W0 = 1 mm, as shown in Fig. 6. Considering a symmetry of the deformation, only one-quarter of the specimen is analyzed. Thus, the boundary conditions for the one-quarter are vl = 0 and/)2 = 0 vz = 0 and p~ = 0
on X~ = 0
P , = p2 = 0
on X1 = Wo on X 2 = Lo.
on X2=O
Vl = 0 and v2 = C
The end displacement (elongation) U is obtained from U = f Udt, and the tensile force is l,V0 . . defined by P = f_woP2dXl on X2 = Lo; 1.e. in the actual analyses, P is obtained as the summation of the nodal forces in 2"2 -direction at X2 = L0. In the plane strain analyses, the unit out-of-plane thickness is considered. The finite element mesh for the re.gion analyzed is also shown in Fig. 6. The applied displacement rate tJ is specified so that (U/Lo)/~o = 1. We employ again the three constitutive models used in Section IV: that is, the noncoaxial model with isotropic or kinematic hardening; the softening model with kinematic hardening; and the porous model with kinematic hardening. All the material properties are the same as those used in Section IV. V.2 Results for local formulation In the computational results shown in this subsection, the nonlocality parameter cs in eqn (18) is set to be zero and eqn (30) is used for the porous model, so that we examine here effects of plastic spin in the conventional local formulations. Figure 7 shows overall tensile load-elongation curves for the noncoaxial model with isotropic hardening. Results are shown for the plastic spin coefficient a = 0, 1 and 3. No difference between the results is observed until U/Lo ~- 0.16. However, subsequent load 2.0-
i-5-
f
~l.Oa~ a-O
0.5-
00
'
O'. i
'
O'. 2
'
0'.3
Fig. 7. Curves of tensile load versus elongation for the noncoaxial model with isotropic hardening.
Roles of plastic spin in shear banding U / Lo
=0.19
U / Lo
a =(I
=0.19
685 U / Lo
a =1
=0.19
a =3
Fig. 8. Deformed meshes and contours of constant equivalent plastic strain eP for the noncoaxial model with isotropic hardening. The interval of contour lines is 0.1. drop behavior is very sensitive to the amount of plastic spin. For a = 1 and 3, there is no sharp drop in load, and the curves are smooth. In the case of the noncoaxial model with kinematic hardening, though the drop in load for a = 0 becomes more sharp, the tendency of the overall load-elongation behavior is almost identical to the one for the isotopic hardening case. Therefore, the results for kinematic hardening are omitted here. Figure 8 shows contours of constant equivalent plastic strain e P and deformed meshes that correspond to the load--elongation curves shown in Fig. 7. The results for kinematic hardening are almost identical to isotropic hardening, too. Hence, they are not shown here. In the: figure, the very sharp shear bands are formed in the case of a = 0. However, the shear bands are made to fade notably with increasing amount of plastic spin. For a = 3, the shear bands eventually disappear. This tendency is quite similar to that for the case of a corner theory of plasticity (Kuroda [1995a]). Tensile load-elongation curves for the softening model with kinematic hardening are shown in Fig. 9. The noticeable effect of plastic spin on load drop behavior is observed, too.
2.0-
1.5a=3 ~ 1.0a=l 0.5-
0
'
0.1 '
'
0'.
2
Fig. 9. Curves of tensile load versus elongation for the softening model with kinematic hardening.
686
M. Kuroda U/L o
=0.16
U/L o
a =0
=0.16
U/L o
a =1
=0.16
a =3
Fig. 10. Deformed meshes and contours of constant equivalent plastic strain eP for the softening model with kinematic hardening. The interval of contour lines is 0.1. In the case of a = 0 and 1, an "oscillatory" behavior emerges at the final stages of l o a d elongation curves. This can be attributed to the fact that the shearing mode in the shear band yields oscillatory behavior due to the small amount of plastic spin. Figure 10 shows contours of constant eP and deformed meshes that correspond to Fig. 9. The sharpness of the shear bands declines with increasing amount of plastic spin. However, even for the case of assuming the large plastic spin coefficient (a = 3), the shear bands are still visible clearly. Figure 11 shows tensile load-elongation curves for the porous model with kinematic hardening. It is noted that the overall load--elongation behavior is very insensitive to the amount of plastic spin in this case. This feature is very different from those for the other constitutive relations, though the sharp drop in load is delayed very slightly by the increase in a. Figure 12 shows deformed meshes and contours of constant macroscopic equivalent plastic strain that is gP J'(2D p : D p/3)1/2dt. Even though the large value of the plastic spin coefficient, a = 3, is assumed, we can observe the shear band formation that is nearly equivalent to the one observed for a = 0. :
2.0-
1.5-
~
S
t~l.O -
a =3 a=l
a=O
~0.5-
0
I
0
1
.1
I
U/ Lo
0I
.2
Fig. 11. Curves of tensile load versus elongation for the porous model with kinematic hardening.
687
Roles of plastic spin in shear banding U / Lo =0.17
a =0
U / L o -0.17
U / L o =0.17
a =1
a =3
Fig. 12. Defo~maedmeshes and contours of constant macroscopic equivalent plastic strain gP for the porous model with kinematic hardening. The interval of contour lines is 0.1. V.3. R e s u l t s f o r n o n l o c a l f o r m u l a t i o n Figure 13 shows c o n t o u r s o f constant EP and deformed meshes for the noncoaxial model with the nonlocal parameter Cs = 1 N. Results are shown for a = 0 and 3. In these calculations, isotropic hardening is assumed. F o r a = 0, the shear b a n d width is wider than the one for the local case (Fig. 8). The shear b a n d width spans over two rows o f elements in this case, while in the case o f local model the b a n d width is essentially determined by the size o f ,one element. We can n o w see a clear influence o f the nonlocality based on the gradient concept. F o r a = 3, no shear b a n d appears as in the case o f the local model (Fig. 8). The identical effects o f plastic spin are observed, whether nonlocality exists or not. Figure 14 shows the tensile load--elongation curves that correspond to Fig. 13. In the case o f a = 0, the d r o p in load is delayed due to the nonlocal effect, c o m p a r e d with the local case. O n the other hand, the curve corresponding to a = 3 is almost identical to the one for the local case (Fig. 7). The critical effect o f plastic spin also can be seen f r o m the figure. C o n t o u r s o f constant ¢P and deformed meshes for the softening model with cs = 2 N are depicted in Fig. 15. In this case, the shear b a n d width for a = 0 spans over at least four U / L o =0.21
a=O
U / L o =0.21
a=3
Fig. 13. Defi~rmed meshes and contours of constant equivalent plastic strain eP for the nonlocal noncoaxial model with isotropic hardening. The nonlocal parameter cs= 1 N. The interval of contour lines is 0.1.
688
M. Kuroda 2.0-
1.5-
a=3
~I:~1.0 a~O
0.5-
i
I
i
i
0.1
i
0.2
i
0.3
Fig. 14. Curves of tensile load versus elongation for the nonlocal noncoaxial model with isotropic hardening. The nonlocal parameter Cs= 1 N.
rows o f elements. The sharpness o f the shear bands fades when the large a m o u n t o f plastic spin, a = 3, is assumed. The fundamental tendency o f the effect o f plastic spin observed here is the same as the one observed in the case o f the local model (Fig. 10). Figure 16 shows the tensile l o a d - e l o n g a t i o n curves that correspond to Fig. 15. The difference between results for a = 0 and 3 becomes smaller in c o m p a r i s o n with the local case (Fig. 9). Figure 17 shows contours o f constant matrix plastic strain eP and meshes for the porous model with the characteristic length l = 0.075 mm. (Contours o f gP are very similar to this figure.) In this case, due to the nonlocal effect, the shear bands are not formed intensively, but they are visible in the contour maps. The load-elongation curves corresponding to Fig. 17 are depicted in Fig. 18. F r o m both figures, we can see a less significant effect o f plastic spin. Vl. DISCUSSION In the case o f the noncoaxial model without plastic spin (a = 0), a significant decrease in shear stress after the attainment o f its peak value is observed during simple shear deformation, U / Lo
=0.20
U / Lo
=0.20
a =0 a =3 Fig. 15. Deformed meshes and contours of constant equivalent plastic strain s p for the nonlocal softening model with kinematic hardening. The nonlocal parameter c~= 2 N. The interval of contour lines is 0.1.
