Multiphase composites with extremal bulk modulus

doi:10.1016/S0022-5096(99)00043-5

Journal of the Mechanics and Physics of Solids

Volume 48, Issue 3, March 2000, Pages 461-498

https://doi.org/10.1016/S0022-5096(99)00043-5 Get rights and content

Abstract

This paper is devoted to the analytical and numerical study of isotropic elastic composites made of three or more isotropic phases. The ranges of their effective bulk and shear moduli are restricted by the Hashin–Shtrikman–Walpole (HSW) bounds. For two-phase composites, these bounds are attainable, that is, there exist composites with extreme bulk and shear moduli. For multiphase composites, they may or may not be attainable depending on phase moduli and volume fractions. Sufficient conditions of attainability of the bounds and various previously known and new types of optimal composites are described. Most of our new results are related to the two-dimensional problem. A numerical topology optimization procedure that solves the inverse homogenization problem is adopted and used to look for two-dimensional three-phase composites with a maximal effective bulk modulus. For the combination of parameters where the HSW bound is known to be attainable, new microstructures are found numerically that possess bulk moduli close to the bound. Moreover, new types of microstructures with bulk moduli close to the bound are found numerically for the situations where the aforementioned attainability conditions are not met. Based on the numerical results, several new types of structures that possess extremal bulk modulus are suggested and studied analytically. The bulk moduli of the new structures are either equal to the HSW bound or higher than the bulk modulus of any other known composite with the same phase moduli and volume fractions. It is proved that the HSW bound is attainable in a much wider range than it was previously believed. Results are readily applied to two-dimensional three-phase isotropic conducting composites with extremal conductivity. They can also be used to study transversely isotropic three-dimensional three-phase composites with cylindrical inclusions of arbitrary cross-sections (plane strain problem) or transversely isotropic thin plates (plane stress or bending of plates problems).

Introduction

The paper is devoted to the long-standing problem of bounds on effective properties of elastic composites comprised of isotropic phases. We aim to achieve three goals.

In Section 2, we investigate sufficient conditions for the attainability of the Hashin and Shtrikman, 1963, Walpole, 1966 bounds. These are the simple conditions on phase properties and volume fractions that guarantee existence of multiphase composites with a bulk or shear modulus exactly equal to the bounds. For the bulk modulus problem, these attainability conditions were found by Milton (1981). Here we formulate analogous conditions for the shear modulus problem. When these conditions are met, we demonstrate several microstructures that achieve the bounds.

The goal in Section 3 is to search for two-dimensional three-phase composites with a maximal bulk modulus by using a numerically based topology optimization method. We start with the situation where the bound is known to be optimal. The goal is to check the reliability of the inverse homogenization procedure (Sigmund, 1994a, Sigmund, 1994b, Sigmund, 1995, Sigmund, 1999a, Sigmund and Torquato, 1996, Sigmund and Torquato, 1997) by solving a problem with known answer, that is, with known optimal bound. The result of the check is more than satisfactory. The procedure not only found a microstructure with an extremal bulk modulus but this microstructure was of a new (although similar to the known) type. Then we used the same procedure to explore parameter regions where the optimal structures were not known. The parameters of the problem were slowly changed and the evolution of the optimal design was studied. New structures were found for the range of parameters where the attainability conditions for the Hashin–Shtrikman–Walpole (HSW) bulk modulus upper bound are not met. The bulk modulus of each of these new structures is higher than the bulk modulus of any previously known composite with the same phase moduli and volume fractions. The HSW bulk modulus bounds apply not only to isotropic composites, but also to composites with square symmetry (in two dimensions) or cubic symmetry (in three dimensions) of the elastic tensor. In our numerical and analytical study of two-dimensional structures we require only square symmetry of the effective tensor. However, given material with square symmetry one can easily make an isotropic composite with the same value of effective bulk modulus. Alternatively, the numerical procedure can solve the problem with some extra computational effort.

Section 4 is devoted to the theoretical study of two-dimensional three-phase composites with a maximal bulk modulus. We found several new types of three-phase structures based on a new class of extremal two-phase composites proposed recently by one of the authors (Sigmund, 1999b) and on the result of our numerical experiments in Section 3. The microstructures of two of the new types have an effective bulk moduli exactly equal to the HSW bulk modulus bounds in a range of parameters where more traditional approaches have failed to deliver optimal composites. When the volume fraction of the stiff phase is low, we found another new type of composite which has an effective bulk modulus that is smaller than the bound but larger than the bulk modulus of any previously known composite with the same phase moduli and volume fractions.

