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Theoretical latent hardening of crystals in double-slip—II. F.C.C. crystals slipping on distinct planes
1983. J. Mech.Phys.SolidsVol. 31,No. 3,pp.231-250, Printedin GreatBritain.
0022~5096/83/030231-20 $03.00/O Pergamon PressLtd. 0 1983.
THEORETICAL LATENT HARDENING OF CRYSTALS IN DOUBLE-SLIP-II. F.C.C. CRYSTALS SLIPPING ON DISTINCT PLANES K. S. HAVNER Department
of Civil Engineering, North Carolina Raleigh, NC 27650, U.S.A.
State University,
and S. A. SALPEKAR~ Corporate
Consulting and Development Company Raleigh, NC 27622, U.S.A.
Ltd.,
(Received 28 October 1982)
ABSTRACT THE LINE of investigation of theoretical latent hardening in axially-loaded cubic crystals initiated by HAVNER and SHALABY (1977), based upon their “simple theory” of rotation-dependent crystal anisotropy (and continued through various other works), is completed here with the analysis of finite double-slip on distinct planes in f.c.c. crystal lattices. In conjunction with a previous study by the present authors of double-slip on a common plane (HAVNER and SALPEKAR, 1982), the genera1 theoretical result is that any f.c.c. crystal which initially deforms homogeneously from the virgin state in double-slip, under either tensile or compressive axial loading, will continue in that mode of deformation to very large shears (y = 1.5 and beyond). The paper also includes a comparative review of various hardening rules (including the “simple theory”), with regard to both theoretical predictions and mathematical issues, that may be found instructive.
1.
FOREWORD
IN HAVNER and SALPEKAR (1982) [Part I of this two-part series, here designated (HS)] the anisotropic latent hardening of axially-loaded f.c.c. crystals, finitely and homogeneously deformed by double-slip on a common slip plane, is determined according to the “simple theory” of hardening proposed by HAVNER and SHALABY (1977). Other analyses of axially-loaded cubic crystals in single-slip and higher order multiple-slip positions, based upon this theory, have been carried out by HAVNER and SHALABY (1978), in the three-part series HAVNER, BAKER and VAUSE (1979), HAVNER and BAKER (1979), and VAUSE and HAVNER (1979) [henceforth designated (I), (II), (III) respectively], and in HAVNER (198 1, 1982b). In the present paper we complete the analysis of theoretical hardening of virgin f.c.c. crystals under tensile or compressive axial loading, according to the “simple theory”, with the solution of finite double-slip on distinct planes.
t Former U.S.A.
address
: Department
of Civil Engineering,
North Carolina
231
State University,
Raleigh, NC 27650,
K. S. HAVNERand S. A. SALPEKAR
232
2.
REVIEW OF CRYSTAL HARDENING RULES
Following the notation in (HS) and earlier works, we let zk denote the critical strength in the kth crystallographic slip system, b,, nk denote unit vectors in the slip and normal directions respectively of that system, and jj denote the slip-rate in the jth critical system, with all jj > 0. The “simple theory” of anisotropic latent hardening then may be expressed [HAVNER and SHALABY, 1977, equation (23)] ?, = hxjj-2 j
tr N,aR
= x(Ltr i
Nkaj)yj,
with N, = {(b 0 n)k}sym, aj = {a(b 0 n-n
h > 0,
(2.1.1)
0 b)j}sym.
(2.1.2)
Here h is a hardening parameter ; o is Cauchy stress ; t2 is the relative spin of material and lattice ; {. . .}symsignifies the symmetric part of a tensor product ; the summation onj is over the n critical systems (in each of which tr Nja = rj) ; and k ranges over all systems, critical or not. (In a slip system that is momentarily non-critical, tr N,a < zk.) The initial value of zk in a statistically isotropic virgin crystal is r0 for all crystallographically equivalent systems (e.g. the set of 24 systems { 11 l} (i10) in f.c.c. crystals). Obviously, it is the rotation-dependent? second term in (2.1.1) that produces the anisotropic part of the hardening, the physical motivation for which is discussed at length in (I). The simple theory has been shown to account for various qualitative features of finite deformation experiments in single-slip and 6- and &fold multiple-slip positions [see (I), (II), (III) and HAWER, 1981, 1982a, b]. In particular, (2.1) is a universal theory of the common phenomenon of “overshooting” in both f.c.c. and b.c.c. crystals in single slip. Three other specific hardening rules of demonstrated usefulness relevant to finite distortion of cubic crystals are: (i) Taylor’s classic theory of isotropic hardening (TAYLOR and ELAM, 1923, 1925); (ii) an empirically-based 2-parameter rule (see NAKADA and KEH, 1966, or JACKSON and BASINSKI, 1967); and (iii) the hardening rule recently proposed by PEIRCE, ASARO and NEEDLEMAN(1982) which for convenience we shall call the P-A-N rule. These are given by : (i) Taylor’s
rule : Z, = HE yj for all k,
(ii) 2-parameter
(2.2)
rule : Z, = H c yl + H’ x1;,, m 1
with n, = nk, (iii) P-A-N
H > 0;
% #
nkj
(2.3.1) (2.3.2)
H’>H>O;
rule : t, = h 1 yj - tr N,aR - tr flkaB (2.4)
= x(h-3
tr N,ai+i’tr
a,Nj)?ij,
h > 0,
i
where R, 1 [(b @ n)Jskew (the skew-symmetric part of the dyadic product), and d is the Eulerian strain-rate (“rate of deformation” tensor) of the crystalline material relative to t Here is meant, of course, the rotation of gross crystalline material relative to the underlying not rigid body rotation with respect to an arbitrary reference frame.
atomic lattice,
Latent hardening of crystals in double-slip-11
233
an effectively rigid lattice (i.e. a lattice experiencing infinitesimal strains that are neglected here in comparison with the finite crystallographic slips ofinterest). The latter two rules are anisotropic, whereas Taylor hardening is (of course) isotropic; and the simple theory may be either isotropic or anisotropic, dependent upon stress state (relative to lattice axes) and the resulting degree of symmetry of the deformation. It may be noted that the P-A-N rule (2.4) incorporates one-half of the relative spin dependence introduced as a source of anisotropy by HAVNER and SHALABY(1977) in the simple theory (2.1), and adds another anisotropic contribution dependent upon the relative strain-rate 8.t This latter term is reminiscent of, but not equivalent to, the translational (or “kinematic”) hardening rule adopted in BUDIANSKY and WV (1962), namely 2, = 2H tr N,d = 2H c tr (NJVj)jj, with H the Taylor modulus (i.e. the slope j of the resolved shear stress vs slip curve in single slip). Another so-called “kinematic hardening” rule for crystals has been suggested by WENG (1979); in terms of the Taylor modulus, Weng’s rule may be expressed ?, = H tr (b @ n),J& -R)
= H 1 (bk - bj) (nk * nj)jj.
The obvious linear combination of isotropic with translational hardening was mentioned by HAVNER (1971). However, all such theories that are symmetric positivedefinite, for 5 or fewer active systems, in the matrix of physical hardening moduli Hkj(?, = c Hkjyj) give results that are qualitatively inconsistent with experiment for finite distortion of crystals from both single and multiple-slip axial-load positions. (This is discussed in some detail in HAVNER, 1981, section 4.) Consequently, such theories need not concern us further. Another valuable perspective on the various hardening rules is a mathematical one. Only the simple theory and the P-A-N rule, of all those proposed (including Taylor hardening), lead to the possibility ofminimum principles in self-adjoint boundary value problems of rate type, either for inhomogeneously deformed single crystals or polycrystalline solids. This is because only these two fall within the general class of hardening rules that permit such mathematical features. This class is defined for the logarithmic strain measure (at constant pressure, to first order in infinitesimal lattice strain) in HAVNER and SHALABY [ 1977, equation (20)] as +, = C (hk. - tr N,aj)lij,
h,j = hj,.
(2.5)
For the “simple theory” (2.1) obviously all hki = h, hence the name, whereas for the P-A-N rule, from (2.4) and (2.5), hkj = h+S tr (N,aj+ Njak). (This equivalence is obscured by PEIRCE et al. (1982) as they use a single symbol to represent two different sets of moduli in their equations (3.31) and (3.32). In their 2-parameter equation h is
t PEIRCE et al. (1982) also
“independent”
hardening
propose a 2-parameter equation that is merely the linear superposition of the rule Of KOITER (1953) with their basic theory. Koiter’s rule is ir = H c akjfj (that is,
zero latent hardening). Koiter did not propose usefulness in that context.
this relation
for crystal
mechanics,
and it is oi questionable
234
K. S. HAVNERand S. A. SALPEKAR
replaced by qh + (1 - q)h lSkj, the latter term being the Koiter contribution.) It is also desirable that the matrix (hkj) be non-negative-definite as well as symmetric. [As shown in HAVNER, 1982a, equation (4.28), this pertains to a material-stability condition suggested by HILL (1968) ; also see the brief discussion that closes HILL and HAVNER, 1982, pp. 21-22.1 The simple theory obviously has this characteristic. Whether the P-A-N rule also is non-negative-definite is not self-evident and apparently has not been determined. Uniqueness criteria and minimum principles for crystalline solids at finite strain, based upon symmetric hki, may be found in HAVNER (1977) (also see HILL, 1978, and the brief review in HAVNER, 1982a, section 4.4). The generalization of (2.5) to a broad class of strain measures has been given by HILL and HAVNER (1982). In the present notation, for the ith system tr Nia(n-Cb)
?i = Chijjj-2 j
=
C(hij-tr .i
iV,aj+2C
tr NifJiVi)jj,
(2.6)
where c is a measure-dependent parameter (zero for the logarithmic measure). As remarked in HILL and HAVNER (1982, p. 21), the simple theory “is unique as a theory of rotation-dependent crystal anisotropy” in that “there can be no other choice of strain measure and associated parameters hij which will both ensure self-adjointness of boundary-value problems and result in isotropic hardening under conditions of zero relative spin Q (when such hardening is to be expected).” The P-A-N rule, although incapable of predicting isotropic hardening in 6- or 8-fold multiple slip (for which see the discussion in WONSIEWICZ and CHIN (1970, p. 2721) of experiments on aluminum), is useful in that it apparently is, by design, a theory of limited “overshooting” (i.e. eventual double-slip after axis rotation beyond the symmetry line). [The empirical rule (2.3) can also give limited overshooting, depending upon the relative values of H and H’.] PEIRCE et al. (1982) employ their theory, in conjunction with an ad hoc plane strain model and the finite element method, in a comprehensive study of shear band formation in single crystals. However, the P-A-N rule has not been evaluated explicitly, so far as we know, for either the f.c.c. or b.c.c. lattice geometry. Hence, its predictions for homogeneously deformed cubic crystals in single and multiple-slip are not yet determined.
