## Abstract

An anisotropic, viscoplastic power law is introduced and applied to analysis of the Goss and Brass orientations, and compared with predictions from rate-independent theory and experiment. The structure of the new power law is so chosen that it has the capability of approaching rate-independent results for lattice rotation and crystal shearing, after finite rotation about the load axis, in the range of unstable lattice orientations in (110) channel die compression. (Rate-independent predictions of shear and lattice rotation are in good to very good agreement with experiments on aluminium and copper in that range, whereas classic *isotropic* power-law results are not.) It is established that, for sufficiently large power-law exponent *n*, the new anisotropic, elasto-viscoplastic theory predicts: (i) lattice stability in each of the Goss and Brass orientations, consistent with both experiment and rate-independent theory; (ii) zero crystal shear in the Goss orientation, also consistent with both; (iii) finite shear in the Brass orientation, in very good agreement with experiment and rate-independent theory; and (iv) a lateral-constraint stress that remains essentially elastic in both orientations, as predicted by rate-independent theory and close to experimental measurements for aluminium and copper in the Brass orientation.

## 1. Introduction

Face-centred cubic (fcc) crystals in channel die compression have been studied experimentally and their finite-deformation responses analysed over a span of half-a-century. Beginning with the classic work of Chin *et al*. [1], orientations with a (110) compression axis have been of particular importance to investigate because that load direction is stable relative to the crystal lattice and also is a preferred normal texture in sheet-rolling of fcc polycrystalline metals. Moreover, among the many experimental investigations within the family of (110) loading, the ’Brass’ and ’Goss’ orientations

In this paper, a new *anisotropic* viscoplastic power law is introduced and applied to analysis of the Goss and Brass orientations, and its predictions compared with rate-independent theory and experiment. The structure of the evolution of the power law’s reference stresses is geometrically based in the (five) different slip-system interactions in fcc crystals. Moreover, inequalities among the parameters defining that evolution are given a deductive basis from an analysis (appendix A) of the failure of the classic *isotropic* power law to give satisfactory predictions (for any exponent *n*) of lattice rotation and crystal shear in the range of orientations for which there is finite lattice rotation about the load axis (as shown in [12]). By contrast, rate-independent theory gives good agreement with experiments on fcc crystals in that range—both classic (isotropic) Taylor hardening and the experimentally and geometrically based anisotropic hardening introduced in Havner [13], which has been applied in Havner [12,14–21] over the full range of orientations in (110) channel die compression.

The paper is organized as follows. Section 2 contains general equations in (110) channel die compression. Section 3 provides a brief review of rate-independent theory. The new anisotropic power law is introduced in §4 and appendix A. A comprehensive viscoplastic analysis of the Goss orientation is presented in §5. Section 6 contains kinematic and stress equations in the Brass orientation, followed by the anisotropic power-law analysis of that orientation in §7 (with detailed calculations for aluminium and copper). Comparisons with rate-independent theory and experiment are included in both §§5 and 7. The results are concisely summarized in §8.

## 2. General equations in (110) compression

With *X*, *Y* and *Z*, the loading, lateral constraint and channel-axis directions (figure 1), and **ι**,**κ**,**μ** unit vectors in the respective directions (such that **ικμ** constitutes a rigid orthogonal triad fixed in the channel frame), the general kinematic and constraint equations in (110) compression may be expressed (eqns (12), (30) and (9) in [20])

where **b**_{j}, **n**_{j} are unit vectors in the slip and slip-plane normal directions, respectively, of the *j*th system, *X*-direction, with *λ* (less than 1) the corresponding ‘spacing stretch’. The first line in equation (2.1) expresses the ideal constraints (assuming, say, a hardened steel die that is ’rigid’ relative to the well-lubricated crystal, with Teflon coating or other). *χ*_{x}, *χ*_{y} are the angles of shear in the *YZ*- and *XZ*-planes (figure 1); and *ϕ* is the clockwise orientation about (110) load-axis *X* of lateral-constraint direction *Y* with respect to the *ω*_{x},*ω*_{y},*ω*_{z} from any orientation in a channel die, sans lattice–strain contributions.)

