Elsevier

European Journal of Mechanics - A/Solids

Volume 85, January–February 2021, 104072
European Journal of Mechanics - A/Solids

Morphology-dependent Hashin–Shtrikman bounds on the effective properties of stress-gradient materials

https://doi.org/10.1016/j.euromechsol.2020.104072Get rights and content

Highlights

  • Variational bounds on the effective elastic moduli of stress-gradient materials are derived.

  • In many cases of interest, the third-order trial strain-polarization needs not be considered.

  • Resulting bounds are sensitive to the two-point microstructural correlations.

Abstract

Stress-gradient materials are generalized continua with two generalized stress variables: the Cauchy stress field and its gradient. For homogenization purposes, we introduce an extension to stress-gradient materials of the principle of Hashin and Shtrikman. The variational principle is first stated within the framework of periodic homogenization, then extended to random homogenization. Contrary to the usual derivation of the classical principle, we adopt here a stress-based approach, much better suited to stress-gradient materials. We show that, in many cases of interest, the third-order trial eigenstrain may be discarded, leaving only one (second-order) trial eigenstrain in the functional to optimize. For N-phase material, the bounds are very similar in structure to their classical counterpart. One notable difference is the fact that, even in the case of isotropy, the bounds depend on some additional microstructural parameters (besides the usual volume fractions).

Introduction

The complementary elastic energy of stress-gradient materials depends on the stress and its gradient. This class of generalized materials was first introduced by Forest and Sab (2012) as an alternative to strain-gradient materials. Since then, other similar models have been proposed, which mostly differ by the boundary conditions; see e.g. Polizzotto (2018) and references therein. Such models are relevant for nanocomposites exhibiting “negative” size-effect (Tran et al., 2018) and foams (Hütter et al., 2020).

The model of Forest and Sab (2012) (including the boundary conditions) was fully justified mathematically by Sab et al. (2016). It was subsequently extended to finite strains by Forest and Sab (2017). Homogenization of heterogeneous, stress-gradient materials as homogeneous, Cauchy materials was then considered in the work by Tran et al. (2018), where a simplified stress-gradient model, akin to the simplified strain-gradient model of Altan and Aifantis (1992, 1997), was also proposed. The converse homogenization problem (Cauchy material at the microscale, stress-gradient material at the macroscale) has recently been addressed by Hütter et al. (2020), who offer a microscopic interpretation of the generalized strain that is work-conjugate to the gradient of the stresses.

The present paper builds upon the results obtained by Tran et al. (2018). More specifically, we propose a variational framework that allows the derivation of rigorous bounds on the macroscopic, classical stiffness of microscopically heterogeneous, stress-gradient materials, within the framework of both periodic and random homogenization. The variational principle that we introduce is an extension to stress-gradient materials of the celebrated principle of Hashin and Shtrikman (1962a). The classical version of this principle is usually presented within a strain-based approach, where eigenstresses are used as trial functions. A stress-based approach, where eigenstrains are used as trial functions, was found to be much better suited to stress-gradient materials and was therefore adopted. Both strain- and stress-based approaches are known to be equivalent in the classical case. For stress-gradient materials, we also verified this equivalence, since in early versions of the work presented here, we used a strain-based approach (Tran, 2016). Note that the variational principle of Hashin and Shtrikman was extended to strain-gradient materials by Smyshlyaev and Fleck (1994).

This paper is organized as follows. Sec. 2 provides an overview of the stress-gradient model of Forest and Sab (2012) and recalls how periodic, heterogenous stress-gradient materials can be homogenized as Cauchy materials through the so-called corrector problem (Tran et al., 2018). In the stress-based formulation of the principle of Hashin and Shtrikman (1962a), the trial function is an eigenstrain; we therefore discuss eigenstrained, stress-gradient materials in Sec. 3. For homogeneous materials, we then introduce in Sec. 4 the Delta operator that maps the eigenstrains to the stresses. This operator is the key ingredient of the Lippmann–Schwinger equation, which is introduced in Sec. 5 as an equivalent, alternative formulation of the corrector problem. Then, the principle of Hashin and Shtrikman (1962a) is stated in Sec. 6, first as the weak form of the Lippmann–Schwinger equation, then (under some conditions) as a variational principle.

These results, which are first introduced within the framework of periodic homogenization, are then extended to random homogenization.

In Sec. 7, we define the apparent and effective compliances of random composites. Then, in Sec. 8, we extend the principle of Hashin and Shtrikman (1962a) to the random case, and derive an operational formula for the resulting bounds on the effective compliance. This formula is simplified in Sec. 9, where it is shown that in some circumstances, it is possible to discard the third-order eigenstrain, the minimization problem being reduced to one trial field only. In Sec. 10, we specialize the general framework to N-phase materials with piece-wise constant trial eigenstrains; the resulting bounds are very close in spirit to the classical bounds of Hashin and Shtrikman (1962b). Finally, these bounds are evaluated numerically in Sec. 11 for a simple (boolean) microstructure.

Section snippets

Nomenclature

The space of second-order, symmetric tensors is denoted T2; I2T2 is the second-order identity tensor and I4 is the fourth-order identity tensor over T2I2=δijeiejandI4=12(δipδjq+δiqδjp)eiejepeq,in cartesian coordinates. The double contraction “:” defines a scalar product over T2a:b=aijxijfor all a,bT2.

It will be convenient to introduce the classical fourth-order spherical and deviatoric projection tensors J4=13I2I2 and K4=I4J4, such thatJ4:K4=K4:J4=0,J4:J4=J4andK4:K4=K4.

