Pergamon
International Journal of Plasticity, Vol. 11, No. 1, pp. 4164, 1995 Copyright © 1995 Elsevier Science Ltd Printed in the USA. All rights reserved 07496419/95 $9.50 + .00
07496419(94)000387
AN ENERGYBASED LOCALIZATION THEORY: II. E F F E C T S O F T H E D I F F U S I O N , I N E R T I A , AND DISSIPATION NUMBERS
H. P. CHERUKURIand T. G. SHAWKI University of Illinois at UrbanaChampalgn
Abstract The basic framework for an energybased theory o f localization in dynamic viscoplasticity was recently developed by CIIERtrKUPaand SrtAWKI [1994]. In this framework, the total kinetic energy serves as a single parameter for the characterization of the full localization history. A characteristic evolution profile of the kinetic energy was shown to correspond to a localizing deformation. Here, the energybased characterization of localization is implemented toward the improved understanding of the mechanics o f shear band formation. In particular, the influence o f three primary dimensionless groups on localization is examined. These groups are referred to as the inertia number, the diffusion number, and the dissipation number. The limits of applicability o f the quasistatic assumption as well as the adiabatic deformation assumption are also addressed. Computational evidence indicates that the dissipation number plays a significant role in determining the material localization sensitivity.
1. INTRODUCTION
The failure of engineering materials during modern severe operating conditions has been a focal point for engineering research in the past decade. Among the phenomena that are intimately connected to catastrophic structural failure is the localization of plastic flow into narrow spatial bands (known as the shear bands). This observation explains the current interest in resolving various questions related to the onset and the late time behavior of shear localization in dynamic deformations. SHAWraand CLIFTON[1989] presented a review of the different mechanisms believed to explain shear band formation in a wide spectrum of materials and loading rates. SHAWKI[1992] presented a review of the various characterizations of shear localization in dynamic viscoplasticity. An energybased framework for the analysis of shear localization in dynamic viscoplasticity is recently developed by SrIAWKI[1994a, 1994b] for the prelocalization regime and later extended by CrmRtrKugi and SrIAWKI[1994] to encompass the complete localization history. This framework uses the system total kinetic energy as a single parameter that is capable of characterizing the full localization process. Table 1 presents a chronological summary of the development of the energybased localization theory. SItAWKI[1994a] illustrated, through a timedependent linear perturbation analysis, that a positive rate of change of the kinetic energy of absolute perturbations is a necessary and sufficient condition for the linear solutionincluding the effects of spatial perturb a t i o n s  t o deviate from the underlying homogeneous solution. This result represented the early indication to the possible role of kinetic energy toward the characterization of localization. In fact, in a subsequent article, SHAWra[1994b] used the energybased onset criterion to reproduce a large number of necessary localization conditions that were ear41
42
H.P. CHERUKURIand T. G. SHAWKI
Table 1. A chronological summary of developments in the energybased localization theory Development
Methodology
Authors
Year
Recognition of the positive kinetic rate as a necessary condition for the onset of localization
Timedependent, linear stability analysis
SHAWKI,T.G.
[1988, 1994a]
Derivation of analytic initiation conditions for large classes of materials based on the energy criterion
Timedependent, linear stability analysis
SHAWKI,T.G.
[1988, 1994b]
Early numerical observations of the connection between final failure and the evolution of the system kinetic energy A review of the various characterizations of shear localization in dynamic viscoplasticity Early numerical results confirming the role of the kinetic energy throughout the entire localization history
Threedimensional, nonlinear finite element solutions
ZBIB, H.M. &
[1992]
Linear and nonlinear analysis
SHAWKI,T.G.
[1992]
Onedimensional, nonlinear finite difference solutions
SHAWKI,T.G., SIIERIF,R.A. & CHERUKURI,H.P.
[1992]
A formal analysis of the energybased theory of localization including details of the used numerical algorithm
Onedimensional nonlinear analysis and finite difference solutions
CHERUKURI,H.P. SrtAWKI,T.G.
[1994]
Implementation of the energybased theory toward a consistent parametric study of the effects of the inertia, diffusion, and dissipation dimensionless numbers on localization
Onedimensional, nonlinear finite difference solutions of the governing equations for simple shearing motion
CHERUKURI, H.P. & SHAWKI,T.G.
JUBRAN, J,S.
This article
lier derived by other authors using drastically different viewpoints o f localization. The success o f this approach within the linear stability context motivated further pilot numerical studies by ZBIB and JUBRA_~ [1992] and by SHAWKI et al. [1992]. These pilot studies confirmed expectations that the system kinetic energy plays a significant role as far as characterizing the complete localization history. CrERUKURI and SHAWKI [1994], in part I o f this work, have presented the formal energybased f r a m e w o r k along with example calculations t h r o u g h which a n u m b e r o f critical times were defined. In the foregoing work, three dimensionless groups are identified and termed the inertia n u m b e r , the diffusion number, and the dissipation n u m ber. Further details regarding these groups are provided in the next section. Furthermore, CHERUKURI and SrIAWKI [1994] presented the details o f a finite difference scheme used for the numerical integration o f the system o f nonlinear equations governing the dynamic, onedimensional simple shear o f a thermallysensitive, viscoplastic material. A convergence analysis o f the finite difference algorithm was presented along with a discussion that illustrates the implementation o f the energybased characterization o f localization t o w a r d finding sufficient convergence conditions. A n example calculation was presented to aid the definitions o f various critical strains along the localization history. In this article, we take advantage o f the a f o r e m e n t i o n e d developments and c o n d u c t a consistent parametric study o f the effects o f the three dimensionless groups (the iner
Energybased localization theory, Part II
43
tia, diffusion, and dissipation numbers) on localization. Moreover, a numerical examination of the effect of the applied strain rate on localization is presented. The numerical method used throughout this work is that described by CImRUKtr~ and SrtawKi [1994]. I1. THE MODEL
Fig. 1 illustrates an infinite homogeneous plate of thickness H in the ~ direction while it extends indefinitely in the other two cartesian coordinate directions. The upper face of the plate is subjected to a constant velocity, while the lower face is fixed. The considered problem is onedimensional in the sense that all the quantities are only functions of space, ~?, and time, L The system of governing equations is given by (see SrL~W~I and CLIFTON [1989] for further details)
a# as. ~o a~  a~
(1)
as.
ai = ro ~ "~P = 6(8.,~,(9)
+ r, 8.~P,
or
8 = ~(~P,~,(~)
(3) (4)
where ,~o =
R 5o~'
fl 6
=
.
