Hashin-Shtrikman Bounds Research Articles

Thermal conductivity and Young's modulus of cubic-cell metamaterials

It is shown that the popular quasi-laminate solutions for the relative (thermal) conductivity of cellular materials with cubic cells (closed or open) are in conflict with the upper Hashin-Shtrikman bound and thus inadmissible. However, numerical modeling can be used to estimate the porosity dependence of conductivity and leads to results that are below the upper Hashin-Shtrikman bound, as required for cubic and isotropic materials. The numerical results predict differences of up to 0.073 relative property units (RPU) between closed- and open-cell materials. For porosities approaching 100%, i.e. infinitely thin walls, both the analytical and the numerical solution for closed cubic cells approach the upper Hashin-Shtrikman bound. Moreover, the porosity dependence of the relative conductivity for closed cubic cells is very close to the upper Hashin-Shtrikman bound in the whole range of porosities and is correlated to the relative Young modulus via a cross-property relation based on the Hashin-Shtrikman bounds.

Hashin-Shtrikman bounds with eigenfields in terms of texture coefficients for polycrystalline materials

The Hashin-Shtrikman bounds accounting for eigenfields are represented in terms of tensorial texture coefficients for arbitrarily anisotropic materials and arbitrarily textured polycrystals. This requires a short review of the Hashin-Shtrikman bounds with eigenfields, an investigation of the polarization field determined by the stationarity condition and, finally, the analysis of the resulting expressions of the Hashin-Shtrikman bound of the effective potential. The resulting expressions are given naturally in terms of symmetric second-order tensors and minor and major symmetric fourth-order tensors. These properties induce, based on the tensorial Fourier expansion of the crystallite orientation distribution function, a dependency of all Hashin-Shtrikman properties in terms of solely the second- and the fourth-order texture coefficients. This is a new result, which is not self-evident, since an alternative formulation of the polarization field would alter the implied algebraic properties of the Hashin-Shtrikman functional. The results obtained by the polarization field, determined through the stationarity condition of the Hashin-Shtrikman functional, are discussed and demonstrated with an example for linear thermoelasticity in which bounds for elastic and thermoelastic properties are illustrated.

Representation of Hashin–Shtrikman Bounds in Terms of Texture Coefficients for Arbitrarily Anisotropic Polycrystalline Materials

The present work generalizes the results of Bohlke and Lobos (Acta Mater. 67:324–334, 2014) by giving an explicit representation of the Hashin–Shtrikman (HS) bounds of linear elastic properties in terms of tensorial Fourier texture coefficients not only for cubic materials but for arbitrarily anisotropic linear elastic polycrystalline materials. Based on the HS bounds as given by Walpole (J. Mech. Phys. Solids 14(3):151–162, 1966) and tensor functions for the representation of the crystallite orientation distribution function, it is shown that the HS bounds are represented in terms of the exact same second- and fourth-order texture coefficients which appear in the consideration of the Voigt and Reuss bounds. The derived representations in terms of tensorial texture coefficients are valid for all symmetry classes in elasticity and present expressions highly attractive for the description of physical quantities in terms of tensorial variables. In order to make these results also accessible for the community of quantitative texture analysis, transformation relations between experimentally obtained Bunge’s or Roe’s coefficients and the tensorial texture coefficients are given. The representation of the present work offers a finite and low dimensional parametrization of the fully anisotropic Hashin–Shtrikman bounds, which can be used in inverse materials design problems in order to explore the set of possible materials properties or for the determination of optimal microstructural influence with respect to prescribed material properties. Examples for orthotropic polycrystals of cubic materials and transversely isotropic polycrystals of hexagonal materials (showing the connection and applicability of the results also to fiber orientation distributions) are discussed. Finally, an implementation in Mathematica® 11 of the HS bounds for arbitrarily anisotropic materials is offered, such that readers can reproduce all the results of this work and use them for their own purposes.

