## Abstract

A set of geometrically based FCC crystal slip-systems hardening inequalities is analytically investigated in (110) channel die compression for all lateral constraint directions between and , following previous analyses of the other two distinct orientation ranges in (110) compression. With all critical slip systems active, it is proved that these inequalities uniquely predict initial lattice stability and finite crystal shearing only in the horizontal channel plane, consistent with experiments for this range of orientations. (The earlier analyses had predicted load-axis stability in both orientation ranges, and lattice stability in one, also commonly found experimentally.) Moreover, it is established that the lateral constraint stress predicted by the hardening inequalities will be less than that given by classic Taylor hardening as this stress evolves with deformation. It is further shown, taking into account experimental stress–strain curves and latent hardening experiments for aluminium and copper, that lattice stability generally can be expected to very large deformations, except perhaps for lateral constraint orientations near the end of the range, which result is consistent with experiment. In appendix A, the possibilities of solutions with a critical slip system inactive are investigated, and predictions of a power law rate-dependent plasticity model are analysed for comparison with the results based on the hardening inequalities.

## 1. Introduction

In Havner (2005), an investigation of a diverse set of classic experimental studies (1960–1982) of finite deformations of FCC crystals in high-symmetry axial loading was undertaken. The range of behaviours included axis stability and axisymmetric deformation in 〈111〉 and 〈100〉 orientations, axis rotation towards a 〈111〉 orientation in coplanar double-slip in 〈110〉 loading and load-axis rotations towards (from an initial few degrees misalignment) or away from 〈111〉 and 〈100〉 orientations (with a reduced number of slip planes from the standard three and four). The experimental studies reviewed and analysed were: Kocks (1960, 1964) in aluminium; Nakada *et al*. (1964) in gold; Ramaswami *et al*. (1965) in silver and a gold–silver alloy; Keh (1965) in iron (a BCC metal, of course); Y. Nakada at Bell Labs in aluminium, briefly reported upon in Wonsiewicz & Chin (1970); and Vorbrugg *et al*. (1971), Ambrosi *et al*. (1974), Ambrosi & Schwink (1978) and Franciosi & Zaoui (1982), all in copper.

Use was made in Havner (2005) of experimental information, rigorous kinematic analyses of separate and coincident rotations among lattice, material and loading frames in the various (initial) orientations, and a geometric perspective on relationships among slip-systems hardening moduli. In each experiment analysed, the relative hardening necessary for (or consistent with) the corresponding crystal deformation and lattice rotation (where observed) was determined from the kinematics and the changing resolved shear stresses. A set of hardening inequalities was found (eqn (104) in Havner 2005) that encompassed all the individual responses from multiple-slip experiments and was also consistent with classic latent hardening experiments in single slip (Jackson & Basinski 1967 and Franciosi *et al*. 1980 among others).

In Havner (2007*a*), I began an investigation of the applicability of these hardening inequalities to FCC crystals in the very different experimental configuration of channel die compression, specifically (110) loading. As discussed at some length in the Introduction to that paper (which contains an extensive listing of channel die experiments), this family of orientations in the channel die test is of particular importance for investigation because (110) has long been established as a stable normal direction and preferred texture orientation in sheet rolling of polycrystalline metals. (Although my students and I carried out various analyses of channel die compression during the 1980s, a review of which may be found in ch. 5 of Havner (1992; including the theory of ‘minimum plastic spin’, Fuh & Havner 1989), these analyses were primarily focused on the following three specific crystal hardening theories: classic Taylor hardening (Taylor & Elam 1923, 1925); the ‘simple theory’ of Havner & Shalaby (1977, 1978); and the ‘P.A.N. rule’ of Peirce *et al*. (1982).)

As shown analytically in Havner (2007*a*), there are three kinematically distinct orientation ranges of the lateral constraint direction (plus three singular orientations) in (110) compression, resulting in different combinations of critical slip systems in each range (apparently first clearly identified by Skalli *et al*. 1983, fig. 4) and necessitating separate and independent analyses of both initial and subsequent responses. In Havner (2007*a*,*b*), all constraint directions between and , named ‘range I’, are investigated to large deformations. It is established therein that the hardening inequalities, in combination with experimental stress–strain curves and experimentally based relative values of the moduli, correctly predict the load-axis stability, finite shearing in the horizontal channel plane only and lattice rotation about the (110) load axis that are well-documented experimentally (e.g. Skalli *et al*. 1983; Skalli 1984; Wrobel *et al*. 1996).