5o~
(5)
Eqn (1) expresses the balance of linear momentum, eqn (2) is the kinematic compatibility equation, eqn (3) represents the energy balance, and eqn (4) provides alternative representations of the material thermal viscoplastic response. Furthermore, 0 is the particle velocity in the ~ direction, b is the shear stress 8.ey, (~ is the absolute temperature, "~P is the plastic strain,/2 is the elastic shear modulus, P0 is a constant material density, /( is a ,constant heat conduction coefficient, 6 is the specific heat, and/~ E [0,1 ] is a nonnegative scalar, which expresses the amount of plastic dissipation that is converted to heat.
x=H
I
~=o
"~ V0
T
~:3
~//////////////////////////~////////////////////////////.~, Fig. 1. Schematic of a onedimensional simple shearing motion.
44
H.P. CHERUKURIand T. G. SHAWKI
It is convenient to express eqns (1) through (4) in a dimensionless form. Nondimensionalisation is conducted so that the form of the governing equations remains unchanged. The dimensionless quantities are defined by
V=Voo' K

/('0o 
boH~'
0o
=
#o~o2 ,

bo
c
/~
' ~0o ~,
#
=
0= o,
 ,
bo
(6)
~ao
r o  ~o6Hi7,° ,
rl
/~o~ 0 •
In eqn (6), quantities with the subscript "0" denote appropriately selected reference values (e.g. values that the dimensional field variables have in a homogeneous deformation at the time that a perturbation is introduced). The characteristic time i0  H/I;'o is the time required to obtain a unit shear strain at the nominal strain rate 4)0  Vo/H.
The foregoing dimensionless quantities are used in the remainder o f this work. At this point, it is useful to note that the dimensionless groups p0, ro, and r I are analogous to the dimensionless numbers commonly encountered in fluid mechanics, namely, the Reynolds number, inverse of the Peclet number, and the ratio of Eckert number and Reynolds number, respectively. In the current context, we rename the foregoing quantities to the inertia number for Po, diffusion number for r0, and dissipation number for rl. II. 1. Physical interpretation o f the dimensionless numbers The inertia number Po can be rewritten as the ratio of inertial effects to viscous effects; that is
Po~2 Po = Oo
PoVoH ao/~o
inertial effects viscous effects
(7)
Thus, larger values of the applied strain rate correspond to larger inertia numbers. In the following sections, we illustrate that larger inertia numbers result in a delay in localization initiation as long as the other dimensionless numbers remain constant. Small inertia numbers correspond to deformations in which viscous effects dominate inertial effects, which is analogous to the creeping flow o f fluid mechanics. In this case, inertial effects can be ignored in the analyses predicting the onset of severe localization (an approximation commonly referred to as the quasistatic approximation). The Kolsky bar testsin generalfall in the domain o f sufficiently small inertia numbers.* Thus, for these tests, the quasistatic assumption appears to be a reasonable approximation for both analytical a n d / o r numerical investigations o f the onset o f severe localization. The diffusion rgzmber is the inverse of Peclet number (which is the product o f the Reynolds and Prandtl numbers). Thus, it can be rewritten as *Further explanation of the implication of "sufficientlysmall" inertia numbers is presented in section IV.
Energybased localization theory, Part II
0o ro = o o2
45
(8)
We note that the Prandtl number is the ratio of some measures of viscous and diffusion effects. The larger (smaller) the Prandtl (diffusion) number, the weaker is the diffusion effect• Furthermore, smaller diffusion numbers are associated with a higher tendency toward adiabatic behavior. However, we shall illustrate that, regardless of how large diffusion number is, the assumption of an adiabatic deformation breaks down once severe localization takes place• The dissipation n u m b e r can be expressed in the form
t3 0 A
rl = t3ot?O°
~o000
(9)
Thus, the dissipation number is the ratio o f the thermal energy produced, per unit volume, during the time interval of length io over the energy required to raise the temperature of a unit volume by 00. Hence, for two different solids subjected to the same applied strain rate at the same test temperature, because the heat capacity is the same, the material with the higher yield stress has a higher dissipation number. For example, the hotrolled steel HRS1020 has a yield stress that is onethird that of HY100 steel. In this. case, the dissipation number for HY100 steel is 3 times that of the HRS steel. Although the energy required to raise the temperature by a finite amount is the same for both steels, the dissipation in HY100 steel is more than that in HRS steel, indicating that localization might occur faster in the former than in the latter. This observation is confirmed through our numerical results in the sections to follow. Here, we note that a similar observation has been experimentally confirmed by ROGERS [1979]. II.2. A u x i l i a r y c o n d i t i o n s The considered boundary conditions consist o f thermally insulated and velocitycontrolled boundaries. This is expressed mathematically as follows: v ( 0 , t ) = 0,
v(1,t) = 1,
q ( O , t ) = q ( 1 , t ) = 0;
q
0_< t < 0% 0O
O <<_t < oo.
(10) (11)
Ox '
The assumption o f insulated boundaries is expected to closely simulate the thermal state associated with the Kolsky bar type tests• A consequence of the insulated boundary conditions is that the posed boundaryvalue problem has a spatially homogeneous solution as opposed to the case o f isothermal boundary conditions for which a homogeneous solution does not exist• The initial conditions are chosen to correspond to a slightly perturbed homogeneous state. The perturbation is assumed to model geometric and/or material inhomogeneities. The set o f considered initial conditions is given by v(x,O) = x,
(12)
46
H . P . CHERUKURI and T. G. SnAWKX
qP(x,O) = 1,
(13)
3' (x,O) = 3"0,
(14)
O(x,O) = 1 + e f ( x ) ,
(15)
a(x,O) = ~b(1, 70, 1 + ¢f(x)),
(16)
for 0 _ x < 1. In eqn (15), f ( x ) is of order unity and represents the perturbation shape which is selected to be consistent with the boundary condition (11). Furthermore, a vanishing e corresponds to an initial homogeneous state. III. D I F F U S I O N N U M B E R A N D L O C A L I Z A T I O N
This section examines the effect o f the diffusion number, r0, on the evolution of localization. First, we review the localization criteria derived by SnAWKI [1994b] using the linear perturbation analysis while accounting for the timedependence of the homogeneous solution. III. 1. Localization criteria based on a linear perturbation theory We restrict the present discussion to the special class of materials modeled by the empirical powerlaw given by n
or = ( 0 ) ~'
(17)
(~/p)m,
where the exponents v, n, and m are real constants. For a strainindependent, power law material (i.e.; n = 0), the necessary condition for the onset o f localization is given by, see SHAWKI [1994b]
Iv + m~
r l ~     ~ ) +ro~2[1 + r l ( 1  u ) t ]
<0,
forv<
1,
(18)
where t _ T, while T denotes the time before which the linear solution represents an acceptable approximation to the nonlinear solution and ~k = k~', (k = 1,2,3 . . . . ). The foregoing inequality indicates that, in the absence of heat conduction (ro = 0), the necessary localization condition reduces to v + m < 0,
(19)
for strain rate hardening materials (i.e. m > 0). Comparison o f the inequalities (18) and (19) illustrates the stabilizing effect of the diffusion number, ro, as far as the onset of localization is concerned. Furthermore, for a given net softening effect (i.e. a given negative value of v + m), there exists a critical wavelength below which the initial disturbances will decay and no localization is anticipated. Now we note that if the diffusion number, ro, is sufficiently large, the necessary condition for localization (18) reduces to
1 +rl(1v)
t
v < 1,
(20)
Energybasedlocalizationtheory, Part II
47
which is never realized, implying no tendency for localization. This conclusion can be extended to the case of a strain dependent material response (n :# 0) because sufficiently large diffusion implies a nearly constant temperature as evidenced by examining the energy balance eqn (4). In such case, the powerlaw constitutive eqn (17) reduces to e =
p)m,
(21)
for which the necessary condition for localization obtained by SrlAWKI [1994b] is n+m<0,
m>0.
