## Abstract

A variational method is developed to estimate the macroscopic constitutive response of composite materials consisting of aggregates of viscoplastic single-crystal grains and other inhomogeneities. The method derives from a stationary variational principle for the macroscopic stress potential of the viscoplastic composite in terms of the corresponding potential of a linear comparison composite (LCC), whose viscosities and eigenstrain rates are the trial fields in the variational principle. The resulting estimates for the macroscopic response are guaranteed to be exact to second order in the heterogeneity contrast, and to satisfy known bounds. In addition, unlike earlier ‘second-order’ methods, the new method allows optimization with respect to both the viscosities and eigenstrain rates, leading to estimates that are fully stationary and exhibit no duality gaps. Consequently, the macroscopic response and field statistics of the nonlinear composite can be estimated directly from the suitably optimized LCC, without the need for difficult-to-compute correction terms. The method is applied to a simple example of a porous single crystal, and the results are found to be more accurate than earlier estimates.

## 1. Introduction

Macroscopic samples of metals and minerals usually appear in the form of aggregates of large numbers of one or more types of single-crystal grains and other inhomogeneities, such as voids and cracks, which are distributed with random positions and orientations in the sample. It is then of scientific and technological value to be able to characterize the *effective* or *average* response of such macroscopic material samples from the properties of their constituents and known statistical information about their distribution, or microstructure. In addition, it is also of interest to be able to extract information about the statistics of the stress and strain fields in the constituents of these composite materials. These problems—which are difficult in general—become especially challenging when the physical mechanisms of deformation are nonlinear, as in metal-forming operations or polar ice flows. The objective of this work is to develop homogenization techniques to characterize the macroscopic response and field statistics in viscoplastic polycrystals and composites.

The simplest and perhaps most commonly used homogenization procedure in polycrystalline plasticity is the uniform strain-rate approximation of Taylor [1]. In the specific context of viscoplastic polycrystals, Hutchinson [2] has shown that the Taylor approximation provides a rigorous upper bound incorporating first-order statistical information in the form of the orientation distribution function. In addition, this author made use of the ‘incremental’ self-consistent approximation of Hill [3] to generate improved estimates by incorporating additional information about the average shape of the grains.

Improved bounds of the Hashin–Shtrikman [4] type for viscoplastic polycrystals were first developed by Dendievel *et al.* [5], making use of a generalization of the Hashin–Shtrikman variational principles for nonlinear media due to Willis [6,7], as well as by deBotton & Ponte Castañeda [8] by means of the variational method of Ponte Castañeda [9]. This second method makes use of a linear comparison composite (LCC), whose properties are determined by means of a suitable variational principle, leading to a ‘secant’ linearization of the constitutive response of the grains, evaluated at the *second moments* of the stresses in the grains [10,11]. In addition, the linear comparison variational method has the advantage that it allows the use of other types of bounds and estimates for the LCC. In particular, Nebozhyn & Gilormini [12] proposed ‘variational linear comparison’ self-consistent estimates for viscoplastic polycrystals and demonstrated that they improved on the earlier ‘incremental’ self-consistent estimates [3] by showing that these last estimates violate rigorous bounds for sufficiently small rate-sensitivity exponents, especially when the single-crystal grains are highly anisotropic.