Roles of plastic spin in shear banding
689
2.0-
1. S. ~ f ' - ' ~1.0a=0
0.50
i
0'.1
0
i
0'. 2
U/ Lo
Fig. 16. Curve,s of tensile load versus elongation for the nonlocal softening model with kinematic hardening. The nonlocal parameter cs= 2 N. even though isotropic hardening is assumed, as shown in Fig. 1. This is a common feature of constitutive relations in which the direction of plastic strain rate depends on that of stress rate such as the comer theories of plasticity (for example, Christoffersen & Hutchinson [1979]; Gotoh [198:5]). Kuroda [1995a] suggested a possibility of existence of oscillatory or risingsinking characteristics of stress response in simple shear on the basis of the predictive feature of-models with the directional dependency of the plastic strain rate on the stress rate. However, it is confirmed that, at least for metals, the strong decrease in shear stress during simple shear is completely inconsistent with the experimental evidence, as shown in Section IV. The decrease in shear stress can be eliminated by the plastic spin, based on the concept of the noncoaxiality between the', plastic strain rate and the stress. It is possible for the noncoaxial model with the plastic spin to produce a monotonic stress response to simple shear, which is consistent with the experimental evidence. On this point, it seems that effects of plastic spin are quite equivalent, whether in the noncoaxial model (a kind of vertex-type theory) or in the porous model (a kind of flow theory).
U / Lo
=0.175
a =0
U / Lo
-0.175
a =3
Fig. 17. Deformedmeshes and contours of constant matrix plastic strain e P for the nonlocal porous model with kinematic hardening. The characteristiclength 1= 0.075 mm. The intervalof contour lines is 0.1.
690
M. Kuroda
2.01.5-
a'3
f
~l.0¢.q
~0.5-
I
I
0.1
I
U/ Lo
I
0.2
Fig. 18. Curves of tensile load versus elongation for the nonlocal porous model with kinematic hardening. The characteristic length l= 0.075 mm.
However, on predictions of shear banding, the effects of plastic spin in the noncoaxial model and in the porous model are significantly different. In the case of the noncoaxial model, it is observed that the plastic spin which is increased large enough to yield a monotonic stress response to simple shear makes the shear bands vanish completely (Figs 8 and 13). This predictive feature of the plastic spin does not change, whether the nonlocality exists or not. On the contrary, in the case of the porous model, the shear band formation is only faded slightly, even when a sufficiently large amount of plastic spin is assumed (Figs 12 and 17). What is the cause of such a difference? It is known that the noncoaxiality between the plastic strain rate and the stress manifests itself in an apparent softening, in the presence of rotation of material. This is referred to as geometric softening (see, for example, Zbib [1993]). Not only does such geometric softening reduce a critical strain for shear localization but it also accelerates significantly the shear band development. The directional dependency of the strain rate on the stress rate in the noncoaxial model induces directly the geometric softening. The back stress also contributes to the noncoaxiality, and therefore, it induces and accelerates the geometric softening, too. In fact, it is known that the noncoaxiality due to the back stress evolution yields an effect that is similar to a vertex formation at yield surface, which is often identified as a rounded vertex effect (Tvergaard [1978]). The stiffness of materials reduced by such a geometric softening is recovered dramatically by the plastic spin, as can be seen clearly from the numerical results for the noncoaxial model. In fact, the large amount of plastic spin yields a monotonic stress response to simple shear, as shown in Figs 1 and 2, and also it restrains completely the shear band formation, as shown in Figs 8 and 13. When a large amount of plastic spin cancels the geometric softening effect, the noncoaxial model becomes a model without any softening effect. Here we should provide a word of caution 'noncoaxial models may not be able to predict any shear localization when one determines the value of the plastic spin coefficient on the basis of the "monotonicity of shear stress in simple shear" that is fairly consistent with the experimental results.' This caution is pertinent to the vertex or corner theories of plasticity being rate-independent, as could be seen from the numerical results in Kuroda [1995a]. We should recognize that this is a difficulty in
Roles of plastic spin in shear banding
691
phenomenological theories of plasticity with the directional dependency of plastic strain rate on the :stress rate. On the other hand, in the case of the porous model, the shear banding behavior seems to be goveFaed mainly by the macroscopic softening of material due to void growth. In fact, the numerical results shown in Figs 11 and 18 imply that the stiffness reduced by such a material softening is no longer recovered by the plastic spin. Consequently, the effect of plastic spin becomes less significant in the localization process, though the shear band formation i~ faded very slightly by the increased amount of plastic spin (Figs 12 and 17). This fading is attributed to the recovery of the geometric softening associated with the back stress evolution. Meanwhile, in the case of simple shear problem, no void grows because no hydrostatic pressure exists. Therefore, the stresses can increase monotonically if a large amount of plastic spin is assumed (Fig. 4). Consequently, the porous plasticity model with a relatively large amount of the plastic spin can reproduce both of the monotonic stress in simple shear and the shear localization in tensile problems. We see no inconsistency in these predictive features of the porous model with the plastic spin. This supports a soundness of the assumption that the macroscopic material softening effect due to pressure-sensitive damage, rather than the noncoaxial effect, plays a dominant role in the macroscopic shear banding at room temperature. The effect of the plastic spin on shear localization for porous material has also been investigated by Tvergaard and van der Giessen [1991]. However, their results and the present finite element result are not directly comparable. Tvergaard and van der Giessen's analysis assumed art infinite initial "band type" imperfection, while the present full numerical analysis requires no band type imperfection and the inhomogeneity due to the diffuse necking only triggers the localization. Tvergaard and van der Giessen [1991] observed that the onset of' localization was noticeably delayed by an increased amount of plastic spin. On the contrary, the present full numerical analysis without any initial imperfection has shown a very smallLeffect of plastic spin on shear banding in the diffuse necking region. The softening model is usually consistent with heating or high-temperature conditions without material damage. In this model, the geometric softening by the noncoaxiality due to the back stress evolution is recovered by the plastic spin, but the true material softening is no longer recovered. The numerical results (Figs 9, 10, 15 and 16) reflect these facts. VII. CONCLUDING COMMENTS
In this paper, the effects of plastic spin on shear banding have been studied systematically. We have observed critical or negligible effects of plastic spin, depending on the types of constitutive: modeling. The observations are summarized and interpreted as follows. The development of shear bands needs at least one softening effect. In the case of the stress rate dependent noncoaxial model, only the geometric softening is possible. The porous model contains the macroscopic material softening due to void growth and the geometric ,;oftening due to the back stress evolution. In the softening model, the true material softening and the geometric softening due to the back stress are active. The numerical results indicate that the geometric softening is cancelled by an increased amount of plastic spin. Therefore, if we assume a large amount of plastic spin on the basis of the experimental evidence for the simple shear deformation, the noncoaxial model becomes a model without any softening effect. Consequently, the noncoaxial model can no longer predict any shear band formation. This is a common difficulty in plasticity
692
M. Kuroda
models in which the direction o f the plastic strain rate is d e t e r m i n e d with that of the stress rate, a n d is subjected to further investigations. O n the other h a n d , the p o r o u s m o d e l with large a m o u n t of the plastic spin has succeeded in predicting b o t h the stress behavior in large simple shear a n d the shear b a n d i n g in tension problem. This is because the macroscopic softening due to the pressure-sensitive void d a m a g e still r e m a i n s as a d o m i n a n t factor o f the localization, even t h o u g h the plastic spin cancels completely the geometric softening arising from the back stress evolution. The softening m o d e l with large a m o u n t o f plastic spin m a y be used for predicting the material b e h a v i o r with n o c o n t r a d i c t i o n u n d e r h i g h - t e m p e r a t u r e conditions.