In Havner (2008), all constraint directions from (commonly called the ‘brass’ orientation) to (the ‘Goss’ orientation) are investigated, again to finite deformations. For this family of lattice orientations, named ‘range III’, it is proved analytically that the hardening inequalities predict the *lattice stability* that is commonly observed in this range, again with finite shearing only in the horizontal plane (see Chin *et al*. 1966*a*; Wonsiewicz & Chin 1970; Driver & Skalli 1982; Kocks & Chandra 1982; Skalli *et al*. 1983; Skalli 1984; Driver *et al*. 1994). Moreover, again taking into account experimental data, it is shown that lattice stability should continue to large strains for all orientations in range III.

In this paper, the middle range of constraint directions between and , named ‘range II’, is investigated using the same hardening inequalities as before (eqn (104) of Havner 2005). In several respects, this orientation range has proven to be the most difficult to fully analyse, partly because, although a range of only slightly more than 19.47° (whereas ranges I and III are each more than 35.26°), it contains two kinematic sub-ranges, which the others do not. The paper is organized as follows.

Section 2 reviews the classic general hardening law, the geometric interactions among slip systems in FCC crystals, and the hardening inequalities. Section 3 contains general kinematics and stress rate equations for (well lubricated) channel die compression, and the equations for the resolved shear stresses in the 12 slip systems of FCC crystals. The range II channel die constraints and their sub-range consequences are given in §4. Section 5 contains the specific hardening rules and consistency conditions for critical slip systems in range II. In §6, a proof of uniqueness of solution with all critical systems active is presented. Its consequences for lattice stability, evolution of the constraint stress and finite crystal shearing are given in §7. The continuation of lattice stability, taking into account experimental stress–strain data from two orientations at the extremes of range II, and including the investigation of hardening of the latent slip systems, is analysed in §8. The main body of the paper ends with a closing discussion in §9, which also takes into account the analyses in appendix A. In appendix A*a*, the sub-ranges of range II are investigated without the assumption that all critical systems are active. In appendix A*b*, the consequences of the ‘standard’ form of a rate-dependent power law in (110) channel die compression are analysed.

## 2. Hardening law and inequalities

We adopt the perspective that, for moderate temperatures and strain rates, the *k*th slip system is momentarily inactive unless the resolved (Cauchy) shear stress *τ*_{k} (the ‘Taylor–Schmid stress’) equals a critical value , the current *critical strength* of that system. The critical strengths in all *N* crystallographic slip systems are considered to evolve with the deformation according to the classic law (Hill 1966, eqn (4))(2.1)where the *H*_{kj} are the non-constant slip-systems hardening moduli (deformation dependent, and at a more fundamental level dislocations dependent); d*γ*_{j} is an increment in crystallographic slip in the *j*th slip system; and *n* is the number of active systems at any stage. Moreover, a *critical* slip system () cannot be active unless the time derivative of its resolved shear stress *on a lattice co-rotational frame* equals the rate-of-change of its critical strength (a concept first clearly expressed by Hill (1966, p. 96); see also Havner (1972)). This may be stated , the *consistency condition* in an active system (for exact equations with unrestricted lattice straining, see Hill & Havner 1982 or Havner 1992, ch. 3).

Herein the moduli are distinguished only by the five categories of geometric slip-system interactions in FCC crystals. The possible interrelations between systems *k* and *j* may be defined as follows (Havner 2005):

coplanar (

*n*_{k}=*n*_{j}), including self-hardening;orthogonal (

*b*_{k}.*b*_{j}=0);collinear (

*b*_{k}.*b*_{j}=±1,*n*_{k}≠*n*_{j});cross slip (

*b*_{k}.*n*_{j}or*n*_{k}.*b*_{j}zero, but not both); and‘conjugate’ or other fully interacting cross slip (no zero scalar products between the

*k*,*j*systems);

where *b*_{k}, *n*_{k} denote unit vectors in the slip and slip-plane normal directions, respectively, of the *k*th slip system. The specification of these five categories, in some cases with identical names, dates at least to Kocks (1964). The interactions have other names in the context of dislocation theory, in particular with reference to dislocation ‘junctions’ (e.g. Franciosi *et al*. 1980, p. 279). (Note that in the hardening theory of Bassani & Wu (1991), contributions of slip in other systems to the hardening of a given *active* system are defined by these same interactions; see their eqn (3.10). However, in comparing with the experiments of Wu *et al*. (1991), Bassani & Wu adopt a *diagonal* hardening matrix such that *in*active systems do not harden. For a comparative study of Bassani & Wu's model and four other hardening rules (including the simple theory of Havner & Shalaby 1977) against texture experiments, from the literature, on polycrystalline copper bars in finite-strain torsion, see Lin & Havner (1996).)