3.
LATENT HARDENING IN DOUBLE-SLIP
Consider the [OOl] stereographic projection for f.c.c. crystals in axial tension shown in Fig. 1. The 24 stereographic triangles are identified as slip systems in Table 1 (with a bar above a number designating the opposite sense of slip). We choose the initial orientation of the loading axis such that a2 = (111) [iOl] is slip-system 1 in tension. (In compression, all slip systems exchange position with their opposite-sense systems, whence triangle a2 becomes system a2 = (111) [ lOi], etc.). Then, the locus of all doubleslip positions is the boundary of triangle a2 (Fig. 1). The symmetry-line segments [01 l]-[ill], [il l]-[OOl], and [OOl]-[01 l] (as one proceeds counterclockwise around a7) will be designated sides 1, 2 and 3 respectively.
Latent
hardening
of crystals
235
in double-slip--II
fro01
FIG. 1. Standard
[OOl] stereographic
3.1 Initial positions on symmetry
projection
line [01 i]-[il
of f.c.c. crystals.
l] (side 1)
Side 1 obviously is distinctly different from the other two as it corresponds to doubleslip on the same plane; consequently, there is a fundamental qualitative difference as regards axis position in the resultant deformation.? For equal double-slip from an initial axis position (tension or compression) along side 1, the limiting positions from pure kinematics lie outside triangle a2 ; they are [Zl l] in tension, which lies beyond [il l] on the continuation of symmetry line [01 l]-[ill], and [ll l] in compression. As regards the hardening theories, both the simple theory (2.1) and the 2-parameter rule that is, axis-rotation beyond the higher-symmetry (2.3) predict “overshooting”, position [ill] (tension) or [01 l] (compression), the latter rule by empirical design and the former as established in (HS). Taylor hardening (2.2) of course does not predict overshooting, and the P-A-N rule (2.4) has not been evaluated for this case as yet. hardening, would predict (Positive-definite hardening rules, such as “kinematic” slipping on other systems before the loading axis reached the higher symmetry position in either tension or compression.)
TABLE
Plane Direction System
[oil] al
1. Designation
(111) [ioil [ii01 a2
a3
[oiil bl
of slip systems in jIc.c. crystals (iil) [Toil [ii01 b2
b3
(ill) -coil-j [ioil cl c2
[ii01
[oiil
i:%\
[ii01
c3
dl
d2
d3
t The issue of single vs double-slip from all double-symmetry positions, for each of the hardening (2.1H2.3), is comprehensively investigated in HAVNER (1982b) and reviewed in (HS), section 2.
rules
236
K. S. HAVNER
and
S. A. SALPEKAR
Explicit solutions for latent hardening of all systems, according to the simple theory, for double-slip along side 1 are given in (HS). We include here, for subsequent comparison with results for sides 2 and 3, only the equation for f, as a function of the equal slips y [HS, equation (3.15)] :
G = [2H
+ (,,,2
j;H
dy)(A,+B,y
-&)]‘.
(3.1)
A,, B,, c2 are constants that depend upon the particular slip-system and/or initial axis position (along [01 l]-[ill]) and whether the loading is tensile or compressive. In active systems 1,2 (respectively a?!, a3 in tension and a2, a3 in compression) both A,, B, are zero, and tr N,a throughout the deformation for all latent systems and initial axis positions. [Refer to (HS) for final equations, including specific expressions for the constants, and their numerical illustration.]
3.2 Initial
positions
on symmetry
line [il l]-[OOl]
(side 2)
Along this side of stereographic triangle a2 (Fig. l), the range of double-slip orientations (excluding corners) evidently is z1+ z2 = 0, z3 > z2 > 0, where z is a unit vector coincident with the axial load. Consider first the case of tensile loading of a long, slender crystal, and let CJdenote the uniaxial Cauchy stress at any stage of deformation. Then d = ozQz, with bl = (both (2. l),
,J = 46 r&z
+
4~J,
(3.2)
z, = z0 + 2 f H dy, the critical strength of active systems a2 = (111) [iOl] and (111) [Oll]. The evolutionary equations for critical strengths in all systems latent and active) in equal double-slip g, = j2 = j, from the simple theory are Z, = (2h - tr N,a)j,
where 1, 2 signify systems a2, bl respectively.
a = a, + a2,
(3.3)
For axial loading
aj = 2{baj}sym= ~C(~~bj){~O~j}sym-(~~~j){~Obj}sy~l~ whence, upon substituting we find
the second equation - 2121,
a=
r, 2(r, + z&3
!
of (3.2) and the respective
2z;-1;
(sym)
(2N,,+N,,-NN,,+3N,,),
4r*r,
(3.5) I
and simplification,
*-(I.,,-N,,),? 2
unit vectors,
- 214 + z:
2& -2z,z,
There follows from (3.3) and (3.5) after rearrangement fk = 2hj-z,
(3.4)
3
1?j. (3.6)
237
Latent hardening of crystals in double-slip-Ii
In active systems a2, bl, f, = fz = 2Hj and (2N,,+N13-_N,,+3N,,), = - 1/(2J6), k = 1,2. Thus
= J6/4,
(IV,, -N&
(3.7) Consequently, we may write f, = 2Hj+z, A,i
=
2E2- Eg A,-----35
(J6/4)+ 3(N,,
-N2& i
& = l/(2J6)-4(2N12+N,3--23+3N3,),,f (with of course A, = B, = 0 for active systems 1,2). There remains to determine the two algebraic fractions in direction cosines I 2r z3 as functions of double-slip y. Let 2 denote the material stretch in loading direction z. From the general kinematic solution for (proportional) double slip on distinct planes [see HAVNER, 1979, equation (34)], we have for yI = y2 = y: AZ= zO+(l/c)([KI --fW)W,
exp(cy)-K2
ew@y)+&
exp(-cy)+K:
exp(-cy)--0s
cos s,!&b,
ICl?lbI,
(3.9)
where $9 is the angle between z. (initial position of the load-axis) and active slip-plane normal nj, and I<, = &cos I/?:- K cos iji),
(3.10.1)
K2 = f(cos *: + Jc cos I&
(3.10.2)
K = c,&,
c = (c,,c,,)‘~~,
cij = -b,*nj.
For systems 1,2 = a2, bl these reduce to : cl2 = czl = - 2/,/6, c = 2/J6, K K, = cos r@ = cos I,@= (l/,/3)& K, = 0. Thus lz = z. + (1/,/2)&exp
(cy) - 11 (b, + b2).
(3.10.3) - 1, and (3.11)
Consequently, %z,= +(22; - 1:) + $E$ exp (cy), II, = 2: exp (cy),
(3.12)
and the required algebraic fractions are 21,-l, .3z3
01 3z2 -=_=-Y> exp tcy) t2 + z3
21; - li 3c2 f exp (c~) c,+exp(cy) ’ ” =T’
(3.13)
(Note that jc2/ G l/3, equalling l/3 in the corner positions [OOl] and [ill], and that c2 = 0 at [i12], which is the limit position for axis rotation in equal double-slip along side 2.) Therefore, the final equations for rates of change of all critical strengths according to the simple theory are
6‘ = {2H+(r,+2j)
dp)[A,&
+ B~~‘eex$ojj,
(3.14)
238
K. S. HAVNER
and S. A. SALPEKAR
with m = 3. (The reason for introducing m will become apparent later.) The striking difference in form between the equations for side l(3.1) and side 2 (3.14) may be noted. Before dealing with the integration of (3.14), we turn to the case of compression of a short, squat crystal (with well-lubricated end faces). The active systems 1, 2 then are a2 = (111) [lOi] and bi = (iii) [OTT] (the slip directions being the opposites of those in tensile loading), and 0 = -J~-c,/[(z, + z&J < 0. It is obvious from (3.4) that aI, a2 remain the same functions of z2, z3 (as both Q and the bj change sign), hence (3.5) and (3.6) apply as well to compressive loading along [ill]-[OOl]. However, N,,), reverse signs in active systems. Thus, (2N1,+N1s-%,+3N,3)k and (K(3.7) and (3.8) are replaced by (3.15) and
A, = (J6/4)+(2N,,+N,,-N,,+3N33)k,
(3.16)
B, = 1/(2J6) - (N, 3 - N,,),. Again, we must express the algebraic fractions involving direction cosines z2, z3in terms of y. [These will be different from (3.13) because the kinematics in compression are different from the tension case.] Now let 2 denote the spacing stretch (i.e. the stretch of the spacing of specimen end planes). Then, as the area1 stretch of the loading faces equals the inverse of the spacing stretch at constant volume, the general kinematic solution for I in equal double-slip on distinct planes for compressive loading [see HAVNER, 1979, equation (24)] can be written AK11
=
zo-(l/c){[c,
exp (c~) - C2 exp
( - CY) -(l/4
+lc[Cl
~0sz&n,
exp(cy)+C,
where xy is the angle between z0 and active slip-direction 2) and c, with c and c = 2/,,/6, K
K =
cos x;-j,
c,
= i[cos
exp(-cy)-cos
xYln2},
(3.17)
bj (note that cos ~9 < 0,j = 1,
+(1/K)
cos x!],
(3.18)
= i[cos
x7+(1/K)
defined l,andC,
as in (3.10.3). For systems 1, 2 = a2, bi : c, z = c2, = 2146, (I; + zi), C, = 0. Consequently, = cos x’: = cos x: = -(l/J2)
a- ‘I = r0 + (J3/2) (1; + 1:) [exp (cy) - l] (n, + n2).
