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Shear bands as surfaces of discontinuity
J. Mcd~. Phw Solids. Vol. 42, No. 4, pp. 697 709, 1994 Copyright 7.1 1994 Elsewer Science Ltd PrInted in Great Bntain. All rights reserved 0022m5096194 $6.00+0.00
Pergamon
0022-5096(93)E0009-F
SHEAR
BANDS
AS SURFACES W. E.
Department
of Engineering
OF DISCONTINUITY
OLMSTEAD
Sciences and Applied Mathematics. Evanston, IL 60208. U.S.A.
Northwestern
University.
and S. Department
of Applied
Mechanics
NEMAT-NASSER
and
L. NI
and Engineering Sciences, University
of California
at San Diego.
La Jolla, CA 92093, U.S.A.
ABSTRACT A THEORETICAL CHAKACTEKIZATION
of a shear band as a surface of discontinuity is developed. For the onedimensional problem of unidirectional shearing of a slab, expressions for the jump discontinuities across a possible shear band are derived. This provides a simplified formulation of the problem. which yields a pair of equations for the evolution of stress and temperature along the surface representing the shear band. Two illustrative examples are examined.
1.
INTRODUCTION
THE
PURPOSE OF THIS investigation is to present a theoretical characterization of shear bands which exploits their extreme thinness. We propose to take advantage of this very small scale by treating the narrow zone of localization as a surface of discontinuity, across which are allowed jumps in velocity, stress gradient and temperature gradient. The thin scale of shear bands associated with high-strength metals is well justified by experimental observations, as reported for example in MARCHAND and DUFFY (1988) and BEATTY et ul. (1990). An example taken from Beatty et al. is shown in Fig. 1. An examination of adiabatic shear bands suggests three basic length scales : (i) a submicron scale to measure the microstructure within the band; (ii) a scale of several hundred micrometers to measure the variation of continuum field quantities within a boundary layer adjacent to the shear bands; and (iii) a scale of a few centimeters or more to measure the sample geometry. This suggests that the shear band itself can be viewed as a strain discontinuity, since its thickness is orders of magnitude less than the boundary layer. These observations have motivated the analysis to follow. Results are developed within the context of a one-dimensional formulation of the
697
698
w.
E.
OLMSTLAI)
PI
t/l.
problem for unidirectional shearing of a slab, which has been used in a variety of previous investigations (e.g. CLIFTON et al., 1984; WRIGHT and BATRA, 1985 ; WRIGHT and WALTER, 1987 ; BURNS, 1990: WAI.TER, 1992). Within this one-dimensional theory, we determine appropriate expressions for the jump discontinuities across the shear band. This concept leads to a simplified reformulation of the problem which permits us to focus directly on the evolution of stress and temperature in the shear band. While our formulation does not give a detailed view within the localization zone, it does instead provide the centerline behavior as the process evolves toward band formation. Our approach utilizes the same empirical models of flow stress cmploycd in other phenomenological approaches to the problem. Such approaches typically avoid the detailed microscopic dynamics of recrystalization through extensive dislocation activities that occur during localization (see BEATTY rt ~1.. 1990). From this perspective. our treatment of the shear band as a surface of discontinuity, with its concomitant simplicity, may be an attractive alternative to other approaches. In previous investigations, the usual practice has been to seed the formation of the shear band through introducing a spatial perturbation in one of the field variables. By introducing a surface of discontinuity, we can circumvent the need for such perturbations. We will illustrate, through specific examples, two different modes for initiation of the localization process. In each case, we can follow the evolution of the centerline values of stress and temperature, from a given initial state to a stage of thermal softening which leads to the stress collapse associated with band formation. Our analysis is based upon the one-dimensional formulation of the unidirectional shearing of a slab. In dimensional form, consider the governing equations of momentum. elasticity and energy, respectively as fiVj = SC,
(1.1)
Si=~(Vi-~). fic,.Ti=kTii+ASl+.
(1.2)
-L<$
i>O,
(1.3)
where V( y, t^, is the velocity, S( 1;, t^, is the shear stress and T( 9, t^, is the temperature in a slab of width 2L centered at _$= 0. The material constants 6, p, c,. and k are the density, elastic shear modulus, specific heat and thermal conductivity, respectively. The constant A converts plastic work into thermal energy. The plastic strain rate i‘ is defined implicitly by a flow rule, s = G”(T, i-, l-). For convenience,
(1.4)
we assume that (I .4) can be resolved into an expression
of the form
i- = F(S, T). The boundary
conditions
at the ends of the slab are
T(iL,i) with appropriate
compliance T(f,
(1.5)
= T,
V(kL,
of the stress and deformation.
0) = T,(.C),
S(_P,O) = So,
(I .6a, b)
i) = *v,
The initial conditions
V(?‘.O) = V,,(f).
are
(1.7%c)
Shear bands as surfaces
FIG. I. Scanning
electron
micrograph
showing
ofdiscontinuity
699
adiabatic shear band in high strength steel [from BElATTY et Gil. (1990)].