Let *f*, *g* denote the ‘true’ compressive load and lateral constraint stresses corresponding to a uniform stress and deformation state in the crystal (well away from the end faces). The general equations for resolved shear stress *τ*_{k} and its rate-of-change in the *k*th slip-system, for any orientation, are (eqns (2) and (9) in [14], and eqn (8) in [20])
**ι** and **κ**. In (110) compression, the resolved shear stresses in the 12 fcc crystal slip-systems

with κ_{1} the direction cosine of the *Y* -axis with the [100] lattice direction. (Note that opposite-sense systems *b*<2, and opposite-sense systems *b*<1. Over the 90° range of distinct lattice orientations in (110) compression, *b* goes from *ϕ*=0) to 0 at *ϕ*=90°), the Goss orientation. In the Brass orientation *ϕ*=54.74°).)

In the rate-independent analyses of finite plastic deformation (neglecting lattice straining) in Havner [15–19], for the orientation ranges defined by Roman numerals in figure 2 and the singular orientations **ι**=(110) (also see eqn (8)_{1} of [14]). Then, from

Lastly, from the transformation matrix from lattice to channel axes, expressed in terms of orientation parameter *b* (eqn (3) in [20]), the lattice strains in any cubic crystal (bcc as well as fcc) are
*s*_{11}, *s*_{12} and *s*_{44} are the cubic elastic compliances in standard notation (with *s*_{A}, zero if isotropic). (As easily confirmed, *ε*_{xx}+*ε*_{yy}+*ε*_{zz}=−(*s*_{11}+2*s*_{12})( *f*+*g*), with *s*_{11}+2*s*_{12} the inverse bulk modulus or *isotropic* ‘linear compressibility’ of cubic lattice systems [24], pp. 146–147.)

## 3. Rate-independent theory

We briefly review rate-independent theory because the structure of that theory provides a helpful perspective in the formation of the anisotropic power law. In rate-independent theory, the ’critical-strength’ rate in the *k*th system is determined from the slip-rates in all *n* systems through the hardening moduli *H*_{kj} ([25], eqn (4); also see [26], eqns (7.4)–(7.5)):
*H*_{I} through *H*_{V}. The inequalities among the slip-systems moduli, first introduced in Havner [13], are

These inequalities were deduced therein from analyses of a range of diverse experimental behaviours in high-symmetry tensile-load orientations of fcc crystals: Kocks [27] on aluminium (Al); Nakada *et al*. [29], gold (Au); Ramaswami *et al*. [30], silver (Ag) and an Au–Ag alloy; Y. Nakada at Bell Labs, also Al (as reported in [2], p. 2721); and Vorbrugg *et al*. [31], Ambrosi *et al*. [32], Ambrosi & Schwink [33] and Franciosi & Zaoui [34], all copper (Cu); plus Keh [35] on bcc iron (Fe). The inequalities also reflect classic latent-hardening experiments on crystals deformed in single slip: Kocks [27], Al and Ag; Kocks & Brown [36], Al; Jackson & Basinski [37], Cu and Franciosi *et al*. [38], Cu and Al. Moreover, in subsequent analyses of the very different loading configuration of fcc crystals in (110) channel die compression [14–19], the inequalities have been shown to either uniquely predict or be consistent with various finite-deformation crystal experimental responses of Al, Cu and a nickel (Ni) alloy.