The evolving moduli are labelled *H*_{I} to *H*_{V}, corresponding to the above geometric interrelations between pairs of slip systems. The interactions among the 10 possible active systems in (110) channel die compression are given in table 1. No specific equations are proposed for the individual moduli (e.g. their dependence on dislocation density or temperature as well as evolving slips). We adopt only the following *hardening inequalities* among these moduli (Havner 2005, eqn (104)):(2.2)These inequalities reflect classic latent hardening experiments on crystals deformed in single slip (Kocks 1964; Kocks & Brown 1966; Jackson & Basinski 1967; Franciosi *et al*. 1980). As briefly described in §1, they were shown in Havner (2005) to either specifically predict or be consistent with a wide range of diverse experimental behaviours (from the literature) in high-symmetry tensile loading of aluminium, gold, silver, a gold–silver alloy and copper crystals. Moreover, in Havner (2007*a*,*b*, 2008), they were applied to the analysis and prediction of finite crystal shearing and (where observed) lattice rotation for two lattice orientation ranges in (110) channel die compression of FCC crystals.

## 3. Kinematics, stress and stress rate equations in channel die compression

For a crystal in the channel die compression test, we denote loading, lateral constraint and channel axis directions by *X*, *Y* and *Z*, respectively, and designate unit vectors normal to the positive *X* and *Y* crystal faces by ** ι**,

**, and a unit vector coincident with the positive channel axis**

*κ**Z*by

**. Thus,**

*μ***constitutes a right-handed orthogonal triad subsequently to be defined relative to the lattice axes [100], [010] and [001].**

*ικμ*As in the channel die analyses in Havner (2007*a*,*b*, 2008; and earlier ones reviewed in Havner 1992, ch. 5), it is assumed that a crystal is well lubricated, that the lubrication is renewed throughout a test (e.g. Ferry & Humphreys 1996, p. 1294) and that the crystal is initially rectangular with free ends. Correspondingly, the non-zero (Cauchy) stresses are defined by *σ*_{xx}=−*f* and *σ*_{yy}=−*g* (initially everywhere, but only away from the free ends of a long crystal after finite deformation), with *f*, *g* the ‘true’ compressive stresses in the loading and constraint directions, from which the stress tensor is(3.1)

The resolved shear stress in the *k*th slip system then may be expressed(3.2)Its rate-of-change in the lattice frame is(3.3)where , are the lattice co-rotational derivatives of ** ι**,

**, given by (Havner 2007**

*κ**a*, eqn (7))(3.4)in which use has been made of the channel die constraints(3.5)Here

*e*

_{L}=−ln

*λ*(figure 1), the logarithmic true compressive strain, and(3.6)the Eulerian strain rate (disregarding the very small lattice strains in comparison with the finite slip deformations that are our focus).

In (110) channel die compression, we have (in Miller index notation)(3.7)where *ϕ* is the clockwise orientation of *Y* with respect to (or the anticlockwise orientation of relative to *Y*; figure 2). We designate the (maximum) 12 slip systems *a*1, , , *c*2, , *b*3, , *d*2, , *b*1, *c*3, and *d*3 by 1–12 in order (with a bar above indicating the opposite sense of slip from that defined in table 2). From equation (3.2), the following are the resolved shear stresses:(3.8)

Lattice orientations *∞*>*b*>2 (0<*ϕ*<35.26°) and 1≥*b*>0 (54.74°≤*ϕ*<90°), respectively, labelled ranges I and III, were investigated in Havner (2007*a*,*b*, 2008). For the remaining range 2>*b*>1 (35.26°<*ϕ*<54.74°), labelled range II, one finds from equation (3.8) that *τ*_{1}>*τ*_{9}, *τ*_{5}>*τ*_{3}>0 and . (Opposite sense systems , rather than 7, 8 are positively stressed in range II.) Thus, because the three channel constraints, equation (3.5), require that at least three slip systems be active for the onset of finite plastic deformation, the resolved shear stresses in systems 1, 2, 5 and 6 (*a*1, , and *b*3) all must attain the initial critical strength, *τ*_{0}, from which (equation (3.8); also Chidambarrao & Havner 1988, eqn (5.1))(3.9)