REFERENCES
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Swift,H.W., "Length Changes in Metals under Torsional Overstrain," Engineering, 163, 253. Hill, R., "Some Basic Principles in the Mechanics of Solids Without a Natural Time," J. Mech. Phys. Solids, 7, 209. 1970 Wang, C.C.,"A New Representation Theorem for Isotropic Functions: An Answer to Professor G.F. Smith's Criticism of My Papers on Representation Theorem for Isotropic Functions, Part. 1 and Part. 2," Arch. Rational Mech. Anal., 36, 166. 1971 Mandel, J., "Plasticit6 Classique et Viscoplasticit6," Courses and Lectures, No. 97, ICMS, Udine, Springer. 1975 St6ren,S. and Rice, J.R., "Localized Necking in Thin Sheets," J. Mech. Phys. Solids, 23, 421. 1977 Gurson, A.L., "Continuum Theory of Ductile Rupture by Void Nucleation and Growth--I. Yield Criteria and Flow Rules for Porous Ductile Media," ASME J. Engng Materials Technol., 99, 2. 1978 Tvergaard, V., "Effect of Kinematic Hardening on Localized Necking in Biaxially Stretched Sheets," Int. J. Mech. Sci., 20, 651. 1979 Christoffersen,J. and Hutchinson, J. W., "A Class of Phenomenological Comer Theories of Plasticity," J. Mech. Phys. Solids, 27, 465. 1981 Tvergaard, V., "Influence of Voids on Shear Band Instabilities Under Plane Strain Conditions," Int. J. Fracture, 17, 389. 1982 Nagtegaal,J.C. and de Jong, J.E., "Some Aspects of Anisotropic Work Hardening in Finite Strain Plasticity," in Lee, E.H. and Mallet, R.L. (eds.), Proceedings of Workshop on Plasticity of Metals at Finite Strain, Division of Applied Mechanics, Stanford University, Stanford, p. 55. 1983 Dafalias, Y.F., "Corotational Rates for Kinematic Hardening at Large Plastic Deformations," ASME J. Appl. Mech., 50, 561. 1983 Lee, E.H., Mallett, R.L. and Wertheimer, T.B., "Stress Analysis for Anisotropic Hardening in FiniteDeformation Plasticity," ASME J. Appl. Mech., 50, 554. 1984 Aifantis, E.C., "On the Microstructure Origin of Certain Inelastic Models," ASME J. Engng Mater. Technol., 106, 326. 1984 Johnson, G.C. and Bammann, D.J., "A Discussion of Stress Rates in Finite Deformation Problems," Int. J. Solids and Structures, 20, 725. 1984 Montheillet, F., Cohen, M. and Jonas, J.J., "Axial Stresses and Texture Development During the Torsion testing of Al, Cu and a-Fe," Acta Metall., 32, 2077. 1984 Peirce,D., Shih, C.F. and Needleman, A., "A Tangent Modulus Method for Rate Dependent Solids," Comput. Struct., lg, 875. 1984 Tvergaard, V. and Needleman, A., "Analysis of the Cup-Cone Fracture in a Round Tensile Bar," Act. Metall., 32, 157. 1985a Dafalias, Y.F., "The Plastic Spin," ASME J. Appl. Mech., 52, 865. 1985b Dafalias, Y.F., "A Missing Link in the Macroscopic Constitutive Formulation of Large Plastic Deformation," in Sawczuk, A. and Bianchi, G. (eels.), Plasticity Today, Modelling, Methods and Applications. Elsevier Appl. Sci., New York, pp. 135-151. 1985 Gotoh, M., "A Class of Plastic Constitutive Equations with Vertex Effect, I and II," Int. J. Solids and Structures, 21, 1101. 1985 Meat, M.E. and Hutchinson, J.W., "Influence of Yield Surface Curvature on Flow Localization in Dilatant Plasticity," Mech. Mater., 4, 395. 1986 Becket,R. and Needleman, A., "Effect of Yield Surface Curvature on Necking and Failure in Porous Plastic Solids," J. Appl. Mech., 53, 491. 1988 Bazant, Z.P. and Pijaudier-Cabot, G., "Nonlocal Continuum Damage, Localization Instability and Convergence," ASME J. Appl. Mech., 55, 287.
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Zbib, H.M. and Aifantis, E.C., "On the Concept of Relative and Plastic Spins and Its Implications to Large Deformation Theories. Parts I and II," Acta Mech., 75, 15. 1989 Szabt,, L. and Balla, M., "Comparison of Some Stress Rates," Int. J. Solids Struct., 25, 279. 1990 Neale, K.W. and Shrivastava, S.C., "Analytical Solutions for Circular Bars Subjected to Large Strain Plastic: Torsion," ASME J. Appl. Mech., 57, 298. 1991 Tvergaard, V. and van der Giessen, E., "Effect of Plastic Spin on Localization Predictions for a Porous Ductile Material," J. Mech. Phys. Solids, 39, 763. 1991 Zbib, H.M., "On the Mechanics of Large Inelastic Deformations: Noncoaxiality, Axial Effects in Torsion and Localization," Acta Mech., 87, 179. 1992 Aifantis, E.C., "On the Role of Gradient in the Localization of Deformation and Fracture," Int. J. Engng, Sei., 311, 1279. 1992 Paulun, J.E. and Pecherski, R.B., "On the Relation for Plastic Spin," Arch. Appl. Mech., 62, 376. 1992 van der Giessen, E., Wu, P.D. and Neale, K.W., "On the Effect of Plastic Spin on Large Strain ElasticPlastic Torsion of Solid Bars," Int. J. Plasticity, 8, 773. 1992 Yang, W., Cheng, L. and Hwang, K., "Objective Corotational Rates and Shear Oscillation," Int. J. Plasticity, 8, 643. 1993 Zbib, H.M., "On the Mechanics of Large Inelastic Deformations: Kinematics and Constitutive Modeling," Acta Mech., 96, 119. 1994 Anand, L. and Kalidindi, S.R., "The Process of Shear Band Formation in Plane Strain Compression of FCC Metals: Effects of Crystallographic Texture," Mech. Mater., 17, 223. 1994 Kuroda, M., "Finite Element Simulations of Large Elasto-Plastic Deformation with Different Spin Tensors," Mech. Res. Comm., 21, 517. 1994 Leblond, J.B., Perrin, G. and Devaux, J., "Bifurcation Effects in Ductile Metals with Nonlocal Damage," ASME J. Appl. Mech., 61, 236. 1994 Zbib, H.M., "Size Effects and Shear Banding in Viscoplasticity with Kinematic Hardening," in Batra, R.C. and Zbib, H.M. (eds.), Material Instabilities--Theory and Application, ASME, New York, pp. 19-33. 1995a Kuroda, M., "Plastic Spin Associated with a Corner Theory of Plasticity," Int. J. Plasticity, l l, 547. 1995b Kuroda, M., "An Elasto-Viscoplasticity Model Involving Noncoaxiality and Plastic Spin," in Tanirnura, S. and Khan, A.S. (eds.), Dynamic Plasticity and Structural Behaviors--Proceedings of Plasticity '95: The F'ifth International Symposium on Plasticity and Its Current Applications. Gordon and Breach, Luxemburg, pp. 919-922. 1995 Tvergaard, V. and Needleman, A., "Effects of Nonlocal Damage in Porous Plastic Solids," Int. J. Solids Structures, 32, 1063.