## 4. A geometrically based anisotropic power law

The general power law of viscoplastic crystal slip may be expressed ([39], eqn (2.4))
*k*th slip-system and 1/*n* is the rate-sensitivity parameter. Hutchinson [39], focusing on creep of metals at relatively high homologous temperatures, took *τ*_{R} is adopted that is the same in all systems but changes with the slips, corresponding to the *isotropic* power law
*χ*_{x} (figure 1). Consider the lattice rotation from an initial orientation *ϕ*_{o}=0.81° (*b*_{o}=100) into *b*=2). This is analysed in Havner [12] for both rate-independent hardening (§5 there) and the isotropic power law (§6). For sufficiently large *n*, the power-law results can be made to agree closely with rate-independent theory for the *initial* lattice rotation and shear rates. However, after the lattice rotation of more than 34° into *half* the rate-independent result (requiring a larger strain than for rate-independent theory to attain that rotation), and the power-law shear-rate 2*d*_{yz} is *two-and-a-half or more times* the rate-independent rate (see tables 9 and 10 of [12]). The integrated rate-independent equations for finite lattice rotation and shear in range I ([20], eqns (40) and (41); also see the algebraically equivalent eqns (3.19)–(3.20) of [40]), which disregard lattice strains, give good to very good predictions in comparison with experimental results for Al and Cu of Skalli [6], Butler and Hu [41], and Wróbel *et al*. [42]. (See §5.3 of [12] for detailed comparisons and discussion.) Consequently, because of the great differences between the viscoplastic and rate-independent results after rotation into *cannot predict well either lattice rotation or crystal shear after large strain in range I* (for whatever power *n*, with or without lattice elasticity)’ [12], p.1949. Moreover, the corresponding constraint-stress rates in viscoplastic theory are problematic as they vary greatly (even as to sign in Al) with *n* and the precise value of *g*/*f* at the onset of finite deformation (see tables 6 and 7 of [12]).

In the Brass orientation, by contrast, the isotropic power law can give very good results in comparison with experiment (and the closely comparable rate-independent results), predicting lattice stability, accurate finite shear (as a function of compressive strain), and a consistent constraint stress so long as lattice elasticity is included ([12], §7.2). (The Goss orientation had not been analysed heretofore using a viscoplastic power law.)

There are two reasons for the great difference in isotropic power-law results between these two cases (as compared with experiment and rate-independent theory). They are: (i) that there is finite lattice rotation in range I, and (ii) that only systems *anisotropically*. The objective, of course, would be to achieve results closer to the rate-independent and experimental values in range I, but in such a way that the very good predictions from the isotropic power law in the Brass and Goss orientations would not be affected negatively by the change to anisotropy.

A general equation for reference-stress rates in terms of slip rates (analogous to equation (3.1) for critical-strength rates in rate-independent theory) was given by Peirce *et al*. [43]. However, they then adopted a simple, two-parameter form that distinguished only between coplanar and non-coplanar slip-system interactions (i.e. one parameter for case (I), equation (3.2), and a second for the other four cases). Their two-parameter viscoplastic theory also was applied by Asaro & Needleman [44], Harren *et al*. [45], and others.

Here we shall consider the evolution of reference stresses *five* moduli *h*_{α} (labelled *h*_{I} through *h*_{V}), one for each of the five geometric interactions among fcc crystal slip systems defined in equation (3.2) and table 2. We thus propose an anisotropic viscoplastic theory in (110) channel die compression defined by (equivalent to equation (4.1))
*τ*_{k} given by equation (2.5) as before, and the reference stresses *b*≥0), and systems 3, 4 replaced by *b*≥0) (in each case because the opposite-sense systems become positively stressed in the respective ranges). Moreover, the anisotropy is assumed to evolve only with finite deformation; hence the slip-system reference stresses *h*_{α} to achieve the objective stated above?

A deductive argument, based in range I, is presented in appendix A proposing that the same inequalities should hold among the *h*_{α} as among the (rate-independent) slip-systems hardening moduli *H*_{α}, equation (3.3) (which inequalities have substantial support from axial-load experiments on fcc crystals, as discussed in §3). Thus, we adopt the following inequalities for the *h*_{α},

## 5. Viscoplastic analysis of the Goss orientation

From equations (2.1), (2.3)–(2.6), evaluated at *b*=0, and the direction cosines of the slip systems from table 1, the kinematic and stress equations applicable to any fcc crystal, and to both rate-independent theories and viscoplastic power laws in the Goss orientation, are
*χ*_{y}≡0 from *d*_{xz}≡0, equation (2.1)), and
*g*_{E} and lateral strain-rate *d*_{yy}, there because the (passive) lateral-constraint stress *g* cannot be tensile. Also, as noted following equation (2.3), opposite-sense systems *g*.