## 4. Consequences of the channel die constraints in range II

With only systems 1, 2, 5 and 6 potentially active at the outset, the constraint equations are (from equations (3.5) to (3.7), or reduced from general eqn (6) in Havner 2007*a*)(4.1)As in previous papers in this series of (110) channel die analyses, we define . Equation (4.1) then may be rewritten as(4.2)

From the non-negative slip rates, there are two kinematic sub-ranges in range II: 2>*b*>4/3 and 4/3>*b*>1, with a singular intermediate orientation *b*=4/3 corresponding to *ϕ*=46.69°. We have the following limits on the magnitude of *x*:(4.3)In the first sub-range, both systems 5 and 6 (, *b*3) and at least one of systems 1 and 2 (*a*2, ) must be active to satisfy the channel die constraints, with the converse true in the second sub-range. At *b*=4/3, only one of each pair necessarily must be active from kinematics alone. Of course, *x*=0 always satisfies the constraints, in which case and .

## 5. Hardening rules and consistency conditions

From table 1 and the general hardening law, equation (2.1), the critical strength rates in the four potentially active systems in range II are(5.1)Thus, from the consistency conditions in critical systems (table 2) and a considerable amount of algebra, we have (with the equality satisfied in an active system)(5.2)where(5.3)and(5.4)This last set of equations is equivalent to eqn (5.5) in Chidambarrao & Havner (1988).

Equation (5.2) requires *H*_{1}≥*d*_{1} and *H*_{5}≥*d*_{5} (at least one of which must be an equality). Therefore, by combining , we can eliminate the unknown rate-of-change *g*′ of the constraint stress to obtain(5.5)an inequality we shall make use of throughout the range.

## 6. A proof of uniqueness with all critical slip systems active

We consider the possibility of a non-unique solution *x*≠0 in range II *with all critical slip systems active*. (The latter automatically is the case in all systems for the rate-dependent equations investigated in appendix A*b*.) With systems 1, 2, 5 and 6 active, *H*_{1}=*d*_{1} and *H*_{5}=*d*_{5} (from equation (5.2)), whence equation (5.5) is an equality. Moreover, if *x*≠0 is assumed, we further must have *h*_{1}=*a*_{1} and *h*_{5}=*a*_{5}, from which (respectively; making use of equation (3.9), , at the onset of the finite deformation)The difference between these equations can be rearranged asFrom the hardening inequality *H*_{IV}≥*H*_{III}, it is seen that the combined second and third terms are *not less than* (1/4)(2+*b*)*H*_{III}. Thus, from (equation (2.2)) and the last equation above (a consequence of our initial assumption that *x*≠0), we have the ordered sequencea contradiction. Thus, from the hardening inequalities, *there is no initial solution x*≠0 *with all four slip systems active*. We have only the solution *x*=0, corresponding to equal slip rates within pairs, unless at least one in critical systems. (Possible solutions with one or more inactive critical slip systems are analysed in appendix A*a*.)

## 7. Consequences of all critical slip systems active in range II

With only systems 1, 2, 5 and 6 critical in range II, the general equations for and on lattice axes in (110) channel die compression (eqn (8) in Havner 2007*a*) reduce to(7.1)which obviously satisfy and (from ** ι**=(110), ) as required. Thus, the only solution consistent with the hardening inequalities when all critical slip systems are active,

*x*=0 (from §6), gives(7.2)corresponding to

*lattice stability*in range II. Henceforth, we shall consider all slip systems to be active and so adopt equation (7.2) and

*x*=0, with(7.3)from equation (4.2). Moreover, the previously defined kinematic sub-ranges are no longer relevant, and all equations beginning with equation (7.2) apply throughout range II, 2>

*b*>1. (The solution at the limit

*b*=1, the brass orientation, with only systems 1 and 2 active, is precisely that in Havner (2008) for the corresponding limit orientation in range III, wherein slip systems 9 and 10, rather than 5 and 6, otherwise are active.)

### (a) Evolution of the constraint stress

With all systems active, *H*_{1}=*d*_{1} and *H*_{5}=*d*_{5} from equation (5.2). Consequently,(7.4)from equation (5.5), and we have the following equation for *g*′ (from *H*_{5}=*d*_{5}):(7.5)If constraint *g*, as the loading stress *f*, were determined experimentally, equations (7.4) and (7.5) would constitute a set of two independent equations for the four evolving moduli *H*_{I}, *H*_{III}, *H*_{IV} and *H*_{V}. If, further, *H*_{IV}/*H*_{I} and *H*_{V}/*H*_{I} had essentially constant ratios during an interval of the finite deformation, explicit equations for the evolution of *H*_{I} and *H*_{III} in terms of *f*, *g* and the fixed lattice orientation could be expressed. Unfortunately, there are apparently no range II orientations for which the constraint stress has been determined experimentally. Consequently, rather than speculating at present about moduli ratios, it seems more useful to obtain a bound on the evolving constraint stress, in particular comparing it with the explicit prediction of classical Taylor hardening (following the precedent in Havner 2007*a*).

