## Abstract

Estimates for the effective resistivity of nonlinear polycrystals are obtained using the ‘linear comparison’ homogenization scheme of DeBotton and Ponte Castañeda (DeBotton & Ponte Castañeda 1995 *Proc. R. Soc. A* **448**, 121–142). Computing the effective properties of linear composites, with the same microstructure as the nonlinear composite, is an essential part of this scheme. The classical self-consistent method is employed for this purpose. An important characteristic of these estimates, for polycrystals with field thresholds, is that they satisfy the recent bound of Garroni and Kohn (Garroni & Kohn 2003 *Proc. R. Soc. A* **459**, 2613–2625), which dramatically improves upon the classical Taylor upper bound at large crystal anisotropy. In addition, the estimates also satisfy the Hashin–Shtrikman bounds, which are more restrictive than the Garroni–Kohn bound at small crystal anisotropy. Interestingly, the scaling exponents for the linear comparison estimates are found to be independent of the constitutive nonlinearity. This last observation provides an explanation for the relative weakness of an earlier linear comparison bound obtained by Garroni and Kohn.

## 1. Introduction

In this work, we are primarily concerned with the effective resistivity of random nonlinear polycrystals with overall isotropic behaviour. Thus, the statistical distribution of grains is taken to be isotropic in both space and orientation so that the polycrystal behaviour is isotropic even for highly anisotropic and nonlinear single-crystal behaviour. The classical Taylor and Sachs estimates provide easy upper (UB) and lower bounds (LB), for effective resistivity. However, these bounds can be far from optimal as they do not use microstructural information beyond the one-point statistics of the crystal orientations (or crystallographic texture). Methods with demonstrated capabilities to significantly improve upon the classical bounds include: (i) the translation method and (ii) the linear comparison method.

The translation method is a powerful method that was developed initially for linear composites (see Milton 2002) and has been used recently to obtain dramatic improvements upon the Taylor UB for some model, two-dimensional, ideally plastic polycrystals (Kohn & Little 1998; Nesi *et al*. 2000; Goldsztein 2001), as well as for three-dimensional polycrystals in the context of dielectric breakdown (Garroni & Kohn 2003). Although this method is difficult to implement for general types of nonlinearities, and may be of limited value in applications, such as in plasticity, it provides rigorous results against which the predictions of other more general methods may be checked for accuracy.

The linear comparison method is a variational procedure that can be used easily and conveniently to extend bounds and estimates for linear composites to *nonlinear* composites with identical microstructures. It was originally proposed by Ponte Castañeda (1991) for nonlinear composites with isotropic phases, and generalized for viscoplastic polycrystals by DeBotton & Ponte Castañeda (1995). This method can provide improved predictions, over the classical uniform-field bounds, for nonlinear composites by making use of microstructural information beyond the one-point statistics. It was first used to obtain bounds of the Hashin–Shtrikman (HS) type for nonlinear polycrystals by DeBotton & Ponte Castañeda (1995). Earlier bounds for polycrystals were given by Willis (1994), using a nonlinear generalization of the HS variational method by Talbot & Willis (1985). However, it is now well understood (Milton 2002) that the HS bounds—although exact to second order in the heterogeneity contrast—are not optimal even for linear polycrystals, especially at large contrast. Owing to this, although improving on the classical bounds, the HS bounds are also not very useful in applications. On the other hand, it has been found that the use of the classical self-consistent (SC) estimates—which can also incorporate information on crystal orientations and average grain shapes—together with the linear comparison method provides realistic predictions, at least for statistically isotropic nonlinear polycrystals. Thus, it was found that the SC predictions for the effective flow stress of model two-dimensional viscoplastic polycrystals satisfy the Kohn–Little bound (Ponte Castañeda & Nebozhyn 1997), and agree reasonably well with the results of full-field numerical simulations (Lebensohn *et al*. 2004). However, the main advantage of the linear comparison method is that it is quite general and has been found—together with a slight generalization, called the ‘second-order’ method—to be very useful in plasticity (Nebozhyn *et al*. 2001; Lebensohn *et al*. 2007).

In this paper, we compute SC estimates for the effective resistivity behaviour of statistically isotropic, three-dimensional, power-law polycrystals, using the linear comparison method. These SC estimates will be shown to satisfy all the bounds, including the recent ‘translation’ UB of Garroni & Kohn (2003), in the limit of threshold-type behaviour for the electric field. Furthermore, it will be shown that at large crystal anisotropy, the scaling of the SC estimates is actually sharper than that of any of the known bounds.

