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Temporal evolution of shear band thickness
J. Mrch. Phys. Solids. Vol. 45, No. 3, pp. 345- 359. 1997 ‘(;1997Elsevier Science Ltd Printed in Great Britain. All rights reserved
Pergamon
PI1 : SOO22-5096(%)00098-l
TEMPORAL
EVOLUTION
OF SHEAR
0022-5096/97 $17.00+0.00
BAND THICKNESS
J. A. DILELLIO and W. E. OLMSTEAD Department
of Engineering
Sciences and Applied Mathematics, IL 60208. U.S.A.
Northwestern
University,
Evanston,
(Received9 May 1996; in revised,form 10 September 1996)
ABSTRACT Boundary layer methods are used to derive expressions for the jump discontinuities in velocity, stress gradient and temperature gradient across a shear band. This analysis also yields an expression for the temporal evolution of the thermal boundary layer width, which can be interpreted as a measure of shear band thickness. The jump discontinuities are used to formulate a simplified version of the boundary value problem for the half-slab geometry. Calculated results for an illustrative example reveal a shear band thickness which evolves by initially narrowing and then later widening as diffusion becomes significant. (cj 1997 Elsevier Science Ltd. All rights reserved Keywords
: B. elastic-viscoplastic
material,
1.
C. asymptotic
analysis
INTRODUCTION
An analysis of the one-dimensional shear band model by Olmstead et al. (1994) was carried out under the assumption that the narrow zone of localization could be treated as a line of discontinuity, across which there are jumps in the velocity, stress gradient and temperature gradient. That analysis derived expressions for the jumps based upon the simplifying assumption that the plastic strain rate function could be scaled in inverse proportion to a constant thickness of the shear band. Our purpose in this investigation is twofold. First, we will refine the line of discontinuity analysis by deriving improved expressions for the jumps through consideration of the shear band as a boundary layer which has a small, but non-zero, thickness. While the results derived here are similar to those of Olmstead et al. (1994) there are significant differences. Second, we determine from the analysis an expression for the boundary layer width as it evolves in time. This evolving width is interpreted as a measure of the time dependent thickness of the shear band. This expression provides an explicit dependence of the shear band thickness on the stress and plastic strain rate at the center of the localized zone. Calculations for an illustrative example reveal that the thickness of the shear band varies in time by first narrowing as plastic deformation begins to occur and then later widening as diffusional effects become significant. This temporal evolution of the shear band thickness is qualitatively consistent with observations discussed by Bai (1990). 345
J. A. DrLELLIOand W. E. OLMSTEAD
346
Our analysis is based upon the standard one-dimensional formulation of the unidirectional shearing of a slab as found in Wright and Batra (1985) and Wright and Walter (1987). In dimensional form, the governing equations of momentum, elasticity and energy are given, respectively, by @Vi = Sp,
(1)
Si = ji(V$-I?),
(2)
~c,Ti=kT~~+ASF,
-L<$
i>O.
(3)
Here V($, t^) is the velocity, S(p, 0 is the shear stress and T(g, i) is the temperature in a slab of width 2L centered at p = 0. The material constants j, p, c, and k are the density, elastic shear modulus, specific heat and thermal conductivity, respectively. The Taylor-Quinney coefficient A provides the fraction of plastic work that is converted into thermal energy. For the plastic strain rate function i-, we use the form (4) which ignores hardening effects. This particular form has been previously considered by Walter (1992), Bayliss et al. (1994) and Olmstead et al. (1994). The boundary conditions at the ends of the slab are T(+L,i)
= T,
S,(kL,f)
=O,
V(fL,f)
= )I?
(5)
Here we have chosen an isothermal end condition; however, a no-flux condition could have been specified instead. The initial conditions are
Here the initial stress is taken to be constant, while the initial temperature and velocity satisfy T&g)
= T&j),
Fo(fL)
= 7,
voo(-g
= -v&j>,
vo(+L)
= +P.
(7)
It is consistent with the above formulation to impose the symmetry conditions T(p,i)
= T(-$,i),
S($,i)
= S(-p,i),
V(j,i)
= -V(-$,t^).
(8)
In the analysis to follow, it is assumed that the zone of shear localization is a boundary layer centered about the line of symmetry 9 = 0. To non-dimensionalize (l)-(7), we introduce the variables
(9)
347
Shear band evolution and
parameters f=
s*/pi‘*,
I=
(ks*~~c~~~*)‘~2,I/* = it-*,
A = AS*2/@,&
p = kI?*/c,S*,
a=
cirii,
p,= i_,(s*)l’“e”‘“/~*.
(10)
in these scalings we treat S*, Y* and I* as arbitrary, so that the analysis to follow retains some additional generality. In individual cases, such as the example in Section 5, specific values of these parameters must be assigned. The problem (l)-(B) then takes the form PV* = $7
O,=O,+Asj,
(11)
L
t > 0,
--;
(13) (14)
0
t =o I(+c t =o (*4I’) t (-+k1’) ’
s.
1')
’
v
=&V/F,
@(Y,O)= e,(y), KY,O) = 30, U(Y,O) = ho.
(15)
(16)
To analyze (ll)-(16) by boundary layer methods, it is necessary to identify an appropriate small parameter which can be used to scale the basic thickness of the boundary layer. We will use the strain rate sensitivity parameter m which is typically small for high-strength metals that exhibit shear localization effects. Thus, we assume that m=c,
O<.c<< 1,
(17)
where E will be the small parameter basis for the scalings and expansions of the boundary layer analysis. In Section 2, we will use bounda~ layer methods to derive expressions for the jump discontinuities in the velocity, stress gradient and temperature gradient across the shear band. In Section 3, the results of that analysis are shown to yield also an expression for the temporal evolution of the shear band thickness. In Section 4, the expressions for the jump discontinuities are used to formulate the boundary value problem for the half-slab geometry. In Section 5, we will solve a specific problem previously considered by Walter (1992), Bayliss et al. (1994) and Olmstead et al. (1994). It will be seen from the results of that solution how the shear band thickness narrows during the initial localization and then later widens as thermal diffusion becomes significant.