More accurate estimates for the macroscopic response of viscoplastic polycrystals were given by Liu & Ponte Castañeda [13] (see also [14]) by means of the ‘second-order’ homogenization method of Ponte Castañeda [15] and Liu & Ponte Castañeda [16]. The ‘second-order’ method makes use of more general LCCs incorporating suitably selected eigenstrain rates, leading to an improved ‘generalized secant’ approximation of the nonlinear constitutive relations and ensuring that the resulting estimates are exact to second order in the heterogeneity contrast, and thus in agreement with the perturbation expansions of Suquet & Ponte Castañeda [17]. In spite of the improved accuracy of the ‘second-order’ homogenization estimates [16] relative to the earlier ‘variational’ estimates [8], the ‘second-order’ estimates have certain undesirable features, arising from the lack of optimality of the eigenstrains, which hamper their efficient application in practice. These include the facts that the macroscopic constitutive relation and fields statistics cannot be obtained directly from the LCC, and the existence of a ‘duality gap’ (i.e. the estimates resulting from the primary and complementary variational statements are different)—which strongly suggests the possibility of further improvements in the accuracy of its predictions. In this work, we propose a new variational method, where both the viscosities and eigenstrain rates of the constituent phases of the LCC are generated by consistent optimization procedures. This leads to ‘full stationarity’ for the resulting estimates, which are still exact to second order in the contrast, but have all the advantages of the earlier ‘variational’ estimates in that the macroscopic constitutive relation and fields statistics of the nonlinear composite can be conveniently expressed in terms of the corresponding quantities for the suitably optimized LCC. Finally, we show explicitly by means of a simple application for a porous single crystal that the new ‘fully optimized’ second-order estimates do not exhibit a duality gap and provide more accurate estimates than the earlier second-order estimates [16].

In this paper, scalars are denoted by italic Roman or Greek letters (e.g. *a*, *α*), vectors by boldface Roman letters (**b**), second-order tensors by boldface italic Roman or boldface Greek letters (** C**,

**) and fourth-order tensors by double-struck letters (**

*α*## 2. Theoretical background on nonlinear homogenization

In this work, we are interested in heterogeneous materials occupying regions of space *Ω* that comprised several (anisotropic) crystalline and possibly other phases, which in turn occupy subregions *Ω*^{(r)} (*r*=1,…,*N*) of *Ω* that are distributed randomly in location and orientation. The phases could correspond to different orientations of the same crystalline material, or they could correspond to altogether different materials. An example of the first type of heterogeneous material would be a standard polycrystal consisting of aggregates of grains of the same single-crystal material with varying orientations. An example of the second type would be a porous single crystal consisting of voids, or vacuous inclusions, that are distributed uniformly in a single-crystal matrix with given uniform orientation. More general examples would include two-phase polycrystals, porous polycrystals and inclusion-hardened polycrystals. Here, we will make use of indicator functions *χ*^{(r)}, defined to be equal to 1 if the position vector **x** is in *Ω*^{(r)} and zero otherwise, to describe the location of the various phases. Note that ^{(r)} to denote volume averages over the composite (*Ω*) and over phase *r* (*Ω*^{(r)}), respectively.

For simplicity, the constitutive behaviour of the single-crystal phases will be taken to be viscoplastic. Then, for a given stress ** σ**, the

*local*constitutive response is defined by

**is the Eulerian strain rate, and**

*ϵ**u*and

*u*

^{(r)}are the stress potentials for the composite and for the

*r*th crystalline phase, respectively. The convex functions

*k*=1,…,

*K*

^{(r)}) characterize the response of the

*K*

^{(r)}slip systems in phase

*r*, consisting of a certain crystalline material with orientation

**Q**

^{(r)}, and depend on the resolved shear (or Schmid) stresses

*k*th system in a crystal with orientation

**Q**

^{(r)}. Note that the Schmid tensors

*μ*_{(k)}for a ‘reference’ crystal via

*m*

_{(k)}=1/

*n*

_{(k)}(0≤

*m*

_{(k)}≤1) and (

*τ*

_{0})

_{(k)}>0 are, respectively, the strain-rate sensitivity and reference flow stress of the

*k*th slip system, and

*γ*

_{0}is a reference strain rate. Note that the limits as

*n*

_{(k)}tends to 1 and

*n*

_{(k)}and (

*τ*

_{0})

_{(k)}could also depend on the type of material for composites with multiple material phases.