Shear bands as surfaces
of discontinuity
Were the initial stress state S, is taken to be constant. initial temperature distribution with the properties T,,(-3) and an initial velocity VU-j) It is consistent conditions
= T”(P),
701
We allow for a non-constant
T,(+L)
(1.8a,b)
= E
profile with the properties = - v,,(Q),
V,,(I!IL) = I!ZK
with the above formulation
7(?;,t^) = T(-_9,&
S($,i)
d”,i VII(P) > 0.
of the problem
= S(-j:,t^),
(1.9ax)
to impose the symmetry
V(_G,t^)= -V(-~?,fi.
(1.lOa-c)
In our analysis to follow we assume that the shear localization is centered about the line of symmetry 3 = 0. Moreover, our non-dimensionalization of the slab width is chosen in order to locate the ends of the slab at an (essentially) infinite distance from the centerline. We introduce the non-dimensional scalings 6, = (T-
i;)/ir,
s = S/S*,
t = f/t,
y = “$11, y = f/P,
u = V/V*, (I.llapf)
and the parameters t=
1 L= (~~*/~~~,~~*)I~~,
s*/$*,
>_= AS”‘&,@,
y* = jr*,
y = ki‘*/c,.S*.
(1.12a-e)
We then can express (I. 1)-( I .3) as (1.13)
P’, = sj 3 s, = 1’).-7, 8, = (I,.,.-t nsj,
(1.14)
-CC
t > 0,
(1.15)
where the ends of the slab have been placed at y = k a, under the assumption L/I >>1. The plastic strain rate relation (1.5) is replaced by the non-dilnensional j = F(s, 0). The boundary
conditions
compliance
(1.16)
become
O(Fc0,f)=O, with appropriate
that form
l’(+X,t)=
of the stress. The initial conditions
q_r, 0) = H,,(.Y),
S(.l’, 0) = so,
(1.17a, b)
*P/v*, are
u( 4’, 0) = &I( y).
(1.18a-c)
where U-Y)
= Q”(Y) 3 0,
r&--y)
= -c,,(4’),
Oo(i Co) = 0, P<,(i_ 30) = i y/v*.
(f.l9a-d)
702
The symmetry
W. E. OLMSTFAII c'to/.
conditions
(I. IO) become
(I( j’, t) = O( -“l’, t).
7 -.
S( j‘. t) = s( -_?‘, t)
DERIVATION
OF THE JUMP
V(j‘, t) = -1.(-J’,
t).
(I .20a c)
CONDITIONS
The key feature of this investigation is to treat the very narrow zone of shear localization as a surface of discontinuity along the line of symmetry. This idealization of the problem squeezes all the internal structure of the shear band into an infinitesimally thin domain. Thus, the behavior of the stress and temperature along the surface, as determined through this approach. can be regarded as representative values of these quantities in the shear band. A convenient interpretation of this surface of discontinuity is that it corresponds to a velocity slip across the zone of localization. The magnitude of this slip initially will be negligible. but it will increase significantly as the system progresses toward the formation of a shear band. Other (equivalent) interpretations of the surface 01 discontinuity correspond to jumps in the temperature gradient and stress gradient. For the one-dimensional problem considered here, we introduce the surface of discontinuity along the line _I’= 0. Across J‘ = 0, we allow for finite jump discontinuities in the values of velocity, temperature gradient and stress gradient. The values of temperature and stress will be taken to remain continuous across _t’= 0. It will be seen that the allowed jumps are related to a strong localization of the plastic strain rate along the surface of discontinuity. To account for the strong localization of $ along ~3= 0, we define the limit
(2.1) It will be seen here that our allowed discontinuities in velocity. temperature gradient and stress gradient can be cxpresscd in terms of ,f’(r). Without the surface of discontinuity, the limit (2.1) would typically vanish. It follows from (I. l4), (I. 15) and (I. l3), respectively, that (1’) s r(O-, I) -P(O
, t) = lim
r, d_t
(2.2)
(H,.)
E
H,(O', 1)-ft),(O
.
t) = )iz
’ O,, d_r
(2.3)
Shear hands as surfaces of discontinuity (s,)
= ,s>.(o+,
703
t) -s, (O-) t) = pf’(t).
(2.4)
that limit While ,f’(t) is defined as a limit operation in (2.1), we can approximate zone. Taking 26 as a by utilizing the extreme thinness of the shear localization characteristic width of the zone, we approximate (2.1) by .f’(t) z
(2.5)
Furthermore, we consider the typical circumstances in which the functional 3 contains a very large multiplicative parameter, so that i’ = F(s, (!I) = BF,,(s, U), We scale the large parameter
B >> 1.
form of
(2.6)
B in terms of the width 26 as B=;;,
This allows (2.5) to be expressed
0 < 6 << 1.
(2.7)
as
(2.8) By retaining
only the leading
order contribution
from (2.8), it follows that
.f’(t) = BoFo[.r(O, t), @(O,t>l.
3.
(2.9)
FORMULATION OF THE HALF-SLAB PROBLEM
Given the symmetry conditions (1.20), together with our localization along _r = 0, we now pose an appropriate problem for the half-slab, 0 < J < CCI.An important implication of localizing 2; along the centerline, r = 0, is that its contribution to partial differential equations (1.14) and (1.15) when considering only the half-slab, is relegated to the boundary conditions at _V= O+. The governing equations for the stress and temperature in the half-slab problem are then expressed as
(3.1) B,(y, r) = H,.,.(y, t),
0< y <
Go,
f >
0.