In their paper, Garroni & Kohn (2003) show that the UB for effective resistivity derived using the translation method gives a superior scaling law than a bound derived using the linear comparison method in conjunction with the bound by Avellaneda *et al*. (1988), which is optimal for *linear* isotropic polycrystals. It will be shown here that polycrystals with statistically isotropic microstructures (which have isotropic overall behaviour even in the nonlinear case) form only a subclass of the set of all linear polycrystals with isotropic overall behaviour. This will help explain why the bound derived using the linear comparison method in conjunction with the linear bound of Avellaneda *et al*. (1988) turns out to be less sharp than the bound derived directly using the translation method.

## 2. Mathematical formulation of the problem

We consider a polycrystal that is composed of a large number of perfectly bonded single crystals distributed randomly in space with varying orientations. A *representative volume element* of the polycrystal occupies a domain *Ω* and can be thought of as an *N*-phase composite, where each phase occupies a subdomain *Ω*^{(r)} (*r*=1, …, *N*) and is constituted of crystals with random orientation *Q*^{(r)} (relative to some fixed reference frame). Note that, in practical applications, *N* is large (*N*→∞). Below, the angular brackets 〈.〉 and 〈.〉^{(r)} will be used, respectively, to denote volume averages over the polycrystal (*Ω*) and over phase *r* of the polycrystal (*Ω*^{(r)}).

The electric field ** E** and the current density

**at any point**

*J***in**

*x**Ω*are related by(2.1)where the

*Χ*

^{(r)}denote the characteristic functions for the phases (i.e. if

**∈**

*x**Ω*

^{(r)}, and 0 otherwise) and the

*w*

^{(r)}(

*r*=1, …,

*N*) denote the corresponding phase energy-density functions, which are assumed to be convex and such that

*w*

^{(r)}(

**0**)=0 and

*w*

^{(r)}(

**)≥**

*E**α*|

**| as |**

*E***|→∞ (for some**

*E**α*>0). Note that the functions

*w*

^{(r)}are in general not quadratic, corresponding to a nonlinear constitutive response.

Following Talbot & Willis (1985), the effective response of the composite is then given by(2.2)where and denote the average current density and the average electric field, respectively, and(2.3)is the *effective energy-density function* of the polycrystal. In this expression, the scalars represent the volume fractions for the phases, or *crystallographic texture*, while denotes the set of admissible (trial) electric fields(2.4)

In this work, it is assumed that the constitutive behaviour for the single-crystal grains is characterized by the following form for the phase energy-density functions(2.5)where the *ψ*_{(k)} (*k*=1, …, *K*) are assumed to be convex even functions of their arguments, and the vectors describe the geometry of the crystals with orientation *Q*^{(r)} in terms of geometric parameters *μ*_{(k)}, depending on the lattice of the reference crystal. For example, for a crystal with a *triclinic* lattice, the vectors *μ*_{(k)} may correspond to the three symmetry axes (not necessarily orthogonal). On the other hand, for an *orthotropic* crystal, these vectors would correspond to the three Cartesian vectors aligned with the symmetry axes of the crystal lattice, *e*_{k} (*k*=1, 2, 3). In the latter case, the , where *E*_{p} is the *p*th Cartesian component of the electric field ** E**. (While

*K*=3 is a natural choice for systems with three symmetry axes, it is useful to allow for the possibility of

*K*>3 for other systems, such as hexagonal closed-packed (HCP) lattices that have four natural symmetry axes, three in the basal plane and one orthogonal to it, or if additional symmetries, such as additional symmetry axes in cubic crystals, were required.)

Although the methodology to be developed in this work is applicable more generally, our attention in this work will be focused on functions *ψ*_{(k)} of the power-law type with the same nonlinearity exponent *n*≥0, i.e.(2.6)where the (*E*_{0})_{(k)} denote the nonlinear resistivities of the single crystal associated with the *K* symmetry directions. This model recovers the classical linear conductivity model for *n*=1, when the conductivity tensor for phase *r* is given by , but our interest for more general values of *n*>0 is in part motivated by analogous models in polycrystalline plasticity.

With the choice (2.6) for the functions , the overall behaviour of the polycrystal is also of power-law type with the same exponent *n* (see Ponte Castañeda & Suquet 1998). In addition, for *statistically isotropic* microstructures (i.e. isotropic distribution of orientations *and* of the phases in space), the effective energy density is only a function of the magnitude of the applied electric field. Therefore, for the statistically isotropic, power-law polycrystals considered here, the effective energy density is of the form(2.7)where is the *effective nonlinear resistivity* of the composite.

The nonlinear single-crystal and polycrystal resistivities, (*E*_{0})_{(k)} and , involved in forms (2.6) and (2.7), respectively, correspond to single-crystal and effective thresholds in the electric field in the limit as *n*→∞ (Garroni & Kohn 2003). However, these quantities are not physically meaningful in the opposite limit as *n*→0, corresponding to thresholds in the current. For this reason, use will be made here of the variables (*E*_{0})_{(k)} and only for values of *n*≥1, while for values of 0≤*n*≤1, use will be made instead of the nonlinear conductivities(2.8)With these definitions, the variables (*J*_{0})_{(k)} and can be interpreted as single-crystal and polycrystal, current threshold values, in the limit as *n*→0.