InternationalJournal of Plasticity,Vol. 12, No. 5, pp. 671~93, 1996 Copyright © 1996 ElsevierScienceLtd Printed in Great Britain.All rights reserved 0749-6419/96 $15.00+ .00 P l h S0749-6419(96)00024-1
ROLES OF PLASTIC SPIN IN SHEAR BANDING Mitsutoshi Kuroda Department of Civil Engineering, Ashikaga Institute of Technology, Omae, Ashikaga, Tochigi 326, Japan (Received in final re visedform 30 January 1996)
Abstract--In this paper, the effects of plastic spin on shear banding and simple shear are examined systematically. Three types of plastic constitutive model with plastic spin are considered: (i) a noncoaxial model in which the direction of the plastic strain rate depends on that of the stress rate; (ii) a strain-softening model based on the J2 flow theory; and (iii) the pressure-sensitive porous plasticity model. AlL1the constitutive models are formulated in viscoplastic forms and in conjunction with nonlocal concepts that have been recently focused and discussed. First, behavior in simple shear is examined by numerical analyses with the aforementioned constitutive models. Moreover, some experimental evidences for stress response to simple shear are shown; that is, several large torsion tests of metal tubes and bars are carried out. Next, finite element simulations of shear banding in plane strain tension are performed. A critical effect of plastic spin on shear banding is observed for the noncoaxial model, while an almost negligible effect is observed for the porous model. The identical effects of plastic spin are observed, whether nonlocality exists or not. Finally, we discuss the relationship between the behavior in simple shear and the shear band formation. It is emphasized that this is a critical issue in predicting shear banding in macroscopic grounds. Copyright © 1996 Elsevier Science Ltd
I. INTRODUCTION M u c h o f the w o r k in finite strain plasticity in the p a s t few decades has focused o n predicting flow l o c a l i z a t i o n in the f o r m o f " s h e a r b a n d s " . W h e n ductile m a t e r i a l s are d e f o r m e d at very high strain rates, the t h e r m a l m a t e r i a l softening due to local h e a t i n g m a y be a n i m p o r t a n t trigger in initiating a n d d e v e l o p i n g the localization. H o w e v e r , the s a m e m a t e r i a l m a y also u n d e r g o the shear l o c a l i z a t i o n at low strain rate, where the t h e r m a l effects are a l m o s t negligible. F o r p r e d i c t i o n s o f the shear l o c a l i z a t i o n in this c i r c u m s t a n c e , one o f the m o s t significant c o n t r i b u t i o n s in p h e n o m e n o l o g i c a l g r o u n d s is the c o n c e p t o f vertex theories in which the d i r e c t i o n o f the plastic strain rate d e p e n d s o n t h a t o f the stress rate (e.g. St6ren & Rice [1975]; Christoffersen & H u t c h i n s o n [1979]; G o t o h [1985]). A n o t h e r i m p o r t a n t c o n t r i b u t i o n is the c o n c e p t o f v o i d d a m a g e theories, such as the G u r s o n m o d e l ( G u r s o n [1977]). T h o u g h these two types o f plasticity t h e o r y h a v e quite different c o n c e p t u a l b a c k g r o u n d s , b o t h o f t h e m are often e m p l o y e d for the same p u r p o s e : t h a t is, p r e d i c t i o n o f a realistic l o c a l i z a t i o n b e h a v i o r in ductile materials. Since N e g t e g a a l a n d de J o n g [1982] have f o u n d an unrealistic o s c i l l a t o r y stress b e h a v i o r in a n analysis o f simple shear, stress b e h a v i o r in shear d e f o r m a t i o n a n d its p r e d i c t i o n have been also actively investigated ( D a f a l i a s [1983]; Lee et al. [1983]; J o h n s o n & B a m m a n n [1984]; Z b i b & A i f a n t i s [1988]; S z a b 6 & Balla [1989]; Y a n g et al. [1992]; K u r o d a [1994]). 671
672
M. Kuroda
Any theory of plasticity is required to predict a realistic stress response to simple shear deformation. One of the most significant contributions on this subject is considered to be the concept of the plastic spin that was originally suggested by Mandel [1971] and developed by Dafalias [1985a, 1985b] and others. The plastic spin concept succeeded not only in reproducing monotonic stress responses to simple shear (Dafalias [1985a, 1985b]), but also in simulating an interesting phenomenon called the Swift's [1947] effect (Zbib [1991 ]; Paulun & Pecherski [ 1992]). However, a possibility of existence of oscillatory stress behavior in simple shear or in fixed-end torsion has been suggested by Montheillet et al. [1984]. In many previous theoretical works, a lack of experimental data documenting large torsion behavior has been pointed out. At present, the large torsion (or simple shear) behavior seems to be still an issue. In these circumstances, some workers have examined the effects of plastic spin on the shear localization behavior. Tvergaard and van der Giessen [1991] investigated the effects of plastic spin on flow localization for porous ductile material, using a simplified plane strain analysis introducing the "band" type initial imperfections. Zbib [1994] studied the influence of plastic spin on the shear band formation in a special rigid plastic softening material during a plane strain compression. Kuroda [1995a] proposed a plastic spin relation associated with a corner theory of plasticity, and investigated its effect on the shear localization in plane strain tension. These studies showed that plastic spin stabilizes the localization phenomena. However, these results cannot be compared directly, because the problems and constitutive models employed to investigate the effects of plastic spin are essentially different. Information on the effects of plastic spin on shear localization has been obtained, as mentioned above. The following important questions, however, still remain: (1) Are the effects of plastic spin equivalent for any constitutive model? If not, how are they different? (2) Can plasticity theories with the plastic spin reproduce both the realistic stress response to simple shear and the localization behavior? (3) What is the realistic stress response to simple shear? The objective of this paper is to answer the foregoing questions. The effects of plastic spin on shear banding and simple shear are systematically examined for several theories of plasticity. The paper is arranged as follows. Section II provides constitutive relations employed in this investigation. Three types of constitutive models with plastic spin are considered: (i) a noncoaxial model in which the direction of the plastic strain rate depends on that of the stress rate (a vertex-like model); (ii) a strain-softening model based on the J2 flow theory; and (iii) the porous plasticity model (Gurson [1977]; Mear & Hutchinson [1985]). All the constitutive models are formulated in viscoplastic forms, and in conjunction with nonlocal concepts that have been recently focused on and discussed, (e.g. Bazant & Pijaudier-Cabot [1988]; Aifantis [1984, 1992]). The first two models are provided as the particular cases of a generalized constitutive model proposed by Kuroda [1995b]. In Section III, the numerical method, namely, finite element formulation, is presented. In Section IV, behavior in simple shear is examined by numerical analyses with the aforementioned three constitutive models. Moreover, some experimental evidences for stress responses to simple shear are shown; i.e. several large torsion tests of metal bar and tube are carried out. In Section V, finite element simulations of shear banding in plane strain tension are performed. Critical or negligible effects of plastic spin on
Roles of plastic spin in shear banding
673
shear band formation are observed, depending on the types of constitutive modeling. Moreover, nonlocal effects are examined. Finally, we discuss the relationship between the simple shear behavior and the shear band formation. It is emphasized that this is a critical issue in predicting shear banding on macroscopic grounds.
II. C O N S T I T U T I V E R E L A T I O N S
II.1 General aspects Assuming a small elastic deformation, we can express the results of the kinematics (Dafalias [1985a, 1985b]), L = Le + Lp
(1)
( L ) s = O = D e -'l- O p
(2)
(L)a : W = to q- W p
(3)
where L is the velocity gradient, (L)s and (L)a denote the symmetric and antisymmetric parts of L, and the superscripts e and p denote the elastic and plastic parts, respectively. Usually, D is called the rate of deformation tensor or the strain rate tensor, and W the continuum spin tensor. Within the theoretical framework considered here, to is an instantaneous rigid body spin or spin of the substructure. The elasticity relation is assumed to be ov r = C : De = C :
(D - D p)
(4)
where or is the Cauchy stress, C is the constant elastic moduli defined with the Young's modulus E and the Poisson's ratio v. The superposed V denotes the objective (co-rotational) rate of any second order tensor a with respect to to, i.e. ~7 a :
a -- ~l)a + a o 0 ;
w :
W - W p.
(5a, b)
In this investigation, for any constitutive relation for Dp, the plastic spin W p is assumed to arise from the noncoaxiality between the Cauchy stress tr and the plastic rate of deformation D p (Kuroda [1995a]). This can be represented as W p = r/(o'D p -- DPo")
(6)
where r/is a scalar valued function of the stress and structure variables. The objective rate defined in eqn (5a) is reduced to the classical Jaumann objective rate when W p = 0. 11.2 .4 Generalized viscoplastic model involving stress rate dependent noncoaxiality and kinematic tiardening Motivated by the deformation theory (St6ren & Rice [1975]), the following relation for DP is employed as a basis:
674
M. Kuroda
D p = iPp -k- ~
(7)
- oe
where ( )' denotes deviatoric quantities, 6- = o- - a, ~ = p : b ' = (3G/#e)(6"' : D - ~ekP), p = 36-'/(2~e), #e = V / ~ ( 6 - ' : 6-,)1/2, o~ is the back stress, G=E/{2(1 +u)}, and ~P is the effective plastic strain rate usually assumed as a function of the effective plastic strain e p : f ~ Pdt and the effective equivalent stress ~e. The modulus H will be specified later by utilizing a strain rate hardening function. In this model, the noncoaxiality between D p and 6- arises from the second term on the right-hand side of eqn (7). By using eqns (4) and (7), the inverted relation for the stress rate ~r can be expressed as =
I) -
(8)
C - --H+3GH C + -~ ~ - --}~ (I ~9 ,)
+ ~2e2to" ® 6-')
P = C : p = _ _ 3G6-, ere
(9) (I0)
where I is the unity tensor. The evolution law for the back stress o~ is taken to be given by a Prager-Ziegler's shift rule ~x' : bp~ p.
(1 I)
The plastic spin relation is derived on the basis of the generalized concept (eqn (6)); that is, the noncoaxiality between D p and o-, W p = v(~D p - DPo") =
( o ' ~ - o'o-) -.]--=-ere
) (o[,o- - o'o[) .