(3.2)
Here, (I. 13) has been used to eliminate the velocity. The jump conditions (2.3)-(2.4), along with the imposed symmetry, yield the boundary conditions along y = Of, as
704
oI.MSTlZAI~ ef ul.
W. E.
;s
J.B,,
3 s(0, t)F,,[s(O, t), H(0, r,],
O,(O,t) = - * (0, t),f’(t) = s, (0, t) =
‘1f’(t)
=
PB,, d 2
dt
At the outer edge of the slab we require
that
6(x3, 1) = 0. The initial conditions
.s(x,
It is straightforward to obtain expressions which satisfy (3. I) 43.5). It follows that
- !j
~--I 2(7ct) I;2
t) =
(3.421, b)
.S,).
are
6( _I‘. 0) = fJo( I’),
H(J’, t) =
(3.32~ b)
~,,bux 0. fi(O.f)]
S(j’, 0) = .Y().
(3.5a. b)
for O( y. t) and s( J, t), in terms of ,f’(t),
-,f’(O)],
(3.6)
f’)]s(O, r’),f’(r’) dr’.
(3.7)
N(t - JY.l’)[,W&.I’)
s I
,)
{exp[-(?~-5)‘/4t]+exp[-(?:+~)‘/4tli(~0(5)d~
’ [n([_f),
’ 2 exp [ -_19’/4(t-
Here it is seen that the stress and temperature are determined throughout the halfslab when ,f’(r) is known. In turn, ,f‘(t) is known from (2.9) once the centerline values of stress and temperature have been determined. Equations for the determination of ~(0, t) and H(0, t) follow by setting J’ = 0 in (3.6) and (3.7), so that
6(0, t) =
i.B,
2
’ s (1
[n(r ~
01
jp4, j-m iF,,[.s(O, t). H(0, t)]-F”[.S,,,
e”(o)];..
(3.8)
“‘~(0, t’)F,)[s(O, t’), II(0, t’)] dt’ I
+(7rt)
“?
I
0
exp (- <‘/4r)ti,,(t)
dl.
(3.9)
Thus, the half-slab problem is reduced to the solution of the coupled equations (3.8) and (3.9) for ~(0, t) and fl(0, t). This is particularly useful since it is these quantities, namely the stress and temperature along the centerline of the localization zone. which are of primary interest.
Ccmsider two examples to illustrate the appiication of the basic results derived in Section 3. Each of these examples represents a different mode of initiating the shear localization process. In the first example. an imposed initial velocity profile is used as the driving mechanism. This demonstrates how the formulation can yield stress collapse without an imposed perturbation in the initial state of a field variable. For the second example, we introduce a perturbation in the initial temperature state as the driving mechanism. This illustrates the versatility of the approach. For both examples, consider the plastic strain rate function
This form is one of several used by WALTER (1992). We also specify an initial velocity profile of the form z-,“(J))= ay,
c! 2 0.
(4.2)
Tncorporating (4.1) and (4.2) into (3.8) and (3.9) obtain
s ?‘
+
(7ct)- If2
0
exp (- 5”/4t)H,(5) d5.
In computations, we assign values to the material parameters which are similar to those used by WRIGHT and WALTER (1987) and WALTER (1992) for a moderately high-strength steel. In SI units, we take ;c;= x000, jr=XxfO’O,
k = SO, CP= 500, S”=6xIOX,
F=roo.
(4,5a-f)
For the ptastic strain rate function (4.1), we specify i_* z.z Is-‘,
B,=2,
6=2~10-~,
IV=40,
/3=2.4.
(4.6a-e)
The value of cn depends upon the reference state chosen for each example. Based on the above parameter values, we determine from (I. 12) the approximate value of p=
IO-iO,
A= IO 2,
(4.7a, b)
for use in the numerical solution of (4.3) and (4.4). It also follows that I = 3 x ID -4,
W. E.
706
OLMST~AI)
cc trl.
of L/l >> I is valid for any slab width which is significantly
so that the assumption larger than 0.6 mm. In the first example, (4.2). We specify
the driving
‘U= 500,
mechanism
Oo( J) = 0,
is the imposed
.v,, = I,
initial
C()= 0.
velocity
profile
(4&mmd)
The value of c( is consistent with a nominal strain rate of 500 s ‘. The choice of n,, and .s(,is consistent with uniform initial temperature and stress. To solve (4.3) and (4.4) numerically we determine .s(O, t) from (4.3) by solving this nonlinear algebraic equation with Newton’s method at each incremental time step. This information is then used in a numerical integration of (4.5) to obtain 0(0, 1) for the next time step. The results of our computations for this example arc displayed in Figs 2 and 3. It is seen that during the elastic stage, there is a linear build-up of stress, consistent with the imposed initial velocity profile. The temperature rises very slowly from its initial value during this stage. Once there has been a sufficient increase in temperature, thermal softening occurs, whereupon the stress passes through a maximum and begins to decrease. This is followed by a rapid collapse of stress along with a steep rise in temperature. as is typical of shear band formation. In our second example, the driving mechanism is a spatial perturbation in the initial temperature state. We specify c( = 0,
Oo(j,) = O.lC 0’5”.