(3.19)
Thus, in compression,
AK’z, =
zi,
ilX’z, = -z$+(z$+z$ exp(cy),
(3.20)
Latent
and the required
algebraic
hardening
of crystals
fractions
239
in double-slip-II
become 12
c2
=p
12+ 13 21, - 13
(3.21.1)
exp (CY)’ 3c2 + exp W
(3.21.2)
c2 + exp (CY) ’
13
4
c2=--
(3.21.3)
1; + 1; ’
so that the equation for <, is again precisely (3.14) (with of course A,, B, and c2 differently defined in tension and compression). Note that the dependence upon initial position z0 along side 2 is expressed solely through the parameter c2 for both the tensile and compressive loading cases. [We consider this an improvement in form upon both equation (3. l), from (HS), for double-slip on side 1 and the general equation for latent strengths in single slip in (I), each of which equations is expressed in parameters having dual-dependence upon position and slip-system.] Also note from (3.21.3) that Jc21< l/2 in compression (equalling l/2 in corner position [ill]) and that c2 = 0 at [OOl], which is the limit position for axis rotation in equal double-slip in compression along side 2. We now address the integration of evolutionary equations (3.14), which apply in both tension and compression as shown, restricting consideration here to the linear approximation z, = 50 + 2Hy of an active hardening curve. [Note also that the solution for constant H may be used in a highly accurate stepwise approximation of an arbitrary experimental curve, as was done for copper crystals in single slip in (I).] The requisite integrals are
y s
c,dy
r0
~
0 exp (cy)
= (ro/c)c2C1
-ew--cy)l,
y mc2 + exp (cy) dy = (ro/c) m(q) TO s 0 c2 + exp (CY) 1
2H
2H
YCZY = (2H/c2)c2[1 -(l s 0 exp (CY)
y,mc2 + exp (cy) s o c2 + exp (CY)
- (m - 1) In
+cy) exp (-cy)],
y dy = (2H/c2)
im(cy)2 -(m - 1) In [c2 + exp (cy)]
(3.22)
Y
+(m- 1) s0
ln Cc2 + exp (cr)lc dy
with ’ In [c2 + exp (cy)]c dy = $(cY)~ s0
f
t-c21
nz
II=1 x
this last a rapidly convergent series. Thus, according to the simple theory, corresponding
[l-exp(-racy)] the final equations to linear hardening
as
lc2l < 1,
for latent strengths in active systems 1,2,
K. S. HAVNER and S. A. SALPEKAR
240
are c2A,[1-exp(-cy)]
rk = r,0+2Hy+(r,/c)
+b{m(cy)-(ml) In [Cz:~~I(C?)]})+(2H/c2)(c2A,11-(i+c?) x exp (- cy)] + B, &2m - 1) (cy)’ -(m - 1) In [cz + exp (cy)]
-(m-l)
f
II=1
(3.23)
y[l-exp(-nc;)]]),
with r: = r0 for a crystal in the virgin state at y = 0, m = 3 for an initial loading position along side 2, c = 2/J6, and (restating the previous expressions) : Tension along [il l]-[OOl]
: (3.24.1)
c2 = (22; - z$/(32$, A, = (J6/4)+3(N,,-N& Compression
axis
B, = 1/(2J6)-4(2N,,+N,,-N,,+3N,,),;
(3.24.2)
:
along [ii l]-[OOl]
(3.24.3)
c2 = -z;/(z;+I;,,
(3.24.4)
B, = l/(246)-(N,,-NJ,.
A, = (&4)+(2N,,+N,,-Nz,+3N&
We shall establish in the next section that (3.23) also applies to equal double-slip along side 3 ([OOl]-[Oil]) of stereographic triangle a2 in both tension and compression, with m = 2 for that side and A,, B,, c2 of course differently defined.
3.3 Initial positions on symmetry
line [OOl]-[01
l] (side 3)
For this side of the stereographic triangle, the range of orientations (excluding corners) is given by zI = 0, z3 > z2 > 0; and active slip systems 1, 2 in tension are respectively a2 and c2 = (ill) [loll. Equations (3.2)-(3.4) apply once more, and we find 0
0
0
212% 1:-z: - 2Z,l,
(sym)
I .
(3.25)
Thus, from (3.3) and (3.25), r f, = 2hj-r,
2(N,,-N,,)+ 1
In the active systems k= 1,2;hence
?I = fz = 2Hj
2h = 2H+z,
+ 2(N2a)xy 2
3
and 2(N22-N33)k
= -2146,
1 j.
2(N&
(3.26) = l/,/6,
(3.27)
Latent
hardening
of crystals
in double-slip--II
241
and fk = 2Hj+z,
-A,
A, = (2/46) + 2W,,
-&
(3.28.1)
+ Bky),.
-N&k,
B, = (l/J6)
- 2W23)k
(3.28.2)
(with A, = B, = 0 for active systems 1, 2). To express the algebraic fractions involving 12, l3 as functions of y, we again use the kinematic solution (3.9) in tension. For systems 1, 2 = aZ, ~2, the constants (3.10) are: c = 2146, IC= - 1 (all as on side 2), and K, = cos $y = cos I@ cr* = c2r = -2/J6, = (l/J3) (&!+ I:), K, = 0. Thus, from (3.9), II = lo + (l/2) (I: + 1:) [exp (cy) - 11 (b, + bJ,
az,= z;, [Note the similarity fractions are then
%z3= -
z:+(1:+ z$ exp (cy).
with (3.20) for side 2 in compression.]
12
-----__ 12
+
CZ 13
exp
13 ---= (cd
(3.29)
’
l2
2~2 c2
l3
+ +
exp
(3.30) The required
algebraic
(CY)
exp (cy) ’ 1 >
(3.31)
1:
c2zzp z;+z3
0’ i
with lczl < l/2 (equalling l/2 in corner position [01 11) and c2 = 0 at [OOl], the limit position of axis rotation in tension along side 3. Upon substitution of (3.31) and z, = z. +2s H dy into (3.28.1), we recover the general form (3.14) of side 2, now with m = 2. Turning to compressive loading along side 3, the active systems are a2 (as for side 2) _and c2 = (ill) [loll, and e = - 46 z,/[(zz + z3)zJ < 0. Again, a1 and a, as functions of 12, z3 do not change, hence (3.25) and (3.26) also hold in compression; but (IV,, -N,,), and (IV,,),, k = 1,2, reverse signs. Thus, (3.27) and (3.28) are replaced by 2k=2H+q
and
(
LA--J6
12+13
1
13-12
J6
. 13
(3.33.1)
+ Bll
A, = (2/&l + 4(N,,)k, (Ak = Bk = 0 for active systems a2, As before, the algebraic fractions in compression. For systems 1, 2 = = 2146, c = 2146, IC= 1, and C, from (3.17)
Bk = (I/ J6)-(N22
(3.32)
“?,
-N33h
(3.33.2)
~2.) are determined using the kinematic solution (3.17) a2, c2, constants (3.10.3) and (3.18) are : cl2 = c21 = cos xy = cos $j’ = -(l/J2)1:, Cz = 0. Hence,
%- ‘z = z. + (J3/2)zg[exp (cy) - l] (n, + n,),
(3.34)
242
K. S.HAVNER and S.A. SALPEKAR
and K1z2 = -(zi-z:)+zi [Compare become
with (3.12) for side 2 in tension.]
___
1_lz,
exp(cy),
= z: exp(cy).
The algebraic
13-12
c2 =---_--_
212
213
expb)'
zz+z3
(3.35)
fractions
in (3.33.1) thus
2c2+evW c2+expW
'
(3.36)
z; - 1; c”=--~’ 1
so that ]c2] < l/2 (c2 = rotation in compression we again obtain (3.14) strengths in double-slip applies to double-slip in before), and :
l/2 at along (with along tension
[OOl]), and c2 = 0 in the limit position [Oil] for axis side 3. Finally, upon substitution of (3.36) into (3.33.1) m = 2). Thus, the general equation (3.23) for critical side 2 (corresponding to linear active hardening) also or compression along side 3, with m = 2, c = 2/J6 (as
Tension along [OOl]-[01 A, =
B,
W46)+2W22-~33ho
Compression
along [OOl]-[01
A, = (2/J6) + 4(N2& 4.
c2 = - zi/(zi + z$,
l] :
l] :
=
(3.37.1) (3.37.2)
W$+W,,hc;
c2 = - (1: - z$/(2z$,
(3.37.3) (3.37.4)
I% = (l/J6)-(N22-N,,),.
NUMERICAL~LLUSTRATIONOF LATENTHARDENING
Representative examples of theoretical latent hardening for initial axis positions along side 1 of stereographic triangle a2 (Fig. l), including demonstrations of “overshooting” in both tension and compression, may be found in (HS). Here we illustrate the general equation (3.23) for latent hardening, according to the simple theory, for axial loading along each of sides 2 and 3. As “overshooting” is excluded by kinematics alone for these latter ranges of load position, this phenomenon is not at issue with any hardening rule. It remains necessary, however, to compare each rk and resolved shear stress tr N,a to confirm that the simple theory does not predict slipping on an initially latent system at a later stage of deformation. The resolved shear stresses for loading on sides 2 and 3 are
Jh tr Nka = 2 tz2+ z3jz3tr (Nkr 0 4
(4.1)
the + sign applying in tension. Upon substitution of (3.9) in tension, or (3.17) in compression, and the linear active hardening relation r, = z,+ 2Hy into (4.1) we obtain tr N,a = (r,+2Hy)
D,+E,A
+ F, ew (CY)
'a c,+exp(cy)
I’
(4.2)
Latenthardeningof crystalsin double-slip--II
with cz as previously (3.37.3)] and :
243
defined for the different load cases [see (3.24.1), (3.24.3), (3.37.1),
Tension along [il l]-[OOl]
(side 2):
D, = ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E, = -(3+/2)(2N,, Compression
along
+N,&
F, = (4&/3)(N,,-NI~+N&
I
:
[ii l]-[OOl]
D,c = -,,‘W&
E, = 2$3&,
--N,3 +N,d,,
F, = +W,,+~& Tension along [OOl]-[01
(4.4)
I
l] (side 3):
D, = &W,, Compression
(4.3)
E, = - 4W,
along [OOl]-[01
I+ 2~,,),,
F, = - $W,,),
;
(4.5)
l] :
D, = ($9) F, = - (&2)
W,, -2N&
E, = -2$W,A,
WI I + 2N23h.