In the equation for *β*_{E} in the Goss orientation, the denominator *s*_{12} always negative) can be either positive or negative, dependent upon the relative magnitudes of the crystal compliances. Cu *contracts* in the *X*-direction (negative *β*_{E}), in contrast with Al, which wants to expand laterally (*β*_{E}>0), hence presses against the channel walls and develops a compressive constraint stress *g*=*β*_{E}*f*>0 in the elastic range. Upon calculating *β*_{E} in the Goss orientation for each of Al, Cu, Ni, Au, Ag and (bcc) Fe, using the *s*_{11}, *s*_{12}, *s*_{44} values in Nye [24], one finds that *β*_{E} *is negative for all but Al*. Correspondingly, there would be no elastic constraint stress in the Goss orientation for any one of Cu, Ni, Au, Ag or Fe.

Experimental data are readily available for finite deformation in the Goss orientation for Cu and Al. The experimental curve of Wonsiewicz & Chin [2] for Cu in this orientation (curve 5 in their fig. 4), together with its very close analytical representation
*et al*. [9] for an A1-1pct. manganese (Mn) crystal in the Goss orientation are shown in figure 4, together with the analytical representation ([19], fig. 5 and eqn (11))
*e*_{L}=1, except at 0.2 and 0.3 (with a maximum difference of 5 pct).

For the anisotropic power law, from eqns (4.3) and (5.2),
*τ*^{R}_{o} at the onset of finite deformation. Thus, initially,
*τ*^{R}_{o}=*τ*_{R}). Then, from equation (5.1),
*d*_{yz}=0. From equations (5.6) and (5.7), the initial rates of change of the reference stresses then reduce to
*not* develop within either double-pair of systems with equal resolved shear stresses (i.e. 1, 2 with 9, 10, and *the lattice is stable with no crystal shearing*, consistent with channel die experiments in the Goss orientation ([2], p. 2717; [8], pp. 1294–1295; [9], p. 649 and [11], p.749). (For an anisotropic rate-independent analysis of the Goss orientation neglecting lattice straining [19].)

Consider the lateral constraint. From *d*_{yy}≤0 (equation (5.1)) and equation (5.10),
*g*=0 and the crystal contracts laterally according to *β*_{E}>0, as in Al, *g* is positive, equation (5.11) becomes an equality, and *g* evolves from its initial value *g*_{E} by
*g*/*f*=*β*_{E} (0.271 in Al), the rate of decrease is the same as with the isotropic power law, and extremely small for large *n*. The *n*th power of the stress ratio will continue to decrease as *g* decreases. Although ^{5} MPa in Al), *β*^{n}≡(*g*/*f*)^{n} is extremely small for large *n*(less than 10^{−28} in Al at the onset of slip for *n*≥50). Moreover, it is likely that *q*_{α}. Thus, we may expect *g* to negligibly decrease below its initial elastic value; and Al (or any fcc metal or alloy with *β*_{E}>0) will remain laterally constrained essentially elastically in the Goss orientation.

## 6. Kinematic and stress equations in the Brass orientation

The kinematic equations in the Brass orientation (from equation (2.1), evaluated at *b*=1, and direction cosines from table 1) are
*d*_{yy}=0 were used to simplify the equations for *d*_{yz}, and *b*=1 become (from equation (2.6))
^{−3} or less in Al and Cu) compared with 1 in the denominator of the resultant equation for

## 7. Viscoplastic analysis of the Brass orientation

For Cu, the stress–strain curve of Wonsiewicz & Chin ([2], fig. 1, curve ‘4’) in the Brass orientation is very closely represented by
*et al*. ([7], fig. 2, curve ‘B’) are well represented by Havner ([17], p. 1968)

### (a) The onset of finite deformation

At the onset of finite deformation, with all *β*_{o}≡(*g*/*f*)_{o}=*β*_{E} ([21], §5.1), there is no difference between the isotropic and anisotropic power laws as regards slip-rates, which are
*n*≥100 the power law gives results for each of *g*′, finite shearing-rate *ϕ*′ that differ from the (corrected) rate-independent results (see the 2015 Corrigendum to [12]) by at most 1 in the fourth significant figure. (The prime superscript signifies differentiation with respect to *e*_{L} rather than time). In evaluations, the elastic compliances in Nye [24] are used, with respective (*s*_{11}, *s*_{12}, *s*_{14}) values for Al and Cu of (1.59, −0.58, 3.52) and (1.49, −0.63, 1.33), units 10^{−5}/MPa, which give *s*_{A}=0.41 and 1.455 (10^{−5}/MPa), respectively, from equation (2.6)).