.s,, = 1.1,
C,, = 50.
(4.9a d)
This specification is intended to roughly correspond to a case considered by WALTER (1992). The initial temperature state n,, represents a perturbation which is strongly
STRESS (N=40, c&00.
P=2.4,h=102, p= IO'")
1.5
FIG. 2. Stress evolution
I” the shear band for rl > 0. (I,, = 0.
Shear bands as surfaces
707
of discontinuity
TEMPERATURE (N&J,a&00, p=2.4, Ll 0 2, p=l O~‘O) 40.0
30.0
I 1
! I 5
2o.o
1
10.0
FIG. 3.Temperature
evolution
in the shear band for x > 0, O,, = 0
localized at J = 0. The choice of x = 0 essentially eliminates any elastic response of the system. WALTER (1992) achieves that effect through increasing the value of p in (4.5) by a factor of IO’. The choice of s0 corresponds to an initial stress slightly larger than the yield strength of the material. The value of co is selected to allow a slight positive bias to the initial plastic strain rate. The numerical solution of (4.3) and (4.4) for this second example is carried out as in the previous case. The results are displayed in Figs 4 and 5. Since the elastic buildSTRESS (N=40,
~0.
p=2.4,
h=102,
p=lO”)
1.5
1.0
=‘
d B
0.5
0.0
-i
/-_, 0.0
-1
0.2
0.4
.
0.6
0.6
t
FIG. 4. Stress evolution
in the shear band for CA= 0, O,, > 0.
I.10
70x
w.
E.
OLMSTEAD
c’[
al
TEMPERATURE (N-40,
a=O, 8~2.4,
h=iO’,
p=lO
“)
‘________1 60’o
r--
0.0
L0.0
-: 0.2
0.4
_lI
~~~--
0.6
0.8
1 1.0
FIG. 5. Temperature evolution in the shear band for ,CI= 0. II,, > 0
up of stress has been suppressed (i.e. c( = 0), there is a gradual decrease in stress away from the initial value, until finally there is enough thermal softening to cause a precipitous stress collapse. This is accompanied by a dramatic growth in temperature. Qualitatively, these results are similar to those of WALTER (1992) ; however, some differences in parameter values and initial conditions do not permit a quantitative comparison.
5.
CONCLUDING
REMARKS
The goal here has been to introduce a new viewpoint for the investigation of the standard one-dimensional problem for shear band formation. The key feature of the approach is to represent the narrow zone of localization by a surface of discontinuity, across which there can be jumps in the values of velocity, temperature gradient and stress gradient. The problem can then be reduced to that of a half-slab with appropriate boundary conditions to account for the presence of the localized zone. Our analysis of the reformulated half-slab problem, as illustrated by specific examples, demonstrates that it can predict an evolution of behavior which leads to the dramatic temperature growth and stress collapse associated with shear band formation. We regard the viewpoint introduced here as a viable alternative, complementary to other methods used to analyse shear localization phenomena. While our approach does not yield all the small-scale details of the localization process, it gains through a significant simplification of the overall problem. Different modes of initiation can be implemented in our formulation, as is illustrated. It is also clear that the approach can accommodate more sophisticated phenomenological models of plastic straining, including those with hardening effects.
Shear bands as surfaces of disc~~~inui~~
709
ACKNOWLEDGEMENTS
The authors would like to acknowledge the input stimulating discussions about the phenomenon of particular aspects of our approach presented here. the National Science Foundation NSF MSS90-2167 at the University of California, San Diego.
of T. W. Wright and J. W. Walter through shear bands in general, as well as various This work was supported in part through I and AR0 contract DAAL 03-92-G-01 08
REFERENCES BEATTY. J., MEYER, L. W., NEMAT-NASSER, S. and MEYERS, M. A. (1992) Formation of controlled adiabatic shear bands in AJSI 4340 high strength steel. In S~~nt-~-JVnz~ and HEIDIStrength Plle~~~~~~na in ‘~ffteri~~~ (ed. M. A. MEYERS, L. E. MCRR, K. P. STAUDHAM~~~R), pp. 59-68. Marcel Dekker, New York. BURNS, T. J. (1990) A mechanism for shear band formation in the high strain-rate torsion test. J. i4ppi. Me&. 112, 836844. CLIFTON, R. J., DUFFY, J., HARTLEY, K. A. and SHAWKI, T. G. (1984) On critical conditions for shear band formation at high strain rates. Scriptu Met. 18, 443448. MAKCHAND, A. and DUFFY, J. (1988) An experimental study of the formation process of adiabatic shear bands in a structural steel. J. Mach. Phq’s. So&/s 36, 25 l-283. WALT.ER, J. W. (1992) Numerical experiments on adiabatic shear band formation in one dimension. ht. J. Plusticity 8, 657--693. WRIC;H,T,T. W. and BATRA, R. C. (1985) The initiation and growth of adiabatic shear bands. ht. J. Plasticity 1, 205-2 12. WRIC;NT, T. W. and WALTER, 3. W. (1987) On stress collapse in adiabatic shear bands. J. Mecl?. Phyr. Solids 35, 70 l-720. WRIGHT, T. W. (1990) Approximate analysis for the forination of adiabatic shear bands. .I. M&. P/zrs. Sdids 38, 5 15p530.