To this end, we first give a *less* simple equation for *g*′ than equation (7.5) by subtracting *H*_{5}=*d*_{5} from *H*_{1}=*d*_{1} to obtain(7.6)For Taylor hardening, with all hardening moduli equal, the term in wavy brackets *is a zero identity*. Thus, the constraint stress *g*_{T} for Taylor hardening evolves according to (first given in Chidambarrao & Havner 1988, eqn (5.15))(7.7)Thus, since *b* is constant (from lattice stability),(7.8)This may also be deduced from equation (3.9), , because the resolved shear stresses in the four active slip systems must increase equally for Taylor hardening, hence the ratio between *g*_{T} and *f* remains constant during deformation for a fixed lattice orientation.

We now substitute *f*′ from equation (7.7) into equation (7.6) to obtain(7.9)Thus, from of the hardening inequalities,(7.10)except at the outset, when they both equal *g*_{0}. Thus, we have a comparable result as in range I (*∞*>*b*>2) and can make an identical statement (Havner 2007*a*, p. 618): ‘*the Taylor hardening constraint stress is an upper bound to the expected constraint stress, and always exceeds it except at the beginning of finite deformation*’. (This is in contrast to range III, wherein a (compressive) constraint stress develops according to the hardening inequalities, which stress remains zero for Taylor hardening (Havner 2008, §7).) As also remarked in Havner (2007*a*), if the constraint stress were determined experimentally, these predictions could be assessed against experiment to either lend support or partially refute the general hardening inequalities as applied to (110) channel die compression.

### (b) Finite crystal shearing

We denote angles of possible finite shear in the horizontal (*YZ*) and vertical (*XZ*) channel planes (figure 1) by *Χ*_{x}, *Χ*_{y} (the anticlockwise rotation about the *x*-axis and clockwise rotation about the *y*-axis of material lines initially coincident with the *Y*- and *X*-directions, respectively). The correct equations for their evolution are (Havner 2007*a*, eqn (11); equivalent to eqn (5.15) in Havner & Chidambarrao 1987)(7.11)We have and , with(7.12)in full vector form. Thus, from equations (3.6) and (7.12) and table 2, with only slip systems 1, 2, 5 and 6 active(7.13)(in which the equations have been simplified using the three channel die constraints, equation (4.1)).

Remarkably, from equations (7.11)_{1} and (7.13)_{2}, the initial rate of shearing in the horizontal channel plane, , is independent of the solution for *x*! (i.e. it is independent of whether or not all four slip systems are active in the sub-ranges). Moreover, it is identical with that in range III (Havner 2008, §7).

For all systems active, d_{xz}=0 from equation (7.3), and there is no finite shearing in the vertical channel plane. Then, as the lattice does not rotate (equation (7.2)), *b* is constant. Therefore, in the horizontal plane (from equations (7.11)_{1} and (7.13)_{2}), , whose solution is(7.14)This is also identical with the range III result in Havner (2008, eqn (27)). Hence, as noted therein (§7), for the common limit point of the two ranges, the brass orientation *b*=1 (, ), the equation reduces to an equivalent one first derived by Chin *et al*. (1966*b*, eqn (36)), which gave excellent agreement (see their fig. 6) with experimental measurements of Chin *et al*. (1966*a*) for a Permalloy crystal. Moreover, this prediction of finite shear in the horizontal channel plane is consistent with measurements of Driver & Skalli (1982) on an aluminium crystal in the brass orientation (see their fig. 4(a)). (In contrast to ranges II and III, in range I the lattice rotates about the (110) load axis, as is well-confirmed experimentally. Hence, *b* changes with deformation and the solution is much less simple than equation (7.14), for which see eqns (39) and (42) in Havner 2007*a*.)

## 8. Analysis of continued lattice stability

To investigate whether the lattice orientation is predicted to remain stable, with all initially critical systems active, requires two analyses. First, the proof in §6 must be revisited because it made use of the relationship between *f*_{0} and *g*_{0} at the onset of the finite deformation, and that relationship changes with compressive strain *e*_{L} (equation (7.9)). Second, we must compare the changing resolved shear stresses (equation (3.8)) in the initially non-critical but positively stressed slip systems with their evolving critical strengths (according to table 1) to determine whether they subsequently may become critical.