(12)
It should be noted that, in spite of the existence of the two skew tensor generators o-~"- ~ro- and ao- - o'o~, this relation for plastic spin has only one scalar valued function 7. The plastic spin defined in eqn (12) does not vanish even if we assume pure isotropic %r v hardening, because of the existence of the term o-o- - o'o-. Dafalias [1983, 1985a, 1985b] provided the plastic spin relation in the form W p = ~(¢xo"- o-~x) on the basis of the representation theorem (Wang [1970]) for skew-symmetric tensor-valued isotropic functions depending on two tensor variables. Flow theories with kinematic hardening is the case. This plastic spin concept is contained in eqn (12) as the particular case for H ~ c~. It is also noted that the definition of Wp in eqn (6) does not require any particular internal variable. The merits of the present concept were discussed with respect to a corner theory of plasticity (Kuroda [1995a]). Zbib and Aifantis [1988] and Zbib [1991,1993] presented a similar concept for the plastic spin on the basis of somewhat different insights, in the absence of kinematic hardening. Difference between the basic ideas by Zbib [1993] and by Kuroda [1995a] was discussed in Kuroda [1995a]. Considering eqns (3) and (5), the classical Jaumann stress rate 6-is related to the present objective stress rate ~r as follows:
Roles of plastic spin in shear banding
675
0".~ O" -- Wo" + o ' W = ~I -- WPff + o ' W p.
(13)
Using eqns (12) and (13), the constitutive relation (8) can be written finally in the form 6"= [ C + S + Q ]
:D-(P+R)~
p
(14)
with
(s)uk, = 37 ( Iimaj, - ~rimI:, ) ( Imoanp -- amoI, p )-Copkt Q =
3~__A®P, 25eH
A = ( t x t r - trtx)cr - t r ( t x t r - trot)
at/A. R = 25e
(15) (16a, b) (17)
The functions ~ p and H, the kinematic hardening coefficient b, and the plastic spin coefficient r~will be specialized below. The strain rate hardening response is assumed to be represented by the power law including a nonlocal term based on the gradient concept (Aifantis [1984,1992]): O'e ~ B;
\e0/
--Cs~72E p
(18)
and therefore eP = eo~,
(19a, b)
with e: = (1 - r)av + rgtrv.
(20)
Here cs is a material constant, V 2 denotes the Laplacian operator, ~0 a reference strain rate, m is a strain rate sensitivity parameter, ov is a reference strength, r is an isotropic hardening parameter, and g is the strain hardening function that is defined as g = (1 + eP/eo) n,
(21)
where n and e0 are material constants. It is clear that when Cs = 0 this gradient-dependent formulation corresponds to the conventional local formulation. The characteristic of strain softening is introduced by ~v --
ffYO 1 + (eP/el) 2
(22)
where cry0 is an initial "yield" strength, and el is a softening parameter. If r = 1, pure isotropic hardening is assumed, and pure kinematic hardening corresponds to r = 0.
676
M. Kuroda
The coefficient H in (7) is defined to be -O'e H -/3eP '
are =
V/~(O.,
: fit)l/2
(23a, b)
In the original deformation theory,/3 is taken to be unity. In that case, however, an elastic unloading-like behavior cannot be represented. (This has been discussed in Kuroda [1995a].) In the present model,/3 is determined by using • defined in eqn (19) as /3=
1 ~
for • > 1 for~
(24)
By introducing/3, the unloading-like behavior can be represented because/3 ~ 0 when O'e ~( K • If we set [3 = 0 (namely, H ~ ec), the present model is reduced to the viscoplastic version of J2 flow theory. The kinematic hardening coefficient b in (11) is determined as b -- (1 - r) 2(5e + CsV2ep) rd(gav) 3• [ deP
day] depJ'
(25)
according to the condition of coincidence of the uniaxial behavior for isotropic/kinematic combined hardening with the one for pure isotropic hardening at constant plastic strain rate, i.e. =
- & =
3<)
(26)
where ~r is the uniaxial stress for pure isotropic hardening given as O" ---- go'y
--Cs~72~ p.
(27)
Finally, the plastic spin coefficient r/in eqn (12) is assumed to be = a/Se
(28)
where a is taken as a constant (nondimensional) for simplicity. If a = 0, the objective stress rate ~r is reduced to the Jaumann stress rate 6". 11.3 Porous plasticity m o d e l The flow potential for Gurson's model with isotropic/kinematic hardening is expressed in the form (Mear & Hutchinson [1985]) ~)(or, o~, OfF,f) =
5e2
/'1 : ~ + 2ql f c o s h [ w - - - ] - 1 - (q, f ) 2 = 0 \ LO'F /
(29)
where f is the void volume fraction and ql is a material parameter introduced by Tvergaard [1981], and ~rF is the effective matrix flow strength associated with a kinematic
Roles of plasticspin in shear banding
677
hardening characteristic. In this porous plasticity model, o- represents the macroscopic Cauchy stress and tx the corresponding back stress. Although a function accounting for the effects of rapid void coalescence into the flow potential (eqn (29)) was introduced by Tvergaard and Needleman [1984], this function is omitted here. In this investigation, we assume that the increase in void volume fraction arises only from the growth of existing voids for simplicity. The matrix material is taken to be plastically incompressible. In the conventional formulation, the rate of the void volume fraction is given by J'= (1 - f ) I : D p.
(30)
We also consider here a nonlocal damage effect in terms of the integral condition on the rate of increase of the void volume fraction, which is assumed to be given by the following expression (Bazant & Pijaudier-Cabot [1988]; Leblond [1994]; Tvergaard & Needleman [1995]):
et al.
j'(x) = A(x)
[I -f(y)]l : DP(y)qo(x - y)dV(y)
(31)
with
A(x)=/v~(X-y)dV(y),
qo(x - y) = exp -
(32a, b)
where V is the volume under consideration and I is a material characteristic length. If we employ the gradient concept (Aifantis [1992]), the simplest form of the rate of the void volume fraction may be written, for example, a s j ' = (1 - f ) I : DP + cV2f. However, some preliminary calculations showed that the weight function-type expression in eqn (31) yields more appropriate results in the case of this porous model. Using the flow potential (eqn (29)) and the flow rule, the macroscopic plastic rate of deformation Dp takes the form
D p = h ~-~,Oq'
A - (1 --f)O'F~ pO~ ~-:--
(33)
0o" The matrix plastic strain rate k P is assumed to be
~P=ko(~rr~ '/m, ~G=(1--r)avo+rgayo
(34a, b)
\~GJ
where the ,;train hardening function g is the same as eqn (21). The evolution equation for aF is obtained from the consistency condition ~ = 0. Substituting eqn (33) into eqn (4) gives the following stress rate-strain rate relation tr=C:D-AC:0t
r
(35)
678
M. Kuroda
Following Mear and Hutchinson [1985], the evolution equation for the back stress is taken to be given by
o o(o
"dg
~x= ( l - r )
0ti)-~ -1
6- : ~-~j
_ :
2
A q av0(1 - g) 0~
(1 - f)CrF#~G
(36)
Based on eqn (6), the plastic spin associated with this porous theory is given by Wp
:
r/(tyOp _ Opo. ) = a_~ (txo" -- O'IX) = r/(txO p -- DPct).
(37)
o" F
where the scalar valued function rl is assumed to be r / = a / c r F. Consequently, this relation is equivalent to that provided by Dafalias [1983, 1985a, 1985b] and employed by Tvergaard and van der Giessen [1991]. Considering eqns (13) and (37), the constitutive relation (35) can be written in terms of the Jaumann rate ~- as b=C:D-A
OO 3r# \ C:~--~+~-~FA ) .
(38)
IlL NUMERICAL M E T H O D
An updated Lagrangian finite element method is used. The FE equation is derived on the basis of the rate-type principle of virtual work (Hill [1959]) with neglect of the body force effect
f
s : 6LdV=/p.