Pergamon
0022-5096(93)E0009-F
SHEAR
BANDS
AS SURFACES W. E.
Department
of Engineering
OF DISCONTINUITY
OLMSTEAD
Sciences and Applied Mathematics. Evanston, IL 60208. U.S.A.
Northwestern
University.
and S. Department
of Applied
Mechanics
NEMAT-NASSER
and
L. NI
and Engineering Sciences, University
of California
at San Diego.
La Jolla, CA 92093, U.S.A.
ABSTRACT A THEORETICAL CHAKACTEKIZATION
of a shear band as a surface of discontinuity is developed. For the onedimensional problem of unidirectional shearing of a slab, expressions for the jump discontinuities across a possible shear band are derived. This provides a simplified formulation of the problem. which yields a pair of equations for the evolution of stress and temperature along the surface representing the shear band. Two illustrative examples are examined.
1.
INTRODUCTION
THE
PURPOSE OF THIS investigation is to present a theoretical characterization of shear bands which exploits their extreme thinness. We propose to take advantage of this very small scale by treating the narrow zone of localization as a surface of discontinuity, across which are allowed jumps in velocity, stress gradient and temperature gradient. The thin scale of shear bands associated with high-strength metals is well justified by experimental observations, as reported for example in MARCHAND and DUFFY (1988) and BEATTY et ul. (1990). An example taken from Beatty et al. is shown in Fig. 1. An examination of adiabatic shear bands suggests three basic length scales : (i) a submicron scale to measure the microstructure within the band; (ii) a scale of several hundred micrometers to measure the variation of continuum field quantities within a boundary layer adjacent to the shear bands; and (iii) a scale of a few centimeters or more to measure the sample geometry. This suggests that the shear band itself can be viewed as a strain discontinuity, since its thickness is orders of magnitude less than the boundary layer. These observations have motivated the analysis to follow. Results are developed within the context of a one-dimensional formulation of the
697
698
w.
E.
OLMSTLAI)
PI
t/l.
problem for unidirectional shearing of a slab, which has been used in a variety of previous investigations (e.g. CLIFTON et al., 1984; WRIGHT and BATRA, 1985 ; WRIGHT and WALTER, 1987 ; BURNS, 1990: WAI.TER, 1992). Within this one-dimensional theory, we determine appropriate expressions for the jump discontinuities across the shear band. This concept leads to a simplified reformulation of the problem which permits us to focus directly on the evolution of stress and temperature in the shear band. While our formulation does not give a detailed view within the localization zone, it does instead provide the centerline behavior as the process evolves toward band formation. Our approach utilizes the same empirical models of flow stress cmploycd in other phenomenological approaches to the problem. Such approaches typically avoid the detailed microscopic dynamics of recrystalization through extensive dislocation activities that occur during localization (see BEATTY rt ~1.. 1990). From this perspective. our treatment of the shear band as a surface of discontinuity, with its concomitant simplicity, may be an attractive alternative to other approaches. In previous investigations, the usual practice has been to seed the formation of the shear band through introducing a spatial perturbation in one of the field variables. By introducing a surface of discontinuity, we can circumvent the need for such perturbations. We will illustrate, through specific examples, two different modes for initiation of the localization process. In each case, we can follow the evolution of the centerline values of stress and temperature, from a given initial state to a stage of thermal softening which leads to the stress collapse associated with band formation. Our analysis is based upon the one-dimensional formulation of the unidirectional shearing of a slab. In dimensional form, consider the governing equations of momentum. elasticity and energy, respectively as fiVj = SC,
(1.1)
Si=~(Vi-~). fic,.Ti=kTii+ASl+.
(1.2)
-L<$
i>O,
(1.3)
where V( y, t^, is the velocity, S( 1;, t^, is the shear stress and T( 9, t^, is the temperature in a slab of width 2L centered at _$= 0. The material constants 6, p, c,. and k are the density, elastic shear modulus, specific heat and thermal conductivity, respectively. The constant A converts plastic work into thermal energy. The plastic strain rate i‘ is defined implicitly by a flow rule, s = G”(T, i-, l-). For convenience,
(1.4)
we assume that (I .4) can be resolved into an expression
of the form
i- = F(S, T). The boundary
conditions
at the ends of the slab are
T(iL,i) with appropriate
compliance T(f,
(1.5)
= T,
V(kL,
of the stress and deformation.
0) = T,(.C),
S(_P,O) = So,
(I .6a, b)
i) = *v,
The initial conditions
V(?‘.O) = V,,(f).
are
(1.7%c)
Shear bands as surfaces
FIG. I. Scanning
electron
micrograph
showing
ofdiscontinuity
699
adiabatic shear band in high strength steel [from BElATTY et Gil. (1990)].
Shear bands as surfaces
of discontinuity
Were the initial stress state S, is taken to be constant. initial temperature distribution with the properties T,,(-3) and an initial velocity VU-j) It is consistent conditions
= T”(P),
701
We allow for a non-constant
T,(+L)
(1.8a,b)
= E
profile with the properties = - v,,(Q),
V,,(I!IL) = I!ZK
with the above formulation
7(?;,t^) = T(-_9,&
S($,i)
d”,i VII(P) > 0.
of the problem
= S(-j:,t^),
(1.9ax)
to impose the symmetry
V(_G,t^)= -V(-~?,fi.
