Adiabatic Shear Bands in Simle and Diolar Plastic Materials T W \-1RIGHT us Army Ballistic Research Laboratory Aberdeen Proving Ground, MD 215 R C BATRA University of Missouri-Rolla Rolla, Missouri 6541 Summary A simle version of thermo/viscolasticity is used to model the formation of adiabatic shear bands in high rate deformation of solids The one dimensional shearing deformation of a finite slab is considered Equations are formulated and numerical solutions are found for diolar lastic materials These solutions are contrasted and comared with revious solutions for simle materials Introduction Adiabatic shear is the name given to a localization henomenon that occurs during high rate lastic deformation such as machining, exlosive forming, shock imact loading, or ballistic enetration The rocess is usually described as being initiated by thermal softening in cometition with rate effects and work hardening Heat generated by lastic work softens the material so that eventually stress falls with increasing strain When that occurs, the mate:rial becomes unstable locally and tends to accumulate essentially all of any additional imosed strain in a narrow band In turn the local heating increases, and the rocess is driven further As the localization intensifies, substantial gradients of temerature build u so that in the later stages of develoment heat flows out of the band thus tending to offset the thermal buildu in the band In two revious aers Wright and Batra [1,2] have described the results of comutations that simulate the formation of a single shear band from a local temerature inhomogeneity in a simle material Strain gradients in the calculations reach aroximately 2 er ~m, an exerimental evidence [3] indicates K Kawata,J Shioiri (Eds) Macro- and Micro-Mechanics of High Velocity Deformation and Fracutre lutam Symosium on MMMHVDF Tokyo, Jaan August U-15, 1985 @ Sringer-Verlag Berlin Heidelberg 1987
19 gradients that ultimately are orders of magnitude larger Therefore, it has seemed worthwhile to reformulate the theory to include gradient effects This has been accomlished by modifying the diolar theory due to Green, McInnis, and Naghdi [4] to include a rate effect Formulation of the Problem In order to concentrate on fundamentals, the rocess has been idealized as one-dimensional shearing of a finite block of material Accordingly the three dimensional theory of Green, et ai [4] is summarized here for one dimension In addition, the yield function is taken to deend on the lastic arts of strain rate and gradient of strain rate as well as the usual variables A one dimensional shearing motion can be exressed as x = X + u(y,t), y = Y, z = Z for -H ~ Y ~ +H If it is suosed that on any fixed surface Y = const, surface tractions L do work against the velocity x and hyertractions a do work against the velocity gradient X'Y' then the one dimensional exressions for balance of linear momentum, energy, and entroy may be written as equations (2) T,y A + b = X '" '" In these equations band c are the simle and diolar body forces resectively, U is the internal energy, q is heat flux, r is the volumetric suly of energy, T is temerature, T) is secific entroy, and is mass density, which is constant The suerimosed dot and the connna followed by Y indicate differentiation with resect to time t and the material coordinate Y, resectively Following Green", et al [4], it will be convenient to define another stress by the equation A S :: T + O'y + C (3 Equations (2) hold for any diolar material, either elastic or
191 lastic, for motions of the tye given by (1) Now define shear strain and shear strain gradient by y :: X,y d :: Y'y = X'yy and suose that these can both be decomosed into elastic and lastic arts Y = Ye + Y, d = d e + d D None of Ye' Y, de or d are necessarily gradients, but of course their sums Y and d must be the gradients of x and y, resectively Next let K be a measure of lastic work hardening Finally, in the same sirit as classical lasticity, a scalar yield or loading function f is assumed to exist, but here it is taken to deend on lastic strain rates, as well as stresses and temerature, f (s, a, T, (5) In general this and other lastic constitutive functions could deend on elastic and lastic strains as well, but that ossibility will be ignored in this aer unless exlicitly stated otherwise The static yield surface is given by (5) with y = d =, and for quasi-static deformations a contirluity argument for neutral loading, as advanced in [4], leads to the following reduced forms for the lastic rates y ::; Aa d = A8 where a and 8 are constitutive functions that deend only on s, a, T, and K The multilier A is roortional to f = f s + f a s a + ftt, where the subscrits denote artial differentiation with resect to the arguments, and y = d = in evaluating the derivatives of f Now let it be assumed that (6) holds even when y, d, so that in the general case (5) becomes