From the resolved shear stresses in equation (6.4) and *g*<*f*, only slip-systems 1, 2 *rate-independent theory* in the Brass orientation; and the lateral constraint in equation (6.1) reduces to *g*=*β*_{E}*f*. Upon disregarding all lattice strains, the rate-independent solution for finite shear in the Brass orientation is *b*=1). As noted there (p. 82), Chin *et al*. [46] derived an equivalent equation (their eqn (36)) and showed excellent agreement with experimental measurements of Chin *et al*. [1] on a Permalloy crystal (4pct Mo, 17pct Fe, 79pct Ni). Also see the discussion in Havner [18] of experimental results from Driver *et al*. [7] for finite shearing of an Al crystal.

Although lattice straining makes only a negligible difference in the results for rate-independent theory, *its inclusion is essential for the power-law in the Brass orientation*, as first shown in Havner [21]. Upon setting all compliances equal to zero, the lateral constraint *d*_{yy}=0 gives *B*_{n}=0 in equation (7.5), which forces a unique relationship between stress-ratio *β*_{o} and power-law exponent *n*: _{1} and line following). For *β*_{o}→0.75, already equalling 0.745 at *n*=50. Experimental ranges for *β* in the Brass orientation are 0.30–0.36 in Cu ([3], fig. 5) and 0.25–0.38 in Al ([9], fig. 5, after 10pct strain). (To my knowledge these are the only experiments in which the lateral-constraint stress was determined in the Brass orientation.) The corresponding *β*_{E} ratios are 0.2788 (Cu) and 0.3367 (Al) (equation (6.4)), comparable with the experimental values. Alternatively, if one adopts *β*_{E} for *β*_{o} in the above equation for *n*, *an extreme rate-sensitivity results*: *n*=0.559 in Al and 0.469 in Cu. Consider its effects.

Without lattice straining, from the lateral-constraint equation (7.5) reduces to
*ϕ*′ equation is identically eqn (37)_{2} in [21]. The _{3} of [12], corresponding to that noted above for the elastic/viscoplastic shearing-rate.) These equations are entirely satisfactory for large *n*, respectively approaching 0 and *g* is substantially in error (more than twice too large, as noted above). By contrast, with *n* values for Al and Cu (0.559 and 0.469, respectively) corresponding to *β*_{o}=*β*_{E}, equation (7.6) gives large and unrealistic *ϕ*′ values of 0.657 (Al) and 0.673 (Cu), rather than 0 (in the limit), and much-too-large values for

### (b) Power-law analysis as anisotropy develops

Guided by the differences in the *r*_{α}≡*H*_{α}/*H*_{I} in anisotropic rate-independent theory (from latent-hardening experiments on Al and Cu noted in §3), we consider the differences among the *q*_{α}≡*h*_{α}/*h*_{I} to be at most a few tenths. In addition, the difference between *n*≥100 in both Al and Cu, [12], §7.2, an accuracy far greater than that of the elastic compliance values). Thus, we expect only a very small increase in *β* with increasing strain; and as *β*=*β*_{E} initially (0.3367 in Al and 0.2688 in Cu), we may assume *β*<1/2. Then, from equations (4.4) and (7.3), we find that, for *n*≥50, we may write the following simplified equations for the differences among the reference stress rates at the outset:
*n*≥50. All the positive differences are from inequalities (4.5), with the exception of the last one, which is justified as follows.