(1.lOa-c)
In our analysis to follow we assume that the shear localization is centered about the line of symmetry 3 = 0. Moreover, our non-dimensionalization of the slab width is chosen in order to locate the ends of the slab at an (essentially) infinite distance from the centerline. We introduce the non-dimensional scalings 6, = (T-
i;)/ir,
s = S/S*,
t = f/t,
y = “$11, y = f/P,
u = V/V*, (I.llapf)
and the parameters t=
1 L= (~~*/~~~,~~*)I~~,
s*/$*,
>_= AS”‘&,@,
y* = jr*,
y = ki‘*/c,.S*.
(1.12a-e)
We then can express (I. 1)-( I .3) as (1.13)
P’, = sj 3 s, = 1’).-7, 8, = (I,.,.-t nsj,
(1.14)
-CC
t > 0,
(1.15)
where the ends of the slab have been placed at y = k a, under the assumption L/I >>1. The plastic strain rate relation (1.5) is replaced by the non-dilnensional j = F(s, 0). The boundary
conditions
compliance
(1.16)
become
O(Fc0,f)=O, with appropriate
that form
l’(+X,t)=
of the stress. The initial conditions
q_r, 0) = H,,(.Y),
S(.l’, 0) = so,
(1.17a, b)
*P/v*, are
u( 4’, 0) = &I( y).
(1.18a-c)
where U-Y)
= Q”(Y) 3 0,
r&--y)
= -c,,(4’),
Oo(i Co) = 0, P<,(i_ 30) = i y/v*.
(f.l9a-d)
702
The symmetry
W. E. OLMSTFAII c'to/.
conditions
(I. IO) become
(I( j’, t) = O( -“l’, t).
7 -.
S( j‘. t) = s( -_?‘, t)
DERIVATION
OF THE JUMP
V(j‘, t) = -1.(-J’,
t).
(I .20a c)
CONDITIONS
The key feature of this investigation is to treat the very narrow zone of shear localization as a surface of discontinuity along the line of symmetry. This idealization of the problem squeezes all the internal structure of the shear band into an infinitesimally thin domain. Thus, the behavior of the stress and temperature along the surface, as determined through this approach. can be regarded as representative values of these quantities in the shear band. A convenient interpretation of this surface of discontinuity is that it corresponds to a velocity slip across the zone of localization. The magnitude of this slip initially will be negligible. but it will increase significantly as the system progresses toward the formation of a shear band. Other (equivalent) interpretations of the surface 01 discontinuity correspond to jumps in the temperature gradient and stress gradient. For the one-dimensional problem considered here, we introduce the surface of discontinuity along the line _I’= 0. Across J‘ = 0, we allow for finite jump discontinuities in the values of velocity, temperature gradient and stress gradient. The values of temperature and stress will be taken to remain continuous across _t’= 0. It will be seen that the allowed jumps are related to a strong localization of the plastic strain rate along the surface of discontinuity. To account for the strong localization of $ along ~3= 0, we define the limit
(2.1) It will be seen here that our allowed discontinuities in velocity. temperature gradient and stress gradient can be cxpresscd in terms of ,f’(r). Without the surface of discontinuity, the limit (2.1) would typically vanish. It follows from (I. l4), (I. 15) and (I. l3), respectively, that (1’) s r(O-, I) -P(O
, t) = lim
r, d_t
(2.2)
(H,.)
E
H,(O', 1)-ft),(O
.
t) = )iz
’ O,, d_r
(2.3)
Shear hands as surfaces of discontinuity (s,)
= ,s>.(o+,
703
t) -s, (O-) t) = pf’(t).
(2.4)
that limit While ,f’(t) is defined as a limit operation in (2.1), we can approximate zone. Taking 26 as a by utilizing the extreme thinness of the shear localization characteristic width of the zone, we approximate (2.1) by .f’(t) z
(2.5)
Furthermore, we consider the typical circumstances in which the functional 3 contains a very large multiplicative parameter, so that i’ = F(s, (!I) = BF,,(s, U), We scale the large parameter
B >> 1.
form of
(2.6)
B in terms of the width 26 as B=;;,
This allows (2.5) to be expressed
0 < 6 << 1.
(2.7)
as
(2.8) By retaining
only the leading
order contribution
from (2.8), it follows that
.f’(t) = BoFo[.r(O, t), @(O,t>l.
3.
(2.9)
FORMULATION OF THE HALF-SLAB PROBLEM
Given the symmetry conditions (1.20), together with our localization along _r = 0, we now pose an appropriate problem for the half-slab, 0 < J < CCI.An important implication of localizing 2; along the centerline, r = 0, is that its contribution to partial differential equations (1.14) and (1.15) when considering only the half-slab, is relegated to the boundary conditions at _V= O+. The governing equations for the stress and temperature in the half-slab problem are then expressed as
(3.1) B,(y, r) = H,.,.(y, t),
0< y <
Go,
f >
0.