(22)
For strainrate hardening solids, localization is only possible for materials that exhibit sufficient strain softening such that the condition (22) is met. Computational experiments involving isothermal deformations of strain softening solids indicate that the strain rate may blow up in finite time leading to zero width band. This issue is discussed further in later sections. It is useful to note that "sufficiently large" diffusion numbers correspond to deformations in which diffusion effects are much stronger than viscous effects (while the inertia number is kept fixed). The numerical results shown in the following sections as well as the linear stability results by SHAWKI[1994b] indicate that the diffusion number plays a major role in determining the shear band thickness. This observation has been first noted by MERZER [1982] through his numerical solutions of the fully nonlinear system. II1.2. Shear band width and the diffusion n u m b e r This section discusses the role of the diffusion number toward setting a length scale for the evolving localized zones. It is important to note that, during high rates of loading of viscoplastic materials, the band "width" evolves continuously. Moreover, there is no clear boundary that separates the localizing zone from the remainder of the body. Hence, the evolution of smooth localizing solutions gives rise to a degree of uncertainty as far as the shear band width is concerned. As a result, a variety of definitions for the band width have been proposed. WRICIITand WALTER[1987] define the shear band width as the width of the region from the position of maximum strain rate (in the localized zone) to the point, on either side of this position, at which the strain rate drops by 10%. SrmRn~ and SrmwKi [1992], WRIGHT[1992], DODDand Bat [1985] considered steady state solutions of the simple shearing motion and obtained approximate expressions for the band width. GIOtAand ORTIZ [1992] considered the steadystate behavior of a twodimensional halfspace problem subjected to constant velocity at the boundary. They defined the band width as the width of the region near the surface bounded by the free surface and the surface at which the velocity is 0.99 times the boundary velocity. In the context of the energybased theory, CrmRUKtrm and SrlAWKI[1994] Part I of this paperhave presented a number of critical times during the late stages of evolution. Of particular interest is the socalled stabilization time, tstb, at which the plastic strain rate attains its maximum value within the localizing zone. CnERUKtmIand SltaWXI [1994] noted that the width of the evolving band decreases monotonically until tstb is reached. For times greater than tstb, the shear band width increases. We select the shear band width to correspond to the value it attains at the time t = t~tb. Further details on this se]lection are provided by CnERUKtmI and SrmwKi [1994].
48
H.P. CHERUKURIand T. G. SHAWKI
At this point, an attempt is made to rationalize the observed, late time strain rate drop for times greater than tsto. For this purpose, we derive the time rate of change of the flow stress, using the constitutive eqn (4b), to obtain the identity
O0"
at=
S102"~p
O0
O"/p
(23)
+s2 +s3 07'
where
Sl ~
30 05/p'
S2 
00 00'
S3 
aa O'r p"
(24)
Taking advantage o f eqn (3), the identity (23) takes the instructive form Sl Og[p
O~ 
CPqP
[
02006
roSz Ox z
~
]
(25)
where
Cp  S3 + rl aS2,
(26)
denotes the slope o f the adiabatic stressstrain curve at constant strain rate. Expression (25) indicates that the increase (decrease) of the plastic strain rate, for a strainrate hardening material, depends on whether the right side of eqn (25) is positive (negative). For thermally softening materials, the term $2 is always negative. Moreover, within the localizing zone, the second temperature gradient as well as the slope Cp are negative. Therefore, the right side o f eqn (25) consists o f a positive term (Cp'~ p) that is being offset by the negative term enclosed in square brackets (noting the stress drop associated with severe localization). Examination o f eqn (25), in view o f the foregoing remarks, implies that sufficiently large diffusion may have the capacity to retard the continued growth of the shear band strain rate. It is also evident that the "strength" of the diffusion term increases as the band width decreases. Hence, we anticipate that there exists a critical minimum band width at which the diffusion term in eqn (25) balances the first term leading to a maximum value o f the plastic strain rate. The attainment of such a maximum strain rate was illustrated by CHERUKURI and SnAWrd [1994] through late time numerical solutions of the fully nonlinear system o f eqns (1) through (4). The time at which diffusion effects begin to retard further localization was referred to as the stabilization time tstb. Finally, we note that at t = tstb, the strain rate attains the local maximum value (at the band center) given by
.p %nax =
Ov Ox
020 roS2 Ox 2
Cp+~
(27)
Eqn (27) was derived using eqns (25) and the kinematic compatibility eqn (2). Fig. 2 shows the spatial distribution o f 020/0x 2 for two different, late times while considering two different values o f the diffusion number ro. Figure 2(a) corresponds to the case study presented by CrmRtrgURI and S~WKI [1994] for a coldrolled steel
Energybased localization theory, Part II
10000
49
100
0
0
T
2~0 O 1o0o0 0x 2 20000 30000 o.o
0.2
0.4
0.6
o20 Ox 2
100 200 300
0.8
4OO
1.0
0.0
0.2
0.4
x
0.6
0.8
1.0
0.6
0.8
1.0
X
100
100 0
O20 Ox 2
100
02'~0 100 OX2
200
200
300
o.0
0.2
0.4
0.6
o.g
300
1.0
0.0
x
0.2
0.4 x
Fig. 2. A c o m p a r i s o n o f the effect o f the diffusion number o n the heat flux for t w o different late times [Case (a}: ro = 1.38 x 10  3 ( C R S  1 0 1 8 ) , h = 0.5 a n d tstb = 0 . 6 , C a s e ( b ) : ro = 1.0 x 10  2 , tl = 1.1 and tstb = 1.3].