(4.6)
I
(Of course, D, = 1, E, = F, = 0 for k = 1,2 in each case.) For comparison, shear stress along side 1 is [(HS) equation (4.4)] :
the resolved
tr N,P = b. + 2ffy)CDL + by + F& + 1~11, with D,, E,, F, differently defined and, moreover, dependent upon initial axis position as well as slip system. We follow the precedent of HAVNER and SHALABY(1977) and separately compare the coefficients of z0 and H in (4.2), as functions of y, with those in the hardening law (3.23) (setting all 7: = z. in the latter corresponding to an initial statistically isotropic crystal). Also, in the figures (beginning with Fig. 2) the initial loading axis position along side 2 or 3 is given by the Euler angle 6, between z0 and lattice direction [OOl]. The corresponding values of c2 in terms of B0 are readily found to be : side 2--tension
side 2-compression side 3--tension
c2 = *(J2
: :
:
side 3-compression
:
tan B. - l),
c2 = -tan
0,/(J2
+ tan e,),
c2 = -tan
8,/(1+
tan e,),
c2 = -t(
1 -tan
0,).
(4.7) 1
(The maximum values of B,, are 54.74” and 45” for sides 2 and 3 respectively.) The various (Nij)k required for determination of the A,, B,, D,, E,, F, of each system for the four loading cases are, of course, defined by the b,, nk of Table 1. The predictions of the simple theory have been extensively illustrated for numerous cases of axial loading in previous works [see HAVNER and SHALABY, 1977,1978, (I), (II), (III), and (HS)]. As the qualitative features of the various results tend to be similar in form from one case to another, in single and double-slip, we present here only a relatively few examples for double-slip along sides 2 and 3. These examples are representative of the many more cases illustrated in SALPEKAR(1982), which contains
244
K. S. HAVNER and
S. A. SALPEKAR
~ ,
d2,di c2 ,cT
i/
45
3.6
0.9
7--T---7--1
0
FIG. 2. Critical
0.25
strength
0.5
0.75
1.0
1.25
1.5
vs slip in tension for OO = 45” ; coefficient
of H (side 2).
40 figures for these two sides. As in previous studies [e.g. (HS)], all results are displayed to large shears y = 1.5. In Figs 2-7 are shown various curves for the coefficients of critical strength ~~ and “net strength” zk - tr N,a in all slip systems for side 2. It is noted from Fig. 3 that, for tension and B0 = 45”, the net z,-coefficients in systems cl, c2 becomes slightly negative for a shear y > 1.3. This also occurs from other initial positions in tension along side 2 at values of y greater than one. However, the corresponding values of the H-coefficients are both positive and greater in magnitude than the z,-coefficients, as illustrated in Fig. 4 for 8, = 45”. Thus, because the value of H is expected to be much larger than z0 in any linear approximation of an experimental resolved shear stress vs slip curve, the resultant net strength z,-tr N,a will be positive in systems cl, c2 as well. In compression, the coefficients of both z0 and N are positive for all initial positions along side 2, as illustrated in Figs 6 and 7. It may also be noted from Figs 2 and 5 that, for the dominant H-contribution, the simple theory predicts the majority of latent systems to harden more than the active ones as slip proceeds. Representative plots of H-coefficients of both critical and net-strengths for side 3 are shown in Figs 8-l 1. [Plots of the less significant z0 coefficients are excluded merely to restrict the total number of figures. These coefficients are always positive, to y = 1.5, for all k and both critical strength zk and net strength zk - tr N,a (see SALPEKAR, 1982, Figs 2.64,2.66,2.68,2.70,2.72,2.74,2.76 and 2.78).] Again, as on side 2, it is seen in Figs 8 and 10 that only a minority of latent systems harden less than the active ones according to the simple theory.
Latent hardening
FIG. 3. Critical
strength
minus resolved
0
0.25
of crystals
in double-slip-II
245
shear stress vs slip in tension for Q0 = 45” : coefficient
0.5
0.75
1.0
1.25
of To (side 2).
1.5
Y FIG. 4. Critical
strength
minus resolved
shear stress vs slip in tension for Q0 = 45” : coefficient
of H (side 2).
246
K. S. HAVNEK and
S. A. SALPEKAR
4.5 a3,b3 d2,dT b2,oi bl,a?? cl& c3@,d3,dg c2,cl
3.6 -
a2,bT(active) al,bz dl,d? b3,a%
0.9 -
0-i 0
O.h5
,7--i
05
0.75
-__,
1.0
I 25
1.5
Y FIG. 5. Critical
0
strength
/
0
vs slip in compression
I 0.25
I Q5
7 0.75
for B, = 15”: coefficient
/ 1.0
/ 1.25
of H (side 2).
c2,ci 02, bT(octive) 1.5
Y FIG. 6. Critical
strength
minus resolved
shear stress vs slip in compression (side 2).
for 8, = 1.5”: coefficient
of rO
Latent
hardening
of crystals
in double-slip--II
247
6.0
I 4.5 b E ."g 5 3.0 0
1.5
0 0
0.25
05
075
1.0
1.5
1.25
Y FIG. 7.Critical
strength
minus resolved
shear stress vs slip in compression (side 2).
for 6, = 15”: coefficient
5.0-
I .g 3.0 E .P 0 % 0" 2.0
1.0
0 0
0.25
05
0.75
1.0
1.25
1.5
r FIG. 8. Critical
strength
vs slip in tension for 610= 15” : coefficient
of H (side 3).
of H
248
K. S. HAVNER and
S. A.
SALPEKAK
75
, a2,c2 1 /’ ,,,‘,,
6.0 !
- -
ai,ci,bl,dl
I
17rc;.9. Critical
strength
minus resolved
shear stress vs slip in tension for 0, = 15” : coefficient
of H (side 3)
I 5.0
! /
0
0.25
0.5
a75
1.0
125
1.5
7 FIG. 10. Critical
strength
vs slip in compression
for B0 = 30’
: coefficient of If (side 3).
of crystalsin double-slip-II
Latent hardening
249
8.0
aI,cl,bl,dl a3,c3 z
4.8
bZ,ds
% E a .P =: g 3.2 u
bS,dz
b3,d3 b2,d2 aJ,cf
1.6
ai,ci,bi,di
,
0 0
0.25
0.5
0.75
1.0
1.25
,a2,c2(active) 1.5
Y FIG. 11. Critical
strength
minus resolved
shear stress vs slip in compression (side 3).
5.
for 8, = 30”: coefficient
of H
CLOSURE
With this paper the line of investigations of axially-loaded single crystals that began with HAVNER and SHALABY (1977) is brought to an end. The qualitative features of finite distortion of cubic crystals that are represented by the simple theory, as cumulatively established in HAVNER and SHALABY (1977, 1978), (I), (II), (III), HAVNER (1981, 1982b), (HS), SALPEKAR (1982) and here, may be concisely summarized as follows : (i) The simple theory predicts “overshooting” from all single-slip positions in (virgin) cubic crystals and from all double-slip positions in f.c.c. crystals for which this phenomenon is kinematically possible. (ii) The theory predicts that the majority of latent systems harden more than the active systems (or system) whenever the loading axis rotates with respect to the underlying atomic lattice (viz. either single or double-slip). (iii) In 4,6 or g-fold symmetry positions (HAVNER, 1981,1982b; SALPEKAR, 1982) the simple theory is consistent with axis stability, and its net effect is the prediction of isotropic hardening for equal multiple-slip (for which the axis is stable). Finally, we note once more that the simple theory is unique as a theory of rotationdependent crystal anisotropy that satisfies Hill’s material-stability condition and leads to self-adjoint boundary value problems for inhomogeneous deformation of crystals and crystalline aggregates (HAVNER, 1982a; HILL and HAVNER, 1982).
250
K. S. HAVNERand S. A. SALPEKAR ACKNOWLEDGEMENT
This work was supported in part by the United States National Mechanics Program, through Grant MEA-7808154.
Science Foundation,
Solid
REFERENCES BUDIANSKY, B. and Wu, T. T.
1962
HAVNER, K. S.
1971 1977 1979 1981 1982a
HAVNER, K. S. and BAKER, G. S. HAVNER, K. S., BAKER, G. S., and VAUSE, R. F. HAVNER, K. S. and SALPEKAR, S. A. HAVNER, K. S. and SHALABY, A. H. HILL, R.
Proc.
1982b 1979
4th U.S. Natn. Cong. Appl. Mech., p. 1175. ASME, New York. Int. J. Solids Struct. 7. 719. Acta Mech. 28, 139. J. Mech. Phys. Solids 27,415. Proc. R. Sot. Lond. A 378,329. Mechanics of Solids, The Rodney Hill 60th Anniversary Volume (edited by H. G. Hopkins and M. J. Sewell), p. 265. Pergamon Press, Oxford. Mech. Matl. 1, 97. J. Mech. Phys. Solids 27, 285.
1979
Ibid. 27, 33.
1982
Ibid. 30, 379.
1977 1978 1968 1978
HILL, R. and HAVNER, K. S. JACKSON, P. J. and BASINSKI, Z. S. KOITER, W. T. NAKADA, Y. and KEH, A.S. PEIRCE, D., ASARO, R. J. and NEEDLEMAN,A. SALPEKAR, S. A.
1982 1967
Proc. R. Sot. Lond. A 358,47. J. appl. Mech. 45, 500. J. Me&. Phys. Solids 16, 315. Advances in Applied Mechanics, Vol. 18 (edited by C.-S. Yih), p. 1. Academic Press, New York. J. Mech. Phys. Solids 30, 5. Can. J. Phys. 45,707.
1953 1966 1982
Quart. appl. Math. 11, 350. Acta Metall. 14,961. Ibid. 30, 1087.
1982
A
TAYLOR, G. I. and ELAM, C. F. VAUSE, R. F. and HAVNER, K. S. WENG, G. J. WONSIEWICZ, B. C. and CHIN, G. Y.
1923 1925 1979
Theoretical Investigation of Axially Loaded F.C.C. Crystals in Multiple Slip Positions at Finite Strain. Ph.D. Thesis. North Carolina State University. Proc. R. Sot. Land. A 102,643. Ibid. A 108, 28. J. Mech. Phys. Solids 27, 393.