## 3. Variational linear comparison method

In this section, we present the ‘variational linear comparison’ method originally proposed by Ponte Castañeda (1991) for nonlinear composites and first applied in polycrystalline plasticity by DeBotton & Ponte Castañeda (1995). For completeness, an abbreviated derivation of the method is given in the context of nonlinear conductivity described in the previous section.

First, define the ‘comparison’ functions(3.1)which measure the nonlinearity of the conductor defined by expression (2.5) relative to a linear conductor with a quadratic energy function and conductivities . Note that these functions are essentially (see DeBotton & Ponte Castañeda 1995) the Legrendre–Fenchel transforms (convex polars) of the functions , with . In other words, .

Assuming that the functions *ψ*_{(k)} have *super-quadratic* growth at infinity, it follows from the definition (3.1) of the functions *V*_{(k)} that the original constitutive functions *ψ*_{(k)} may be approximated as(3.2)Therefore, using expression (2.5), it is deduced that(3.3)where the second-order conductivity tensor ^{(r)} is given by(3.4)Note that expression (3.3) gives the energy function of phase *r* in the nonlinear polycrystal, as defined by relation (2.5), in terms of the energy function of phase *r* of a linear comparison medium defined by(3.5)where ^{(r)} is given by expression (3.4).

Use of expression (3.3) for the functions *w*^{(r)} and expression (3.5) in expression (2.3) for the effective energy function , and interchanging the order of the supremum and the minimum can be shown (for details, see DeBotton & Ponte Castañeda 1995) to lead to(3.6)where denotes the effective energy density of a *linear comparison composite* (LCC) with the same microstructure (i.e. the same *Χ*^{(r)}) as the original nonlinear polycrystal, defined by relations (2.1), but with quadratic phase energy-density functions , as given by (3.5). Using the definition (2.3) of the effective energy function and the linearity of the LCC problem, may be expressed in the form(3.7)where is the effective conductivity tensor of the LCC, which depends on the phase conductivities ^{(r)}, as determined by (3.4) in terms of the variables . Although, by construction, the LCC has the same microstructure as the original nonlinear polycrystal, it should be emphasized that the optimized LCC, determined by the solution of (3.6), will, in general, *not* be a true polycrystal *except* when the principal conductivities of all phases happen to be all identical (i.e. except when for all *r*).

The variational estimate (3.6) is valid under the assumption that the functions *ψ*_{(k)} have super-quadratic growth. This would be the case, for example, for the power-law material behaviour defined by relation (2.6) with *n*>1. On the other hand, if the functions have *sub-quadratic* growth (e.g. for power-law phases with 0≤*n*<1), similar arguments could be used to obtain the following estimate for the effective energy function of the nonlinear polycrystal:(3.8)where the supremum in the definition (3.1) of the functions *V*_{(k)} must be replaced by an infimum. These estimates are valid for *K*≥3, although it follows from the recent work of Idiart & Ponte Castañeda (2007) in the analogous context of viscoplasticity that these bounds could be improved for values of *K* strictly greater than 3. In the application of this work, we will be interested only in the case where *K*=3 and the above estimates will suffice for our purposes.

For power-law polycrystals with *n*>1 (in expression (2.6)), the variational bound (3.6) can be shown (see Ponte Castañeda & Suquet (1998) for details) to simplify to(3.9)which generalizes an earlier result by Suquet (1993) for power-law composites with isotropic phases. It must be pointed out that this form is well posed numerically even in the limit as *n*→∞. When the polycrystal is statistically isotropic, the effective energy function must be of the form (2.7), and the r.h.s. of the estimate (3.9) becomes independent of the direction of . In this case, the magnitude of may be factored out, and a simpler expression may be obtained for the effective nonlinear resistivity , which is independent of (although the optimal values of and consequently of will depend on the orientation of ).

For power-law polycrystals with 0≤*n*<1, it is best to work with the nonlinear conductivities (*J*_{0})_{(k)}, and a completely analogous expression is obtained for with the exception that the estimate is now a UB and the supremum must be changed into an infimum(3.10)Similarly, for statistically isotropic polycrystals, the above expression can be used to obtain a simpler expression for the effective nonlinear conductivity , which, as already mentioned, becomes an effective threshold in the current for *n*=0.