(3.2)
Here, (I. 13) has been used to eliminate the velocity. The jump conditions (2.3)-(2.4), along with the imposed symmetry, yield the boundary conditions along y = Of, as
704
oI.MSTlZAI~ ef ul.
W. E.
;s
J.B,,
3 s(0, t)F,,[s(O, t), H(0, r,],
O,(O,t) = - * (0, t),f’(t) = s, (0, t) =
‘1f’(t)
=
PB,, d 2
dt
At the outer edge of the slab we require
that
6(x3, 1) = 0. The initial conditions
.s(x,
It is straightforward to obtain expressions which satisfy (3. I) 43.5). It follows that
- !j
~--I 2(7ct) I;2
t) =
(3.421, b)
.S,).
are
6( _I‘. 0) = fJo( I’),
H(J’, t) =
(3.32~ b)
~,,bux 0. fi(O.f)]
S(j’, 0) = .Y().
(3.5a. b)
for O( y. t) and s( J, t), in terms of ,f’(t),
-,f’(O)],
(3.6)
f’)]s(O, r’),f’(r’) dr’.
(3.7)
N(t - JY.l’)[,W&.I’)
s I
,)
{exp[-(?~-5)‘/4t]+exp[-(?:+~)‘/4tli(~0(5)d~
’ [n([_f),
’ 2 exp [ -_19’/4(t-
Here it is seen that the stress and temperature are determined throughout the halfslab when ,f’(r) is known. In turn, ,f‘(t) is known from (2.9) once the centerline values of stress and temperature have been determined. Equations for the determination of ~(0, t) and H(0, t) follow by setting J’ = 0 in (3.6) and (3.7), so that
6(0, t) =
i.B,
2
’ s (1
[n(r ~
01
jp4, j-m iF,,[.s(O, t). H(0, t)]-F”[.S,,,
e”(o)];..
(3.8)
“‘~(0, t’)F,)[s(O, t’), II(0, t’)] dt’ I
+(7rt)
“?
I
0
exp (- <‘/4r)ti,,(t)
dl.
(3.9)
Thus, the half-slab problem is reduced to the solution of the coupled equations (3.8) and (3.9) for ~(0, t) and fl(0, t). This is particularly useful since it is these quantities, namely the stress and temperature along the centerline of the localization zone. which are of primary interest.
Ccmsider two examples to illustrate the appiication of the basic results derived in Section 3. Each of these examples represents a different mode of initiating the shear localization process. In the first example. an imposed initial velocity profile is used as the driving mechanism. This demonstrates how the formulation can yield stress collapse without an imposed perturbation in the initial state of a field variable. For the second example, we introduce a perturbation in the initial temperature state as the driving mechanism. This illustrates the versatility of the approach. For both examples, consider the plastic strain rate function
This form is one of several used by WALTER (1992). We also specify an initial velocity profile of the form z-,“(J))= ay,
c! 2 0.
(4.2)
Tncorporating (4.1) and (4.2) into (3.8) and (3.9) obtain
s ?‘
+
(7ct)- If2
0
exp (- 5”/4t)H,(5) d5.
In computations, we assign values to the material parameters which are similar to those used by WRIGHT and WALTER (1987) and WALTER (1992) for a moderately high-strength steel. In SI units, we take ;c;= x000, jr=XxfO’O,
k = SO, CP= 500, S”=6xIOX,
F=roo.
(4,5a-f)
For the ptastic strain rate function (4.1), we specify i_* z.z Is-‘,
B,=2,
6=2~10-~,
IV=40,
/3=2.4.
(4.6a-e)
The value of cn depends upon the reference state chosen for each example. Based on the above parameter values, we determine from (I. 12) the approximate value of p=
IO-iO,
A= IO 2,
(4.7a, b)
for use in the numerical solution of (4.3) and (4.4). It also follows that I = 3 x ID -4,
W. E.
706
OLMST~AI)
cc trl.
of L/l >> I is valid for any slab width which is significantly
so that the assumption larger than 0.6 mm. In the first example, (4.2). We specify
the driving
‘U= 500,
mechanism
Oo( J) = 0,
is the imposed
.v,, = I,
initial
C()= 0.
velocity
profile
(4&mmd)
The value of c( is consistent with a nominal strain rate of 500 s ‘. The choice of n,, and .s(,is consistent with uniform initial temperature and stress. To solve (4.3) and (4.4) numerically we determine .s(O, t) from (4.3) by solving this nonlinear algebraic equation with Newton’s method at each incremental time step. This information is then used in a numerical integration of (4.5) to obtain 0(0, 1) for the next time step. The results of our computations for this example arc displayed in Figs 2 and 3. It is seen that during the elastic stage, there is a linear build-up of stress, consistent with the imposed initial velocity profile. The temperature rises very slowly from its initial value during this stage. Once there has been a sufficient increase in temperature, thermal softening occurs, whereupon the stress passes through a maximum and begins to decrease. This is followed by a rapid collapse of stress along with a steep rise in temperature. as is typical of shear band formation. In our second example, the driving mechanism is a spatial perturbation in the initial temperature state. We specify c( = 0,
Oo(j,) = O.lC 0’5”.
.s,, = 1.1,
C,, = 50.