1979 1970
Int. J. Solids Struct. 15, 861. Metall. Trans. 1, 2715.
0022~5096/83/030231-20 $03.00/O Pergamon PressLtd. 0 1983.
THEORETICAL LATENT HARDENING OF CRYSTALS IN DOUBLE-SLIP-II. F.C.C. CRYSTALS SLIPPING ON DISTINCT PLANES K. S. HAVNER Department
of Civil Engineering, North Carolina Raleigh, NC 27650, U.S.A.
State University,
and S. A. SALPEKAR~ Corporate
Consulting and Development Company Raleigh, NC 27622, U.S.A.
Ltd.,
(Received 28 October 1982)
ABSTRACT THE LINE of investigation of theoretical latent hardening in axially-loaded cubic crystals initiated by HAVNER and SHALABY (1977), based upon their “simple theory” of rotation-dependent crystal anisotropy (and continued through various other works), is completed here with the analysis of finite double-slip on distinct planes in f.c.c. crystal lattices. In conjunction with a previous study by the present authors of double-slip on a common plane (HAVNER and SALPEKAR, 1982), the genera1 theoretical result is that any f.c.c. crystal which initially deforms homogeneously from the virgin state in double-slip, under either tensile or compressive axial loading, will continue in that mode of deformation to very large shears (y = 1.5 and beyond). The paper also includes a comparative review of various hardening rules (including the “simple theory”), with regard to both theoretical predictions and mathematical issues, that may be found instructive.
1.
FOREWORD
IN HAVNER and SALPEKAR (1982) [Part I of this two-part series, here designated (HS)] the anisotropic latent hardening of axially-loaded f.c.c. crystals, finitely and homogeneously deformed by double-slip on a common slip plane, is determined according to the “simple theory” of hardening proposed by HAVNER and SHALABY (1977). Other analyses of axially-loaded cubic crystals in single-slip and higher order multiple-slip positions, based upon this theory, have been carried out by HAVNER and SHALABY (1978), in the three-part series HAVNER, BAKER and VAUSE (1979), HAVNER and BAKER (1979), and VAUSE and HAVNER (1979) [henceforth designated (I), (II), (III) respectively], and in HAVNER (198 1, 1982b). In the present paper we complete the analysis of theoretical hardening of virgin f.c.c. crystals under tensile or compressive axial loading, according to the “simple theory”, with the solution of finite double-slip on distinct planes.
t Former U.S.A.
address
: Department
of Civil Engineering,
North Carolina
231
State University,
Raleigh, NC 27650,
K. S. HAVNERand S. A. SALPEKAR
232
2.
REVIEW OF CRYSTAL HARDENING RULES
Following the notation in (HS) and earlier works, we let zk denote the critical strength in the kth crystallographic slip system, b,, nk denote unit vectors in the slip and normal directions respectively of that system, and jj denote the slip-rate in the jth critical system, with all jj > 0. The “simple theory” of anisotropic latent hardening then may be expressed [HAVNER and SHALABY, 1977, equation (23)] ?, = hxjj-2 j
tr N,aR
= x(Ltr i
Nkaj)yj,
with N, = {(b 0 n)k}sym, aj = {a(b 0 n-n
h > 0,
(2.1.1)
0 b)j}sym.
(2.1.2)
Here h is a hardening parameter ; o is Cauchy stress ; t2 is the relative spin of material and lattice ; {. . .}symsignifies the symmetric part of a tensor product ; the summation onj is over the n critical systems (in each of which tr Nja = rj) ; and k ranges over all systems, critical or not. (In a slip system that is momentarily non-critical, tr N,a < zk.) The initial value of zk in a statistically isotropic virgin crystal is r0 for all crystallographically equivalent systems (e.g. the set of 24 systems { 11 l} (i10) in f.c.c. crystals). Obviously, it is the rotation-dependent? second term in (2.1.1) that produces the anisotropic part of the hardening, the physical motivation for which is discussed at length in (I). The simple theory has been shown to account for various qualitative features of finite deformation experiments in single-slip and 6- and &fold multiple-slip positions [see (I), (II), (III) and HAWER, 1981, 1982a, b]. In particular, (2.1) is a universal theory of the common phenomenon of “overshooting” in both f.c.c. and b.c.c. crystals in single slip. Three other specific hardening rules of demonstrated usefulness relevant to finite distortion of cubic crystals are: (i) Taylor’s classic theory of isotropic hardening (TAYLOR and ELAM, 1923, 1925); (ii) an empirically-based 2-parameter rule (see NAKADA and KEH, 1966, or JACKSON and BASINSKI, 1967); and (iii) the hardening rule recently proposed by PEIRCE, ASARO and NEEDLEMAN(1982) which for convenience we shall call the P-A-N rule. These are given by : (i) Taylor’s
rule : Z, = HE yj for all k,
(ii) 2-parameter
(2.2)
rule : Z, = H c yl + H’ x1;,, m 1
with n, = nk, (iii) P-A-N
H > 0;
% #
nkj
(2.3.1) (2.3.2)
H’>H>O;
rule : t, = h 1 yj - tr N,aR - tr flkaB (2.4)
= x(h-3
tr N,ai+i’tr
a,Nj)?ij,
h > 0,
i
where R, 1 [(b @ n)Jskew (the skew-symmetric part of the dyadic product), and d is the Eulerian strain-rate (“rate of deformation” tensor) of the crystalline material relative to t Here is meant, of course, the rotation of gross crystalline material relative to the underlying not rigid body rotation with respect to an arbitrary reference frame.
atomic lattice,
Latent hardening of crystals in double-slip-11
233
an effectively rigid lattice (i.e. a lattice experiencing infinitesimal strains that are neglected here in comparison with the finite crystallographic slips ofinterest). The latter two rules are anisotropic, whereas Taylor hardening is (of course) isotropic; and the simple theory may be either isotropic or anisotropic, dependent upon stress state (relative to lattice axes) and the resulting degree of symmetry of the deformation. It may be noted that the P-A-N rule (2.4) incorporates one-half of the relative spin dependence introduced as a source of anisotropy by HAVNER and SHALABY(1977) in the simple theory (2.1), and adds another anisotropic contribution dependent upon the relative strain-rate 8.t This latter term is reminiscent of, but not equivalent to, the translational (or “kinematic”) hardening rule adopted in BUDIANSKY and WV (1962), namely 2, = 2H tr N,d = 2H c tr (NJVj)jj, with H the Taylor modulus (i.e. the slope j of the resolved shear stress vs slip curve in single slip). Another so-called “kinematic hardening” rule for crystals has been suggested by WENG (1979); in terms of the Taylor modulus, Weng’s rule may be expressed ?, = H tr (b @ n),J& -R)
= H 1 (bk - bj) (nk * nj)jj.
The obvious linear combination of isotropic with translational hardening was mentioned by HAVNER (1971). However, all such theories that are symmetric positivedefinite, for 5 or fewer active systems, in the matrix of physical hardening moduli Hkj(?, = c Hkjyj) give results that are qualitatively inconsistent with experiment for finite distortion of crystals from both single and multiple-slip axial-load positions. (This is discussed in some detail in HAVNER, 1981, section 4.) Consequently, such theories need not concern us further. Another valuable perspective on the various hardening rules is a mathematical one. Only the simple theory and the P-A-N rule, of all those proposed (including Taylor hardening), lead to the possibility ofminimum principles in self-adjoint boundary value problems of rate type, either for inhomogeneously deformed single crystals or polycrystalline solids. This is because only these two fall within the general class of hardening rules that permit such mathematical features. This class is defined for the logarithmic strain measure (at constant pressure, to first order in infinitesimal lattice strain) in HAVNER and SHALABY [ 1977, equation (20)] as +, = C (hk. - tr N,aj)lij,
h,j = hj,.
(2.5)
For the “simple theory” (2.1) obviously all hki = h, hence the name, whereas for the P-A-N rule, from (2.4) and (2.5), hkj = h+S tr (N,aj+ Njak). (This equivalence is obscured by PEIRCE et al. (1982) as they use a single symbol to represent two different sets of moduli in their equations (3.31) and (3.32). In their 2-parameter equation h is
t PEIRCE et al. (1982) also
“independent”
hardening
propose a 2-parameter equation that is merely the linear superposition of the rule Of KOITER (1953) with their basic theory. Koiter’s rule is ir = H c akjfj (that is,
zero latent hardening). Koiter did not propose usefulness in that context.
this relation
for crystal
mechanics,
and it is oi questionable
234
K. S. HAVNERand S. A. SALPEKAR
replaced by qh + (1 - q)h lSkj, the latter term being the Koiter contribution.) It is also desirable that the matrix (hkj) be non-negative-definite as well as symmetric. [As shown in HAVNER, 1982a, equation (4.28), this pertains to a material-stability condition suggested by HILL (1968) ; also see the brief discussion that closes HILL and HAVNER, 1982, pp. 21-22.1 The simple theory obviously has this characteristic. Whether the P-A-N rule also is non-negative-definite is not self-evident and apparently has not been determined. Uniqueness criteria and minimum principles for crystalline solids at finite strain, based upon symmetric hki, may be found in HAVNER (1977) (also see HILL, 1978, and the brief review in HAVNER, 1982a, section 4.4). The generalization of (2.5) to a broad class of strain measures has been given by HILL and HAVNER (1982). In the present notation, for the ith system tr Nia(n-Cb)
?i = Chijjj-2 j
=
C(hij-tr .i
iV,aj+2C
tr NifJiVi)jj,
(2.6)
where c is a measure-dependent parameter (zero for the logarithmic measure). As remarked in HILL and HAVNER (1982, p. 21), the simple theory “is unique as a theory of rotation-dependent crystal anisotropy” in that “there can be no other choice of strain measure and associated parameters hij which will both ensure self-adjointness of boundary-value problems and result in isotropic hardening under conditions of zero relative spin Q (when such hardening is to be expected).” The P-A-N rule, although incapable of predicting isotropic hardening in 6- or 8-fold multiple slip (for which see the discussion in WONSIEWICZ and CHIN (1970, p. 2721) of experiments on aluminum), is useful in that it apparently is, by design, a theory of limited “overshooting” (i.e. eventual double-slip after axis rotation beyond the symmetry line). [The empirical rule (2.3) can also give limited overshooting, depending upon the relative values of H and H’.] PEIRCE et al. (1982) employ their theory, in conjunction with an ad hoc plane strain model and the finite element method, in a comprehensive study of shear band formation in single crystals. However, the P-A-N rule has not been evaluated explicitly, so far as we know, for either the f.c.c. or b.c.c. lattice geometry. Hence, its predictions for homogeneously deformed cubic crystals in single and multiple-slip are not yet determined.