(CRS1018). Figure 2(b) corresponds to the same case except that the diffusion number is larger by an order of magnitude, while all the other dimensionless numbers are kept constant. The plotting time tl is smaller than the critical time tcr (at which the kinetic energy evolution profile passes through an inflection point) and; therefore, t~ is less than tstb. It is evident that the smaller the diffusion number, the larger is that heat flux gradient at the center of the band. Therefore, it seems reasonable to conclude that heat conduction effects assume a rather significant role during the later stages of localization. Such effects must be included in the model even if the diffusion number is numerically small. It is interesting to observe that within the core of the localized zone (below the dashed lines in Fig. 2), the temperature rise is caused by plastic dissipation (since ~20/~X2 is negative), whereas outside the core the temperature rise is caused by heat conduction (since ~20/0x2 is positive). Finally, it is useful to note that the diffusion number introduces a thermal length scale, which is given by i thermal A thermal 
H

x/~o
(28)
Thus, if the shear band width is set by the above thermal length scale (whatever the definition of the band width may be) then the band width must be proportional to A thermal; i.e., w = Athermatf(.."
),
(29)
50
H . P . CHERUKORI and T. G, SHAWKI
Table 2. Material properties for a CRS1018 steel Symbol
Description
Value
/( t50
Thermal conductivity Mass density Specific heat Flow stress Elastic shear modulus
54 W/(m °K) 7800 kg/m 3 500 J/(kg °K) 436 MPa 81 GPa
60 /2
where f ( . . . ) is a dimensionless function that depends on the inertia number, the dissipation number, the shear modulus and the loading conditions at the stabilization time t~tb. It should be noted that eqn (29) is postulated in view of our numerical observations and is supported by the results of the steadystate shear band model derived by SrmRIF and SnAWKI [1992]. 111.3. Localization and the diffusion number The effect of the diffusion number on localization is explored through several numerical experiments corresponding to different values of r0 while keeping P0 and r~ fixed. The data for the cold rolled steel remain the same as that used by Crml~trKtrRI and SrIAWKI [1994] except for changing r0. The numerical values for the material parameters are given by Table 2. Furthermore, the specimen length is taken to be 2.5 ram. In Fig. 3, the stress at the moving boundary (x = 1) is plotted against the nominal strain for various values of ro. In all the cases,/~(t) reaches a maximum just when the stress begins to drop sharply, again confirming its validity as a single scalar quantity for characterizing severe localization. As expected, the smaller the diffusion number r0, the faster is the evolution of localization and the sharper the stress collapse. Comparison of the behavior of CRS steel with that corresponding to the smaller diffusion number r0 = 105 indicates that the assumption of an adiabatic deformation results in extremely conservative predictions of the critical localization strains (an error that can be as high as a 100%!). In Fig. 4, the evolution of displacement profiles is shown for various values of r0. In all of the considered cases, the maximum plotting time corresponds approximately to the time tstb at which the strain rate attains a maximum. Clearly, smaller diffusion numbers correspond to larger displacement gradients. Once severe localization occurs, the rigid motion of material surrounding the band is quite evident. Fig. 5 illustrates the dependence of various field quantities on the diffusion numberA In Fig. 5 (a), the critical times tcr and tstb are plotted against the diffusion number. The difference between tcr and t~tb, denoted by A trot, provides the duration of severe localization. Larger values of A hoe indicate longer durations for localization completion. This is due to the fact that large diffusion numbers allow the locally generated heat to be conducted away from the band at faster rates, thus reducing the severity of localization. Furthermore, for small diffusion numbers (near adiabatic deformations), there is virtually no distinction between the critical times tcr and tstb; i.e. Attoc ~ O. We also point ~The smallest tested value of the diffusion number is 105. Smaller values of the diffusion number result in great numerical difficulties and the loss of band resolution.
Energybased localization theory, Part II
51
1.1
1.0
o'(1, 0
0.9
0.8 r o = 105
[ ,I
0.7 0.0
0.~
1.5
1.0
t
8 6
ro =
4
K(t)
1°i' 
Oo
I CRS101S
/
2
1,013s x 103
0
r 0 = 8 X 10  3
2 0.0
0.5
1.0
1.5
t
Fig. 3. Effect of the diffusion number on the farfield stress profile at the moving boundary.
out that the diffusion numbers typically used in Kolsky bar experiments correspond to near adiabatic deformations. Fig. 5 (b) indicates that the temperatures 0or and Ostb decrease with increasing diffusion numbers. Moreover, for small diffusion numbers, even though Atto~ is very small, the corresponding differences in temperatures Ocr and Ostb are quite substantial, which
1.5
. . . . . . . . .
1.5
I
.
.
.
.
.
.
.
.
.
1.0
1.0
u(x, t)
u(x, t)
0.5
0.5
0.0
0.0 0.0
0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
x
X 0.5
0.8
r o = 1.38 x 10  3
0.4
0.6 0.3
u(x,t)
u(x, t)
0.4
0.2
0.2
0.1 0.0
0.0 0.0
0.2
0.4
0.6 X
0.8
1.0
0.0
0.2
0.4 X
Fig. 4. Displacement profiles for different values of the diffusion number.
52
H.P. CHERUKURIand T. G. SHAWKI
300
2.4 0(0.5, tstb)
2.2
200
2.0 1.8 0(0.5, tc,)
1.6
m
m
I00
1.4 1.2
(b)
1.0
0 0.002
0.0
0.004
0,006
0.008
0.002
0.0
0.01
0.004
ro
0.006
0.008
0.01
0.008
0.01
r0
1.4
(d)
(a)
1.2
I
tstb
yP(0.5, t stb) 6
1.0 ea
0.8
5
0.6
4
0.4 0.0
0.002
0.004
0.006
0.008
0.01
0.0
0.002
0.004
r0
0.006 ro
Fig. 5. The dependence of band field variables at the center of the band and the critical localization times on the diffusion number. (The smallest tested value of r0 is 105.)
is qualitatively consistent with reported experimental findings. Fig. 5 (c) and (d) illustrate the dependence of the maximum strain rate (given by eqn (27)) and the central plastic strain, respectively, on the diffusion number. As expected through the examination of eqn (27), the maximum strain rate decreases while the central plastic strain increases for increasing values of the diffusion number. I V. T H E
INERTIA
NUMBER
AND
LOCALIZATION
This section examines the effect of the inertia number P0, given by eqn (7), on localization while keeping both the diffusion and dissipation numbers fixed. Typical values of the inertia number, for Kolsky bar experiments, are of the order of 10 3. For this class of deformations, we illustrate that inertial effects may be ignored, which is consistent with observations made by WRIGHT and WALTER [1987]. However, for applications such as plateimpact tests or the high strain rate end of Kolsky bar tests, the quasistatic assumption can not be made. IV. 1. Linear perturbation analysis and inertia number The linear stability theory is a useful tool for the derivation of necessary conditions for the onset of localization. Mathematical difficulties associated with the implementation of such tools often lead to the assumption of quasistatic deformations for which analytical results may be obtained. The quasistatic assumption is argued to be reasonable as far as the onset of localization is concerned (see SHAWKI [1994b1). As a result, criteria for the onset of localization obtained through linear perturbation analyses do not provide information regarding the effects of the inertia number Po.