Assuming the appropriate *separation of length scales*, the effective behaviour of the composite can be described by the *effective stress potential* (e.g. [2,11])
*c*^{(r)}=〈*χ*^{(r)}〉 denote the volume fractions of the given phases and
*macroscopic stress* in the composite, and it can be related to the *macroscopic strain rate*

Assuming convexity of the stress potentials *u*^{(r)} of the phases, it is also possible to define a dual formulation in terms of the strain-rate potentials *w*^{(r)} for the phases, which may be obtained by means of the Legendre transformation (see below). The dual formulation is described by the macroscopic strain-rate potential

In addition to the macroscopic constitutive relation for a nonlinear composite, it is also of interest to characterize the statistics of the stress and strain-rate fields in the composite, as described, in particular, by the first and second moments of the fields in the phases of the composite. Thus, the first moments, or averages, of the stress and strain-rate fields over phase *r* are defined via *r* are given by 〈** σ**⊗

**〉**

*σ*^{(r)}and 〈

**⊗**

*ϵ***〉**

*ϵ*^{(r)}. Here, we will also make use of the statistical quantities

*k*in phase

*r*.

Analytical expressions for the first and second moments of the stress and strain-rate fields in the phases of nonlinear composites have been given by Idiart & Ponte Castañeda [18]. The idea is to perturb the potential of the given phase by means of suitable terms that are linear, or quadratic in the appropriate field, and to make use of the above variational formulation to generate corresponding estimates for the homogenized potentials of the perturbed problem, which can then be differentiated with respect to the coefficients of the perturbing terms. Thus, the first moment, or average, of the stress field in phase *r* of the nonlinear composite may be obtained via the identity (see proposition 3.1 of [18])
*η*^{(r)} is a constant, symmetric, second-order tensor, and

Similarly, the second moment of the stress field in phase *r* of the nonlinear composite may be obtained via the identity (see corollary 3.3 of [18])

where

In addition, the first moment, or average, of the strain-rate field in phase *r* of the nonlinear composite may be obtained via (see proposition 3.4 of [18])
*τ*^{(r)} is a constant, symmetric, second-order tensor, and

Idiart & Ponte Castañeda [18] have shown that, when the ‘variational’ (secant) linear comparison method [9] is used to estimate the effective potential of the nonlinear heterogeneous material in terms of the potential of a certain LCC, the first and second moments of the fields over the phases can be estimated directly from the corresponding moments in the LCC. On the other hand, Idiart & Ponte Castañeda [18] have also shown that, when the ‘generalized secant’ second-order method [15] is used to estimate the effective potential of the nonlinear heterogeneous material, the resulting estimates for the first and second moments of the fields over the phases of the nonlinear heterogeneous material do not, in general, coincide with the corresponding first and second moments in the LCC; certain additional correction terms are needed on account of the lack of full stationarity in the ‘generalized secant’ linear comparison method [15]. One of the primary objectives of this work is to develop fully optimized estimates of the generalized secant type for crystalline aggregates, such that the first and second moments of the fields over the phases of the nonlinear heterogeneous material can also be estimated directly from the corresponding moments in the LCC—without the need for correction terms. This will resolve an outstanding issue with the generalized secant estimates of Liu & Ponte Castañeda [16], which were obtained by an appropriate adaptation of the work of Ponte Castañeda [15] for polycrystalline aggregates (see also [19]).