Weaker ‘relaxed’ bounds, which are easier to compute, can be obtained from (3.9) and (3.10) by restricting the set of optimization variables to the subset where for all *r*. In this case, for statistically isotropic microstructures, the LCC is isotropic and its effective conductivity is of the form , where is the second-order identity tensor. It follows from (3.9) that, for power-law polycrystals with *n*>1, the relaxed bound for the effective resistivity is given by(3.11)where and is the effective nonlinear conductivity of the LCC in terms of the crystal conductivities *σ*_{(k)}. A completely analogous expression may be obtained from (3.10) for power-law polycrystals with 0≤*n*<1, but is not included here for conciseness.

## 4. Effective conductivity of linear polycrystals

The variational linear comparison estimates (3.6) and (3.8), as well as (3.9) and (3.10), for nonlinear and power-law polycrystals require corresponding estimates for the LCCs with the same microstructure as the original nonlinear polycrystals. However, as already emphasized in the previous section, the best estimates for nonlinear polycrystals would require corresponding estimates for *N*-phase linear polycrystals (i.e. each crystal in the LCC can, in principle, have different properties and is not simply a rotated version of the same basic crystal). In this section, we recall the generalized HS bounds and SC estimates of Willis (1977) for the effective conductivity of linear composites and polycrystals and compare them with the Voigt UB and the LB of Avellaneda *et al*. (1988) for isotropic polycrystals.

Before proceeding, however, it is useful to emphasize that the Willis and the Avellaneda *et al*. estimates make different assumptions about the polycrystal microstructures. Thus, as will be discussed in more detail below, the Willis estimates assume that the microstructures are random with prescribed one- and two-point statistics. In particular, we will be primarily interested in statistically isotropic microstructures, leading to isotropic overall behaviour for the polycrystal. On the other hand, the Avellaneda *et al*. microstructures need not be random and, as will be shown below, are strictly larger than those allowed by the Willis estimates. In particular, they include microstructures that are not statistically isotropic, even if the overall behaviour of the polycrystal is assumed to be isotropic.

Willis (1977) made use of the HS variational principles to obtain the following estimate for the effective conductivity tensor of a polycrystal with prescribed one- and two-point statistics:(4.1)In this expression, the volume fractions *c*^{(r)} characterize the orientation distribution function (ODF; i.e. the one-point probability of the crystal orientations), while the microstructural tensor depends on the two-point probabilities of the crystal orientations, which, for simplicity, have been assumed here to be identical for all pairs of orientations. In particular, when the microstructure is statistically isotropic, all the *c*^{(r)} become equal (i.e. all orientations are equally probable) and the two-point probability functions become independent of orientation (i.e. equiaxed grains). In this case, the tensor simplifies to the surface integral(4.2)The expression (4.1) then reduces to the celebrated HS UB (LB), when the reference tensor is chosen so that −^{(r)} is as small (large) as possible but *positive* (*negative*) definite for all *r*. On the other hand, this same estimate reduces to the also well-known SC estimate when is chosen to be equal to . Note that there are alternative derivations of these results for ‘cell polycrystals’ (see Milton 2002, §23.10). Also, Helsing (1993) has given improved bounds for linear isotropic polycrystals with equiaxed grains.

In particular, for statistically isotropic polycrystals with orthotropic single crystals with principal conductivities *σ*_{(1)}, *σ*_{(2)} and *σ*_{(3)}, such that(4.3)the polycrystal is isotropic and the effective conductivity can be computed analytically. In the following, analytical expressions are provided for in terms of the principal conductivities *σ*_{(k)} for the various bounds and estimates.

Thus, the HS LB for statistically isotropic, orthotropic polycrystals is given by (Hashin & Shtrikman 1963)(4.4)while the corresponding HS UB is given by(4.5)On the other hand, the corresponding SC estimate, is the unique positive root to the equation (Helsing & Helte 1991)(4.6)In the limit when *σ*_{(1)}≫*σ*_{(3)} (recall *σ*_{(1)}≥*σ*_{(2)}≥*σ*_{(3)}), this estimate can be shown to simplify to(4.7)

In a separate development, Avellaneda *et al*. (1988) made use of the translation method to obtain an LB for the effective conductivity of polycrystals. In contrast with the Willis estimates, this bound involves no hypothesis whatsoever about the microstructures involved. It depends only on the principal conductivities of the reference single crystal and contains information about neither the ODF (i.e. the volume fractions *c*^{(r)}) nor the two-point probabilities of the orientations (i.e. the grain shape). For uniaxial crystals, Avellaneda *et al*. also showed that this bound is optimal using the Schulgasser and other microstructures, while Nesi & Milton (1991) demonstrated the optimality of the bound more generally, making use of sequentially laminated microstructures. In particular, this bound holds for isotropic microstructures, where an LB for the (isotropic) effective conductivity is obtained from the unique positive root of the equation(4.8)This bound will be referred to as an ACLM bound from here on. In the limit when *σ*_{(1)}≫*σ*_{(3)}, this bound can be shown to simplify to(4.9)It should be emphasized that, unlike the HS and SC estimates, the ACLM bound is only available for polycrystals made up of one basic single crystal and not for more general *N*-phase polycrystals. Owing to this, when computing the ACLM bound for nonlinear polycrystals using the linear comparison method (3.9), the set of optimization variables will need to be restricted to the subset where .