Energybased localization theory, Part II
53
IV.2. Numerical results and discussion A number of numerical experiments have been carried out by varying the inertia number po while keeping r0 and rl fixed. This is achieved by varying ~ and H appropriately. The data used is that corresponding to a CRS1018 steel except for the value of po. Fig. 6 shows the evolution of the stress at the moving boundary for various values of the inertia number along with the underlying behavior of the normalized kinetic energy rate. It is evident that rapid stress drop is associated with an extremum value attained b y / < ( t ) . Furthermore, we note that sharper stress drops are observed for larger inertia numbers. Moreover, profiles associated with inertia numbers less than O(10 2 ) seem to be indistinguishable, which indicates that the quasistatic assumption may provide satisfactory description of prelocalization. To further illustrate this issue, we present the evolution o f the band stress as well as the far field stress for two different values of the inertia number as shown in Fig. 7. For the large inertia number, p0 = 1, the initial stress evolution is nearly homogeneous, while a strong stress inhomogeneity develops as soon as localization begins to take place. Further, the band stress drops at a slower rate as opposed to the boundary stress. For the small inertia number, p0 = 105, the two curves are nearly identical, implying a near homogeneous stress distribution. In such case, the quasistatic assumption is expected to provide a satisfactory description of localization history. It is also clear that higher values o f the inertia number delay the onset o f severe localization. The evolution of various field variables is shown in Figs. 8 and 9 for small and large values of the inertia number po, respectively. Comparison o f the velocity profiles for the two cases indicates that, for large inertia numbers, three distinct spatial regions may be distinguished: (1) a central region where severe localization takes place, (2) a neighboring region in which the deformation becomes progressively more rigid, and (3) an
1.1 1.0 0.9 ~ o(1, t) 0.8
28~7510
0.7 0.6
, .,"~ ii Q0 = tl.1
0
1
t
2
1.5 2.86 x 104 1.0
105
K(t) e~ 0.5
Q0 = 0.1 00=1
~).5
0
1
t
2
Fig. 6. Evolution of the boundary stress for various values of the inertia number.
54
H . P . CHERUKURI and T. G. SFIAWKI
1.1
1.0
0.9
or(l, t) 0.8
Band & Boundary
13a~ ~ d ~ "x"
0.7
0.6 1
2
t
Fig. 7. Stress evolution at the band center and the moving boundary for two values o f the inertia number,
outer region where the deformation is nearly homogeneous. The second region expands into the third region as the deformation continues until it completely absorbs it. For small inertia numbers, on the other hand, the inhomogeneous deformation takes place simultaneously throughout the deforming body. This is also clear from the strain rate plots in the two figures. In the case of small p0, the strain rate increases in the perturbed region, while simultaneously decreasing outside this region. On the other hand, for large P0, there seems to be a wavelike behavior through which region 2 expands into region 3. Further, the larger the inertia number, the smaller is the maximum strain rate, and consequently the wider is the shear band.
19
3.0 P0 = 105
15
2.5
11 0 2.0
r ~
7 1.5 1.0 i 0.0
3 1
w
0.2
0.4
0,6
0.8
1.0
0.0
0.2
0.4
0.2
0.4
X
x
0,6
0.8
1.0
0.6
0.8
1.0
1.00 .... £)0 = 1 0  5
4
0.75
yP
V 0.50
3 2
0.25 0.00 0.0
1 i
0.2
0.4
0,6 x
0,8
1.0
0
0.0
x
Fig. 8. Field variables as functions of position at different times A t = 0.101 and max t = 0.606 (tstb).
E n e r g y  b a s e d l o c a l i z a t i o n theory, Part II
Oo = 1
0.0
0.2
O0 =
0.4
0.6
0.8
1.0
1
0.2
0.0
0.4
x
0.6
0.8
1.0
X
1.00
20
Y
Qo = 1 0.75
v
55
0.50
0.25 ,
~
O0 = 1 16 12
t
,
4
0.00 0.0
0.2
I
\, 0.4
0.6
0.8
0
1.0
0.0
0.2
0.4
x
0.6
0.8
1.0
x
Fig. 9. Field variables as functions o f position at different times with At = 0.425 a n d m a x t = 2.55
(tstb).
Fig. l0 illustrates the dependence of various field variables on the inertia number Po at the times tcr and ts,b. As Po increases, the critical localization time ter (critical localization nominal strain) increases. The difference between tcr and ts,b increases with increasing values of the inertia number. Typically, for Kolsky bar tests, the inertia number
140 120
4
.5,
3
tstb)
~ 8o
i2
S
0.0
0.2
0.4
0.6
0.8
1.0
40 0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
Qo
Q0 3
50 40
cr
YP("
~
2O 10
0.0
0.2
0.4
0.6 Q0
0.8
1.0
0 0.0
0.2
0.4
0.6 Q0
Fig. 10. V a r i a t i o n o f several variables w i t h the inertia n u m b e r .
56
H.P. CHERUKURIand T. G. SHAWKI
is of the order of 1 0  4 , which explains the observed rapid transition from localization to catastrophic failure in those experiments. The maximum strain rate at the center of the band decreases with increasing p0. The present numerical results indicate that for very small values of Po the maximum strain rate remains approximately constant indicating again that the quasistatic assumption is a valid approximation for modelling Kolsky bar experiments.