## 3. Fully optimized linear comparison estimates

In this section, a ‘second-order’ variational technique is derived to estimate the effective behaviour and field fluctuations in composites with viscoplastic crystalline phases. For this purpose, we first derive a generalization of the ‘linear comparison’ transformation [8,9] for the stress potentials of the nonlinear phases. It is recalled that the linear comparison transformation expresses the potential of the nonlinear material in terms of a quadratic comparison potential (corresponding to linear constitutive behaviour), where the viscosity of this comparison medium is determined by a suitable optimization procedure. Building on earlier work [15,16], the generalization consists in the use of a comparison potential including both a linear term as well as the quadratic term, where the coefficients of such a comparison potential correspond to certain eigenstrain rates and viscosities to be determined by a suitable optimization process. We then make use of this transformation to express the macroscopic potential for the nonlinear composite in terms of the macroscopic potential for an LCC with the same microstructure as the original nonlinear composite, but with linear properties characterized by the eigenstrains and viscosities of the slip systems in the various phases. This allows the use of homogenization estimates for the LCC to generate corresponding estimates for the original nonlinear composite by means of a suitable optimization procedure over the properties of the LCC. Different from the earlier ‘second-order’ estimates [15,16], the new ‘second-order’ estimates can be fully optimized over the properties of the relevant LCCs. This endows the new estimates with some advantageous and useful properties, including the facts that the macroscopic constitutive relation and the field statistics of the nonlinear composite can be obtained directly from those of the LCC.

### (a) Generalized Legendre transformation

We first consider an even, scalar-valued, (strictly) convex function *ϕ* of a scalar variable *τ*, such that *ϕ*(0)=0 (e.g. the power-law function defined in (2.3)). Then, under appropriate smoothness hypotheses, its Legendre transform is defined by the single-valued function
*ϕ*^{**}=*ϕ*, or
*γ*=*ϕ*′(*τ*) and *τ*=*ϕ**′(*γ*) are the *unique* inverses of each other. In fact, as is well known, the Legendre transformation can be defined even when the function is not smooth, but still convex. In this case, the stationary operation in expression (3.1) is replaced by a supremum operation, and then *ϕ** is usually referred to as the Legendre–Fenchel transform of *ϕ*.

In this work, we are interested in expressing the macroscopic potential of a nonlinear composite in terms of a linear composite with a quadratic potential. With this goal in mind, we consider the function
*ϕ* has been ‘shifted’ by a quadratic function

As illustrated in figure 1*a* for the special case where *ϕ*(*τ*)=*τ*^{4}/4, the shifted function is not expected to be convex in general. However, following Sewell [20], it is still possible to define the Legendre transform of this function via the expression
*τ* as a function of *γ*, as depicted in figure 1*b* for the example of figure 1*a*. In such cases, as depicted in figure 1*c*, the Legendre transform *ψ** will be a multiple-valued function for a range of values of the variable *γ*. Because of the multi-valued character of the function *ψ**, it is necessary to exercise care in the computation of the double-Legendre transform *ψ***. Nevertheless, it can be shown [20] that, with the proper branch selection, the duality result (*ψ***=*ψ*) still survives in this case.

Thus, using Legendre duality, we can write that
*ψ** must be used to ensure equality.

Now, for given *τ*, we assume that λ is chosen such that the three solutions of the equation (3.5) are labelled and ordered as

Then, for reasons that will become evident further below, it is useful to introduce the functions
*ψ** (i.e. not the ‘correct’ branch of *ψ** recovering the value of the function of the left-hand side of expression (3.6) at *τ*). Note that the labels *b*), satisfying the conditions
*τ* is the ‘correct’ value of the inverse of *γ* in (3.5), in the sense that it is consistent with expression (3.6). Therefore, as illustrated in figure 1*c*, the functions *γ* (and also different from *ψ**(*γ*)).

Next, writing expression (3.6) in the form
*α* is some appropriately selected ‘weight factor’ such that 0<*α*<1, it is possible to generate the ‘relaxed’ estimate
*ψ** (evaluated at *τ*) by the two ‘other’ branches of the function *ψ**, namely *τ*^{2} to the right-hand side of the expression (3.11), so that
*α*. To see this, it is noted that the stationarity conditions with respect to *γ* and λ reduce to
*ϕ* require that *α*<1, and imply equality in expression (3.13).