Much earlier, Schulgasser (1977) had shown the optimality of the Voigt UB for isotropic polycrystals as(4.10)

We conclude this section by showing that the above-described Voigt UB and ACLM LBs—while optimal for the class of *all* isotropic polycrystals—cannot be optimal for the subclass of statistically isotropic microstructures (i.e. with isotropic one- and two-point probabilities). This is easily accomplished by recalling that the HS bounds are exact to second order in the heterogeneity contrast. For simplicity, we assume that the crystals are uniaxial with *σ*_{(2)}=*σ*_{(3)} and introduce the small anisotropy (contrast) parameter *ϵ*=*σ*_{(1)}/*σ*_{(3)}−1. Then, from (4.4) and (4.5), it follows that for small anisotropy contrast,(4.11)which is valid for all statistically isotropic microstructures. (Note that the SC estimate also satisfies this property.) On the other hand, for small contrast, the ACLM LB (4.8) and the Voigt UB (4.10) yield(4.12)Since the estimate (4.11) is exact to *second* order in the anisotropy contrast, it follows that the ACLM and Voigt bounds are only exact to *first* order in the anisotropy contrast, allowing for a finite ‘gap’ in the second-order term. Thus, while the ACLM and Voigt bounds are known to be optimal for the class of all microstructures with the isotropic effective conductivity, for the subclass of statistically isotropic microstructures (i.e. with isotropic one- and two-point probabilities), these bounds are definitely *not* optimal. (On the other hand, the HS bounds, although optimal for small anisotropy contrast, are known to be rather weak at large contrast.) In fact, the ACLM and Voigt bounds include microstructures where both the distribution of orientations *and* the grain shapes are anisotropic, but ‘conspire’ to produce a polycrystal with an isotropic effective conductivity. It is interesting to remark that the situation for polycrystals is different from the well-studied case of two-phase composites with isotropic phases, where the G-closure bounds for all isotropic microstructures and the HS bounds for statistically isotropic microstructures do, in fact, lead to the same result. This makes sense because for two-phase composites with isotropic phases, an isotropic overall response does require an isotropic distribution of the phases. As will be argued more precisely in §5, the use of the ACLM bounds—through the linear comparison method—is not the most appropriate approach to obtain bounds for isotropic nonlinear polycrystals, because these bounds include microstructures that, while leading to an isotropic effective response for linear polycrystals, would also lead to an *anisotropic* response for nonlinear polycrystals. They are still bounds, but they clearly cannot be optimal for isotropic microstructures since they include microstructures that would not be isotropic in the nonlinear context. In a nutshell, this is the reason why the bound derived by Garroni & Kohn using the linear comparison method in conjunction with the ACLM bound cannot possibly be as good as the bound derived via the translation method (which uses isotropy in a much stronger way). It cannot be overemphasized that the bounds for nonlinear composites derived by means of the linear comparison method can only be as good as the corresponding bounds for the LCC.

## 5. Results and discussion

In this section, results are presented for the effective resistivity of power-law orthotropic polycrystals with an overall isotropic response, as defined by relations (2.6) and (2.7). The results are given as a function of the nonlinearity exponent *n* and the principal resistivities, which are ordered such that(5.1)for consistency with (4.3) in the linear case. Use will be made here of the variational linear comparison method discussed in §3, together with the HS bounds and SC estimates recalled in §4 for the linear polycrystals to generate corresponding HS bounds and SC estimates for the nonlinear polycrystals. More specifically, expression (4.1) with the appropriate choice of the reference conductivity for the HS and SC estimates of the LCC will be used, together with expressions (3.9) and (3.10) to obtain estimates for and of the power-law polycrystals with values of *n*>1 and 0≤*n*<1, respectively. It should be emphasized that only one bound of the HS type can be obtained using this procedure; this will be a UB for *n*>1 and an LB for 0≤*n*<1, respectively.

An important objective is to compare these bounds and estimates for the power-law polycrystals with the translation bound of Garroni & Kohn (2003) in the limit of a threshold in the electric field (*n*→∞). In addition, comparisons will be given with the linear comparison bound of these same authors using the ACLM bound (4.8) for the LCC, as well as with the uniform-field UB and LBs, for more general power-law polycrystals with *n*≥0.