J. A. DrLELLIO and W. E. OLMSTEAD
348
2.
JUMP CONDITIONS
VIA BOUNDARY
LAYER ANALYSIS
For the boundary layer analysis, we consider (1 l)-(15) with the small parameter be used as the nominal thickness of the shear band. We will ignore the initial conditions (16) for this part of the analysis, since our goal is only to derive expressions for the jumps in 0,, s, and 2,across the shear band. It is consistent with the physical concept of a shear band to assume that the plastic strain rate k is negligible outside of the boundary layer, Thus, for the outer solution, we set i) = 0 in (12)-(13) and introduce the expansions
m = E to
d(y,t) = @D,(y,0+s@,(y,0+
..‘,
(18)
s(y,t) = Y,(y,t)+sY,(y,0+
.“>
(19)
v(y,0 = v,(y,0+aV,(y,0+
....
(20)
The leading order contributions
satisfy av, Px=ay’
ZPo (21)
(23)
The behavior of @, Y’, and V, at y = 0 will be determined by matching to the inner solution obtained from the boundary layer analysis. For the inner solution, valid in the boundary layer, we introduce the stretched variable (25) In terms of the stretched variable, we investigate the inner solution in terms of the expansions Q,t)
=cPo(r,t)+&(P1(5,t)+...,
(26)
S(Y, t) = $0 (5, t) + $,(5,~)
+ . . .?
(27)
v(y,t) = Q(r,t)+a~1(5,t)+
.‘..
(28)
Thus (1 l)-( 13) take the form a&J
(
p
at+“at+“’
au,
)+35+!3$+
.
..)
(29)
349
Shear band evolution
+A($,+@,
+ . ..)‘j.
(31)
We will assume that i) is 0(&-l), whereupon the leading order asymptotic balance in (29) and (31) yields
w. at
---E
0’
a2qo 0.
ag’=
(32)
The solutions of these equations, which satisfy the required symmetry about 5 = 0, are $0(&t) = Il/o(k
(Po(L t) = cpo(t).
(33)
To achieve the desired scaling of i), we assume in (14) that p. can be expressed as
Here /IOis O(l), while t,k;t:and cpt denote representative values of IC/“(t)and qo(t), respectively. These representative values are selected so that the function ho(t) =
Wo(O/~~l’~“exp {f[eo(O-,n)
(35)
can be treated as O(1). Although this will not be uniformly valid for all time, the scalings are selected to be appropriate when the plastic strain rate function nears its maximum value. While this may imply some discrepancy in the asymptotic balance for very early times, this error is regarded as insignificant to the overall evolution. This allows i), as given by (14), to be expressed asymptotically as li w
th0(O
exp
bcp,
(t,
01.
(36)
It then follows from (31) that the asymptotic balance at O(E-‘) yields a2qI ~ = -~/?o$o(t)ho(t)e”‘P~.
ap
(37)
The solution of (37) gives
cP,(Lt) =
U?I(O,t)-f;log{cosh[~]}, (38)
where d(t) =
[~a~~oIl/o(t)ho(t) eaql(o.‘)]p “2.
(39)
350
J. A. DrLELLIO and W. E. OLMSTEAD
We can now match the outer expansion of the temperature expansion (26) by the matching principle
(18) to the inner
Y=~~~~O[~O(Y,r)+E~*(Y,t)+ ...I = r~~~[(PO(I;,r)+E(P,(5,r)+..,l.
(40)
Using (33)-(38) in the inner solution, (40) yields
~o(+o,r)+E
+... [0,(*0,r)+~(*O,r)i’ 1 = c&(r)+&
‘pl(0,r)+~log2T~
+ .... 1
[
(41)
The implication of (41) is that (42) This indicates a jump discontinuity in the temperature gradient across the thin boundary layer representing the shear band. From (42) we infer that the jump can be expressed as
= ey(o+,t)-ey(o-,r)
=
-&.
By pursuing a similar line of analysis, we could derive expressions for the jumps in the velocity and the stress gradient. However, with (43) established, we can derive the other jumps in a straightforward manner from (1 l)-( 13). In an analogous fashion to the analysis of Olmstead et al. (1994), we can use the field equations (1 l)-( 13) to relate the various jumps. After eliminating i) between (12) and (13), integration across the boundary layer yields
Given the continuity of s(y, t) and B(y, t) at y = 0, it follows that
(e,) = -h(o,
t)
o+ u,dy = -ks(O, t)[o(O+, t)-a(O-,
t)] = -ks(O, t)(u).
s 0Thus, the jump in velocity across the boundary layer is given by (v) =
4
(45)
als(0, t)iqt) .
The jump in the stress gradient follows directly from (11) as (.Y~) = ~,(o+,r)-x,(0-,t)
= p&(O+,r)-u(O-,f)l
=
P-$zJ).
Shear band evolution
351
In view of (45), this yields
(5)=Z&o ,:)6(r)]~
(46)
The jump conditions (43), (45) and (46) are each expressed in terms of J(t), as defined by (39). A more useful expression for s(t) is obtained by evaluating (36) at r = 0 and using ~(0, t) - $o(t). We can then replace (39) by the asymptotically equivalent form (47) Utilizing (47) the jump conditions take the form
(49)
Thus, the jump discontinuities can be expressed in terms of the stress and temperature at y = 0 together with the plastic strain rate function (14). One of the important applications of the jump conditions is in the reformulation of the basic mathematical problem (l l)-(16) as a half-slab problem. This reformulation is presented in Section 4. The other important result of this boundary layer analysis is the relationship of s(t) to the temporal evolution of the shear band thickness. This relationship is derived in Section 3.