We will make use of the representation (3.13) for the slip potentials *u*^{(r)} in expressions (2.1), in the corresponding expression (2.4) for the effective stress potentials

### (b) The linear comparison composite

By analogy with expressions (2.1) for the stress potentials of the phases of the nonlinear composite, we introduce stress potentials for the phases of the LCC by means of the expressions
*r*=1,…,*N*; *s*=1,…,*K*^{(r)}). It is easy to see that the phase potential *r*=1,…,*N*; *s*=1,…,*K*^{(r)}) correspond to the slip tensors that have been defined in the context of expression (2.2).

Differentiation of expression (3.18) with respect to ** σ** shows that the constitutive relation of this material is indeed linear, i.e.

It is also well known (e.g. [11]) that the first and second moments of the stress in the LCC can be obtained from *k* over phase *r* in the LCC.

### (c) Stationary variational estimates

Returning now to the nonlinear composite with stress potentials *u*^{(r)} given by expressions (2.1) in terms of the slip potentials *ϕ* to write
*α*^{(r)} are constant ‘weight factors’ between 0 and 1, and where we have identified the variables λ in expressions (3.13) with the ‘slip viscosities’

By recalling that *r* of the LCC, it is noted that the above expression for *u*^{(r)} can be rewritten more compactly as

It follows, by substituting expressions (3.25) for the potentials *u*^{(r)} in expression (2.4) for the effective potential *s*=1,…,*N*,*n*=1,…,*K*^{(r)}) appearing both in the effective potential **x** within the individual phases of the composite. In this context, however, it should be emphasized that the trial fields ** σ** in the variational statement (2.4) for

Thus, it is noted that, since the trial fields *α*^{(r)} and the variables *V* ^{(r)} (instead of one). As will be seen next, the use of such additional error functions will enable the generation of estimates for *both* the variables *either* the *or* the *V* ^{(r)} in the earlier estimate of Liu & Ponte Castañeda [16] had the implication that the optimization conditions for the variables

Next, we spell out the stationarity conditions associated with the estimate (3.28). We begin with the ‘inner’ stat problems for the variables *ϕ*′ has been used to denote the derivative of *ϕ*. Note that these two conditions imply that

Next, we consider the ‘outer’ optimization problems with respect to the variables *r* in the problem (3.21) for the LCC, which may be computed in terms of _{1}. Also, as a consequence, the quantities *s*=1,…,*N*;*n*=1,…,*K*^{(r)}). Similarly, it is found that the stationarity conditions with respect to the slip compliance ** σ**⊗

**〉**

*σ*^{(r)}corresponds to the second moment of the stress field in phase

*r*of the LCC, which may be computed via identity (3.23)

_{2}, and is therefore also a function of all the variables

It is also worth mentioning that straightforward algebra shows that the conditions (3.31) and (3.32) can be combined to obtain the following result for the standard deviations of the resolved shear stress fluctuations in the phases, as defined by expressions (2.7), namely

The stationarity conditions (3.29), (3.31) and (3.32) provide, at least in principle, a sufficient number of equations to solve for the *fully stationary* variational estimates for *α*^{(r)}; however, the selection of these weight factors does not affect their stationarity properties and will be dictated by certain symmetry requirements to be discussed below in the context of specific cases.) As will be seen next, the new variational estimates (3.28) exhibit several interesting and useful properties as a consequence of their stationarity properties.

*Simplified expression for the macroscopic stress potential.* Use of the stationarity conditions (3.31) and (3.32) in expression (3.28) for the effective potential

*Macroscopic constitutive relation via the linear comparison composite.* Due to the stationarity of the variables

*Field statistics.* As already mentioned, a general procedure is available [26] for computing the moments of the stress and strain-rate field in the phases of the nonlinear composite by means of suitably perturbed nonlinear problems. In particular, the first moment, or average, of the stress field in phase *r* of the nonlinear composite with stress potentials *u*^{(s)} may be obtained via expression (2.8), where *η*^{(r)} is a constant, symmetric, second-order tensor, and *L* in *r* of the LCC, and where *η*^{(r)}, and using the chain rule and the stationarity with respect to the variables *η*^{(r)}=0, it is concluded that