The space of

On eigenstrained stress-gradient materials

Deformation of stress-gradient materials is defined through two strain measures, namely: e and ϕ. It is therefore a priori possible to introduce two eigenstrains: the second-order eigenstrain e*T2(Ω) and the third-order eigenstrain ϕ*T3(Ω). In the present section, we first state the equilibrium of linearly elastic, eigenstrained, stress-gradient as a minimization problem. We then extend Clapeyron's theorem, that delivers the value of the complementary energy.

Homogeneous, eigenstrained materials – the green operator

We now consider a homogeneous stress-gradient material with compliances S0 and M0, occupying the domain Ω=(0,L1)×(0,L2)×(0,L3), subjected to second- and third-order eigenstrains only. In other words, we specialize Problem (25a), (25b), (25c), (25d), (25e), (25f), (25g) to homogeneous materialsσ=0,e=S0:σ+e*ϕ=M0(σ)+ϕ*,e=ϕ+sym(u),σ and uex are Ωperiodic,ϕn is Ωskewperiodic,σ=σ¯.

The solution to the above problem is unique, and depends linearly on the loading parameters, namely: σ¯

The Lippmann–Schwinger equation

In the present section, we derive an integral equation that is equivalent to Problem (21)(23). This integral equation can be seen as an extension to stress-gradient materials of the classical Lippmann–Schwinger equation for linear elasticity (Korringa, 1973; Kröner, 1974; Zeller and Dederichs, 1973). We first rewrite Problem (21)(23) as followsσ=0,e=S0:σ+e*,ϕ=M0(σ)+ϕ*,e=ϕ+sym(u),σ and uex are Ωperiodic,ϕn is Ωskewperiodic,σ=σ¯,e*=(SS0):σ,ϕ*=(MM0)(σ),where we have

The principle of Hashin and Shtrikman

We consider again the setting introduced in Sec. 2.3 for periodic homogenization. We further introduce a homogeneous, reference stress-gradient material with compliance S0 and generalized compliance M0 and eliminate the stresses σ in Problem (53a), (53b), (53c)(SS0)1:e*+Δ0(e*)+Λ0(ϕ*)=σ¯,(MM0)1ϕ*+[Δ0(e*)+Λ0(ϕ*)]=0.

Then, multiplying with the test functions δe* and δϕ* and averaging over the unit-cell Ω delivers the following weak form of the Lippmann–Schwinger equationσ¯:δe*=e*:(SS0)1:

Apparent and effective compliances of random composites

In the present section we show how the effective compliance of a random heterogeneous, stress-gradient material can be derived. Most symbols defined previously keep the same meaning, except that now, the microstructure is no longer periodic, but random: the heterogeneous properties of the composite S(x,ω) and M(x,ω) are random variables, indexed by the realization ω (which fills the whole space R3). We emphasize that Eq. (20) no longer holds. Separation of scales in the sense of Sec. 2.3 does

The principle of Hashin and Shtrikman for random homogenization

We are now in a position to extend the variational principle of Hashin and Shtrikman presented in Sec. 6 to random homogenization. We can first write this principle realization-by-realization, for a fixed-size SVE:12σ¯:Sapp(L,ω):σ¯HS(e*,ϕ*,σ¯,L,ω),subjected toSS0andMM0.

In the above equation, HS(,,σ¯,L,ω) denotes the value of the functional of Hashin and Shtrikman [see Eq. (59)], evaluated for the realization ω, on a unit-cell of size L. It is observed that the critical strain

Simplifications of the bounds

We assume in this section that Reϕ(r)=Reϕ(r). At this point, this assumption should be seen as purely mathematical; it will be shown in Sec. 10 that, for N-phase, isotropic materials and strain polarizations that are constant in each phase, this assumption holds. Under this assumption, Rˆeϕ is real and the last term in Eq. (72) vanishesHS(e*,ϕ*,σ¯)=12σ¯:S0:σ¯+σ¯:E[e*]12E[e*:(SS0)1:e*]12E[ϕ*(MM0)1ϕ*]12(2π)3kR3Δˆ0(k)Rˆee(k)d3k12(2π)3kR3Δˆ0(k)Rˆϕϕ(k)d3k=HS(e*,0,σ¯)+HS(0,ϕ*,0)

Specialization to isotropic, N-phase materials

We assume that the composite is a N-phase material; χα denotes the indicator function of phase α=1,,N: xΩ belongs to phase α if, and only if, χα(x)=1 (note that χα depends on the realization). The microstructure is statistically homogeneous, ergodic and isotropic; fα=E[χα] denotes the volume fraction of phase α. Finally, Sαβ denotes the cross-correlationSαβ(r)=E[χα(x)χβ(x+r)]fαfβ,

which depends on the norm r=||r|| of the lag-vector only. The compliance and generalized compliance of phase α

Applications

We consider in this section two applications of the above results. Both microstructures are distributions of monodisperse, spherical inclusions. The domain covered by the spheres will be referred to as “the inclusions” (index “i”) while the complementary space will be referred to as “the matrix” (index “m”). The total volume fraction of inclusions is f; the radius of the spheres is a and their volume is v=43πa3.

In order to ease comparisons, we use the same numerical values as in the previous

Conclusions and perspectives

Adopting a stress-based approach, we have extended the classical variational principle of Hashin and Shtrikman (1962a) to stress-gradient materials, first for periodic homogenization, then for random homogenization. In both cases, we assumed that the material internal lengths were all of the order of the size of the heterogeneities or smaller, which led to homogenization as classical (rather than stress-gradient) materials.

The functional of Hashin and Shtrikman that we introduced involves two

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work has benefited from a French government grant managed by ANR within the frame of the national program Investments for the Future ANR-11-LABX-022-01.

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