192 f(s,, T, Aa(s,, T, K), AS(s,, T, K)) = K (7) and the criterion for elastic or lastic loading is simly (s,, T,, ) < K, elastic; f(s,, T,, ) > K, lastic (8) To make sense the derivative fa' obtained from (7), must be negative for all values of the other arguments Then if lastic flow is occurring according to (8), A may be found from inversion of (7), and because of the assumed monotonicity of f with resect to A, there will always be a unique solution with A > O With the assumtion that U, s,, T deend on the variables y e de, n and that q deends on the same variables lus the gradient T,y, standard thermodynamic arguments give T = au an' au s = Pay-, e au a = Paa e (9) The energy and entroy laws now reduce to T!l = sy + ad D - q,y + r, gyp + ad where y = d = during elastic deformation and f during lastic deformation (1) Constitutive Equations and Nondimensional Forms For comutational uroses simle constitutive equations have - c - 1) K = ~ (SY + ad) q = -kt,y (11 a = s, s=ho In (11)1 ~ is the usual elastic shear modulus and is a characteristic material roerty with the dimensions of length, To is a reference temerature, and c is secific heat With this simle choice for the internal energy there is no thermoelastic
193 effect and no thermal exansion The stresses and the temerature follow from (9) In (11)2 the function h(k) is the lastic sloe of a reference isothermal shear test, which is chosen to be 1 1-:!: n n n h=-,-k K Here K is the initial yield stress, n is the work hardening exonent, and ~ may be chosen to fit the initial sloe of an emirical curve of stress vs lastic strain Equation (11)2 states that K evolves according to the lastic work done no matter what the conditions of the test Equation (11)3 is Fourier's law and equations (11)4 and (11)5 have been chosen in a simle form that is dimensionally correct and leads to ositive lastic work Finally in (11)6 the function f has been chosen to be made u of three multilicative factors involving the stresses, the temerature, and the lastic rates, resectively In (11)6 a is the coefficient of thermal softening, b is a characteristic time, and m is the strain rate exonent In general the value of could be different in each of the four laces that it aears in (11), but such comlexity is not warranted for the time being since there is no microscoic theory available for guidance In nondimensional form the full set of equations in the absence of body forces and suly of energy may be written as follows: Momentum: Energy Constitutive s = ~(V'V Yield: - y ) y P - d ) D K - K - n ~ (SY = As, d - - A a R, + Q,od) h = K D 1/1 l_l ~ where A > only if (S2 + 2)~ > K(l - a8) = otherwise
In (13) the temerature T has been relaced by the temerature increase e = T - T The nondimensional variables are related to their dimensional (barred) counterarts as follows: Y = Y/H, _ t = tyo (14) Besides m, n, and ~ there are six other nondimensional ara- meters, which are related to their dimensional (barred) counterarts as follows: b = byo' = I/H (15) In (14) and (15) y = v(h,e)/h is the average alied strain rate between the boundaries Y = + H Homogeneous Solutions and Perturbations Equations (13) have homogeneous solutions with v = Y and all other deendent variables indeendent of Y With initial values taken to be s(o) = K(O) = 1 and 8() = a(o) =, the solution for s, K, and 8 is exactly the same as for a simle material (see Wright and Batra [1,2]) and a is identically zero Figure 1 shows several curves for homogeneous solutions of equations (13) obtained for articular choices of the arameters With a = b = there is no thermal softening and no rate effect so the resulting curve is simly the slow, isothermal stress/strain curve, called the reference curve, from which the function h() was derived With a =, but with a finite value for b, the curve shows the isothermal resonse at a high rate of deformation With finite values for both a and b the curve shows the adiabatic resonse at a high rate of deformation The nondimensional arameters chosen here are the same as those listed in [2J, namely = 3928 x 1-5 k = 3978 X 1-3 a = 4973, ~ = 243