Following Garroni & Kohn (2003), for *n*>1 we introduce the nonlinear anisotropy parameters *M* and *N*, such that(5.2)while for 0≤*n*<1, it is more convenient to work with *M*_{J} and *N*_{J}, such that(5.3)Note from (5.1) and (2.8) that *N*≥*M*≥1, while *N*_{J}≥*M*_{J}≥1.

Thus, for the special class of electric-field threshold polycrystals (*n*→∞), the results will be compared with the Garroni–Kohn (GK) bound, given by(5.4)while, for more general *n*≥0, they will be compared with the uniform-field bounds(5.5)and(5.6)The labels ‘Taylor’ and ‘Sachs,’ which are inspired from the corresponding threshold problems in plasticity, are consistent with Garroni & Kohn for *n*>1. For 0≤*n*<1, the Taylor and Sachs bounds for effective nonlinear conductivity can be obtained from (5.5) and (5.6), respectively, by means of expression (2.8). Note, however, that in this case, the Taylor and Sachs bounds for are LBs and UBs, respectively, for the effective nonlinear conductivity. It should be emphasized that while the uniform bounds make use of the isotropy of the ODF, leading to overall isotropic behaviour for the nonlinear resistivity and conductivity, respectively, the GK bound assumes overall isotropy of the polycrystal (with no references to the statistics). In §4, it was suggested that in the linear case it is possible for the one- and two-point probabilities to conspire to yield an overall isotropic result, even when the one- and two-point probabilities are anisotropic themselves. This is less likely to take place in the nonlinear case, where it is not enough to match a finite set of eigenvalues of the conductivity tensor, and instead an infinite number of symmetries would be required to ensure overall isotropy in general. Thus, for nonlinear polycrystals, the Taylor and Sachs bounds are expected to apply for a larger class of microstructures than the GK bound, since the higher point statistics in the Taylor–Sachs bounds need not be isotropic, and polycrystals in this larger class could still be anisotropic (even if the Taylor–Sachs bounds themselves are isotropic).

For values of *n*≥1, results will be presented in the following for the effective resistivity as a function of the grain anisotropy parameter *N*≥1, where the other anisotropy parameter *M* will be chosen according to the rule *M*=*N*^{α} with 0≤*α*≤1. In addition, scaling laws will be provided as *N*→∞ for these results, which will turn out to be of the form(5.7)for some *γ*, with 0≤*γ*≤1. Similarly, for 0≤*n*<1, results will be given for the effective conductivity as a function of the grain anisotropy parameter *N*_{J}≥1, with (0≤*α*≤1), where the relevant scaling law will take the form(5.8)as *N*_{J}→∞. It is noted that these two forms of the scaling law are identical for the linear case (*n*=1). The reason for the two alternative forms for the different ranges of *n* is again to have a physically meaningful measure of effective behaviour in the limits of threshold-type behaviour in the electric field and the current, respectively. However, the two forms can be seen to be equivalent (for a given value of *n*) on account of the expressions in (2.8), provided that *C*_{J} is equal to *C*^{n}.

### (a) Effective resistivity of linear polycrystals (*n*=1)

In this subsection, we provide explicit results for the effective resistivity of linear polycrystals, which will serve as references for the corresponding results for nonlinear polycrystals. For linear polycrystals, since conductivity is simply the inverse of resistivity, the ACLM UB, the HS bounds and the SC estimate for the effective resistivity are evaluated from expression (4.4) through (4.10) by substituting and *σ*_{(k)} with and 1/(*E*_{0})_{(k)}, respectively.

Figure 1 shows plots of the SC estimate and the HS bounds for the effective resistivity , normalized by the grain resistivity (*E*_{0})_{(1)}, of untextured linear polycrystals with equiaxed grains (i.e. with isotropic one- and two-point statistics). These results are compared with the ACLM UB for all isotropic polycrystals, as well as the Taylor (Voigt) UB and the Sachs (Reuss) LB for untextured microstructures (i.e. isotropic one-point statistics). Figure 1*a–c* provides results as a function of the crystal anisotropy *N*, for values of *M*=*N*^{α} with *α*=0, 1/2 and 1, respectively, while table 1 gives the corresponding scaling laws for large values of *N*. Figure 1*d* provides results as a function of *α* for *N*=100 (and *M*=*N*^{α}). The plots show that neither the HS UB nor the ACLM UB can be optimal UBs for statistically isotropic linear polycrystals. While the HS bounds are more restrictive at small crystal anisotropy—in fact, they are exact to second order in the grain anisotropy—the ACLM bound is generally superior at large crystal anisotropy (*N*→∞). This is confirmed for 0≤*α*<1 by the scaling laws for large *N* presented in table 1. However, for the case where *α*=1 (*M*=*N*), the HS UB is found to be more restrictive than the ACLM bound for all values of the crystal anisotropy, including for large values of *N* where the HS UB for the normalized effective resistivity scales as 2*N*/5, as opposed to *N*/2 for the ACLM bound. Thus, even though the HS bounds are known to be non-optimal for isotropic polycrystals with highly anisotropic grains, in this particular case (*α*=1), they are sharper than the ACLM bound. Given the optimality of the ACLM bound for the class of all isotropic polycrystals, the improvement of the HS bound can be explained by the fact that it applies to the more restrictive class of polycrystals with isotropic one- and two-point statistics, consistent with the discussion in §4 for small contrast. Correspondingly, it can also be seen that the HS LB improves generally on the Sachs bound. In particular, for large anisotropy, the scaling laws for the HS and Sachs LBs are similar (∼*N*^{0}), but the coefficient for the HS bound is tighter.