3.
EVOLUTION
OF THE SHEAR BAND THICKNESS
The results of the boundary layer analysis of Section 2 suggest a means for determining the temporal evolution of the thickness of the shear band. By examining the form of the solution near y = 0, we are able to identify a time dependent function that reflects the boundary layer width associated with the thermal effects. We propose that this boundary layer width can be interpreted as a good qualitative measurement of shear band thickness. The inner solution expansion for the temperature is given by (26). The results of (33) and (38) with 5 = Y/Eand E = m allow the inner solution to be expressed as (51)
352
J. A. DtLELLIO and W. E. OLMSTEAD
The form of (51) suggests that a boundary layer thickness can be associated with A(t) = m&t).
(52)
It then follows from (47) that
It is this function that is proposed as a time dependent measure of the shear band thickness in non-dimensional form. A formula analogous to (53) for a steady state band thickness was derived using asymptotic methods by Glimm et al. (1993). Related expressions for constant characteristic widths of shear bands have been found by Sherif and Shawki (1992) and Wright and Ockendon (1992). The explicit behavior of A(t), as given by (53), requires that the stress and temperature at y = 0 be known. Nevertheless, some qualitative behavior of A(t) can be inferred from this expression. It is typically observed that, as the shear band begins to form, there is a large increase in the plastic strain rate, while the stress decreases relatively gradually. From (53), this general behavior suggests that A(t) would be decreasing and hence the shear band width narrowing during this early stage of localization. In the later stage of shear band formation, the stress experiences a dramatic collapse, while the plastic strain rate either decreases or remains nearly fixed. From (53), this would suggest that A(t) would be increasing, thus implying a widening of the shear band during this later stage. This qualitative behavior of A(t) is consistent with both observation and numerical calculations. 4.
FORMULATION
OF THE HALF-SLAB
PROBLEM
We proceed in a manner analogous to Olmstead et al. (1994) to reformulate (1 l)(16) into an appropriate problem for the half-slab, 0 < y < L/Z. With the localization of the plastic deformation to a thin zone along the centerline y = 0, this justifies dropping $Jfrom the leading order outer problem (21)-(23). The influence of $Jis then accounted for in the outer problem by appropriate boundary conditions imposed at y = Of. These boundary conditions follow from the jump relationships which were derived through the matching of the inner and outer solutions presented in Section 2. The governing equations for the outer solution of the half-slab then follow from (2 I)-(23) as
UY, 0 = $y(Y, t), 0 <
Y <
$ t > 0.
(55)
Here f3- @, and s - ‘PO,while the velocity has been eliminated by combining (21) and (22). The velocity is easily recovered, once the stress and temperature are determined.
Shear band evolution
353
The boundary conditions on stress and temperature along y = O+ follow from the jump conditions (48) and (50) under the assumptions of symmetry (8). Thus “2{S(o, t)3[s(O, t), QO, t)]}‘!?,
(56)
(57) with k given by (14) as
MO, 4 WA 01 = V.MO,41 lirnexp
[
;o(o,t)
1 .
(58)
At the outer edge of the slab we require that
O(~,i)=O, sJ(~,i)=o, t>o.
(59)
The initial conditions are
O(Y,0) = e,(y), s(y, 0) = so,
(60)
0 d y < ; .
The formulation of the problem expressed in (54)-(60) constitutes a pair of linear partial differential equations (54)-(55) coupled through the nonlinear boundary conditions (56)-(57). An integral representation of (54)-(60) is given by Ljl sty,
f)
=
g(y,tlyo,O)~b(4’o)dyo--o
s0 l/2
dto,
s
(61)
L/l
QY,
0=
'3~3 4~0, O)~O(Y,)
dye
0
1:2
I
OS
+2 g
G(Y, tl0, fo){.$k ~ohW0, to), WX f0)1}“~ df,,
(62)
0
where g(y, tl0, to) and G(y, tI0, to) are the Green’s functions which satisfy grr-gyy = ~(Y-Yo)~(~-~o)~ g,@,tlYo,bd=O,
,,(~,,,,o,lo)=o,
0 < Y < $
t > to, t>t,,
(63)
(64)
354
J. A. DrLELLIO and W. E. OLMSTEAD
(65) and
G,-G,,=6(y-yo)6(t-t,),
t> to,
O
=o,
G,(O,ho, to) = 0,
G(y,tolyo,to)=O,
(66)
t>t,,
(67)
O<,<$
(68)
Various forms of the solutions to these Green’s function problems are found in Stakgold (1967). Once ~(0, t) and 0(0, t) are found, then (61)-(62) provide the desired solution of (54)-(60). Equations for the determination of ~(0, t) and 13(0,t) follow from setting y = 0 in (61)-(62). This yields
s L/l
s(0, t) =
do,
tbo, 0%(vo)
Go -50
0
I/2 -2
qo,t)
s
ik ()S
I 0
d 9@, tl0, to),,
3b@ 0
to), NO, toI1 1’2 dt s(O, to)
07
(69)
L/l
=
0
‘30, tlyo,
O)~O(YO) dye
(30, tl0, to>{@, tohG(o,
to), e(O, to)1)1’2 dto.
(70)
The determination of ~(0, t) and 0(0, t) is achieved through the simultaneous solution of (69)-(70). A simplified version of (69)-(70) is obtained for the special case in which the nondimensional scaling is such that L/I >> 1, so that (54)-(60) can be posed in the limiting case, 0 < y -C co. This limiting case is consistent with the physical reality that the shear band located at the center of the slab is so thin that it is effectively at an infinite distance from the outer edge. In this limiting case, (69)-(70) take the form
(71)
Shear band evolution
Am 112 r /(rqo, t) = 2 2a OS
355
t’)] - “‘(~(0, t’)y[s(O, t’), 0(0, t’)]) “2 dt’
s a0
+(7tt)-“2
exp(-52/4+%(5)
a.