The argument is made in appendix A that the *h*_{α}, hence *q*_{α}, may be taken to parallel the *H*_{α}, hence *r*_{α}, as regards differences among them. In latent-hardening experiments in axial-loading, the greatest difference among slip-systems has consistently been found to be between coplanar hardening and that of *any* non-coplanar system (II through V, equation (3.2)), rather than between any of the latter systems among themselves (see the papers cited following equation (3.3)). Therefore, *H*_{II}−*H*_{I} is greater than the difference between any two of moduli *H*_{II} through *H*_{V} Hence, by analogy, we take (*q*_{II}−1)>(*q*_{V}−*q*_{IV}). Then, from *q*_{V}>*q*_{IV}>*q*_{II}>1, we have the following continued inequality
*n*.

As anisotropy develops, from the constraints (using equations (4.3), (6.1) and (6.4), together with the simplified equations for *ϕ*′ and *A*_{n} (signifying anisotropy) reduces to *F*_{n}, and *B*_{2}/*B*_{1} reduces to *B*_{n}, equation (7.5)). For subsequent comparison, the rate-independent equations are ([12], eqn (F1) and 2015 Corrigendum)

### (c) Evaluations for Al and Cu

We now numerically assess if the new anisotropic power law can give results in the Brass orientation comparable with rate-independent theory and experiment. In Al at the *onset* of finite deformation, with *g*′_{o}, *ϕ*′ and *n*=100 equal the corresponding rate-independent values ^{−3} (purely elastic), and 0.7057 to at least seven significant figures (table 3). For *n*=50 the differences among the results are in the third significant figures. In Cu, from equation (7.1), *f*_{o}=8.30 MPa, *f*′_{o}=1035 MPa and *g*_{o}=*β*_{o}*f*_{o}=2.314 MPa (with *g*′_{o}=288.6 MPa, *ϕ*′_{o}=0.3549×10^{−2} and *n*=100, the same as the rate-independent values (differing from them by at most 1 in the sixth significant figure). The differences between the theories for *n*=50 are proportionately greater in Cu than in Al (table 3).

Consider a very small strain step *δe*_{L}=0.008 (several times larger than the elastoplastic-transition strain range of order 10^{−3}, [20]). As this is only about a thousandth of the full strain range in the Brass orientation experiments of Driver *et al*. [7] on Al (figure 5) and Wonsiewicz & Chin [2] on Cu (figure 3), we shall apply simple forward-differences to calculate the values to be used in equation (7.10) at the new strain. The probable accuracy of this can be assessed by comparing *f* calculated directly from equation (7.1) or (7.2) after the initial strain step. Either calculation gives 26.71 MPa for Al and 16.58 MPa for Cu, rounded to four significant figures. As the parameters in the equation representing the experimental data (figures 3 and 5) can be defined at best to three significant figures, we shall consider the forward-difference calculations sufficiently accurate.

At the new strain in Al: *f*′=213.0 MPa (from equation (7.2)), *g*=*g*_{o}+*g*′_{o}*δe*_{L}=8.995 MPa and both *β*≡*g*/*f* and *β*=*β*_{o}+*β*′_{o}*δe*_{L} give 0.3368 (rounded). (The original *β*_{o}=*β*_{E} actually is no more accurate than 0.337 as the cubic elastic compliances from which it is calculated are at best accurate to three significant figures.) From equation (7.3), the initial slip-rates ^{−32}*γ*′_{1} for *n*≥50, with *γ*′_{1}=1.2215 (equation (7.9)). For the same strain increment in Cu, *f*′(*δe*_{L})=1033.7 MPa (from equation (7.1)), *g*=*g*_{o}+*g*_{o}′*δe*_{L}=4.658 MPa and *β*≡*g*/*f*=0.2810 (a slight increase) at the new strain. The initial slip-rates ^{−38}*γ*′_{1} for *n*≥50 (equation (7.3)) in Cu, with *γ*′_{1}=1.2156 (equation (7.9)). Thus, for both Al and Cu, we may disregard slip-rates

In the equation *ε*′_{xx}<1 and *B*_{1}>1; but each is approximately 1, as is their ratio. Thus, we shall assume the change in *γ*′_{1} between its initial value and that following the very small strain step *δe*_{L} to be negligible (subsequently to be confirmed). Then, from equations (4.3) and (6.4), we obtain *n* (consistent with equation (7.11)),
*n*≥50.