### (a) Investigation of a theoretical critical strain for possible loss of lattice stability

Let us assume that after some finite interval of straining there is a solution *x*≠0. Then, as in §6, we must have both *h*_{1}=*a*_{1} and *h*_{5}=*a*_{5} from equation (5.2), which give(8.1)with(8.2)from the integration of equation (7.6) up to the critical strain *e*_{cr}. (A bar above signifies the mean value of a modulus in the interval, and the subscript ‘A’ is a mnemonic for anisotropic. would be zero for isotropic Taylor hardening.) Upon eliminating *f*_{cr} and *g*_{cr} between equations (8.1) and (8.2), one obtains (positive from the hardening inequalities)(8.3)Thus, there is a theoretical critical strain at which there could be a change from continued lattice stability with all systems active. However, equation (8.3) is not yet the full solution for that strain because contains integrals of the individual moduli up to *e*_{cr} (i.e. the mean value of a modulus is given by ).

Whereas *h*_{1}=*a*_{1} and *h*_{5}=*a*_{5} only at the critical strain, *H*_{1}=*d*_{1} and *H*_{5}=*d*_{5} from equation (5.2) at all levels of strain up to and including *e*_{cr}. Thus, equation (7.4) for *f*′ holds throughout, from which, upon integrating, we have(8.4)where *h*≡d*f*/d*e*_{L} and .

Although *H*_{α}'s that must be integrated to obtain and are, in general, nonlinear functions of the slips and so vary nonlinearly with increasing strain (thus making equations (8.3) and (8.4) complex equations for the critical value), there is experimental support for taking the *ratios* of the hardening moduli to be essentially constant. This is particularly the case for copper crystals, as discussed in Havner (2008, §8), based upon the classic latent hardening experiments of Jackson & Basinski (1967). Also, there appear to be only relatively small variations with deformation in silver and aluminium, from latent hardening experiments of Kocks (1964) and Kocks & Brown (1966; see discussion in appendix A*a*(i) here). Henceforth, in order to obtain an approximate magnitude for *e*_{cr}, we assume the ratios of the various moduli *H*_{α}, where *α*=I, II, III, IV and V, to the self-hardening modulus *H*_{I} to be constant and denote (from the hardening inequalities)(8.5)with all *r*'s constant. We further define orientation-dependent parameters(8.6)Then, from equations (8.3)–(8.6), we obtain(8.7)as the governing equation for the critical strain at which non-uniqueness is theoretically possible in range II.

Consider the following analytical form of the compressive stress–strain curve:(8.8)with *f*_{0}, *h*_{0}, *h*_{T} and *e*_{0} constants. Although originally intended for representation of uniaxial polycrystalline stress–strain curves when first suggested (without the *f*_{0}) in Havner (1966), it has been shown to well-represent experimental data for various orientations of single crystals of copper (Sue & Havner 1984) and aluminium (Havner & Chidambarrao 1987) in channel die compression and of aluminium in axial tension (Havner & Yu 2005), both with and without the *f*_{0}. Adopting it here, equation (8.7) for the critical strain becomes(8.9)

To be specific, the data points of Skalli (1984) to *e*_{L}=1.0, for an aluminium crystal in a nominal orientation in channel die compression (*b*≈2), are closely fitted by *f*_{0}=58.0, *h*_{0}=558, *h*_{T}=44.5 (all MPa) and *e*_{0}=0.0871 (from eqn (5.16) in Havner & Chidambarrao 1987; see also Chidambarrao 1988, fig. 8). Moreover, experimentally based values for the hardening ratios are *r*_{V}=1.25 and *r*_{IV}=*r*_{III}=1.2 (see discussion in the appendix A*a*(i)), from which (at *b*=2) *r*_{A}=0.3 and *r*_{E}=0.4 from equation (8.6). Thus, for aluminium at the beginning of range II (), equation (8.9) givesfrom which *e*_{cr}=0.350, corresponding to a shear *Χ*_{x} in the horizontal plane of 30.65° (from equation (7.14)). For an aluminium crystal in this orientation, to a strain of approximately 1.0, Skalli *et al*. (1983, p. 297) indicate (i) finite shear only in the horizontal plane (as here), (ii) a lattice rotation of ‘some degrees’ about the (110) load axis (evidently small from a subsequent statement), and (iii) slip only on lattice planes *a* and *b*. The analysis herein permits no lattice rotation up to the theoretical critical strain of 0.35, but conceivably further analysis might permit a lattice rotation beyond that strain (which would require the *c*-plane to become active if ). However, *x*=0 (hence continued lattice stability) is always a theoretically admissible solution consistent with the hardening inequalities. (Many things can contribute to small differences between theory and the finite deformation experiments, of course, including frictional effects along the channel walls not taken into account here.)