Finally, the SC estimates are seen to satisfy all bounds for all crystal anisotropies. In this connection, it is relevant to recall Kröner's (1978) derivation of these estimates ensuring compliance with bounds of all orders for random isotropic polycrystals. Furthermore, the SC estimate scales very differently to any of the bounds at large crystal anisotropy. Thus, as can be deduced from (4.7), the SC estimate scales as , while the ACLM UB, the HS (and Taylor) UB and the HS (and Sachs) LB scale as , *N* and *N*^{0}, respectively. In particular, the SC estimate can yield a finite prediction for the effective resistivity, when *M*∼1, even as *N*→∞. This is in contrast with the ACLM and HS UBs, which would blow up as *N*→∞, regardless of the value of *M*.

### (b) Effective resistivity of nonlinear polycrystals *n*≥1

Figure 2*a–c* shows results analogous to those shown in figure 1*a–c* for nonlinear polycrystals with a threshold in the electric field (*n*→∞), while figure 2*d* presents the results as a function of 0≤*m*=1/*n*<1 for a highly eccentric crystal with *M*=*N*=100. Table 2 summarizes the corresponding scaling laws for the nonlinear polycrystals for large values of *N*. Note that these results include only the HS UB, since an HS LB cannot be obtained by this procedure for values of *n*>1. It should be noted in the context of the scaling laws that the optimal values for in the numerical implementation of the linear comparison procedure (3.9) are such that in the limit of large values of *N*, for the SC estimate, the HS and ACLM bounds. Therefore, the scaling laws for the SC estimate, the HS and ACLM bounds can be obtained analytically by solving the relaxed variational statement (3.11). This is true for all values of *α* and all three estimates with only one exception: the SC estimate for *α*=0 (*M*=1). In this case, the result reported in table 2 for the SC estimate was still obtained from the relaxed variational statement (3.11), and so it is only a UB for the corresponding SC result shown in figure 2*a*. However, it can be shown that the scaling exponent is still the same for the ‘optimal’ and relaxed versions of the result (i.e. zero). In the table, the (numerically obtained) optimal scaling coefficient of the SC estimate is reported (in parentheses) for *n*→∞.

These results show that the scaling exponent *γ* in expression (5.7) for the bounds and estimates for *n*>1 are identical to the corresponding exponents for *n*=1 (although the scaling coefficient *C* does depend on *n*). In other words, the scaling exponents for the bounds and estimates are independent of the nonlinearity exponent *n*. Note that figure 3*d* confirms that the bounds and estimates derived using the linear comparison method and the classical variational principles are all weak functions of nonlinearity parameter *n*. It is evident that at large crystal anisotropy, owing to a superior scaling exponent, the GK bound (5.4) is sharper than the ACLM, HS or Taylor bound. On the other hand, the SC estimate does not violate any of the bounds, including the GK bound. More specifically, the SC estimate scales as , as *N*→∞, while the GK bound scales as (*MN*)^{1/3}. In particular, this means that the SC estimate can remain bounded for *M* bounded, while the GK bound would blow up, regardless of whether *M* is bounded, or not. Note that analogous results have been found (Lebensohn *et al*. 2007) for viscoplastic polycrystals.

### (c) Effective resistivity of nonlinear polycrystals (0≤*n*≤1)

Figure 3*a–c* shows results for *n*=0 corresponding to those shown in figure 2*a–c* for *n*→∞, while figure 3*d* shows results for 0≤*n*<1 corresponding to those shown in figure 2*d* for values of *n*>1. Similarly, table 3 summarizes the corresponding scaling laws for the nonlinear polycrystals with 0≤*n*<1 for large values of *N*_{J}. However, in these figures, the results are shown in terms of the resistivity measure . As discussed at the beginning of this section, this way of exhibiting the results for 0≤*n*<1 gives physically meaningful results in the limit as *n*→0, while it agrees exactly with the results shown in figure 1 and table 1 for *n*=1, and preserves the scaling behaviour (i.e. the exponent in expression (5.8) is the same as in (5.7)) for large values of *N* (albeit expressed in terms of *N*_{J}), allowing meaningful comparisons with the cases of *n*>1. It is also recalled from (3.10) that the ACLM and HS UBs cannot be computed for *n*<1. Instead, the HS LB is provided.