3.
LATENT HARDENING IN DOUBLE-SLIP
Consider the [OOl] stereographic projection for f.c.c. crystals in axial tension shown in Fig. 1. The 24 stereographic triangles are identified as slip systems in Table 1 (with a bar above a number designating the opposite sense of slip). We choose the initial orientation of the loading axis such that a2 = (111) [iOl] is slip-system 1 in tension. (In compression, all slip systems exchange position with their opposite-sense systems, whence triangle a2 becomes system a2 = (111) [ lOi], etc.). Then, the locus of all doubleslip positions is the boundary of triangle a2 (Fig. 1). The symmetry-line segments [01 l]-[ill], [il l]-[OOl], and [OOl]-[01 l] (as one proceeds counterclockwise around a7) will be designated sides 1, 2 and 3 respectively.
Latent
hardening
of crystals
235
in double-slip--II
fro01
FIG. 1. Standard
[OOl] stereographic
3.1 Initial positions on symmetry
projection
line [01 i]-[il
of f.c.c. crystals.
l] (side 1)
Side 1 obviously is distinctly different from the other two as it corresponds to doubleslip on the same plane; consequently, there is a fundamental qualitative difference as regards axis position in the resultant deformation.? For equal double-slip from an initial axis position (tension or compression) along side 1, the limiting positions from pure kinematics lie outside triangle a2 ; they are [Zl l] in tension, which lies beyond [il l] on the continuation of symmetry line [01 l]-[ill], and [ll l] in compression. As regards the hardening theories, both the simple theory (2.1) and the 2-parameter rule that is, axis-rotation beyond the higher-symmetry (2.3) predict “overshooting”, position [ill] (tension) or [01 l] (compression), the latter rule by empirical design and the former as established in (HS). Taylor hardening (2.2) of course does not predict overshooting, and the P-A-N rule (2.4) has not been evaluated for this case as yet. hardening, would predict (Positive-definite hardening rules, such as “kinematic” slipping on other systems before the loading axis reached the higher symmetry position in either tension or compression.)
TABLE
Plane Direction System
[oil] al
1. Designation
(111) [ioil [ii01 a2
a3
[oiil bl
of slip systems in jIc.c. crystals (iil) [Toil [ii01 b2
b3
(ill) -coil-j [ioil cl c2
[ii01
[oiil
i:%\
[ii01
c3
dl
d2
d3
t The issue of single vs double-slip from all double-symmetry positions, for each of the hardening (2.1H2.3), is comprehensively investigated in HAVNER (1982b) and reviewed in (HS), section 2.
rules
236
K. S. HAVNER
and
S. A. SALPEKAR
Explicit solutions for latent hardening of all systems, according to the simple theory, for double-slip along side 1 are given in (HS). We include here, for subsequent comparison with results for sides 2 and 3, only the equation for f, as a function of the equal slips y [HS, equation (3.15)] :
G = [2H
+ (,,,2
j;H
dy)(A,+B,y
-&)]‘.
(3.1)
A,, B,, c2 are constants that depend upon the particular slip-system and/or initial axis position (along [01 l]-[ill]) and whether the loading is tensile or compressive. In active systems 1,2 (respectively a?!, a3 in tension and a2, a3 in compression) both A,, B, are zero, and tr N,a throughout the deformation for all latent systems and initial axis positions. [Refer to (HS) for final equations, including specific expressions for the constants, and their numerical illustration.]
3.2 Initial
positions
on symmetry
line [il l]-[OOl]
(side 2)
Along this side of stereographic triangle a2 (Fig. l), the range of double-slip orientations (excluding corners) evidently is z1+ z2 = 0, z3 > z2 > 0, where z is a unit vector coincident with the axial load. Consider first the case of tensile loading of a long, slender crystal, and let CJdenote the uniaxial Cauchy stress at any stage of deformation. Then d = ozQz, with bl = (both (2. l),
,J = 46 r&z
+
4~J,
(3.2)
z, = z0 + 2 f H dy, the critical strength of active systems a2 = (111) [iOl] and (111) [Oll]. The evolutionary equations for critical strengths in all systems latent and active) in equal double-slip g, = j2 = j, from the simple theory are Z, = (2h - tr N,a)j,
where 1, 2 signify systems a2, bl respectively.
a = a, + a2,
(3.3)
For axial loading
aj = 2{baj}sym= ~C(~~bj){~O~j}sym-(~~~j){~Obj}sy~l~ whence, upon substituting we find
the second equation - 2121,
a=
r, 2(r, + z&3
!
of (3.2) and the respective
2z;-1;
(sym)
(2N,,+N,,-NN,,+3N,,),
4r*r,
(3.5) I
and simplification,
*-(I.,,-N,,),? 2
unit vectors,
- 214 + z:
2& -2z,z,
There follows from (3.3) and (3.5) after rearrangement fk = 2hj-z,
(3.4)
3
1?j. (3.6)
237
Latent hardening of crystals in double-slip-Ii
In active systems a2, bl, f, = fz = 2Hj and (2N,,+N13-_N,,+3N,,), = - 1/(2J6), k = 1,2. Thus
= J6/4,
(IV,, -N&
(3.7) Consequently, we may write f, = 2Hj+z, A,i
=
2E2- Eg A,-----35
(J6/4)+ 3(N,,
-N2& i
& = l/(2J6)-4(2N12+N,3--23+3N3,),,f (with of course A, = B, = 0 for active systems 1,2). There remains to determine the two algebraic fractions in direction cosines I 2r z3 as functions of double-slip y. Let 2 denote the material stretch in loading direction z. From the general kinematic solution for (proportional) double slip on distinct planes [see HAVNER, 1979, equation (34)], we have for yI = y2 = y: AZ= zO+(l/c)([KI --fW)W,
exp(cy)-K2
ew@y)+&
exp(-cy)+K:
exp(-cy)--0s
cos s,!&b,
ICl?lbI,
(3.9)
where $9 is the angle between z. (initial position of the load-axis) and active slip-plane normal nj, and I<, = &cos I/?:- K cos iji),
(3.10.1)
K2 = f(cos *: + Jc cos I&
(3.10.2)
K = c,&,
c = (c,,c,,)‘~~,
cij = -b,*nj.
For systems 1,2 = a2, bl these reduce to : cl2 = czl = - 2/,/6, c = 2/J6, K K, = cos r@ = cos I,@= (l/,/3)& K, = 0. Thus lz = z. + (1/,/2)&exp
(cy) - 11 (b, + b2).
(3.10.3) - 1, and (3.11)
Consequently, %z,= +(22; - 1:) + $E$ exp (cy), II, = 2: exp (cy),
(3.12)
and the required algebraic fractions are 21,-l, .3z3
01 3z2 -=_=-Y> exp tcy) t2 + z3
21; - li 3c2 f exp (c~) c,+exp(cy) ’ ” =T’
(3.13)
(Note that jc2/ G l/3, equalling l/3 in the corner positions [OOl] and [ill], and that c2 = 0 at [i12], which is the limit position for axis rotation in equal double-slip along side 2.) Therefore, the final equations for rates of change of all critical strengths according to the simple theory are
6‘ = {2H+(r,+2j)
dp)[A,&
+ B~~‘eex$ojj,
(3.14)
238
K. S. HAVNER
and S. A. SALPEKAR
with m = 3. (The reason for introducing m will become apparent later.) The striking difference in form between the equations for side l(3.1) and side 2 (3.14) may be noted. Before dealing with the integration of (3.14), we turn to the case of compression of a short, squat crystal (with well-lubricated end faces). The active systems 1, 2 then are a2 = (111) [lOi] and bi = (iii) [OTT] (the slip directions being the opposites of those in tensile loading), and 0 = -J~-c,/[(z, + z&J < 0. It is obvious from (3.4) that aI, a2 remain the same functions of z2, z3 (as both Q and the bj change sign), hence (3.5) and (3.6) apply as well to compressive loading along [ill]-[OOl]. However, N,,), reverse signs in active systems. Thus, (2N1,+N1s-%,+3N,3)k and (K(3.7) and (3.8) are replaced by (3.15) and
A, = (J6/4)+(2N,,+N,,-N,,+3N33)k,
(3.16)
B, = 1/(2J6) - (N, 3 - N,,),. Again, we must express the algebraic fractions involving direction cosines z2, z3in terms of y. [These will be different from (3.13) because the kinematics in compression are different from the tension case.] Now let 2 denote the spacing stretch (i.e. the stretch of the spacing of specimen end planes). Then, as the area1 stretch of the loading faces equals the inverse of the spacing stretch at constant volume, the general kinematic solution for I in equal double-slip on distinct planes for compressive loading [see HAVNER, 1979, equation (24)] can be written AK11
=
zo-(l/c){[c,
exp (c~) - C2 exp
( - CY) -(l/4
+lc[Cl
~0sz&n,
exp(cy)+C,
where xy is the angle between z0 and active slip-direction 2) and c, with c and c = 2/,,/6, K
K =
cos x;-j,
c,
= i[cos
exp(-cy)-cos
xYln2},
(3.17)
bj (note that cos ~9 < 0,j = 1,
+(1/K)
cos x!],
(3.18)
= i[cos
x7+(1/K)
defined l,andC,
as in (3.10.3). For systems 1, 2 = a2, bi : c, z = c2, = 2146, (I; + zi), C, = 0. Consequently, = cos x’: = cos x: = -(l/J2)
a- ‘I = r0 + (J3/2) (1; + 1:) [exp (cy) - l] (n, + n2).