At the other limit of range II, the brass orientation , the data points (extending to *e*_{L}=0.83) and full-range experimental curve B in fig. 2 of Driver *et al*. (1994) are well represented by *f*=25+60*e*_{L}+15.2 tanh (10.13*e*_{L}) MPa (*f*_{0}=25.0, *h*_{0}=214, *h*_{T}=60.0 (all MPa), and *e*_{0}=0.0987). At *b*=1, *r*_{A}=0.1 and *r*_{E}=0.35 from equation (8.6). Thus with the same experimentally based values of the hardening ratios as before, we have (from equation (8.9))which gives *e*_{cr}=3.25! This corresponds to a thickness reduction of 96%(!) and is far beyond any strain achieved in this orientation range in the experiments of Skalli *et al*. (1983), Skalli (1984) and Driver *et al*. (1994). Thus, based on the equations governing the initially active systems, we may conclude that lattice stability should continue (at least in aluminium crystals) through any realistic experimental finite deformation levels for orientations near the end of the range (*b*→1). This is fully consistent with the experiments of Skalli *et al*. (1983), who found the small tendency towards lattice rotation about *X*=(110) at the other end of the range (*b*≈2).

For intermediate orientations in range II, it seems reasonable to consider *e*_{0} in equation (8.9) to be of order-of-magnitude 0.1 (consistent with its empirical values of 0.0871 and 0.0987 at the ends of the range). It would also seem that we may expect *e*_{cr} to lie somewhere between the critical strain values calculated above. In such a case, equation (8.9) can be greatly simplified because and then would equal 1 and 0, respectively, to 3 or more significant figures. Thus, we have (to that approximation) the following very simple equation for *e*_{cr}:(8.10)

Consider *b*=4/3. Then equations (8.6) and (8.10) give or (with *e*_{0}≈0.1). From the values above, the empirical ratios *h*_{0}/*h*_{T} in aluminium at the beginning and end of the range are 12.5 and 3.57, respectively. Thus, even for a ratio at the high end, say 10 for simplicity, *e*_{cr} remains quite large, with an approximate value of 1.3, indicating that the lattice orientation would remain stable through very large deformations.

Now consider copper in (110) channel die compression, again at *b*=1. Sue & Havner (1984) found that *h*_{0}=168 ksi (1158 MPa), *h*_{T}=21.8 ksi (150 MPa) and *e*_{0}=0.190 give extraordinarily close agreement with the experimental stress–strain curve of Wonsiewicz & Chin (1970, fig. 1, curve 4) over an 80% strain range for a copper crystal in this orientation. Also, from the classic latent hardening experiments of Jackson & Basinski (1967) on axially loaded copper crystals, appropriate values of the hardening ratios are (see discussion in Havner 2008, §9) *r*_{V}=1.35, *r*_{IV}=1.32 and *r*_{III}=1.28, from which equation (8.6) give *r*_{E}/*r*_{A}=5.5 at *b*=1. Thus, from equation (8.10), *e*_{cr}=4.23(!), again indicating predicted lattice stability to any achievable finite deformation, as found by Wonsiewicz & Chin (1970) in their experiment to a strain of 0.8. (I am not aware of experimental data for copper crystals in other orientations in range II. Thus, the question of lattice stability near the beginning of the range cannot be investigated at present.)

### (b) Investigation of the latent slip systems

As noted at the beginning of §8, lattice stability could be affected if initially inactive slip systems subsequently became critical. Thus, we must compare the evolving critical strengths with the changing resolved shear stresses in all positively stressed latent systems in range II.

In the active systems, from equations (5.1) and (7.3),(8.11)and in the positively stressed latent systems, from table 1,(8.12)with the bar above 7 and 8 signifying the opposite sense systems from those defined in table 1. (From equation (3.8), the original systems 7 and 8 are negatively stressed in range II, and slip systems 11 and 12 are unstressed throughout (110) channel die compression.)