(72)
0
An analogous version of (71)-(72) was derived by Olmstead et al. (1994) for a different set of jump conditions. The solution of these coupled equations is accomplished by the application of straightforward numerical methods. This is carried out for an illustrative example in Section 5.
5.
ILLUSTRATIVE
EXAMPLE
Here we consider a specific example to illustrate the results derived in Sections 3 and 4. For the numerical solution of (71)-(72), we select parameter values which are similar to those used by Wright and Batra (1985) and Walter (1992) for a moderately high strength steel. For a nominal strain rate of 1000 s-‘, we specify in SI units b = 7860,
k = 49.2,
c, = 473,
p = 8 x IO”,
A = 1,
S* = 8.93 x lo*,
F= 162,
Vy = 1.28x 10-3.
(73)
For the plastic strain rate function, we take F* = 11.2,
y. = 59.18,
m=0.025,
a=0.104.
(74)
Based upon these values, we determine from (10) that p=
1.30x10-‘,
I=
1.65~10-~,
I= 1.15~10-~,
(75)
which are used in the numerical computation of (71)-(72). Based on the determined value of 1, the limiting case of L/Z + co is clearly appropriate for the slab width of 3.47 mm used by Wright and Batra (1985) and Walter (1992). For the initial conditions we take Qo(y) = 0.1e-0~0’5Y2, So = 1,
o,(y) = gkanh
rq]“‘)“‘,
(76)
where ci = 89.6,
r/p
= 2703.
(77)
Here the form of ~~ is selected for purposes of compatibility with the boundary condition at the outer edge in the limiting case of L/l -+ co. This selection replicates the desired linear profile near the centerline, while still providing the proper edge value asy + co. To treat (71)-(72) numerically, we use Newton’s method to solve the nonlinear algebraic equation (71) for ~(0, t) at each time step. This information is then used in the numerical integration of (72) to obtain @O,t) for the next time step.
356
J. A. DILELLIO
and W. E. OLMSTEAD
30.0
20.0
z
0 z
10.0
0.10
0.20
0.30
0.40
t Fig.
1.Centerline
temperature.
The results of our computations for this example are displayed in Figs 1-4. The temporal evolution of the temperature and stress at the center of the shear band (y = 0) are given in Figs 1 and 2, respectively. Although the jump conditions derived here represent a significant theoretical improvement over those used by Olmstead et al. (1994), the resulting behavior of stress and temperature is quite similar. These results are also qualitatively in agreement with the full scale numerical studies carried out by Walter (1992) and Bayliss et al. (1994). The corresponding result for the plastic strain rate is shown in Fig. 3. This behavior is in qualitative agreement with Walter (1992). It should be noted that this qualitative agreement holds even though some of the parameter values in (74)-(75) significantly differ from the order unity size assumed for the asymptotic analysis of Section 3. From the computed behavior of $0, t) and QO, t), shown in Figs 1 and 2, we can then calculate the temporal evolution of the non-dimensional shear band thickness from (53). The result is shown in Fig. 4. We see that initially the band thickness narrows to a minimum width in accord with the dramatic rise of the plastic strain rate toward its maximum value. The band then widens as thermal diffusion becomes significant. From Fig. 4, it is found that min A(t) z 5 x 10P3. Converting this back to a dimensional scale based on the specimen size treated numerically by Walter (1992) and Bayliss et al. (1994), this suggests a minimum band thickness of about 0.6 microns. This can be compared with a value of about 0.3 microns computed from a formula deduced from ad hoc arguments by Bayliss et al. while these values are smaller
357
Shear band evolution
1.4
1.2
0.4
0.2
0.0
0.00
0.30
0.20
0.10
0.40
t Fig. 2. Centerline
stress.
2.0e+06
Itie+
50e+05
\\ o.oe+oa
C
D
-J 0.10
0.20 t
Fig. 3. Centerline
plastic strain rate.
0.30
0.40
J. A. DrLELLIO and W. E. OLMSTEAD
358
0.00
0.20 t Fig. 4. Shear band thickness.
than found in experiments, it must be remembered neglected in the model analyzed here.
0.30
that hardening
effects have been
ACKNOWLEDGEMENT This research was supported by NSF Grant DMS-9401016 and AR0 Grant DAAHO4-96l-0065.
REFERENCES Bayliss, A., Belytschko, T., Kulkarni, M. and Lott-Crumpler, D. A. (1994) On the dynamics and the role of imperfections for localization in thermo-viscoplastic materials. Modelling Sin&. Mater. Sci. Engng 2,941-964. Bai, Y. (1990) Adiabatic shear banding. Res. Mech. 31, 133-203. Glimm, J. G., Plohr, B. J. and Sharp, D. H. (1993) A conservative formulation for largedeformation plasticity. Appl. Mech. Rev. 46, 519-526. Olmstead, W. E., Nemat-Nasser, J. and Ni, L. (1994) Shear bands as surfaces of discontinuity. J. Mech. Phys. Solids 42,697-709.
Sherif, R. A. and Shawki, T. G. (1992) The role of heat conduction during the post-localization regime in dynamic viscoplasticity. Plastic Flow and Creep, AMD Vol. 135/MD-Vol. 3 1, pp. 159-l 73. ASME. Stakgold, I. (1967) Boundary Value Problems of Mathematical Physics. Macmillian, New York.
Shear band evolution
359
Walter, J. W. (1992) Numerical experiments on adiabatic shear band formation in one dimension. ht. J. Plasticity 8, 657-693. Wright, T. W. and Batra, R. C. (1985) The initiation and growth of adiabatic shear bands. Znt. J. Plasticity
1, 205-212.
Wright, T. W. and Walter, J. W. (1987) On stress collapse in adiabatic shear bands. J. Mech. Phys. Solidr 354701-720.