6vdS,
i.e. fj,6LodV=/p,6v,dS
(39)
where V and S are the current volume and surface, s is the nominal stress rate, p is the nominal rate of the force per unit current surface, v is the velocity, and 6v and 6L denote virtual velocity and virtual velocity gradient, respectively. The relation between the nominal stress rate and the Jaumann rate of the Cauchy stress ~ is given by = ~- + Wcr - o'W - Lo- + (I : L)o'•
(40)
The finite element equation is derived by substituting eqn (40) into eqn (39) and by assuming the distribution of the velocity v in a finite element. All the calculations performed in this paper employ the constant strain triangular element. The FE analysis may be performed with the constitutive equation (14) or (38). However, in such a case, a very small time increment is usually required due to numerical unstable problems• In order to perform stable numerical calculations, a powerful rate tangent modulus method (Peirce e t al. [1984]) is employed in the present
Roles of plastic spin in shear banding
679
numerical c,alculations shown below. The procedures for deriving the rate tangent modulus are essentially identical to those in Peirce et al. [1984], Becker and Needleman [1986], Tvergaard and Needleman [1995], and Kuroda [1995b]. Therefore, they are not repeated here. IV. BEHAVIOR IN SIMPLE SHEAR
Simple shear is one of the simplest homogeneous deformation modes. This problem is expressed by xl = X1 + 7X2,
x2 = )(2,
x3 = )(3
(41)
where -~ is the shear strain measure, x is the position vector of the particle in the deformed (current) configuration and X the one in the undeformed (initial) configuration. Generally, constitutive relations aimed at large strain analysis are required to provide a reasonable stress behavior in the simple shear deformation. In fact, validity of various constitutive relations has been actively discussed on the basis of stress responses to simple shear. However, the lack of experimental evidence has been pointed out by many workers. Reliable experimental data for simple shear is essential to evaluate the effectiveness of constitutive relations for large strain analysis. In this section, stress behavior in simple shear is examined by both numerical analyses and experiments. IV.1 Numerical analysis Fixed material constants used in the calculations are assumed to be E = 200 GPa, v=0.333, av0 = 400 MPa, n = 0.1, e0=0.002, E0 = 0.0002/s and m=0.002. The problem specified by (41) is solved by the finite element analysis with two triangular elements. And the applied strain rate is specified as "~ = 10x/-3~0. We consider here the following three types of models: • A stress rate dependent noncoaxial model with strain hardening (eqn (14) with eqns (23) and (24), and el ~ c~ in eqn (22); this model is henceforth referred to simply as the "noncoaxial model"). • A strain-softening model based on J2 flow theory (eqn (14) with H --* oo, and el = 1.5 in eqn (22); this model is referred to as the "softening model"). • A porous model (eqn (38) with ql = 1.5). For the noncoaxial model, both pure isotropic hardening and pure kinematic hardening are considered, since even if isotropic hardening is assumed the plastic spin still exists due to the stress rate dependent term trot-~rtr (as seen in eqn (12)). For the softening model and the porous model, pure kinematic hardening is considered, since the plastic spin arises only from the back stress. As the purpose of the present investigation is to study the effects of plastic spin, such extreme cases are chosen and examined. Though the principal motivation for using a kinematic hardening model is to represent deformation-induced anisotropy, the kinematic hardening model in large strain plasticity analysis has succeeded in reproducing axial effects in torsion or in shear deformation (Paulun & Pecherski [1992]; van der Giessen et al. [1992]). The
680
M. Kuroda a-3
2.0 1.5 (",1
1.0 0.5
-0
a-3
~'- -0. 5 I II
~ -1.0 a-1
-1.5 Fig. 1. Stress responses to simple shear predicted by the noncoaxial model with isotropic hardening. The stress components are normalized by avo/vr3. axial effects have been widely recognized as an i m p o r t a n t and interesting p h e n o m e n o n at large strain since Swift [1947]. Figure 1 shows the c o m p u t e d curves o f the nonzero stress c o m p o n e n t s versus shear strain 7 for the noncoaxial model with isotropic hardening (r = 1). The curves o f n o r m a l stress cr22 =-0.11 and shear stress o12 corresponding to the plastic spin p a r a m e t e r a = 0, 1 and 3 are illustrated. Strong decrease in 0.12 and significant development in 0.22 are observed when no plastic spin is assumed (a = 0). This is a c o m m o n predictive feature o f "vertex-type" constitutive models. The calculation was terminated when 0.12 reached zero, since the situation of 0.12 < 0 seems to be physically meaningless. In the case of a = 3, the
a-3
2.0
a~l
1.5 c'q
1.0
0.5 a-3
~-0.5 I
a~l
n -1.0 -1.5
a~0
Fig. 2. Stress responses to simple shear predicted by the noncoaxial model with kinematic hardening. The stress components are normalized by av0/v~.
Roles of plastic spin in shear banding
681
2.0 1.5 a-3 ~
1.0 0.5 NN~ "0 i
o
~3
i
~' i
I
'
1:1=-0.5 fl ~e~-l. 0 -1.5 Fig. 3. Stress responses to simpleshear predicted by the softeningmodel with kinematichardening. The stress components are normalizedby av0/v~. shear stress a12 increases monotonically. Figure 2 shows the results for kinematic hardening case, and almost the same effect of plastic spin is observed. Figure 3 ,.shows the results for the softening model with kinematic hardening (r = 0). The shear and normal stresses decline even for an increased amount of plastic spin, because the material is assumed to soften essentially. Figure 4 shows the results for the porous model with kinematic hardening. The initial void volume fraction J~ is assumed to be 0.01. Although a12 for a = 0 does not become negative, the oscillatory behavior is observed as well-known since Nagtegaal and de Jong [1982]. The voids do not grow from the initial state because no hydrostatic pressure exists in the simple shear deformation. Therefore, no progress in damage occurs during the simple shear process. Corresponding to this fact, a monotonic stress response is observed for the large amount of plastic spin, a = 3.
2.0
a-3
1.5
a-1
t"q
1.o
0.5
~=-0. 5 i II
~ -1.0
a-O
--
-1.5 Fig. 4. Stress responses to simple shear predicted by the porous model with kinematichardening. The stress components are normalizedby avo/v~.
682
M. Kuroda
IV.2 Experimental evidence The simple shear deformation can be reproduced approximately by a large strain torsion test of thin-walled tubes or solid circular bars with fixed ends. In particular, the deformation in the tube specimen, is almost identical to simple shear deformation. The close relationship between the behaviors in simple shear and in solid bar torsion was studied in detail by Neale and Shrivastava [1990] and by van der Giessen et al. [1992]. In the fixed end torsion tests, we observe usually the development of axial force (normal stress) in addition to the torque (shear stress). Torsion tests were carried out for tube specimens with fixed ends at room temperature and static loading condition. The testing machine was newly designed and assembled. Tube specimens of aluminum-magnesium alloy with outside diameter 8 mm, inside diameter 6 mm, and gage length 15 mm were prepared. They were machined from commercial bars (JIS A 5056B) of diameter 10 mm, and were not annealed. The sample surfaces were sufficiently polished. An example of the test results is shown in Fig. 5. In the figure, -~represents the shear strain at the mid-thickness of the wall of the tube specimen; this is defined by "~= 2NTrRo/16, where/~0 denotes the average value of outside and inside radii, N is the number of revolutions, and lc is the gage length. The reference strength ~rR is taken to be 118 MPa, which is the nominal yield strength for the commercial A1-Mg alloy. The torque T is macroscopically monotonic, and the axial force F shows a saturation. But, a buckling was observed clearly at ~ 1.4. The subsequent torque and axial force have been denoted by the broken lines. For comparison purposes, solid bar specimens of AI-Mg alloy and copper (the diameter 6 mm and the gage length 40 mm) were also tested. Some of them were annealed (3 h at 400°C and 3 h at 700°C for AI-Mg and Cu, respectively). Though the results for the solid bars are omitted here, it was confined that their torsion behavior was very similar to the results for the tube specimens. In the case of the solid bars, the monotonic torque and the saturation of axial force were observed up to failure at 7 -~ 5 (any buckling did not occur). Thus, we see no "softening" behavior during the torsion processes.
2.01.5-
~II~
1.0.5"
Not annealed
0
Fig. 5. Example of fixed-endtorsion test results: torque T and axial force F versus shear strain -~ for a thinwalled tube specimen of AI-Mg alloy. (--Ro = average value of outside and inside radii of the tube, to= initial thickness of the tube.)
Roles of plastic spin in shear banding
683
Although we do not carry out here any parameter-fitting study that corresponds to the above experimental results, the global stress behavior observed in the experiments has been captured semi-quantitatively by the noncoaxial model (Figs 1 and 2) and the porous model (Fig. 4) with the large amount of plastic spin. Anand and Kalidindi [1994] showed a torsion test result for a thin-walled tubular specimen of copper that was annealed at 800°C (1 h). This torsion test was a reverse test in which a direction of twist was reversed at -~= 1.5. The shear stress response up to -~= 1.5 in Anand and Kalidindi [1994] is very similar to the one in Fig. 5. The monotonic increase in shear stress in a fixed end torsion was also shown for a mild steel in the earlier work of Swift [194711. However, in the Swift report, the corresponding response in axial force was not shown. Montheillet et al. [1984] observed that, in solid bar torsion tests with aluminum and copper specimens at room temperature, the axial forces decreased notably after the attainment of a clear peak value and then they increased considerably again. But, the correspondiLng responses in torque at room temperature were not shown in their work. This behavior of axial forces seemed to be "oscillatory" and suggested a possibility of occurrence of stress oscillations in simple shear deformation. In fact, Zbib and Aifantis [1988] tried to simulate this type of oscillatory stress behavior by using a complex back stress model. However, in the present experiments, such "oscillatory" behavior of axial force has n~ot been observed. Meanwhile, at very high temperatures, a clear decline in torque was observed in the experiments by Montheillet et al. [1984], which may correspond to the results for the softening model shown in Fig. 3.