A completely analogous calculation starting with expression (2.10) for the second moment of the stress in the nonlinear composite can be used to show that the second moments of the stress fields in the nonlinear composite can be consistently estimated from the LCC, that is,

In addition, by proposition 3.4 of [26], the first moment, or average, of the strain-rate field in phase *r* of the nonlinear composite may be obtained via expression (2.12), where *τ*^{(r)} is a constant, symmetric, second-order tensor, and *u*_{L}_{τ} given by expression (2.13) with *u*^{(s)} replaced by *τ*^{(r)}, and is stationary with respect to the variables

It is also possible to similarly show by means of corollary 3.7 of [26] that the second moments of the strain-rate field in the nonlinear composite can be consistently estimated from the LCC, that is,

In summary, the full stationarity of the variational estimate (3.28) implies that not only can the macroscopic constitutive relation for the nonlinear composite be estimated directly from the corresponding constitutive relation of the LCC, but, in addition, so can the first and second moments of the stress and strain-rate fields in the nonlinear composite. Since the ‘fully stationary’ estimate (3.28) already requires the computation of the first and second moments of the stress field in the LCC, the fact that these quantities actually also provide estimates for the corresponding first and second moments of the stress field in the actual nonlinear composite is very convenient, in particular, given that the direct computation from the appropriate perturbed potentials is not straightforward. For example, it has been shown [26] that the first and second moments of the stress and strain-rate fields in the earlier versions of the ‘tangent’ and ‘generalized secant’ second-order estimates [15,27,28]—which are not stationary with respect to the variables *special* choices of these parameters. On the other hand, the present fully stationary version of the second-order estimates is similar to the variational bounds [8,9], for which the first and second moments of the stress and strain-rate fields in the LCC can be used to directly estimate the corresponding moments of the fields in the actual nonlinear composite. This is a feature that certainly makes worthwhile the additional complexities associated with the use of multiple error functions in the stationary estimates (3.28).

*Legendre duality.* As we have seen, it is possible to generate consistent estimates for the macroscopic constitutive relation of the composite, as well as for the phase averages and second moments of the stress and strain-rate fields, directly from the optimized LCC. In this sense, the variational statement (3.28) for the effective stress potential

## 4. Applications for porous single crystals

In this section, our objective is to illustrate the main features of the fully optimized second-order (FO-SO) homogenization technique by means of an example. For this purpose, we consider a special class of (two-phase) porous materials with ‘particulate’ microstructures [29], consisting of aligned cylindrical pores (*r*=2) that are distributed randomly and isotropically in a viscoplastic single-crystal matrix phase (*r*=1). We assume that the laboratory frame of reference is described by the orthonormal set of vectors **e**_{i} (*i*=1,2,3), such that the symmetry axes of the cylindrical pores are aligned with **e**_{3}. We further assume that the behaviour of the crystalline matrix is characterized by an *incompressible* stress potential *u*^{(1)} of the form (2.1), where the slip potentials *n*_{(k)}=*n*, and (*τ*_{0})_{(k)}=*τ*_{0} for all *k*. On the other hand, the Schmid tensors *μ*_{(k)} are taken to be of the form (2.2)_{2}, with slip directions **m**_{(k)}=**e**_{3} (for all *k*) and slip normals given by *square* (*K*=2), *θ*_{(k)}=0,*π*/2, and (ii) *hexagonal* (*K*=3), *θ*_{(k)}=0,±*π*/3. The porous material is subjected to anti-plane loadings, and the relevant viscoplastic boundary value problem then becomes a two-dimensional vectorial problem, where the non-zero components of the stress and strain-rate vectors, namely *σ*_{13}, *σ*_{23}, *ϵ*_{13} and *ϵ*_{23}, are functions of *x*_{1} and *x*_{2} only.