IV.3. Shear band width and the inertia number Fig. 10 indicates that the maximum strain rate at the center (at time tstb) decreases with increasing the inertia number. Because the overall strain rate remains constant and the elastic strain rate is much smaller than the plastic strain rate, this implies that the shear band width increases with increasing the inertia number. The present computations indicate that in the limit of a vanishing inertia number, the band width remains finite in the presence of heat conduction. V. APPLIED STRAIN RATE AND LOCALIZATION Examination of eqns (7) and (8) indicates that the inertia and diffusion numbers depend on the applied strain rate, albeit in different ways. For a given material and a given specimen geometry, p0 is proportional to the square of the applied strain rate, q~o, whereas r0 is inversely proportional to ~o. Hence, variations of 00 cause simultaneous variations in the inertia and diffusion numbers. In this section, the effect of varying the applied strain rate on localization is examined.
V. 1. Numerical results and discussion
It is well known that the yield stress of most metals is ratesensitive. The powerlaw given by eqn (21) needs to be modified in order to examine the effects of the applied strain rate. The modified version takes the form
(30) where qff is the dimensionless reference strain rate. Note that the corresponding dimensional strain rate ffff is different in general from the applied strain rate ~o, which is also used to nondimensionalize the governing equations. Here, we take ,~nu to be equal to 1600/s. It is important to emphasize that the powerlaw is an empirical formula that provides a satisfactory description of material response over a specific range of applied strain rates (typically the range of strain rates associated with the Kolsky bar test). Therefore; interpretation of the present results must be conducted with caution. Several numerical experiments have been carried out by changing the applied strain rate. The material data correspond to the CRS1018 steel. The evolution of the kinetic energy rate/~(t) and the boundary stress tr(1, t) are shown in Fig. 11. It is evident that larger values of the applied strain rate correspond to larger yield stresses. Fig. 12 indicates that as ~0 increases, the critical time for the onset of localization tcr decreases. The acceleration of localization caused by increasing applied rates
Energybased localization theory, Part II
1.1
104 4 x 104
1.0
. 6 x 10
o(1, t)
3 x 103
~
0.9
0.8
0.0
. . . . 0.2
0.4
4 x 1034[
2
X 103
0.6
y
3
K(t) Qo
57
0.8
2 x 104 4 x 104
3 x 1 I 0M ~ ,
l 0 1
2 x 103" 0.0
0.2
0.4
1 . 6 x 10! 0.6
0.8
t
Fig. l l . Evolution of the boundary stress and the kinetic energy rate for different applied strain rates.
continues until a critical strain rate is reached. Further increases in the applied rates lead to longer localization times. Hence, there exists a critical strain rate at which the material localizes the fastest. Fun:her examination of Fig. 11 indicates that, when localization takes place, the drop in stress is much sharper for higher strain rates than it is for lower strain rates. It is also
0.6:; A
B
0.60
Qo<~o)
0.55
tcr 0.51)
0:45
~0
I 0.40 0
104
2 × 104
3 × 104
4 x 104
Applied strain rate, ~o Fig. 12. Critical localization time as a function of the applied strain rate.
58
H . P . CHERUKURI and T. G. SHAWKI
useful to note that the initial yield stress is weakly sensitive to the applied strain rate since a 20fold increase in the applied strain rate leads to an increase in the initial yield stress by only 10°70. This behavior is specific to the powerlaw constitutive form and may not adequately describe the observed material response at ultrahigh loading rates. In an attempt to rationalize the behavior shown in Fig. 12, we note that the critical localization time tcr increases with increasing values of the inertia and diffusion numbers. However, we note that Po  ~o2 and ro  1/~bo. The insert in Fig. 12 illustrates the dependence of the normalized inertia and diffusion numbers on the applied strain rate for the considered material and geometry. It is evident that, in regime A, the fast decay in the diffusion number tends to accelerate localization beyond the stabilization effect introduced through the increase of the inertia number. In regime B, the faster increase in the inertia number offsets the effect of diffusion number decay and, therefore, results in the observed increase in the critical localization time. V.2. Shear band width and the applied strain rate The present numerical results show that the maximum band plastic strain rate "Ymax E ~P(0.5, tstb) increases with increasing the applied strain rate. For example, ~m~ is equal to approximately 128 when ~o = 1600/s and is equal to approximately 550 when q~o = 10000/s. Thus, as the applied strain rate increases, the band width decreases. This is indicative of the role that the diffusion number r0 plays in determining the band width. Furthermore, it is important to note that the reduction in band thickness due to the decay in the diffusion number overwhelms the increase in band thickness due to increasing values of the inertia number. Because the band width is inversely proportional to the applied strain rate, sufficiently refined meshes are used to obtain the numerical results for high applied strain rates (e.g. for an applied strain rate of 10000/s, the total number of grid divisions used is 1000). VI. DISSIPATION N U M B E R A N D L O C A L I Z A T I O N
This section examines the effect of the dissipation number rl = / ~ o / ~ J 0 on localization. For Kolsky bar experiments of structural steels at room temperature, the dissipation number has a value of approximately 0.25. On the other hand, for a Tungsten Heavy Alloy, it assumes a value of approximately 1.3. Larger values of the dissipation number lead to the generation of more thermal energy per unit time which tends to accelerate localization. Thus, if the inertia and diffusion numbers are small, localization is expected to take place faster in Tungsten Heavy Alloys than in steels. Guided by earlier observations, it appears that localization is weakly sensitive to sufficiently small values of both the inertia and diffusion numbers. This study illustrates that localization is strongly sensitive to variations of the dissipation number. VI. 1. Localization criteria f r o m linear perturbation analysis For a strainindependent, thermally sensitive power law type material, the necessary condition for localization is given by the inequality (18), which reduces to inequality (19) in the absence of heat conduction effects. The condition (19) implies that for adiabatic deformations, the necessary condition for localization is insensitive to the dissipation number r~. This observation illustrates that the onset of localization may be insensitive to dissipation, while it does not provide information as to the effect of dissipation on
Energybased localization theory, Part II
59
postlocalization. However, in the presence of heat conduction, as inequality (18) indicates, the effect of rl is to counteract the stabilizing behavior of heat conduction. It will be shown in this section that this is consistent with the fully nonlinear results. VI.2. Numerical results and discussion Several numerical experiments are carried out to study the effect of dissipation number on localization. The inertia number and the diffusion number are kept fixed at the values for the case studied in section IV for CRS steel. Fig. 13 illustrates the evolution of the stress at the moving boundary along with the evolution of the normalized kinetic energy rate for two different values of the dissipation number. It is evident that localization develops at a faster rate for the case corresponding to the larger dissipation number. Furthermore, for larger values of rl, thermal softerring overwhelms hardening mechanisms at smaller values of the nominal strains leading to a shorter hardening branch of the stressstrain curve. Moreover, the flow stress exhibits a sharper drop during localization for larger values of the dissipation number. Another interesting observation relates to the smaller critical localization times (for larger dissipation numbers) being associated with larger values of R(tcr). Fig. 14 shows the evolution of the boundary stress as well as the band stress for the two cases presented in Fig. 13. For each of the two cases, the two stresses are almost identical, which is consistent with our earlier observation regarding the validity of the quasistatic assumption for small inertia numbers. Hence, we conclude that the dissipation number has a negligible effect on the spatial distribution of the flow stress. Fig. 15 provides the displacement profiles for the two cases considered. Each computation is carried out up to its associated stabilization time tstb. Here, we note that as rl increases, the nominal strain and hence the maximum displacement at x = 1 decreases. Furthermore, it is evident that the band width increases with decreasing values
1.10 1.05 = ~ 2 2 3 9 5 1.00
a(1, t) 0.95 0.90 0.85 0.80 0.0
',
0.5
1.0
1.5
t 6
~
g(t~2) 2 Qo
r I = 0.7453
r I = 0.22395 2
0.0
0.5
1.0
1.5
t
Fig. 13. Evolution of the boundary stress for various values o f the dissipation number.