195 ISOTHERMAL CURVE a =, b 1" U') U') ~ U') w cr ~ a:: «w I (/) 'ADIABATIC CURVE a #, b # a "'"REFERENCE CURVE a=b=o I - -, 5 AVERAGE 1 STRAIN, YAVE 15 - n?n Fig 1 curves Tyical reference, isothermal, and adiabatic resonse w ~~ "<:( :: zo:: -w «~ O::z ~w (/) () W l- I (/) r- «-J Z D- /1VFR/1GF - -- STRAIN, ~VE Fig 2 Plastic strain rate in the center of the band with a temerature erturbation
196 n = 9, 1/1 = 17, b = 5 X 16, m = 25 The value for ~ is immaterial for the homogeneous resonse as are and k The adiabatic resonse curve is tyical Initially the stress rises elastically above the reference curve, but as the overstress increases, lastic straining sets in, the temerature rises, and the resonse softens relative to the isothermal resonse Eventually thermal softening wins over work and rate hardening, so that the stress asses through a maximum (indicated by P in the figure) and then decreases with further deformation The general character of the homogeneous resonse is well understood and has been reorted many times in the literature,although the way the resonse changes with the various arameters deends on the articular model used for thermo/visco/ lasticity Once eak stress has been assed the material becomes extremely sensitive to inhomogeneities, and the deformation has a strong tendency to localize To examine this behavior for a diolar material and to comare the results to revious calculations for a simle material, calculations have been made for the resonse following a small temerature erturbation At the oint marked I in Figure 1 the comuted homogeneous resonse was modified by adding a smooth temerature bum (height 1 and width 5) to the basic homogeneous resonse After recalculating s so that the yield condition in (13) is still satisfied with the new temerature distribution, the roblem was restarted as a new initial-boundary value roblem with all other initial values as calculated reviously and with boundary values at Y = + 1, taken to be v = + 1, and a = 8,y =, With these boundary values the average strain rate in the stri is the same as in the homogeneous calculation and the material remains adiabatic overall The same aroach was used in [1,2] for a simle material, but now the new material arameter is nonzero Comutations have been made for =, 1-~ and 1-2 After casting the equations into a weak form, solutions were found using the finite element method for satial discretization and an imlicit Crank-Nicolson scheme to march forward in time Previously a forward difference method was used for the time integration
197 [1,2], but the ste size was necessarily very small for the diolar case - - shows the lastic strain rate in the center of the band as a function of the average alied strain The nondimensionalization is such that increments of time are exactly equal to increments of the average alied strain, so the curve may be interreted equally well as a time lot After a brief interval during which the field variables regain their essential balance, the central lastic strain rate begins a slow but accelerating climb Eventually it turns u rather sharly for t = or 1-~ and somewhat less sharly for t = 1-2, the first two cases being virtually indistinguishable Also shown is the result from revious calculations [2] with t =, where the strain rate increases even more dramatically The delay in the resonse for the resent case relative to the revious one with t = is robably due to a nonhysical daming, which is introduced by the Crank-Nicolson scheme as comared to the forward difference method The result for t = 1-2 relative to the other two cases shows the same stiffening effect due to the material length t that was reorted in [2] It is clear that the rate of increase of lastic strain rate can be substantially retarded if t is large enough All cases calculated so far show the develoment of a late stage lateau, but the comuted value is a numerical artifact and is not a hysical result Since v = 1 at the boundary Y = 1, it follows that As shown in Figure 3, the lastic strain rate builds u in the center, but it decreases in the outer region In the lateau region of the calculations, the lastic strain rate is nonzero at the center but falls to zero at Y = 1 With linear interolation the nonzero art of the distribution is triangular, so the level of the lateau is effectively caused by the length scale that is introduced through the comutational grid In the