The fact that the viscous exponent *n* is the same for all the slip systems leads to the effective potential of the polycrystal *n*+1 on the average stress *effective flow stress*, which depends on the porosity *c*=*c*^{(2)}, the nonlinearity *n*, and the direction of loading *u*^{(1)}, and therefore *π*/*K*. Thus, it suffices to restrict attention to loading directions in the range

For additional simplicity, we consider here only loadings that are aligned with one of the slip systems (*γ*_{(1)} and a viscous compliance tensor of the form
*k*=λ/*μ*, which takes the value 1 for isotropic tensors and 0 or infinity for strongly anisotropic tensors. The behaviour of the LCC can then be characterized by estimates of the Willis type [23,24], which for the above-described porous microstructures takes the form [30]
** σ**⊗

**〉**

*σ*^{(1)}. Using this information, the stationarity conditions (3.31) and (3.32) can be easily used to obtain the expressions

*k*may then be obtained by substituting expressions (4.6) and (4.7) into (4.8)

_{2}for

*k*is independent of

*n*, as well as on the geometry of the slip systems (through the angles

*θ*

_{(k)}). Having obtained the value of

*k*, the values of the

However, a simple expression for the effective flow stress of the nonlinear porous material may be obtained from expression (3.34) and is given by
*α*, in the results to be presented next, *α* will be set equal to 1/2, which is the most symmetric choice.

Figure 2 presents plots of the above-described FO-SO estimates for the effective flow stress *m*=1/*n*, for a fixed porosity (*c*=0.25). The results are normalized by the flow stress *τ*_{0} of the matrix slip systems and are given for two different matrix materials: (*a*) ‘square’ symmetry (*K*=2) and (*b*) ‘hexagonal’ symmetry (*K*=3). The results are compared with several bounds and estimates for the effective flow stress of the porous crystals, as described next. The Taylor bound is obtained by making use of a uniform strain rate (*c*. The variational (VAR) upper bounds are obtained by means of the variational linear comparison method of Idiart & Ponte Castañeda [29], making use of the Hashin–Shtrikman–Willis bounds [4,23] for the appropriate LCC, while the relaxed variational (RVAR) upper bounds are obtained by means of the relaxed version of the variational linear comparison method proposed by deBotton & Ponte Castañeda [8], making use of the same bounds for the LCC. Note again that these two bounding methods give the same predictions for *K*=2, but somewhat different answers for *K*=3. The partially optimized second-order (PO-SO) estimates were generated by means of the method proposed by Liu & Ponte Castañeda [16] for the stress potential *m* except *m*=0 and 1, due to the lack of stationarity with respect to the eigenstrain rates in these estimates. Finally, results are also shown for the estimates of Idiart [31] for sequentially laminated microstructures (LAMs). These microstructures are also ‘particulate’ in character, with a well-defined matrix and inclusion phases (i.e. the crystalline and vacuous phases, respectively). They have the property that, in the special case of linearly viscous behaviour (*m*=1), they agree exactly with the Hashin–Shtrikman–Willis bounds [23] for statistically isotropic microstructures (in the plane), as well as with the Willis-type estimates [24] for statistically isotropic distributions of circular voids.