(3.19)
Thus, in compression,
AK’z, =
zi,
ilX’z, = -z$+(z$+z$ exp(cy),
(3.20)
Latent
and the required
algebraic
hardening
of crystals
fractions
239
in double-slip-II
become 12
c2
=p
12+ 13 21, - 13
(3.21.1)
exp (CY)’ 3c2 + exp W
(3.21.2)
c2 + exp (CY) ’
13
4
c2=--
(3.21.3)
1; + 1; ’
so that the equation for <, is again precisely (3.14) (with of course A,, B, and c2 differently defined in tension and compression). Note that the dependence upon initial position z0 along side 2 is expressed solely through the parameter c2 for both the tensile and compressive loading cases. [We consider this an improvement in form upon both equation (3. l), from (HS), for double-slip on side 1 and the general equation for latent strengths in single slip in (I), each of which equations is expressed in parameters having dual-dependence upon position and slip-system.] Also note from (3.21.3) that Jc21< l/2 in compression (equalling l/2 in corner position [ill]) and that c2 = 0 at [OOl], which is the limit position for axis rotation in equal double-slip in compression along side 2. We now address the integration of evolutionary equations (3.14), which apply in both tension and compression as shown, restricting consideration here to the linear approximation z, = 50 + 2Hy of an active hardening curve. [Note also that the solution for constant H may be used in a highly accurate stepwise approximation of an arbitrary experimental curve, as was done for copper crystals in single slip in (I).] The requisite integrals are
y s
c,dy
r0
~
0 exp (cy)
= (ro/c)c2C1
-ew--cy)l,
y mc2 + exp (cy) dy = (ro/c) m(q) TO s 0 c2 + exp (CY) 1
2H
2H
YCZY = (2H/c2)c2[1 -(l s 0 exp (CY)
y,mc2 + exp (cy) s o c2 + exp (CY)
- (m - 1) In
+cy) exp (-cy)],
y dy = (2H/c2)
im(cy)2 -(m - 1) In [c2 + exp (cy)]
(3.22)
Y
+(m- 1) s0
ln Cc2 + exp (cr)lc dy
with ’ In [c2 + exp (cy)]c dy = $(cY)~ s0
f
t-c21
nz
II=1 x
this last a rapidly convergent series. Thus, according to the simple theory, corresponding
[l-exp(-racy)] the final equations to linear hardening
as
lc2l < 1,
for latent strengths in active systems 1,2,
K. S. HAVNER and S. A. SALPEKAR
240
are c2A,[1-exp(-cy)]
rk = r,0+2Hy+(r,/c)
+b{m(cy)-(ml) In [Cz:~~I(C?)]})+(2H/c2)(c2A,11-(i+c?) x exp (- cy)] + B, &2m - 1) (cy)’ -(m - 1) In [cz + exp (cy)]
-(m-l)
f
II=1
(3.23)
y[l-exp(-nc;)]]),
with r: = r0 for a crystal in the virgin state at y = 0, m = 3 for an initial loading position along side 2, c = 2/J6, and (restating the previous expressions) : Tension along [il l]-[OOl]
: (3.24.1)
c2 = (22; - z$/(32$, A, = (J6/4)+3(N,,-N& Compression
axis
B, = 1/(2J6)-4(2N,,+N,,-N,,+3N,,),;
(3.24.2)
:
along [ii l]-[OOl]
(3.24.3)
c2 = -z;/(z;+I;,,
(3.24.4)
B, = l/(246)-(N,,-NJ,.
A, = (&4)+(2N,,+N,,-Nz,+3N&
We shall establish in the next section that (3.23) also applies to equal double-slip along side 3 ([OOl]-[Oil]) of stereographic triangle a2 in both tension and compression, with m = 2 for that side and A,, B,, c2 of course differently defined.
3.3 Initial positions on symmetry
line [OOl]-[01
l] (side 3)
For this side of the stereographic triangle, the range of orientations (excluding corners) is given by zI = 0, z3 > z2 > 0; and active slip systems 1, 2 in tension are respectively a2 and c2 = (ill) [loll. Equations (3.2)-(3.4) apply once more, and we find 0
0
0
212% 1:-z: - 2Z,l,
(sym)
I .
(3.25)
Thus, from (3.3) and (3.25), r f, = 2hj-r,
2(N,,-N,,)+ 1
In the active systems k= 1,2;hence
?I = fz = 2Hj
2h = 2H+z,
+ 2(N2a)xy 2
3
and 2(N22-N33)k
= -2146,
1 j.
2(N&
(3.26) = l/,/6,
(3.27)
Latent
hardening
of crystals
in double-slip--II
241
and fk = 2Hj+z,
-A,
A, = (2/46) + 2W,,
-&
(3.28.1)
+ Bky),.
-N&k,
B, = (l/J6)
- 2W23)k
(3.28.2)
(with A, = B, = 0 for active systems 1, 2). To express the algebraic fractions involving 12, l3 as functions of y, we again use the kinematic solution (3.9) in tension. For systems 1, 2 = aZ, ~2, the constants (3.10) are: c = 2146, IC= - 1 (all as on side 2), and K, = cos $y = cos I@ cr* = c2r = -2/J6, = (l/J3) (&!+ I:), K, = 0. Thus, from (3.9), II = lo + (l/2) (I: + 1:) [exp (cy) - 11 (b, + bJ,
az,= z;, [Note the similarity fractions are then
%z3= -
z:+(1:+ z$ exp (cy).
with (3.20) for side 2 in compression.]
12
-----__ 12
+
CZ 13
exp
13 ---= (cd
(3.29)
’
l2
2~2 c2
l3
+ +
exp
(3.30) The required
algebraic
(CY)
exp (cy) ’ 1 >
(3.31)
1:
c2zzp z;+z3
0’ i
with lczl < l/2 (equalling l/2 in corner position [01 11) and c2 = 0 at [OOl], the limit position of axis rotation in tension along side 3. Upon substitution of (3.31) and z, = z. +2s H dy into (3.28.1), we recover the general form (3.14) of side 2, now with m = 2. Turning to compressive loading along side 3, the active systems are a2 (as for side 2) _and c2 = (ill) [loll, and e = - 46 z,/[(zz + z3)zJ < 0. Again, a1 and a, as functions of 12, z3 do not change, hence (3.25) and (3.26) also hold in compression; but (IV,, -N,,), and (IV,,),, k = 1,2, reverse signs. Thus, (3.27) and (3.28) are replaced by 2k=2H+q
and
(
LA--J6
12+13
1
13-12
J6
. 13
(3.33.1)
+ Bll
A, = (2/&l + 4(N,,)k, (Ak = Bk = 0 for active systems a2, As before, the algebraic fractions in compression. For systems 1, 2 = = 2146, c = 2146, IC= 1, and C, from (3.17)
Bk = (I/ J6)-(N22
(3.32)
“?,
-N33h
(3.33.2)
~2.) are determined using the kinematic solution (3.17) a2, c2, constants (3.10.3) and (3.18) are : cl2 = c21 = cos xy = cos $j’ = -(l/J2)1:, Cz = 0. Hence,
%- ‘z = z. + (J3/2)zg[exp (cy) - l] (n, + n,),
(3.34)
242
K. S.HAVNER and S.A. SALPEKAR
and K1z2 = -(zi-z:)+zi [Compare become
with (3.12) for side 2 in tension.]
___
1_lz,
exp(cy),
= z: exp(cy).
The algebraic
13-12
c2 =---_--_
212
213
expb)'
zz+z3
(3.35)
fractions
in (3.33.1) thus
2c2+evW c2+expW
'
(3.36)
z; - 1; c”=--~’ 1
so that ]c2] < l/2 (c2 = rotation in compression we again obtain (3.14) strengths in double-slip applies to double-slip in before), and :
l/2 at along (with along tension
[OOl]), and c2 = 0 in the limit position [Oil] for axis side 3. Finally, upon substitution of (3.36) into (3.33.1) m = 2). Thus, the general equation (3.23) for critical side 2 (corresponding to linear active hardening) also or compression along side 3, with m = 2, c = 2/J6 (as
Tension along [OOl]-[01 A, =
B,
W46)+2W22-~33ho
Compression
along [OOl]-[01
A, = (2/J6) + 4(N2& 4.
c2 = - zi/(zi + z$,
l] :
l] :
=
(3.37.1) (3.37.2)
W$+W,,hc;
c2 = - (1: - z$/(2z$,
(3.37.3) (3.37.4)
I% = (l/J6)-(N22-N,,),.
NUMERICAL~LLUSTRATIONOF LATENTHARDENING
Representative examples of theoretical latent hardening for initial axis positions along side 1 of stereographic triangle a2 (Fig. l), including demonstrations of “overshooting” in both tension and compression, may be found in (HS). Here we illustrate the general equation (3.23) for latent hardening, according to the simple theory, for axial loading along each of sides 2 and 3. As “overshooting” is excluded by kinematics alone for these latter ranges of load position, this phenomenon is not at issue with any hardening rule. It remains necessary, however, to compare each rk and resolved shear stress tr N,a to confirm that the simple theory does not predict slipping on an initially latent system at a later stage of deformation. The resolved shear stresses for loading on sides 2 and 3 are
Jh tr Nka = 2 tz2+ z3jz3tr (Nkr 0 4
(4.1)
the + sign applying in tension. Upon substitution of (3.9) in tension, or (3.17) in compression, and the linear active hardening relation r, = z,+ 2Hy into (4.1) we obtain tr N,a = (r,+2Hy)
D,+E,A
+ F, ew (CY)
'a c,+exp(cy)
I’
(4.2)
Latenthardeningof crystalsin double-slip--II
with cz as previously (3.37.3)] and :
243
defined for the different load cases [see (3.24.1), (3.24.3), (3.37.1),
Tension along [il l]-[OOl]
(side 2):
D, = ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E, = -(3+/2)(2N,, Compression
along
+N,&
F, = (4&/3)(N,,-NI~+N&
I
:
[ii l]-[OOl]
D,c = -,,‘W&
E, = 2$3&,
--N,3 +N,d,,
F, = +W,,+~& Tension along [OOl]-[01
(4.4)
I
l] (side 3):
D, = &W,, Compression
(4.3)
E, = - 4W,
along [OOl]-[01
I+ 2~,,),,
F, = - $W,,),
;
(4.5)
l] :
D, = ($9) F, = - (&2)
W,, -2N&
E, = -2$W,A,
WI I + 2N23h.