It can be seen from these results that the SC estimates do not violate any of the bounds. Furthermore, the scaling exponents for the SC estimates, as well as those for the Taylor and Sachs bounds, are exactly the same as the corresponding exponents obtained for *n*≥1. (It is also evident from figure 3*d* that the resistivity measure is a weak function of the nonlinearity exponent.) In particular, this confirms that the scaling exponent is preserved by the linear comparison method and, as a consequence, strongly suggests that the differences in the scaling laws observed by Garroni & Kohn (2003) between the translation bound and the linear comparison bound (using the ACLM bound for the LCC) are not due to a failure of the linear comparison method, but due to the limitations, identified earlier, with the ACLM bound. The HS LB improves over the Sachs bound, but only very slightly.

It should be emphasized that all the scaling results shown in table 3, except those identified with the letter ‘a’, were obtained using the appropriate version of the relaxed linear comparison estimate (3.11), since it was found that the fully optimized estimate (3.10) leads to values of such that in the limit of large grain anisotropy. Owing to this, the analytical results shown in table 3 for 0≤*n*<1 are essentially the same as the corresponding results shown in table 2 for *n*>1 once they are put in the same form. In any case, even for the special cases when the fully optimized estimate is better than the relaxed estimate, it does not change the scaling. It only affects the scaling coefficient that could not be computed analytically, but which can be estimated numerically and which was found to be slightly different from the corresponding relaxed coefficient.

## 6. Concluding remarks

SC estimates for the effective resistivity of statistically isotropic power-law polycrystals with orthotropic basic crystals have been obtained using the linear comparison method. These estimates are found to have the following distinctive features. (i) They satisfy all known bounds including the GK bound (2003), which dramatically improves upon the classical Taylor bound at large heterogeneity contrasts. (ii) For large crystal anisotropies (1≤*M*≤*N*→∞, with *M*=(*E*_{0})_{(2)}/(*E*_{0})_{(1)} and *N*=(*E*_{0})_{(3)}/(*E*_{0})_{(1)}), they scale as(6.1)where the coefficient *C*_{n} depends on the nonlinearity *n* and the ratio *M*/*N* (cf. (4.7)). Note that this scaling is quite different from that of the GK bound (5.4), which scales as (*MN*)^{1/3}, or any of the other known bounds. (iii) In particular, they remain bounded even as *N*→∞, provided that *M* is finite. This behaviour for ‘deficient’ basic crystals in the polycrystal is analogous to the corresponding results for isotropic HCP polycrystals in plasticity (Gilormini *et al*. 2001; Lebensohn *et al*. 2007), where the effective flow stress remains bounded, even when one of the five linearly independent slips systems becomes rigid. All these features suggest that, though not rigorous bounds, the SC estimates could be quite accurate and useful for predicting the effective conductivity of nonlinear polycrystals, especially in those cases where the GK bound is not available, e.g. in the study of the effect of morphological texture.

It is important to emphasize that the scaling exponent for the SC estimates, as well as for the HS and ACLM bounds, obtained here by means of the linear comparison method, are found to be independent of the nonlinearity of the basic crystals. This fact is consistent with the physical interpretation of the large anisotropy behaviour of polycrystals in terms of percolation theory (Yu & Dykhne 1983), which strongly suggests that the scaling exponent should be dependent only on the geometry (and not on the specific constitutive behaviour) of the basic crystal. In addition, this fact demonstrates that the weaker scaling exponent noted by Garroni & Kohn for the linear comparison, ACLM-type bound is not a consequence of a deficiency of the linear comparison method for three-dimensional problems (as initially suggested by these authors), but rather a property inherited from the ACLM bound, which, as argued in the body of this paper, is not necessarily the most appropriate bound in the context of the linear comparison method (assuming that the ultimate objective is to obtain bounds for *isotropic* nonlinear polycrystals). One possible way to get around this problem is to make use of improved bounds in the context of the linear comparison method, or alternatively to attempt to apply the combined translation–HS approach of Nesi *et al*. (2000) for nonlinear polycrystals in three dimensions.

## Acknowledgments

The work of V.R. was supported by the Agence Nationale de la Recherche (France) and that of P.P.C. by the National Science Foundation (USA) through grant CMMI-0654063.

## Footnotes

- Received January 17, 2008.
- Accepted March 27, 2008.

- © 2008 The Royal Society