(4.9a d)
This specification is intended to roughly correspond to a case considered by WALTER (1992). The initial temperature state n,, represents a perturbation which is strongly
STRESS (N=40, c&00.
P=2.4,h=102, p= IO'")
1.5
FIG. 2. Stress evolution
I” the shear band for rl > 0. (I,, = 0.
Shear bands as surfaces
707
of discontinuity
TEMPERATURE (N&J,a&00, p=2.4, Ll 0 2, p=l O~‘O) 40.0
30.0
I 1
! I 5
2o.o
1
10.0
FIG. 3.Temperature
evolution
in the shear band for x > 0, O,, = 0
localized at J = 0. The choice of x = 0 essentially eliminates any elastic response of the system. WALTER (1992) achieves that effect through increasing the value of p in (4.5) by a factor of IO’. The choice of s0 corresponds to an initial stress slightly larger than the yield strength of the material. The value of co is selected to allow a slight positive bias to the initial plastic strain rate. The numerical solution of (4.3) and (4.4) for this second example is carried out as in the previous case. The results are displayed in Figs 4 and 5. Since the elastic buildSTRESS (N=40,
~0.
p=2.4,
h=102,
p=lO”)
1.5
1.0
=‘
d B
0.5
0.0
-i
/-_, 0.0
-1
0.2
0.4
.
0.6
0.6
t
FIG. 4. Stress evolution
in the shear band for CA= 0, O,, > 0.
I.10
70x
w.
E.
OLMSTEAD
c’[
al
TEMPERATURE (N-40,
a=O, 8~2.4,
h=iO’,
p=lO
“)
‘________1 60’o
r--
0.0
L0.0
-: 0.2
0.4
_lI
~~~--
0.6
0.8
1 1.0
FIG. 5. Temperature evolution in the shear band for ,CI= 0. II,, > 0
up of stress has been suppressed (i.e. c( = 0), there is a gradual decrease in stress away from the initial value, until finally there is enough thermal softening to cause a precipitous stress collapse. This is accompanied by a dramatic growth in temperature. Qualitatively, these results are similar to those of WALTER (1992) ; however, some differences in parameter values and initial conditions do not permit a quantitative comparison.
5.
CONCLUDING
REMARKS
The goal here has been to introduce a new viewpoint for the investigation of the standard one-dimensional problem for shear band formation. The key feature of the approach is to represent the narrow zone of localization by a surface of discontinuity, across which there can be jumps in the values of velocity, temperature gradient and stress gradient. The problem can then be reduced to that of a half-slab with appropriate boundary conditions to account for the presence of the localized zone. Our analysis of the reformulated half-slab problem, as illustrated by specific examples, demonstrates that it can predict an evolution of behavior which leads to the dramatic temperature growth and stress collapse associated with shear band formation. We regard the viewpoint introduced here as a viable alternative, complementary to other methods used to analyse shear localization phenomena. While our approach does not yield all the small-scale details of the localization process, it gains through a significant simplification of the overall problem. Different modes of initiation can be implemented in our formulation, as is illustrated. It is also clear that the approach can accommodate more sophisticated phenomenological models of plastic straining, including those with hardening effects.
Shear bands as surfaces of disc~~~inui~~
709
ACKNOWLEDGEMENTS
The authors would like to acknowledge the input stimulating discussions about the phenomenon of particular aspects of our approach presented here. the National Science Foundation NSF MSS90-2167 at the University of California, San Diego.
of T. W. Wright and J. W. Walter through shear bands in general, as well as various This work was supported in part through I and AR0 contract DAAL 03-92-G-01 08
REFERENCES BEATTY. J., MEYER, L. W., NEMAT-NASSER, S. and MEYERS, M. A. (1992) Formation of controlled adiabatic shear bands in AJSI 4340 high strength steel. In S~~nt-~-JVnz~ and HEIDIStrength Plle~~~~~~na in ‘~ffteri~~~ (ed. M. A. MEYERS, L. E. MCRR, K. P. STAUDHAM~~~R), pp. 59-68. Marcel Dekker, New York. BURNS, T. J. (1990) A mechanism for shear band formation in the high strain-rate torsion test. J. i4ppi. Me&. 112, 836844. CLIFTON, R. J., DUFFY, J., HARTLEY, K. A. and SHAWKI, T. G. (1984) On critical conditions for shear band formation at high strain rates. Scriptu Met. 18, 443448. MAKCHAND, A. and DUFFY, J. (1988) An experimental study of the formation process of adiabatic shear bands in a structural steel. J. Mach. Phq’s. So&/s 36, 25 l-283. WALT.ER, J. W. (1992) Numerical experiments on adiabatic shear band formation in one dimension. ht. J. Plusticity 8, 657--693. WRIC;H,T,T. W. and BATRA, R. C. (1985) The initiation and growth of adiabatic shear bands. ht. J. Plasticity 1, 205-2 12. WRIC;NT, T. W. and WALTER, 3. W. (1987) On stress collapse in adiabatic shear bands. J. Mecl?. Phyr. Solids 35, 70 l-720. WRIGHT, T. W. (1990) Approximate analysis for the forination of adiabatic shear bands. .I. M&. P/zrs. Sdids 38, 5 15p530.