(4.6)
I
(Of course, D, = 1, E, = F, = 0 for k = 1,2 in each case.) For comparison, shear stress along side 1 is [(HS) equation (4.4)] :
the resolved
tr N,P = b. + 2ffy)CDL + by + F& + 1~11, with D,, E,, F, differently defined and, moreover, dependent upon initial axis position as well as slip system. We follow the precedent of HAVNER and SHALABY(1977) and separately compare the coefficients of z0 and H in (4.2), as functions of y, with those in the hardening law (3.23) (setting all 7: = z. in the latter corresponding to an initial statistically isotropic crystal). Also, in the figures (beginning with Fig. 2) the initial loading axis position along side 2 or 3 is given by the Euler angle 6, between z0 and lattice direction [OOl]. The corresponding values of c2 in terms of B0 are readily found to be : side 2--tension
side 2-compression side 3--tension
c2 = *(J2
: :
:
side 3-compression
:
tan B. - l),
c2 = -tan
0,/(J2
+ tan e,),
c2 = -tan
8,/(1+
tan e,),
c2 = -t(
1 -tan
0,).
(4.7) 1
(The maximum values of B,, are 54.74” and 45” for sides 2 and 3 respectively.) The various (Nij)k required for determination of the A,, B,, D,, E,, F, of each system for the four loading cases are, of course, defined by the b,, nk of Table 1. The predictions of the simple theory have been extensively illustrated for numerous cases of axial loading in previous works [see HAVNER and SHALABY, 1977,1978, (I), (II), (III), and (HS)]. As the qualitative features of the various results tend to be similar in form from one case to another, in single and double-slip, we present here only a relatively few examples for double-slip along sides 2 and 3. These examples are representative of the many more cases illustrated in SALPEKAR(1982), which contains
244
K. S. HAVNER and
S. A. SALPEKAR
~ ,
d2,di c2 ,cT
i/
45
3.6
0.9
7--T---7--1
0
FIG. 2. Critical
0.25
strength
0.5
0.75
1.0
1.25
1.5
vs slip in tension for OO = 45” ; coefficient
of H (side 2).
40 figures for these two sides. As in previous studies [e.g. (HS)], all results are displayed to large shears y = 1.5. In Figs 2-7 are shown various curves for the coefficients of critical strength ~~ and “net strength” zk - tr N,a in all slip systems for side 2. It is noted from Fig. 3 that, for tension and B0 = 45”, the net z,-coefficients in systems cl, c2 becomes slightly negative for a shear y > 1.3. This also occurs from other initial positions in tension along side 2 at values of y greater than one. However, the corresponding values of the H-coefficients are both positive and greater in magnitude than the z,-coefficients, as illustrated in Fig. 4 for 8, = 45”. Thus, because the value of H is expected to be much larger than z0 in any linear approximation of an experimental resolved shear stress vs slip curve, the resultant net strength z,-tr N,a will be positive in systems cl, c2 as well. In compression, the coefficients of both z0 and N are positive for all initial positions along side 2, as illustrated in Figs 6 and 7. It may also be noted from Figs 2 and 5 that, for the dominant H-contribution, the simple theory predicts the majority of latent systems to harden more than the active ones as slip proceeds. Representative plots of H-coefficients of both critical and net-strengths for side 3 are shown in Figs 8-l 1. [Plots of the less significant z0 coefficients are excluded merely to restrict the total number of figures. These coefficients are always positive, to y = 1.5, for all k and both critical strength zk and net strength zk - tr N,a (see SALPEKAR, 1982, Figs 2.64,2.66,2.68,2.70,2.72,2.74,2.76 and 2.78).] Again, as on side 2, it is seen in Figs 8 and 10 that only a minority of latent systems harden less than the active ones according to the simple theory.
Latent hardening
FIG. 3. Critical
strength
minus resolved
0
0.25
of crystals
in double-slip-II
245
shear stress vs slip in tension for Q0 = 45” : coefficient
0.5
0.75
1.0
1.25
of To (side 2).
1.5
Y FIG. 4. Critical
strength
minus resolved
shear stress vs slip in tension for Q0 = 45” : coefficient
of H (side 2).
246
K. S. HAVNEK and
S. A. SALPEKAR
4.5 a3,b3 d2,dT b2,oi bl,a?? cl& c3@,d3,dg c2,cl
3.6 -
a2,bT(active) al,bz dl,d? b3,a%
0.9 -
0-i 0
O.h5
,7--i
05
0.75
-__,
1.0
I 25
1.5
Y FIG. 5. Critical
0
strength
/
0
vs slip in compression
I 0.25
I Q5
7 0.75
for B, = 15”: coefficient
/ 1.0
/ 1.25
of H (side 2).
c2,ci 02, bT(octive) 1.5
Y FIG. 6. Critical
strength
minus resolved
shear stress vs slip in compression (side 2).
for 8, = 1.5”: coefficient
of rO
Latent
hardening
of crystals
in double-slip--II
247
6.0
I 4.5 b E ."g 5 3.0 0
1.5
0 0
0.25
05
075
1.0
1.5
1.25
Y FIG. 7.Critical
strength
minus resolved
shear stress vs slip in compression (side 2).
for 6, = 15”: coefficient
5.0-
I .g 3.0 E .P 0 % 0" 2.0
1.0
0 0
0.25
05
0.75
1.0
1.25
1.5
r FIG. 8. Critical
strength
vs slip in tension for 610= 15” : coefficient
of H (side 3).
of H
248
K. S. HAVNER and
S. A.
SALPEKAK
75
, a2,c2 1 /’ ,,,‘,,
6.0 !
- -
ai,ci,bl,dl
I
17rc;.9. Critical
strength
minus resolved
shear stress vs slip in tension for 0, = 15” : coefficient
of H (side 3)
I 5.0
! /
0
0.25
0.5
a75
1.0
125
1.5
7 FIG. 10. Critical
strength
vs slip in compression
for B0 = 30’
: coefficient of If (side 3).
of crystalsin double-slip-II
Latent hardening
249
8.0
aI,cl,bl,dl a3,c3 z
4.8
bZ,ds
% E a .P =: g 3.2 u
bS,dz
b3,d3 b2,d2 aJ,cf
1.6
ai,ci,bi,di
,
0 0
0.25
0.5
0.75
1.0
1.25
,a2,c2(active) 1.5
Y FIG. 11. Critical
strength
minus resolved
shear stress vs slip in compression (side 3).
5.
for 8, = 30”: coefficient
of H
CLOSURE
With this paper the line of investigations of axially-loaded single crystals that began with HAVNER and SHALABY (1977) is brought to an end. The qualitative features of finite distortion of cubic crystals that are represented by the simple theory, as cumulatively established in HAVNER and SHALABY (1977, 1978), (I), (II), (III), HAVNER (1981, 1982b), (HS), SALPEKAR (1982) and here, may be concisely summarized as follows : (i) The simple theory predicts “overshooting” from all single-slip positions in (virgin) cubic crystals and from all double-slip positions in f.c.c. crystals for which this phenomenon is kinematically possible. (ii) The theory predicts that the majority of latent systems harden more than the active systems (or system) whenever the loading axis rotates with respect to the underlying atomic lattice (viz. either single or double-slip). (iii) In 4,6 or g-fold symmetry positions (HAVNER, 1981,1982b; SALPEKAR, 1982) the simple theory is consistent with axis stability, and its net effect is the prediction of isotropic hardening for equal multiple-slip (for which the axis is stable). Finally, we note once more that the simple theory is unique as a theory of rotationdependent crystal anisotropy that satisfies Hill’s material-stability condition and leads to self-adjoint boundary value problems for inhomogeneous deformation of crystals and crystalline aggregates (HAVNER, 1982a; HILL and HAVNER, 1982).
250
K. S. HAVNERand S. A. SALPEKAR ACKNOWLEDGEMENT
This work was supported in part by the United States National Mechanics Program, through Grant MEA-7808154.
Science Foundation,
Solid
REFERENCES BUDIANSKY, B. and Wu, T. T.
1962
HAVNER, K. S.
1971 1977 1979 1981 1982a
HAVNER, K. S. and BAKER, G. S. HAVNER, K. S., BAKER, G. S., and VAUSE, R. F. HAVNER, K. S. and SALPEKAR, S. A. HAVNER, K. S. and SHALABY, A. H. HILL, R.
Proc.
1982b 1979
4th U.S. Natn. Cong. Appl. Mech., p. 1175. ASME, New York. Int. J. Solids Struct. 7. 719. Acta Mech. 28, 139. J. Mech. Phys. Solids 27,415. Proc. R. Sot. Lond. A 378,329. Mechanics of Solids, The Rodney Hill 60th Anniversary Volume (edited by H. G. Hopkins and M. J. Sewell), p. 265. Pergamon Press, Oxford. Mech. Matl. 1, 97. J. Mech. Phys. Solids 27, 285.
1979
Ibid. 27, 33.
1982
Ibid. 30, 379.
1977 1978 1968 1978
HILL, R. and HAVNER, K. S. JACKSON, P. J. and BASINSKI, Z. S. KOITER, W. T. NAKADA, Y. and KEH, A.S. PEIRCE, D., ASARO, R. J. and NEEDLEMAN,A. SALPEKAR, S. A.
1982 1967
Proc. R. Sot. Lond. A 358,47. J. appl. Mech. 45, 500. J. Me&. Phys. Solids 16, 315. Advances in Applied Mechanics, Vol. 18 (edited by C.-S. Yih), p. 1. Academic Press, New York. J. Mech. Phys. Solids 30, 5. Can. J. Phys. 45,707.
1953 1966 1982
Quart. appl. Math. 11, 350. Acta Metall. 14,961. Ibid. 30, 1087.
1982
A
TAYLOR, G. I. and ELAM, C. F. VAUSE, R. F. and HAVNER, K. S. WENG, G. J. WONSIEWICZ, B. C. and CHIN, G. Y.
1923 1925 1979
Theoretical Investigation of Axially Loaded F.C.C. Crystals in Multiple Slip Positions at Finite Strain. Ph.D. Thesis. North Carolina State University. Proc. R. Sot. Land. A 102,643. Ibid. A 108, 28. J. Mech. Phys. Solids 27, 393.
1979 1970
Int. J. Solids Struct. 15, 861. Metall. Trans. 1, 2715.