Wright, T. W. and Ockendon, H. (1992) A model for fully formed shear bands. J. Mech. Phys. Solids 40, 1217-1226.
Pergamon
PI1 : SOO22-5096(%)00098-l
TEMPORAL
EVOLUTION
OF SHEAR
0022-5096/97 $17.00+0.00
BAND THICKNESS
J. A. DILELLIO and W. E. OLMSTEAD Department
of Engineering
Sciences and Applied Mathematics, IL 60208. U.S.A.
Northwestern
University,
Evanston,
(Received9 May 1996; in revised,form 10 September 1996)
ABSTRACT Boundary layer methods are used to derive expressions for the jump discontinuities in velocity, stress gradient and temperature gradient across a shear band. This analysis also yields an expression for the temporal evolution of the thermal boundary layer width, which can be interpreted as a measure of shear band thickness. The jump discontinuities are used to formulate a simplified version of the boundary value problem for the half-slab geometry. Calculated results for an illustrative example reveal a shear band thickness which evolves by initially narrowing and then later widening as diffusion becomes significant. (cj 1997 Elsevier Science Ltd. All rights reserved Keywords
: B. elastic-viscoplastic
material,
1.
C. asymptotic
analysis
INTRODUCTION
An analysis of the one-dimensional shear band model by Olmstead et al. (1994) was carried out under the assumption that the narrow zone of localization could be treated as a line of discontinuity, across which there are jumps in the velocity, stress gradient and temperature gradient. That analysis derived expressions for the jumps based upon the simplifying assumption that the plastic strain rate function could be scaled in inverse proportion to a constant thickness of the shear band. Our purpose in this investigation is twofold. First, we will refine the line of discontinuity analysis by deriving improved expressions for the jumps through consideration of the shear band as a boundary layer which has a small, but non-zero, thickness. While the results derived here are similar to those of Olmstead et al. (1994) there are significant differences. Second, we determine from the analysis an expression for the boundary layer width as it evolves in time. This evolving width is interpreted as a measure of the time dependent thickness of the shear band. This expression provides an explicit dependence of the shear band thickness on the stress and plastic strain rate at the center of the localized zone. Calculations for an illustrative example reveal that the thickness of the shear band varies in time by first narrowing as plastic deformation begins to occur and then later widening as diffusional effects become significant. This temporal evolution of the shear band thickness is qualitatively consistent with observations discussed by Bai (1990). 345
J. A. DrLELLIOand W. E. OLMSTEAD
346
Our analysis is based upon the standard one-dimensional formulation of the unidirectional shearing of a slab as found in Wright and Batra (1985) and Wright and Walter (1987). In dimensional form, the governing equations of momentum, elasticity and energy are given, respectively, by @Vi = Sp,
(1)
Si = ji(V$-I?),
(2)
~c,Ti=kT~~+ASF,
-L<$
i>O.
(3)
Here V($, t^) is the velocity, S(p, 0 is the shear stress and T(g, i) is the temperature in a slab of width 2L centered at p = 0. The material constants j, p, c, and k are the density, elastic shear modulus, specific heat and thermal conductivity, respectively. The Taylor-Quinney coefficient A provides the fraction of plastic work that is converted into thermal energy. For the plastic strain rate function i-, we use the form (4) which ignores hardening effects. This particular form has been previously considered by Walter (1992), Bayliss et al. (1994) and Olmstead et al. (1994). The boundary conditions at the ends of the slab are T(+L,i)
= T,
S,(kL,f)
=O,
V(fL,f)
= )I?
(5)
Here we have chosen an isothermal end condition; however, a no-flux condition could have been specified instead. The initial conditions are
Here the initial stress is taken to be constant, while the initial temperature and velocity satisfy T&g)
= T&j),
Fo(fL)
= 7,
voo(-g
= -v&j>,
vo(+L)
= +P.
(7)
It is consistent with the above formulation to impose the symmetry conditions T(p,i)
= T(-$,i),
S($,i)
= S(-p,i),
V(j,i)
= -V(-$,t^).
(8)
In the analysis to follow, it is assumed that the zone of shear localization is a boundary layer centered about the line of symmetry 9 = 0. To non-dimensionalize (l)-(7), we introduce the variables
(9)
347
Shear band evolution and
parameters f=
s*/pi‘*,
I=
(ks*~~c~~~*)‘~2,I/* = it-*,
A = AS*2/@,&
p = kI?*/c,S*,
a=
cirii,
p,= i_,(s*)l’“e”‘“/~*.
(10)
in these scalings we treat S*, Y* and I* as arbitrary, so that the analysis to follow retains some additional generality. In individual cases, such as the example in Section 5, specific values of these parameters must be assigned. The problem (l)-(B) then takes the form PV* = $7
O,=O,+Asj,
(11)
L
t > 0,
--;
(13) (14)
0
t =o I(+c t =o (*4I’) t (-+k1’) ’
s.
1')
’
v
=&V/F,
@(Y,O)= e,(y), KY,O) = 30, U(Y,O) = ho.
(15)
(16)
To analyze (ll)-(16) by boundary layer methods, it is necessary to identify an appropriate small parameter which can be used to scale the basic thickness of the boundary layer. We will use the strain rate sensitivity parameter m which is typically small for high-strength metals that exhibit shear localization effects. Thus, we assume that m=c,
O<.c<< 1,
(17)
where E will be the small parameter basis for the scalings and expansions of the boundary layer analysis. In Section 2, we will use bounda~ layer methods to derive expressions for the jump discontinuities in the velocity, stress gradient and temperature gradient across the shear band. In Section 3, the results of that analysis are shown to yield also an expression for the temporal evolution of the shear band thickness. In Section 4, the expressions for the jump discontinuities are used to formulate the boundary value problem for the half-slab geometry. In Section 5, we will solve a specific problem previously considered by Walter (1992), Bayliss et al. (1994) and Olmstead et al. (1994). It will be seen from the results of that solution how the shear band thickness narrows during the initial localization and then later widens as thermal diffusion becomes significant.