Because the lattice has not rotated, the resolved shear stress rates, equation (3.3), reduce to in general, and to for systems 3 through (the corresponding *m*_{k}'s being zero). Then, recalling from equation (3.8) that *τ*_{5}>*τ*_{3}, we have (as the *r*_{k}'s are constant). Moreover, we see from equations (8.11) and (8.12) that (the first inequality making use of *H*_{III}>*H*_{I})(8.13)Thus we have , whence the evolving critical strength in systems 3 and 4 is never attained by their resolved shear stress and those systems remain inactive.

For the remaining positively stressed latent systems, the resolved shear stress rates are (from equations (3.8), (7.4) and (7.5))(8.14)From the first of these and equation (8.12)_{2},(8.15)This can be established as positive for all initial orientations in range II by substituting for *H*_{III} and *H*_{II} from the hardening inequalities *H*_{III}≤*H*_{IV} and *H*_{II}>*H*_{I}. After some algebra, one obtains(Q.E.D.). Thus, as with systems 3 and 4, neither can systems and become critical as deformation proceeds in range III.

Finally, we investigate latent systems 9 and 10. From equations (8.12)_{3} and (8.14)_{2},(8.16)obviously positive for any but an extreme anisotropy (for which there is no experimental support) that would require *H*_{V} to be more than *three times larger* than each of the other moduli. Thus, systems 9 and 10 (, *b*1), as the other latent systems, remain inactive. (For aluminium, adding *H*_{II}=1.15*H*_{I} (consistent with experiments of Kocks & Brown 1966) to the moduli ratios given in §8*a*, we find . For copper, adding *H*_{II}=1.23*H*_{I} (see Havner 2008, §9, p. 80) to the other ratios in §8*a*, we have .)

In summary, none of the initial latent systems will become active during any finite deformation. Hence equation (8.7) from the analysis in §8*a* (or the particular analytical equation (8.9)) remains the governing equation for a critical strain and possible loss of lattice stability in range II.

## 9. Closing discussion

As has been shown, when all critical systems are active, the hardening inequalities from Havner (2005) (equation (2.2) here) uniquely predict initial lattice stability, and a consequent finite shearing only in the horizontal channel plane, for all constraint directions *Y* between and (range II) in (110) channel die compression, as commonly found experimentally. Also, it is established that the lateral constraint stress predicted by the inequalities will be less than that given by classic Taylor hardening. Moreover, taking into account experimental stress–strain curves for aluminium and copper, and hardening ratios based upon classic latent hardening experiments in those metals, it is found that if there were a subsequent small lattice rotation about the (110) load axis after the finite deformation, it should be expected only near the end of the range for the levels of deformation that have been obtained experimentally, consistent with experiments of Skalli *et al*. (1983), who reported such a rotation.

In investigating the application of the hardening inequalities in range I (constraint directions between and ) in Havner (2007*a*,*b*) and range III (constraint directions from to ) in Havner (2008), the assumption that all critical slip systems are active was *not* made. Nevertheless, it was established in those works that the hardening inequalities, together with experimental information on stress–strain curves and experimentally based hardening ratios, uniquely predicted load-axis stability throughout, with lattice rotation about the load axis only in range I, as consistently found experimentally. Consequently, it would seem in order to carry out an analysis for range II without that assumption. This is done in appendix A*a*, where it is necessary to perform separate analyses of two sub-ranges, *Y* between and and between and , and of as it is a singular orientation mathematically.

For the first sub-range, making use of the same experimental stress–strain data and hardening ratios for aluminium as before, it is again shown that lattice stability is the unique solution from the hardening inequalities, at least for this metal. However, in the narrow orientation range (only 8.05°) of the second sub-range, including (*ϕ*_{0}=46.69°), it is found in appendix A*a*(ii),(iii) that other initial solutions are mathematically possible, which would rotate the lattice away from a stable (110) position. However, the hardening inequalities remain *consistent* with lattice stability in the second sub-range, since they uniquely predict it by considering the four critical slip systems to be active.

Owing to the assumption in the main body of the paper that all critical systems are active, it also seems appropriate to investigate the standard power law form of rate-dependent theory, in which all positively stressed slip systems automatically are active. This is done in appendix A*b* for (110) channel die compression. It is shown algebraically that the power law rate-dependent theory never predicts lattice stability *exactly* for any orientation, but that in the brass orientation , for small values of the ‘rate-sensitivity’ parameter, as typically adopted, lattice rotation would be negligible. This also probably would be the case for other orientations in ranges II and III, but that would require specific selection of parameters and numerical studies for confirmation.

## Footnotes

- Received October 15, 2007.
- Accepted March 5, 2008.

- © 2008 The Royal Society