After the strain step *δe*_{L}=0.008, the reference stress ratios in Al will differ only slightly from 1 for any *q*_{α} values between 1 and 2. Consequently, for a specific evaluation, we take *q*_{α}=*r*_{α}, the experiment-based ratios of the various hardening moduli in anisotropic rate-independent theory. From Havner ([12], eqn (16)) (consistent with equation (3.3) and latent-hardening experiments on Al of Kocks [27] and Kocks & Brown [36]), these are: *q*_{V}=1.25, *q*_{IV}=*q*_{III}=1.2, *q*_{II}=1.18. The reference-stress ratios are then (equations (7.11) and (7.12)): *n*≥100, these equal the respective rate-independent values (the equations at the end of §7.2) through eight significant figures, and differ only in the third figure for *n*=50 (This very close agreement between the anisotropic power law and rate-independent theory is because both *B*_{2}/*B*_{1} in equation (7.9) and *A*_{n} in equation (7.10) are negligible for large *n*.)

Finally, we calculate *γ*′_{1} in Al from equation (7.9) using the *f*′, *g*′ and *B*_{1} values after the strain step, to assess the assumption of a negligible change from its initial value. We obtain *γ*′_{1}=1.2215 to five significant figures, the same value as before. (Carrying out all calculations to nine significant figures, one finds that the change is an increase of only one one-thousandth of a per cent). Thus, the assumption is confirmed, and the anisotropic power-law values for *g*′, *ϕ*′ and

In Cu, the *q*_{α}=*r*_{α} ratios are ([12], eqn (19), consistent with latent-hardening experiments on Cu of [37]): *q*_{V}=1.35, *q*_{IV}=1.31, *q*_{III}=1.28, *q*_{II}=1.23. Again assuming the change in *γ*′_{1} to be negligible, equation (7.12) applies (although for Cu the second term in the denominator must be retained), from which *n*=100, as shown in table 3, differing at most by only 1 or 2 in the sixth place (and only in the eighth place for *n*=50, the differences between the anisotropic power-law and rate-independent theory are significantly larger in Cu than in Al.

Calculating *γ*′_{1} for Cu from equation (7.9), following the 0.008 strain step, we obtain *γ*′_{1}=1.2155, a decrease of 1 in the fifth significant figure from the initial value above (a change of less than six one-thousandth of a per cent). Thus, the assumption of negligible change is confirmed for Cu as well as Al, and the anisotropic power-law values for *g*′, *ϕ*′ and

The *g*/*f* ratio in viscoplastic theory will continue to change from *β*_{E} by only very small amounts for *n*≥100, while the slope *f*′ of the loading stress–strain curve rapidly decreases after finite strain to its asymptotic value of 60 MPa in Al (figure 5) and 147 MPa in Cu (figure 3). Consequently, the very small lattice strain contributions to the lattice rotation and shear rates at large strain are approximately 5.8×10^{−5} and −4.1×10^{−4}, respectively, in Al, and 5.0×10^{−4} and 2.6×10^{−4} in Cu. Thus, the lattice rotation-rate becomes negligible (essentially due to the horizontal lattice-shearing), corresponding to experimental lattice stability in the Brass orientation, and (for *n*≥100 in the power law) the shearing-rate is indistinguishable in both theories, to three significant figures, from the rate-independent, rigid/plastic value

## 8. Concluding remarks

A specific new anisotropic power law in crystal elasto-viscoplasticity has been introduced (§4) and shown to give predictions for the Goss and Brass orientations (§§5 and 7) in (110) channel die compression that are comparable (for large exponent *n*) with those of rate-independent theory and consistent with experiment. Both the structure for evolution of the power-law reference stresses (equation (4.4)), based on geometric interactions among fcc crystal slip-systems (table 2), and the defined inequalities among the five evolutionary parameters *h*_{α} (equation (4.5)), are motivated by failure of the classic isotropic power law to give satisfactory results, as compared with rate-independent theory and experiment, for lattice rotation and crystal shear in range I (figure 2) of channel die compression (as established in [12]). Moreover, the inequalities are given a deductive basis (appendix A) from relative relationships among the slip-rates as the lattice rotates (in range I) about the (110) load-axis into lateral-constraint direction

In the Goss orientation, it is shown that the new anisotropic power law (§5) predicts both zero crystal shearing and lattice stability, consistent with experiments on Al and Cu. Moreover, for large *n* in the power law, one may expect that the stress ratio *g*/*f* (constraint to compressive loading) will remain essentially elastically determined in Al, as predicted by rate-independent theory.