Fig. 6. Plane strain tensile specimen and its finite element discretization.
684
M. Kuroda V. EFFECT OF PLASTIC SPIN ON SHEAR BANDING
V. 1 Problem formulation The plane strain tension problem to be analyzed here is illustrated in Fig. 6. The initial configuration of the plane strain specimen is specified by the initial length 2L0 and the initial width 2 W0 = 2L0/3, where W0 = 1 mm, as shown in Fig. 6. Considering a symmetry of the deformation, only one-quarter of the specimen is analyzed. Thus, the boundary conditions for the one-quarter are vl = 0 and/)2 = 0 vz = 0 and p~ = 0
on X~ = 0
P , = p2 = 0
on X1 = Wo on X 2 = Lo.
on X2=O
Vl = 0 and v2 = C
The end displacement (elongation) U is obtained from U = f Udt, and the tensile force is l,V0 . . defined by P = f_woP2dXl on X2 = Lo; 1.e. in the actual analyses, P is obtained as the summation of the nodal forces in 2"2 -direction at X2 = L0. In the plane strain analyses, the unit out-of-plane thickness is considered. The finite element mesh for the re.gion analyzed is also shown in Fig. 6. The applied displacement rate tJ is specified so that (U/Lo)/~o = 1. We employ again the three constitutive models used in Section IV: that is, the noncoaxial model with isotropic or kinematic hardening; the softening model with kinematic hardening; and the porous model with kinematic hardening. All the material properties are the same as those used in Section IV. V.2 Results for local formulation In the computational results shown in this subsection, the nonlocality parameter cs in eqn (18) is set to be zero and eqn (30) is used for the porous model, so that we examine here effects of plastic spin in the conventional local formulations. Figure 7 shows overall tensile load-elongation curves for the noncoaxial model with isotropic hardening. Results are shown for the plastic spin coefficient a = 0, 1 and 3. No difference between the results is observed until U/Lo ~- 0.16. However, subsequent load 2.0-
i-5-
f
~l.Oa~ a-O
0.5-
00
'
O'. i
'
O'. 2
'
0'.3
Fig. 7. Curves of tensile load versus elongation for the noncoaxial model with isotropic hardening.
Roles of plastic spin in shear banding U / Lo
=0.19
U / Lo
a =(I
=0.19
685 U / Lo
a =1
=0.19
a =3
Fig. 8. Deformed meshes and contours of constant equivalent plastic strain eP for the noncoaxial model with isotropic hardening. The interval of contour lines is 0.1. drop behavior is very sensitive to the amount of plastic spin. For a = 1 and 3, there is no sharp drop in load, and the curves are smooth. In the case of the noncoaxial model with kinematic hardening, though the drop in load for a = 0 becomes more sharp, the tendency of the overall load-elongation behavior is almost identical to the one for the isotopic hardening case. Therefore, the results for kinematic hardening are omitted here. Figure 8 shows contours of constant equivalent plastic strain e P and deformed meshes that correspond to the load--elongation curves shown in Fig. 7. The results for kinematic hardening are almost identical to isotropic hardening, too. Hence, they are not shown here. In the: figure, the very sharp shear bands are formed in the case of a = 0. However, the shear bands are made to fade notably with increasing amount of plastic spin. For a = 3, the shear bands eventually disappear. This tendency is quite similar to that for the case of a corner theory of plasticity (Kuroda [1995a]). Tensile load-elongation curves for the softening model with kinematic hardening are shown in Fig. 9. The noticeable effect of plastic spin on load drop behavior is observed, too.
2.0-
1.5a=3 ~ 1.0a=l 0.5-
0
'
0.1 '
'
0'.
2
Fig. 9. Curves of tensile load versus elongation for the softening model with kinematic hardening.
686
M. Kuroda U/L o
=0.16
U/L o
a =0
=0.16
U/L o
a =1
=0.16
a =3
Fig. 10. Deformed meshes and contours of constant equivalent plastic strain eP for the softening model with kinematic hardening. The interval of contour lines is 0.1. In the case of a = 0 and 1, an "oscillatory" behavior emerges at the final stages of l o a d elongation curves. This can be attributed to the fact that the shearing mode in the shear band yields oscillatory behavior due to the small amount of plastic spin. Figure 10 shows contours of constant eP and deformed meshes that correspond to Fig. 9. The sharpness of the shear bands declines with increasing amount of plastic spin. However, even for the case of assuming the large plastic spin coefficient (a = 3), the shear bands are still visible clearly. Figure 11 shows tensile load-elongation curves for the porous model with kinematic hardening. It is noted that the overall load--elongation behavior is very insensitive to the amount of plastic spin in this case. This feature is very different from those for the other constitutive relations, though the sharp drop in load is delayed very slightly by the increase in a. Figure 12 shows deformed meshes and contours of constant macroscopic equivalent plastic strain that is gP J'(2D p : D p/3)1/2dt. Even though the large value of the plastic spin coefficient, a = 3, is assumed, we can observe the shear band formation that is nearly equivalent to the one observed for a = 0. :
2.0-
1.5-
~
S
t~l.O -
a =3 a=l
a=O
~0.5-
0
I
0
1
.1
I
U/ Lo
0I
.2
Fig. 11. Curves of tensile load versus elongation for the porous model with kinematic hardening.
687
Roles of plastic spin in shear banding U / Lo =0.17
a =0
U / L o -0.17
U / L o =0.17
a =1
a =3
Fig. 12. Defo~maedmeshes and contours of constant macroscopic equivalent plastic strain gP for the porous model with kinematic hardening. The interval of contour lines is 0.1. V.3. R e s u l t s f o r n o n l o c a l f o r m u l a t i o n Figure 13 shows c o n t o u r s o f constant EP and deformed meshes for the noncoaxial model with the nonlocal parameter Cs = 1 N. Results are shown for a = 0 and 3. In these calculations, isotropic hardening is assumed. F o r a = 0, the shear b a n d width is wider than the one for the local case (Fig. 8). The shear b a n d width spans over two rows o f elements in this case, while in the case o f local model the b a n d width is essentially determined by the size o f ,one element. We can n o w see a clear influence o f the nonlocality based on the gradient concept. F o r a = 3, no shear b a n d appears as in the case o f the local model (Fig. 8). The identical effects o f plastic spin are observed, whether nonlocality exists or not. Figure 14 shows the tensile load--elongation curves that correspond to Fig. 13. In the case o f a = 0, the d r o p in load is delayed due to the nonlocal effect, c o m p a r e d with the local case. O n the other hand, the curve corresponding to a = 3 is almost identical to the one for the local case (Fig. 7). The critical effect o f plastic spin also can be seen f r o m the figure. C o n t o u r s o f constant ¢P and deformed meshes for the softening model with cs = 2 N are depicted in Fig. 15. In this case, the shear b a n d width for a = 0 spans over at least four U / L o =0.21
a=O
U / L o =0.21
a=3
Fig. 13. Defi~rmed meshes and contours of constant equivalent plastic strain eP for the nonlocal noncoaxial model with isotropic hardening. The nonlocal parameter cs= 1 N. The interval of contour lines is 0.1.