Having described the various bounds and estimates, we can now make the following observations concerning the new fully optimized (FO-SO) estimates developed in this work. First, the FO-SO estimates satisfy all available upper bounds, including the RVAR bounds [8], as well as the tighter (for *K*=3 only) VAR bounds [29], which provide generalizations of the Hashin–Shtrikman–Willis bounds for the nonlinear porous composites. Note that these bounds agree with the Hashin–Shtrikman–Willis bounds for *m*=1. Second, the FO-SO estimates are in very good agreement with—but are not exactly identical to—the LAM estimates [31] for the full range of values of *m*. Once again, these results are known to agree exactly in the linear case (*m*=1), but are not expected to be in perfect agreement for other values of *m*, since they correspond to somewhat different microstructures. The microstructures for the *exact* LAM results shown in figure 2 have isotropic two-point statistics in the transverse plane; however, the voids are not circular, and their centres are distributed anisotropically (albeit in a way that leads to isotropic two-point statistics). On the other hand, the FO-SO estimates make use of *approximate* estimates for the LCC, which do correspond to circular voids whose centres are distributed with isotropic two-point statistics in the transverse plane. Third, the new FO-SO estimates are fairly similar to the earlier PO-SO estimates for *K*=2, except for fairly low values of the strain-rate sensitivity (*m*<0.2), and less similar for *K*=3, when significant differences are observed for values of *m*<0.5. In particular, the FO-SO estimates do not exhibit the ‘bellies’ exhibited by the PO-SO estimates for small values of *m*, when the differences can reach values of about 10%. Interestingly, however, the FO-SO estimates agree exactly with both versions of the PO-SO estimates not only for *m*=1, but also for *m*=0. The difference in the predictions of the FO-SO and PO-SO methods for 0<*m*<1 is expected in view of the fact that the eigenstrain rates in the PO-SO methods are not optimized, leading to a duality gap (i.e. the U and W versions are different), whereas the new FO-SO estimates are fully optimized and therefore exhibit no duality gap. However, it is interesting to further remark that the new FO-SO estimates agree with the PO-SO estimates in the limit as

## 5. Concluding remarks

In this paper, we have developed a variational method for estimating the macroscopic properties of nonlinear composites with viscoplastic crystalline phases in terms of the corresponding properties of suitably designed LCCs. The phases of the LCC are characterized by certain slip viscosities and eigenstrain rates, which in turn play the role of trial fields in a variational statement for the stress potential of the viscoplastic composites. When fully optimized, these slip viscosities and eigenstrain rates are found to be determined by a certain ‘generalized secant’ linearization of the slip potentials of the nonlinear crystals, and to depend on the first and second moments of the resolved shear stresses in the LCC, which in turn depend on the microstructure of the nonlinear composite. The resulting ‘fully optimized’ estimates are exact to second order in the contrast and satisfy all known bounds. In addition, they exhibit several desirable and useful properties that were missing in earlier estimates of the second-order type, such as the estimates of Liu & Ponte Castañeda [16], due to the lack of ‘full stationarity’ with respect to the slip viscosities and eigenstrain rates in the LCC. These properties, which were already present in the earlier—albeit less accurate—variational bounds of deBotton & Ponte Castañeda [8], include the following: (i) the macroscopic response and field statistics of the nonlinear composite can be obtained directly from the corresponding macroscopic response and field statistics in the LCC, thus greatly simplifying the computation of these quantities, and (ii) the lack of a ‘duality gap’, leading to more accurate predictions. By way of an example, the new method was applied to the simple case of a porous single crystal, and the results were compared with known bounds and earlier estimates, including the exact estimates of Idiart & Vincent [19] for porous crystals with ‘sequentially laminated’ microstructures [31]. In this context, it should be noted that, while both methods give very similar predictions for this particular example, the new method developed in this work would apply to more general multi-phase composites, including polycrystals, while the method of Idiart [31] is so far restricted to two-phase systems with ‘particulate’ microstructures. Finally, it should also be noted that, although the new homogenization method developed in this work was presented in the specific context of viscoplastic crystalline composites, the general ideas of the method are expected to be useful for a large number of applications, where homogenized nonlinear properties can be given appropriate variational formulations. These will be explored in future publications.

## Competing interests

We declare we have no competing interests.

## Funding

This work was supported by a Research Award from the Alexander von Humboldt Foundation at the University of Stuttgart, Germany, where the author was kindly hosted by Prof. Christian Miehe, as well as by the National Science Foundation under grant nos. DMS-1108847 and CMMI-1332965.

- Received September 18, 2015.
- Accepted October 29, 2015.

- © 2015 The Author(s)