60
H.P.
CHERUKURI a n d T. G . SHAWKI
1.tO
1.05 1.00 r,t)
= 0.7453
r t = 0.22395
0.95
o
m 0.90
0.85
~x
= 05 andx =1 ]
0.80 0.0
0.5
1.0
1.5
F i g . 14. E v o l u t i o n o f the band and b o u n d a r y stresses for t w o different values o f the dissipation number.
of the dissipation number. This numerical observation is consistent with the linear stability prediction by SHAWKI [1994a] in which the critical wave length of perturbations is given by
f /cr =
]
ro
(31)
r l ( p  1)
for a powerlaw material with n = 0 and p =  v / m > 1. SHAWKI [1994a] showed that the foregoing wavelength threshold represents the minimum below which initial perturbations do not grow due to the early stabilizing effect of heat conduction. Moreover, SImRir and SHAWKI[1992] showed through a latetime steadystate solution that the minimum shear band width is proportional to ~/ro/rl. Hence, results obtained through the linear stability analysis (SHAWKI [1994b]), latetime analytic steadystate (SHERIFand SHAWKI [1992]), and the present numerical solutions confirm the aforementioned functional dependence of shear band thickness.
0.25
= .
1.00 .
=
~
0.20 0.75
u(x,t) 0.15
u(x, t) 0.50
0.10
0.25
0.05 0.00 0.0
0.2
0.4
0.6 X
0.8
1.0
0.00 0.0
0.2
0.4
0.6
0.8
X
F i g . 15. Displacement profiles for t w o different values o f the dissipation number.
1.0
Energybased localization theory, Part II
61
200
0(0.5,t~tb) i
2.4 "~ 150 1.9
100
0(0.5, tcr )
1.4
0.2
0.4
0.6
0.8
50
0.2
0.4
rI
0.6
0.8
0.6
0.8
rI
lO
1.5 1.2
8 ~tb)
0.9
6 0.6 4
0.3 0.0 0.2
2 0.4
rl
0.6
0.8
0.2
0.4
rI
Fig. 16. Variation of several field variables with the dissipation number.
Fig. 16 shows the variation of various field variables with respect to rl at the times
tcr, arid tstb. As rl increases, the critical time tcr for localization (equivalently, the critical nominal strain) decreases in a nearly exponential fashion. Furthermore, the difference between tcr and tstb decreases as r~ increases. Hence, localization takes place at a much faster rate for large values of the dissipation number. Further examination of Fig. 16 indicates that the critical temperature 0or  0(0.5, tc~) is weakly proportional to r~, while the m a x i m u m strain rate increases with r~. On the other hand, the critical strain "Ycr ~ " y ( 0 . 5 , tcr) is inversely proportional to r~.
VII. DISCUSSION We take advantage of the results in previous sections to estimate the localization sensitivit.v of various engineering materials. Therefore, we consider a number of materials that have been reported to localize under favorable experimental conditions. VII. 1. S o m e experimental results and nondimensional numbers Table 3 summarizes the thermomechanical properties of the materials considered for the present discussion. The critical nominal localization strains for the materials in Table 3 at an applied strain rate of approximately (1000/s) are presented in Table 4. For the times that the Kolsky bar tests were conducted on copper specimens, no localization was observed. We note that the H R S steel is the slowest to localize, the (HRC55) is the fastest, whereas the CRS steel and the HY100 localize at about the same nominal strain. Table 5 summarizes the dimensionless values o f the inertia number o0, the diffusion number ro, and the dissipation number r~ for the considered materials at
4340
62
H . P . CHERUKURIand T. G. SHAWKI
Table 3. Thermomechanical properties of selected engineering materials
Material
Density (kg/m 3)
Specific heat (J/kgK)
Thermal conductivity (W/mK)
Shear modulus (GPa)
CRS steel HRS steel HY100 4340 (HRC 44) 4340 (HRC 55) Copper WHA
7800 7800 7800 7800 7800 8960 17140
500 500 500 500 500 385 139
54 54 54 54 54 400 75
81 81 81 81 81 134
Initial yield stress (MPa) 436 261 530 800 (approx.) 1020 (approx.) 200* 900
*Extrapolated from the data given by LINDHOLMet al. [1980].
an applied strain rate of 1000/s., The specimen length is taken to be 0.0025 m for all materials. A comparison of Table 4 and Table 5 confirms the numerical observations of section VI in which smaller critical strains are associated with larger dissipation numbers. Furthermore, we note that the diffusion number is the same for all steels, while the inertia and dissipation numbers assume different values. For structural steels deforming at strain rates lower than 104 per s, the effect of P0 can be practically ignored. Hence, it appears that localization in structural steels is mostly influenced by the dissipation number. This observation explains the small localization strain reported for the HRC 55 as shown in Table 4. Furthermore, experimental observations indicate that the shear band width is the smallest for the HRC 55 steel. Again, this is consistent with the discussion in section VI. The tungsten heavy alloy (WHA) has the largest dissipation number in Table 5. However, its diffusion number is about twice that of the considered steels. Consequently, due to the stabilizing effects of heat conduction, the WHA localizes at nearly the same nominal strain as the 4340 (HRC 55) steel. Examination of the dimensionless numbers for Copper indicates that its inertia number is about five times that of HRS steel, while its diffusion number is about nine times larger than that of the considered *It is important to note that the dimensionless values compiled in Table 5 correspond to the applied strain rate of 1000/s. Furthermore, the values of the dimensionless numbers for the CRS steel which appear throughout the article may differ because of evaluations at different applied strain rates.