688
M. Kuroda 2.0-
1.5-
a=3
~I:~1.0 a~O
0.5-
i
I
i
i
0.1
i
0.2
i
0.3
Fig. 14. Curves of tensile load versus elongation for the nonlocal noncoaxial model with isotropic hardening. The nonlocal parameter Cs= 1 N.
rows o f elements. The sharpness o f the shear bands fades when the large a m o u n t o f plastic spin, a = 3, is assumed. The fundamental tendency o f the effect o f plastic spin observed here is the same as the one observed in the case o f the local model (Fig. 10). Figure 16 shows the tensile l o a d - e l o n g a t i o n curves that correspond to Fig. 15. The difference between results for a = 0 and 3 becomes smaller in c o m p a r i s o n with the local case (Fig. 9). Figure 17 shows contours o f constant matrix plastic strain eP and meshes for the porous model with the characteristic length l = 0.075 mm. (Contours o f gP are very similar to this figure.) In this case, due to the nonlocal effect, the shear bands are not formed intensively, but they are visible in the contour maps. The load-elongation curves corresponding to Fig. 17 are depicted in Fig. 18. F r o m both figures, we can see a less significant effect o f plastic spin. Vl. DISCUSSION In the case o f the noncoaxial model without plastic spin (a = 0), a significant decrease in shear stress after the attainment o f its peak value is observed during simple shear deformation, U / Lo
=0.20
U / Lo
=0.20
a =0 a =3 Fig. 15. Deformed meshes and contours of constant equivalent plastic strain s p for the nonlocal softening model with kinematic hardening. The nonlocal parameter c~= 2 N. The interval of contour lines is 0.1.
Roles of plastic spin in shear banding
689
2.0-
1. S. ~ f ' - ' ~1.0a=0
0.50
i
0'.1
0
i
0'. 2
U/ Lo
Fig. 16. Curve,s of tensile load versus elongation for the nonlocal softening model with kinematic hardening. The nonlocal parameter cs= 2 N. even though isotropic hardening is assumed, as shown in Fig. 1. This is a common feature of constitutive relations in which the direction of plastic strain rate depends on that of stress rate such as the comer theories of plasticity (for example, Christoffersen & Hutchinson [1979]; Gotoh [198:5]). Kuroda [1995a] suggested a possibility of existence of oscillatory or risingsinking characteristics of stress response in simple shear on the basis of the predictive feature of-models with the directional dependency of the plastic strain rate on the stress rate. However, it is confirmed that, at least for metals, the strong decrease in shear stress during simple shear is completely inconsistent with the experimental evidence, as shown in Section IV. The decrease in shear stress can be eliminated by the plastic spin, based on the concept of the noncoaxiality between the', plastic strain rate and the stress. It is possible for the noncoaxial model with the plastic spin to produce a monotonic stress response to simple shear, which is consistent with the experimental evidence. On this point, it seems that effects of plastic spin are quite equivalent, whether in the noncoaxial model (a kind of vertex-type theory) or in the porous model (a kind of flow theory).
U / Lo
=0.175
a =0
U / Lo
-0.175
a =3
Fig. 17. Deformedmeshes and contours of constant matrix plastic strain e P for the nonlocal porous model with kinematic hardening. The characteristiclength 1= 0.075 mm. The intervalof contour lines is 0.1.
690
M. Kuroda
2.01.5-
a'3
f
~l.0¢.q
~0.5-
I
I
0.1
I
U/ Lo
I
0.2
Fig. 18. Curves of tensile load versus elongation for the nonlocal porous model with kinematic hardening. The characteristic length l= 0.075 mm.
However, on predictions of shear banding, the effects of plastic spin in the noncoaxial model and in the porous model are significantly different. In the case of the noncoaxial model, it is observed that the plastic spin which is increased large enough to yield a monotonic stress response to simple shear makes the shear bands vanish completely (Figs 8 and 13). This predictive feature of the plastic spin does not change, whether the nonlocality exists or not. On the contrary, in the case of the porous model, the shear band formation is only faded slightly, even when a sufficiently large amount of plastic spin is assumed (Figs 12 and 17). What is the cause of such a difference? It is known that the noncoaxiality between the plastic strain rate and the stress manifests itself in an apparent softening, in the presence of rotation of material. This is referred to as geometric softening (see, for example, Zbib [1993]). Not only does such geometric softening reduce a critical strain for shear localization but it also accelerates significantly the shear band development. The directional dependency of the strain rate on the stress rate in the noncoaxial model induces directly the geometric softening. The back stress also contributes to the noncoaxiality, and therefore, it induces and accelerates the geometric softening, too. In fact, it is known that the noncoaxiality due to the back stress evolution yields an effect that is similar to a vertex formation at yield surface, which is often identified as a rounded vertex effect (Tvergaard [1978]). The stiffness of materials reduced by such a geometric softening is recovered dramatically by the plastic spin, as can be seen clearly from the numerical results for the noncoaxial model. In fact, the large amount of plastic spin yields a monotonic stress response to simple shear, as shown in Figs 1 and 2, and also it restrains completely the shear band formation, as shown in Figs 8 and 13. When a large amount of plastic spin cancels the geometric softening effect, the noncoaxial model becomes a model without any softening effect. Here we should provide a word of caution 'noncoaxial models may not be able to predict any shear localization when one determines the value of the plastic spin coefficient on the basis of the "monotonicity of shear stress in simple shear" that is fairly consistent with the experimental results.' This caution is pertinent to the vertex or corner theories of plasticity being rate-independent, as could be seen from the numerical results in Kuroda [1995a]. We should recognize that this is a difficulty in
Roles of plastic spin in shear banding
691
phenomenological theories of plasticity with the directional dependency of plastic strain rate on the :stress rate. On the other hand, in the case of the porous model, the shear banding behavior seems to be goveFaed mainly by the macroscopic softening of material due to void growth. In fact, the numerical results shown in Figs 11 and 18 imply that the stiffness reduced by such a material softening is no longer recovered by the plastic spin. Consequently, the effect of plastic spin becomes less significant in the localization process, though the shear band formation i~ faded very slightly by the increased amount of plastic spin (Figs 12 and 17). This fading is attributed to the recovery of the geometric softening associated with the back stress evolution. Meanwhile, in the case of simple shear problem, no void grows because no hydrostatic pressure exists. Therefore, the stresses can increase monotonically if a large amount of plastic spin is assumed (Fig. 4). Consequently, the porous plasticity model with a relatively large amount of the plastic spin can reproduce both of the monotonic stress in simple shear and the shear localization in tensile problems. We see no inconsistency in these predictive features of the porous model with the plastic spin. This supports a soundness of the assumption that the macroscopic material softening effect due to pressure-sensitive damage, rather than the noncoaxial effect, plays a dominant role in the macroscopic shear banding at room temperature. The effect of the plastic spin on shear localization for porous material has also been investigated by Tvergaard and van der Giessen [1991]. However, their results and the present finite element result are not directly comparable. Tvergaard and van der Giessen's analysis assumed art infinite initial "band type" imperfection, while the present full numerical analysis requires no band type imperfection and the inhomogeneity due to the diffuse necking only triggers the localization. Tvergaard and van der Giessen [1991] observed that the onset of' localization was noticeably delayed by an increased amount of plastic spin. On the contrary, the present full numerical analysis without any initial imperfection has shown a very smallLeffect of plastic spin on shear banding in the diffuse necking region. The softening model is usually consistent with heating or high-temperature conditions without material damage. In this model, the geometric softening by the noncoaxiality due to the back stress evolution is recovered by the plastic spin, but the true material softening is no longer recovered. The numerical results (Figs 9, 10, 15 and 16) reflect these facts. VII. CONCLUDING COMMENTS
In this paper, the effects of plastic spin on shear banding have been studied systematically. We have observed critical or negligible effects of plastic spin, depending on the types of constitutive: modeling. The observations are summarized and interpreted as follows. The development of shear bands needs at least one softening effect. In the case of the stress rate dependent noncoaxial model, only the geometric softening is possible. The porous model contains the macroscopic material softening due to void growth and the geometric ,;oftening due to the back stress evolution. In the softening model, the true material softening and the geometric softening due to the back stress are active. The numerical results indicate that the geometric softening is cancelled by an increased amount of plastic spin. Therefore, if we assume a large amount of plastic spin on the basis of the experimental evidence for the simple shear deformation, the noncoaxial model becomes a model without any softening effect. Consequently, the noncoaxial model can no longer predict any shear band formation. This is a common difficulty in plasticity
692
M. Kuroda
models in which the direction o f the plastic strain rate is d e t e r m i n e d with that of the stress rate, a n d is subjected to further investigations. O n the other h a n d , the p o r o u s m o d e l with large a m o u n t of the plastic spin has succeeded in predicting b o t h the stress behavior in large simple shear a n d the shear b a n d i n g in tension problem. This is because the macroscopic softening due to the pressure-sensitive void d a m a g e still r e m a i n s as a d o m i n a n t factor o f the localization, even t h o u g h the plastic spin cancels completely the geometric softening arising from the back stress evolution. The softening m o d e l with large a m o u n t o f plastic spin m a y be used for predicting the material b e h a v i o r with n o c o n t r a d i c t i o n u n d e r h i g h - t e m p e r a t u r e conditions.
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