>,Q w- ~ :: -2 J = 1 ; "YAVE ~ 79, 93, 17-4 2 = 1 ; "YAVE = 79, 93 ~ «:: l- V) u i= V) «-l ~ v ~ -' -1"1~ ~: 4 n&: ~~ 8 "- In t DISTANCE, Y 3 Cross lots of lastic strain for increasing deformation b cn~ cn w Ct: t1 Ct: <1: -1 ~ (5 O2~ 16 Ol? 8-2 ~ = 1 ; YAVE = 79, 88, 96 :-4 --- 1 = 1; YAVE = 87, 99 vā t nl~tfnr~ y 2 4 6 8 1,,, 4 Cross lots of diolar stress for increasing deformation 16 12 \ ~ " \ \, "" "- J=O "" 1 =, 14 8 4 FORWARD DIFFERENCES, = --- CRANK-NICOLSON, i=o, 1-4 - CRANK-NICOLSON, 2= 1-2 --- HOMOGENEOUS CASE! I I~ I I I 7 8 9' 1 11 12 13 AVERAGE STRAIN, r; AVE Fig 5 Stress in a she~r h~nn
199 revious results [1,2] a more comlicated interolation scheme and a finer grid were used, which allowed the lateau to be delayed until the lastic rate reached aroximately 8, as indicated in the figure Although the level of the lateau is nonhysical, the fact that it is very likely fixed by the length scale of the grid suggests that one might exect a true lateau to develo where the level would be determined by the hysical length scales of the roblem Two such hysical lengths, one arising from thermal conductivity and one from viscous stress effects, were identified in [1] and the arameter Q, is a ossible thir,d one All three length scales may be very small and beyond convenient numerical resolution The cross lots of lastic strain rate at increasing values of average strain, as shown in Figure 3, are taken well before the lateau is reached, and so should be accurately reresentative of the rogressive localization that occurs The five crosses on the curves in Figure 2 corresond to the five curves in Figure 3 As the deformation continues, the lastic strain rate builds u in the center, but decreases in the outer regions The relative stiffening of the diolar material can be seen clearly in Figure 3 as well For examle, when the average strain reaches 93, the diolar case with t = 1-2 has reached only half the value for the cases t = or 1-~, and obviously continues to develo at a much slower rate Since the eak in lastic strain rate is lower for larger values of t, naturally the distribution is broader and the rate falls more slowly in the outer regions, as well Temerature lots are not shown, but the results are similar to the revious calculations At first the temerature follows the ath of the homogeneous case quite closely, but as the deformation localizes so does the temerature distribution with a eak forming in the center and a lateau forming in the outer regions as the lastic heating falls to zero In all calculations so far the case for t = 1-4 shows essentially no difference from the simle material The reason for this is shown dramatically in Figure 4, which shows the diolar
2 stress distribution at several values of time The assumed symmetry of the roblem requires that the diolar stress be an odd function of Y, and hence that it vanish at Y = O The eak of the distribution for ~ = 1-2 lies near Y = 2 and moves somewhat toward the center for increasing time The diolar distribution for ~ = 1-4 shows a similar shae, but now the eak value is only about 2% of the eak value for ~ = 1-2 at the same time For the smaller value of ~ the diolar effect is so weak that it has no effect on the calculations rior to the formation of the false lateau Figure 5 shows the comuted values of shear stress as a function of average alied strain At first the stress for each of the erturbed cases follows the homogeneous case quite closely, but eventually it deviates markedly The stress remains nearly constant through the cross section at all times until the strain rate accelerates sharly uwards Then the stress begins to fall, first in the center and then in the outer regions The start of the dro in stress is clearly evident in the comuted results, although in every case the curves have been drawn ast the robable limit of validity (the tick mark on the curve) Here again the stiffening effect of the diolar stress is evident in the delay and rate of dearture from the homogeneous case It is also interesting to note that even though the diolar stress attains significant values, and the stiffening effect is clearly evident in the distribution of lastic strain, there is little difference between the shear stress s and the traction T, as might have been exected from equation (3)
21 References 2 W httw - r7g :'\ o BtRC a ra,-~ adi-abati-c shear bands; Int j P' LL ast 4 c ; L ty ' LL ( ' 98'5) ;;con: ~"c r'e: s s; Wright,T c c W; Batra,R'C: c c'c c ationand C 1 growth ofadiabatrc 8 shear c (1 8 ~ ) cc c cccccc,c rates t S~~ YMAT 5, Par1S 9 Moss,G:L:Shea:stral;ns stral;n rates and temerature changesl;ti adl;abatl;c shear bands In Shock waves and hi h-strain rate P-Q~!:!2~ L Eo e so ew York: Plenum Press 1981-4 Green, AE; MclnnLs, B C; Naghdi, P M continua with simle force diole Int J Eng ~ '7 " " QI "\ "\-"\ LL c Elastic- Sc16X1968) c c lastic