In Section 5 we summarize, discuss, and generalize the results of the paper. In particular, two-dimensional three phase composites with extremal conductivity are discussed. We show that the bulk modulus results for the phases with zero Poisson’s ratio are readily applied to conductivity. Thus, two new types of conducting composites are found. The microstructures of one of the new types have effective conductivity exactly equal to the Hashin–Shtrikman (HS) bound in the range of the parameters where previously known composites were not optimal. Outside of the parameter range where the HS bound is attainable, we found another new type of composite which has effective conductivity that is not equal to the bound but closer to the bound than the conductivity of any previously known composite with the same phase moduli and volume fractions.

The obtained results can be applied to the effective moduli of a transversely isotropic three-dimensional composite with cylindrical inclusions of an arbitrary cross-section. They can also be applied to the effective plane stress or bending moduli of thin plates with the microstructure independent of the coordinate perpendicular to the plate.

Section snippets

Hashin–Shtrikman–Walpole bounds

We deal with the linear elasticity problem. The stiffness of an isotropic elastic material can be described by two moduli, that is, the bulk modulus κ and the shear modulus μ. We consider a composite made of N elastic phases, with bulk moduli κ_i and shear moduli μ_i, taken in the prescribed volume fractions f_i, i=1, …, N. An isotropic composite is characterized by its effective bulk modulus κ_^∗ and its effective shear modulus μ_^∗. For accounts of homogenization theory, see for example Bensoussan

Numerical design of extreme materials

In this section the numerically based material design or ‘inverse homogenization’ procedure is briefly reviewed, and then used for the two-dimensional three-phase bulk modulus optimization problem.

New structures

In this section we consider two-dimensional three-phase elastic composites. We first assume that $κ_{1} ≤κ_{2} ≤κ_{3}, μ_{3} =μ_{max} .$ Then the HSW bulk modulus upper bound (3) is given by $κ_{U} = f_{1} κ_{1} +μ_{3} + f_{2} κ_{2} +μ_{3} + f_{3} κ_{3} +μ_{3}^{−1} −μ_{3} .$ One can check that the attainability condition κ_U≥κ₂ is met if the volume fraction f₃ is sufficiently high, that is, $f_{3} ≥f^{∗}_{3}, f^{∗}_{3} = f_{1} (κ_{3} +μ_{3})(κ_{2} −κ_{1}) (κ_{1} +μ_{3})(κ_{3} −κ_{2}) .$ By using the relation f₁=1−f₂−f₃, condition (38) can also be presented in the form $f_{3} ≥ f ̂^{∗}_{3}, f ̂^{∗}_{3} = (1−f_{2})(κ_{3} +μ_{3})(κ_{2} −κ_{1}) (κ_{2} +μ_{3})(κ_{3} −κ_{1}) .$ The

Interpretation of the numerical results

In this section we compare the results of the numerical optimization of Section 3 with the theoretical results of Section 4. For the first numerical example with the phase moduli and volume fractions given by (28) we have $f_{1} =0.2, f_{2} =0.4, f_{3} =0.4, f^{∗∗}_{3} =0.097, f^{∗}_{3} =0.143, f ̂^{∗}_{3} =0.25.$ As we can see, f₃≥f^∗₃ and the bound κ_U=0.224 is attainable by the Milton, 1981, Lurie and Cherkaev, 1984 schemes. For this example, we found a new one-scale optimal structure shown in Fig. 4, Fig. 5, with the effective

Acknowledgements

L.G. gratefully acknowledges the support by the DCAMM International Graduate Research School and the Danish Research Academy, and the hospitality of the Danish Center for Applied Mathematics and Mechanics. O.S. gratefully acknowledges the support by the Danish Technical Research Council [THOR/Talent-programme: Design of MicroElectroMechanical Systems (MEMS)].

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Multiphase composites with extremal bulk modulus

Abstract

Introduction

Section snippets

Hashin–Shtrikman–Walpole bounds

Numerical design of extreme materials

New structures

Interpretation of the numerical results

Acknowledgements

Comp. Meth. Appl. Mech. Engng

Int. J. Solids Struct.

J. Mech. Phys. Solids

J. Mech. Phys. Solids

J. Mech. Phys. Solids

J. Mech. Phys. Solids

Mech. Mat.

Int. J. Solids Struct.

Mech. Mat.

J. Mech. Phys. Solids

J. Mech. Phys. Solids

J. Mech. Phys. Solids

Optimization of Structural Topology, Shape and Material

Asymptotic Analyses of Periodic Structures

Optimal distribution of multimaterial composites for torsional beams

Struct. Opt.

Numerical experiments of the homogenization method for operators with periodic coefficients

Homogenization and optimal bounds in linear elasticity

Arch. Rational Mech. Anal.

Bounds for effective parameters of multicomponent media by analytic continuation

J. Stat. Phys.

Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods

Comp. Meth. Appl. Mech. Eng.