J. A. DrLELLIO and W. E. OLMSTEAD
348
2.
JUMP CONDITIONS
VIA BOUNDARY
LAYER ANALYSIS
For the boundary layer analysis, we consider (1 l)-(15) with the small parameter be used as the nominal thickness of the shear band. We will ignore the initial conditions (16) for this part of the analysis, since our goal is only to derive expressions for the jumps in 0,, s, and 2,across the shear band. It is consistent with the physical concept of a shear band to assume that the plastic strain rate k is negligible outside of the boundary layer, Thus, for the outer solution, we set i) = 0 in (12)-(13) and introduce the expansions
m = E to
d(y,t) = @D,(y,0+s@,(y,0+
..‘,
(18)
s(y,t) = Y,(y,t)+sY,(y,0+
.“>
(19)
v(y,0 = v,(y,0+aV,(y,0+
....
(20)
The leading order contributions
satisfy av, Px=ay’
ZPo (21)
(23)
The behavior of @, Y’, and V, at y = 0 will be determined by matching to the inner solution obtained from the boundary layer analysis. For the inner solution, valid in the boundary layer, we introduce the stretched variable (25) In terms of the stretched variable, we investigate the inner solution in terms of the expansions Q,t)
=cPo(r,t)+&(P1(5,t)+...,
(26)
S(Y, t) = $0 (5, t) + $,(5,~)
+ . . .?
(27)
v(y,t) = Q(r,t)+a~1(5,t)+
.‘..
(28)
Thus (1 l)-( 13) take the form a&J
(
p
at+“at+“’
au,
)+35+!3$+
.
..)
(29)
349
Shear band evolution
+A($,+@,
+ . ..)‘j.
(31)
We will assume that i) is 0(&-l), whereupon the leading order asymptotic balance in (29) and (31) yields
w. at
---E
0’
a2qo 0.
ag’=
(32)
The solutions of these equations, which satisfy the required symmetry about 5 = 0, are $0(&t) = Il/o(k
(Po(L t) = cpo(t).
(33)
To achieve the desired scaling of i), we assume in (14) that p. can be expressed as
Here /IOis O(l), while t,k;t:and cpt denote representative values of IC/“(t)and qo(t), respectively. These representative values are selected so that the function ho(t) =
Wo(O/~~l’~“exp {f[eo(O-,n)
(35)
can be treated as O(1). Although this will not be uniformly valid for all time, the scalings are selected to be appropriate when the plastic strain rate function nears its maximum value. While this may imply some discrepancy in the asymptotic balance for very early times, this error is regarded as insignificant to the overall evolution. This allows i), as given by (14), to be expressed asymptotically as li w
th0(O
exp
bcp,
(t,
01.
(36)
It then follows from (31) that the asymptotic balance at O(E-‘) yields a2qI ~ = -~/?o$o(t)ho(t)e”‘P~.
ap
(37)
The solution of (37) gives
cP,(Lt) =
U?I(O,t)-f;log{cosh[~]}, (38)
where d(t) =
[~a~~oIl/o(t)ho(t) eaql(o.‘)]p “2.
(39)
350
J. A. DrLELLIO and W. E. OLMSTEAD
We can now match the outer expansion of the temperature expansion (26) by the matching principle
(18) to the inner
Y=~~~~O[~O(Y,r)+E~*(Y,t)+ ...I = r~~~[(PO(I;,r)+E(P,(5,r)+..,l.
(40)
Using (33)-(38) in the inner solution, (40) yields
~o(+o,r)+E
+... [0,(*0,r)+~(*O,r)i’ 1 = c&(r)+&
‘pl(0,r)+~log2T~
+ .... 1
[
(41)
The implication of (41) is that (42) This indicates a jump discontinuity in the temperature gradient across the thin boundary layer representing the shear band. From (42) we infer that the jump can be expressed as
=
-&.
By pursuing a similar line of analysis, we could derive expressions for the jumps in the velocity and the stress gradient. However, with (43) established, we can derive the other jumps in a straightforward manner from (1 l)-( 13). In an analogous fashion to the analysis of Olmstead et al. (1994), we can use the field equations (1 l)-( 13) to relate the various jumps. After eliminating i) between (12) and (13), integration across the boundary layer yields
Given the continuity of s(y, t) and B(y, t) at y = 0, it follows that
(e,) = -h(o,
t)
o+ u,dy = -ks(O, t)[o(O+, t)-a(O-,
t)] = -ks(O, t)(u).
s 0Thus, the jump in velocity across the boundary layer is given by (v) =
4
(45)
als(0, t)iqt) .
The jump in the stress gradient follows directly from (11) as (.Y~) = ~,(o+,r)-x,(0-,t)
= p&(O+,r)-u(O-,f)l
=
P-$zJ).
Shear band evolution
351
In view of (45), this yields
(5)=Z&o ,:)6(r)]~
(46)
The jump conditions (43), (45) and (46) are each expressed in terms of J(t), as defined by (39). A more useful expression for s(t) is obtained by evaluating (36) at r = 0 and using ~(0, t) - $o(t). We can then replace (39) by the asymptotically equivalent form (47) Utilizing (47) the jump conditions take the form
(49)
Thus, the jump discontinuities can be expressed in terms of the stress and temperature at y = 0 together with the plastic strain rate function (14). One of the important applications of the jump conditions is in the reformulation of the basic mathematical problem (l l)-(16) as a half-slab problem. This reformulation is presented in Section 4. The other important result of this boundary layer analysis is the relationship of s(t) to the temporal evolution of the shear band thickness. This relationship is derived in Section 3.
3.