For the Brass orientation, specific calculations are made as power-law anisotropy develops in both Al and Cu (§7b). It is shown that, for *n*≥100, the lattice rotation equals a purely elastic, rate-independent value (to at least five significant figures) due solely to horizontal lattice-shearing (table 3), which is negligible, thereby a result consistent with experimentally established lattice stability in the Brass orientation. Moreover, the crystal shear-rate approaches the rigid/plastic rate-independent value

## Data accessibility

All data are available in the manuscript.

## Authors' contributions

I am sole author on this manuscript.

## Competing interests

I have no competing interests.

## Funding

I have no funding for this work.

## Acknowledgements

I thank Heramb Mahajan, MSc student in Civil Engineering, for preparing the computer-generated figures 3 and 5 from my hand-plotted data and equations.

## Appendix A. A deductive basis for the anisotropic power-law inequalities

To more easily understand the causes of significant differences between predictions of rate-independent theory and the isotropic power law in range I, corresponding to finite lattice rotation, we shall disregard the very small effects of lattice strains in the finite-deformation equations. Then, upon changing to differentiation with respect to *e*_{L}, the equations for *ϕ*′ and *d*_{yy}=0 having been used to eliminate slip-rates *b*=2:
*d*_{yy}=0)
*b*=2 is simply

As systems 7, 8 are unstressed at *b*=2 (from equation (2.3)), *γ*′_{7}=0 for any power law and equations (A2)–(A3) become

For the isotropic power law, it is found that ([12], §6.3.1 and table 9) *n*≥50, each only half the rate-independent value, and *γ*′_{3}. From equation (A4)_{3}, this obviously can be achieved by decreasing *γ*′_{5} and increasing *γ*′_{9}, the latter requiring a decrease in *γ*′_{1} from equation (A3)_{1}. With the slip-rates inversely proportional to the *n*th power of the reference stresses (equation (4.3)), the desired changes are consistent with *γ*′_{7}=0 at *b*=2.

Consider the second of these desired inequalities. From equation (4.4),
*γ*′_{9}→0 for large *n* in the isotropic power law, and equation (A3)_{1} must always be satisfied, we may expect *γ*′_{1}>*γ*′_{9} for moderate changes in the *h*_{α} from equal values. Thus, the inequality is satisfied by *h*_{V}>*h*_{II} and *h*_{III}>*h*_{II}.

The other inequality is (from equation (4.4))
*γ*′_{1} and *γ*′_{5}, and express this as

As we seek a *γ*′_{3} value greater than the limit-result *n*) at *b*=2, we require the right-hand side of equation (A6) to exceed *h*_{III}>*h*_{I}, *h*_{V}+*h*_{IV}>2*h*_{III} and *h*_{II}>*h*_{I}.

The combination of the latter two inequalities, equations (A6) and (A7), may be written

From the above inequalities among the *h*_{α}, the multiplier {2(*h*_{V}+*h*_{IV})−*h*_{III}−3*h*_{I}} is positive, as is each term on the right-hand side, from which

In summary, we have shown that inequalities
*γ*′_{3} is greater than its isotropic power-law value, from which *ϕ*′ will increase and *b*=2, as sought. Equation (A9) is fully consistent with (although slightly less restrictive than)
*h*_{V}>*h*_{IV}≥*h*_{III}>*h*_{II}>*h*_{I}>0 and *precisely parallel the* *inequalities among slip-systems hardening moduli H*_{α}, equation (3.3) (from eqn (104) in [13], deduced therein from analyses of various axial-load experiments on fcc crystals, as noted in §3 here). We, therefore, adopt equation (A10), identically equation (4.5) of §4, as a key part of the proposed anisotropic power law.

- Received August 17, 2015.
- Accepted December 14, 2015.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.