Table 4. Experimentally reported localization strains for a selection of engineering materials Material
Critical localization strain
CRS steel HRS steel HY100 4340 (HRC 44) 4340 (HRC 55) Copper WHA
0.5 1.0 (approx.) 0.45 0.2 0.15 no localization observed* 0.15 (approx.)
*For the times that the tests were made, no band has been observed by Dtlrrv [1992].
Energybased localization theory, Part II
63
Table 5. Typical values of the dimensionless numbers
for selectedengineeringmaterials Material
Inertia number
Diffusion number
Dissipation number
CRS steel HRS steel HY100 4340 (HRC 44) 4340 (HRC 55) Copper WHA
0.0001118 0.0001868 0.0000920 0.0000609 0.0000478 0.0002800 0.0001190
0.0022154 0.0022154 0.0022154 0.0022154 0.0022154 0.0185530 0.0050368
0.3354 0.2008 0.4077 0.6154 0.7846 0.1740 1.1333
steels. The dissipation number is also smaller than that of HRS steel. Hence, Copper is expected to localize at relatively large nominal strains (if any). VIII. CONCLUSIONS
A f:ramework for the analysis of flow localization based on the evolution of the kinetic energy has been proposed for the onedimensional problem o f simple shear with constant •velocity and adiabatic boundary conditions. The existence o f an inflection point in the evolution o f the kinetic energy is shown to be associated with the onset of severe localization. The usefulness o f this framework has been demonstrated through a variety of numerical parametric studies. Here, we summarize the main conclusions regarding the effects of the diffusion, inertia, and dissipation numbers on localization: • The assumption of an adiabatic deformation (ro = 0) provides excessively conserwative estimates of the critical localization strain. In fact, the estimated critical strain using adiabatic conditions can be as small as half o f the actual value. • The stabilizing effect of diffusion is significant during severe localization. • The shear band width is strongly sensitive to the diffusion number. The computed band width approaches vanishing values for adiabatic deformations. Hence, it appears that the thermal length scale introduced by heat conduction is primarily responsible for the observed shear band width for the considered class of deformations. • Effects of the inertia number become important for sufficiently high applied strain rates (or small specimen thickness). Typical values of the inertia number for Kolsky bar experiments on structural steels are of the order of 10 3  10 4. Hence, the assumption of quasistatic behavior appears to a reasonable approximation for these experiments. On the other hand, if Po is of the order of 10 2 or greater, inertia effects must be accounted for. • The critical time for the onset of localization, tcr, the duration o f localizationtofailure, Attoc, and the shear band width, w, exhibit the following qualitative dependencies:
Itcr, Ahoc,wl °~ Ipo,ro,11 •
(32)
Fimdly, we note the usefulness of the energy framework for the considered boundaryvalue problem suggests that an exploration o f its applicability to other deformations is worthy o f merit.
64
H . P . CHERUKURI and T. G. SHAWKI
AcknowledgementsThe support of the National Science Foundation to (TGS) through the Presidential Young Investigator Award # NSF MSS 8957180 PYI is gratefully acknowledged. Computer donations from the Hewlett Packard Company and the NeXT Computer Company to the University of Illinois have made the computations possible. These donations are appreciated. The authors would like to acknowledge the fruitful discussions with Professor Rodney Clifton, Professor Jack Duffy, and E. Andrews.
REFERENCES
1979 1980 1982 1985 1987 1988 1988 1989 1992 1992 1992 1992 1992 1992 1992 1994 1994a 1994b
ROGERS, H.C., "Adiabatic Plastic Deformation," Ann. Rev. Mat. Sci., 9, 283. LINDHOLM, O.S., NAGY, A., JOHNSON, G.R., and HOEGFELDT, J.M., "Large Strain, High Strain Rate Testing of Copper," J. Eng. Mat. Tech., 102, 376. MERZER, A.M., "Modelling of Adiabatic Shear Band Development From Small Imperfections," J. Mech. Phys. Solids, 30, 323. DODD, B., and BAI, Y.L., "Adiabatic Shear Band Width," Mat. Sci. Tech., 1, 38. WRIGHT, T.W., and WALTER, J.W., "On Stress Collapse in Adiabatic Shear Bands," J. Mech. Phys. Solids, 35, 701. MARCHAND, A., and Dur'rv, J., "An Experimental Study of the Formation Process of Adiabatic Shear Bands in a Structural Steel," J. Mech. Phys. Solids, 36, 251. SHAWKI, T.G., "Necessary and Sufficient Conditions for Localization in Thermal Viscoplastic Materials," TAM report No. 489, UIUC. SHAWKI, T.G., and CLIFTON, R.J., "Shear Band Formation in Thermal Viscoplastic Materials," Mech. of Materials, 8, No. 1, 13. DUFFY, J. personal communication. GIOLA, G., and ORTIZ, i . , work in progress. SHAWKI, T.G., "The Phenomenon of Shear Strain Localization in Dynamic Viscoplasticity," Appl. Mech. Rev., 45, No. 3, Part 2, 46. SHAWKI, T.G., SHERn~, R.A., and CHERUKURI,H.P., "Characterization of the Flow Localization History in Dynamic Viscoplasticity," Appl. Mech. Rev., 45, No. 3, Part 2, 149. SrlE~, R.A., and SiiAwra, T.G., "The Role of Heat Conduction During the PostLocalization Regime in Dynamic Viscoplasticity," in ZBIB, H.M., Plastic Flow and Creep, ASME, 135, 159. WRIGHT, T.W., "A Model for the Fully Formed Shear Bands," J. Mech. Phys. Solids, 40, 1217. ZBm, H.M., and JUBRAN, J.S., "Dynamic Shear Banding: A ThreeDimensional Analysis," Int. J. Plasticity, 8, 619. CHERUKOm, H.P., and SHAWKI, T.G., "An EnergyBased Localization T h e o r y  P a r t I: Basic Framework," Int. J. Plasticity, 10, xxx. SHAWKI,T.G., "An Energy Criterion for the Onset of Shear Localization in Thermal Viscoplastic MaterialsPart I: Necessary and Sufficient Initiation Conditions," J. Appl. Mech., 61, No. 3, 530. SHAWK1, T.G., "An Energy Criterion for the Onset of Shear Localization in Thermal Viscoplastic MaterialsPart II: Applications and Implications," J. Appl. Mech., 61, No. 3, 538.
Department of Theoretical and Applied Mechanics College of Engineering University of Illinois at UrbanaChampaign Urbana, IL 61801, USA
(Received in final revised form 11 March 1994)