EVOLUTION
OF THE SHEAR BAND THICKNESS
The results of the boundary layer analysis of Section 2 suggest a means for determining the temporal evolution of the thickness of the shear band. By examining the form of the solution near y = 0, we are able to identify a time dependent function that reflects the boundary layer width associated with the thermal effects. We propose that this boundary layer width can be interpreted as a good qualitative measurement of shear band thickness. The inner solution expansion for the temperature is given by (26). The results of (33) and (38) with 5 = Y/Eand E = m allow the inner solution to be expressed as (51)
352
J. A. DtLELLIO and W. E. OLMSTEAD
The form of (51) suggests that a boundary layer thickness can be associated with A(t) = m&t).
(52)
It then follows from (47) that
It is this function that is proposed as a time dependent measure of the shear band thickness in non-dimensional form. A formula analogous to (53) for a steady state band thickness was derived using asymptotic methods by Glimm et al. (1993). Related expressions for constant characteristic widths of shear bands have been found by Sherif and Shawki (1992) and Wright and Ockendon (1992). The explicit behavior of A(t), as given by (53), requires that the stress and temperature at y = 0 be known. Nevertheless, some qualitative behavior of A(t) can be inferred from this expression. It is typically observed that, as the shear band begins to form, there is a large increase in the plastic strain rate, while the stress decreases relatively gradually. From (53), this general behavior suggests that A(t) would be decreasing and hence the shear band width narrowing during this early stage of localization. In the later stage of shear band formation, the stress experiences a dramatic collapse, while the plastic strain rate either decreases or remains nearly fixed. From (53), this would suggest that A(t) would be increasing, thus implying a widening of the shear band during this later stage. This qualitative behavior of A(t) is consistent with both observation and numerical calculations. 4.
FORMULATION
OF THE HALF-SLAB
PROBLEM
We proceed in a manner analogous to Olmstead et al. (1994) to reformulate (1 l)(16) into an appropriate problem for the half-slab, 0 < y < L/Z. With the localization of the plastic deformation to a thin zone along the centerline y = 0, this justifies dropping $Jfrom the leading order outer problem (21)-(23). The influence of $Jis then accounted for in the outer problem by appropriate boundary conditions imposed at y = Of. These boundary conditions follow from the jump relationships which were derived through the matching of the inner and outer solutions presented in Section 2. The governing equations for the outer solution of the half-slab then follow from (2 I)-(23) as
UY, 0 = $y(Y, t), 0 <
Y <
$ t > 0.
(55)
Here f3- @, and s - ‘PO,while the velocity has been eliminated by combining (21) and (22). The velocity is easily recovered, once the stress and temperature are determined.
Shear band evolution
353
The boundary conditions on stress and temperature along y = O+ follow from the jump conditions (48) and (50) under the assumptions of symmetry (8). Thus “2{S(o, t)3[s(O, t), QO, t)]}‘!?,
(56)
(57) with k given by (14) as
MO, 4 WA 01 = V.MO,41 lirnexp
[
;o(o,t)
1 .
(58)
At the outer edge of the slab we require that
O(~,i)=O, sJ(~,i)=o, t>o.
(59)
The initial conditions are
O(Y,0) = e,(y), s(y, 0) = so,
(60)
0 d y < ; .
The formulation of the problem expressed in (54)-(60) constitutes a pair of linear partial differential equations (54)-(55) coupled through the nonlinear boundary conditions (56)-(57). An integral representation of (54)-(60) is given by Ljl sty,
f)
=
g(y,tlyo,O)~b(4’o)dyo--o
s0 l/2
dto,
s
(61)
L/l
QY,
0=
'3~3 4~0, O)~O(Y,)
dye
0
1:2
I
OS
+2 g
G(Y, tl0, fo){.$k ~ohW0, to), WX f0)1}“~ df,,
(62)
0
where g(y, tl0, to) and G(y, tI0, to) are the Green’s functions which satisfy grr-gyy = ~(Y-Yo)~(~-~o)~ g,@,tlYo,bd=O,
,,(~,,,,o,lo)=o,
0 < Y < $
t > to, t>t,,
(63)
(64)
354
J. A. DrLELLIO and W. E. OLMSTEAD
(65) and
G,-G,,=6(y-yo)6(t-t,),
t> to,
O
=o,
G,(O,ho, to) = 0,
G(y,tolyo,to)=O,
(66)
t>t,,
(67)
O<,<$
(68)
Various forms of the solutions to these Green’s function problems are found in Stakgold (1967). Once ~(0, t) and 0(0, t) are found, then (61)-(62) provide the desired solution of (54)-(60). Equations for the determination of ~(0, t) and 13(0,t) follow from setting y = 0 in (61)-(62). This yields
s L/l
s(0, t) =
do,
tbo, 0%(vo)
Go -50
0
I/2 -2
qo,t)
s
ik ()S
I 0
d 9@, tl0, to),,
3b@ 0
to), NO, toI1 1’2 dt s(O, to)
07
(69)
L/l
=
0
‘30, tlyo,
O)~O(YO) dye
(30, tl0, to>{@, tohG(o,
to), e(O, to)1)1’2 dto.
(70)
The determination of ~(0, t) and 0(0, t) is achieved through the simultaneous solution of (69)-(70). A simplified version of (69)-(70) is obtained for the special case in which the nondimensional scaling is such that L/I >> 1, so that (54)-(60) can be posed in the limiting case, 0 < y -C co. This limiting case is consistent with the physical reality that the shear band located at the center of the slab is so thin that it is effectively at an infinite distance from the outer edge. In this limiting case, (69)-(70) take the form
(71)
Shear band evolution
Am 112 r /(rqo, t) = 2 2a OS
355
t’)] - “‘(~(0, t’)y[s(O, t’), 0(0, t’)]) “2 dt’
s a0
+(7tt)-“2
exp(-52/4+%(5)
a.
(72)
0
An analogous version of (71)-(72) was derived by Olmstead et al. (1994) for a different set of jump conditions. The solution of these coupled equations is accomplished by the application of straightforward numerical methods. This is carried out for an illustrative example in Section 5.
5.
ILLUSTRATIVE
EXAMPLE
Here we consider a specific example to illustrate the results derived in Sections 3 and 4. For the numerical solution of (71)-(72), we select parameter values which are similar to those used by Wright and Batra (1985) and Walter (1992) for a moderately high strength steel. For a nominal strain rate of 1000 s-‘, we specify in SI units b = 7860,
k = 49.2,
c, = 473,
p = 8 x IO”,
A = 1,
S* = 8.93 x lo*,
F= 162,
Vy = 1.28x 10-3.
(73)
For the plastic strain rate function, we take F* = 11.2,
y. = 59.18,
m=0.025,
a=0.104.
(74)
Based upon these values, we determine from (10) that p=
1.30x10-‘,
I=
1.65~10-~,
I= 1.15~10-~,
(75)
which are used in the numerical computation of (71)-(72). Based on the determined value of 1, the limiting case of L/Z + co is clearly appropriate for the slab width of 3.47 mm used by Wright and Batra (1985) and Walter (1992). For the initial conditions we take Qo(y) = 0.1e-0~0’5Y2, So = 1,
o,(y) = gkanh
rq]“‘)“‘,
(76)
where ci = 89.6,
r/p
= 2703.
(77)
Here the form of ~~ is selected for purposes of compatibility with the boundary condition at the outer edge in the limiting case of L/l -+ co. This selection replicates the desired linear profile near the centerline, while still providing the proper edge value asy + co. To treat (71)-(72) numerically, we use Newton’s method to solve the nonlinear algebraic equation (71) for ~(0, t) at each time step. This information is then used in the numerical integration of (72) to obtain @O,t) for the next time step.
356
J. A. DILELLIO
and W. E. OLMSTEAD
30.0
20.0
z
0 z
10.0
0.10
0.20
0.30
0.40
t Fig.
1.Centerline
temperature.
The results of our computations for this example are displayed in Figs 1-4. The temporal evolution of the temperature and stress at the center of the shear band (y = 0) are given in Figs 1 and 2, respectively. Although the jump conditions derived here represent a significant theoretical improvement over those used by Olmstead et al. (1994), the resulting behavior of stress and temperature is quite similar. These results are also qualitatively in agreement with the full scale numerical studies carried out by Walter (1992) and Bayliss et al. (1994). The corresponding result for the plastic strain rate is shown in Fig. 3. This behavior is in qualitative agreement with Walter (1992). It should be noted that this qualitative agreement holds even though some of the parameter values in (74)-(75) significantly differ from the order unity size assumed for the asymptotic analysis of Section 3. From the computed behavior of $0, t) and QO, t), shown in Figs 1 and 2, we can then calculate the temporal evolution of the non-dimensional shear band thickness from (53). The result is shown in Fig. 4. We see that initially the band thickness narrows to a minimum width in accord with the dramatic rise of the plastic strain rate toward its maximum value. The band then widens as thermal diffusion becomes significant. From Fig. 4, it is found that min A(t) z 5 x 10P3. Converting this back to a dimensional scale based on the specimen size treated numerically by Walter (1992) and Bayliss et al. (1994), this suggests a minimum band thickness of about 0.6 microns. This can be compared with a value of about 0.3 microns computed from a formula deduced from ad hoc arguments by Bayliss et al. while these values are smaller
357
Shear band evolution
1.4
1.2
0.4
0.2
0.0
0.00
0.30
0.20
0.10
0.40
t Fig. 2. Centerline
stress.
2.0e+06
Itie+
50e+05
\\ o.oe+oa
C
D
-J 0.10
0.20 t
Fig. 3. Centerline
plastic strain rate.
0.30
0.40
J. A. DrLELLIO and W. E. OLMSTEAD
358
0.00
0.20 t Fig. 4. Shear band thickness.
than found in experiments, it must be remembered neglected in the model analyzed here.
0.30
that hardening
effects have been
ACKNOWLEDGEMENT This research was supported by NSF Grant DMS-9401016 and AR0 Grant DAAHO4-96l-0065.
REFERENCES Bayliss, A., Belytschko, T., Kulkarni, M. and Lott-Crumpler, D. A. (1994) On the dynamics and the role of imperfections for localization in thermo-viscoplastic materials. Modelling Sin&. Mater. Sci. Engng 2,941-964. Bai, Y. (1990) Adiabatic shear banding. Res. Mech. 31, 133-203. Glimm, J. G., Plohr, B. J. and Sharp, D. H. (1993) A conservative formulation for largedeformation plasticity. Appl. Mech. Rev. 46, 519-526. Olmstead, W. E., Nemat-Nasser, J. and Ni, L. (1994) Shear bands as surfaces of discontinuity. J. Mech. Phys. Solids 42,697-709.
Sherif, R. A. and Shawki, T. G. (1992) The role of heat conduction during the post-localization regime in dynamic viscoplasticity. Plastic Flow and Creep, AMD Vol. 135/MD-Vol. 3 1, pp. 159-l 73. ASME. Stakgold, I. (1967) Boundary Value Problems of Mathematical Physics. Macmillian, New York.
Shear band evolution
359
Walter, J. W. (1992) Numerical experiments on adiabatic shear band formation in one dimension. ht. J. Plasticity 8, 657-693. Wright, T. W. and Batra, R. C. (1985) The initiation and growth of adiabatic shear bands. Znt. J. Plasticity
1, 205-212.
Wright, T. W. and Walter, J. W. (1987) On stress collapse in adiabatic shear bands. J. Mech. Phys. Solidr 354701-720.
Wright, T. W. and Ockendon, H. (1992) A model for fully formed shear bands. J. Mech. Phys. Solids 40, 1217-1226.