Analytical homogenization of linear elasticity based on the interface orientation distribution – A complement to the self-consistent approach

We combine a laminate-based approach and an interface orientation average. For linear elasticity, we obtain analytical expressions for the effective stiffness of mixtures of two isotropic phases with isotropic, transversely isotropic, hexahedral and octahedral interface orientation distributions (IOD). The estimates are in accordance with well-established analytical results such as the Hashin–Shtrikman bounds and Hill’s findings for phases with equal shear moduli.For isotropic and transversely isotropic IODs, this approach is compared to the Hashin–Shtrikman bounds and the inclusion-based self-consistent approach. At extremal volume fractions, both approaches coincide up to first order with complementary and mixed Hashin–Shtrikman-bounds. Thus, our approach provides an alternative to the self-consistent approach.Moreover, the effective stiffness inherits any symmetry of the IOD. Therefore, it captures basic features of the morphology-induced anisotropy.For the hexahedral and octahedral IOD, the approach is compared to RVE simulations. To assess the impact of morphological features beyond the IOD, we consider RVEs with different microstructures but equal IOD. Comparing similar microstructures, the error attains a local minimum close to the point of equal arrangement of phases. Further, we find a reasonable agreement if the shear moduli are of the same order of magnitude or if both phases percolate the material.

The sigmoidal average – a powerful tool for predicting the thermal conductivity of composite ceramics

The sigmoidal average between the upper and lower Hashin-Shtrikman bounds is shown to be the appropriate average relation for isotropic two-phase composites consisting of geometrically equivalent grains. This average ensures that the prediction is close the upper bound for low volume fractions of the low-conductivity phase (corresponding to low-conductivity inclusions in a high-conductivity matrix) and vice versa. In the intermediate concentration range the sigmoidal average reflects the fact that the microstructure is bicontinuous and can undergo a percolation-type transition. It is shown that the sigmoidal average of the Hashin-Shtrikman bounds lies automatically within the three-point bounds (Miller bounds) for a practically important class of microstructures (symmetric-cell materials with spherical cells) and that in the infinite-phase-contrast case (porous media) it is close to the exponential relation, which has been very successful in describing the porosity dependence of properties. Theoretical predictions are compared with experimentally measured values for alumina-zirconia composites in the whole range of volume fractions, from pure alumina to pure zirconia. Even after appropriate correction for porosity, essentially all experimental data are below the sigmoidal average. The fact that some values are even below the lower Hashin-Shtrikman bound is indicative of microcracking and / or grain size (interface) effects.

Bounds and self-consistent estimates for elastic constants of polycrystals composed of orthorhombics or crystals with higher symmetries

Methods for computing Hashin-Shtrikman bounds and related self-consistent estimates of elastic constants for polycrystals composed of crystals having orthorhombic symmetry have been known for about three decades. However, these methods are underutilized, perhaps because of some perceived difficulties with implementing the necessary computational procedures. Several simplifications of these techniques are introduced, thereby reducing the overall computational burden, as well as the complications inherent in mapping out the Hashin-Shtrikman bounding curves. The self-consistent estimates of the effective elastic constants are very robust, involving a quickly converging iteration procedure. Once these self-consistent values are known, they may then be used to speed up the computations of the Hashin-Shtrikman bounds themselves. It is shown furthermore that the resulting orthorhombic polycrystal code can be used as well to compute both bounds and self-consistent estimates for polycrystals of higher-symmetry tetragonal, hexagonal, and cubic (but not trigonal) materials. The self-consistent results found this way are shown to be the same as those obtained using the earlier methods, specifically those methods designed specially for each individual symmetry type. But the Hashin-Shtrikman bounds found using the orthorhombic code are either the same or (more typically) tighter than those found previously for these special cases (i.e., tetragonal, hexagonal, and cubic). The improvement in the Hashin-Shtrikman bounds is presumably due to the additional degrees of freedom introduced into the available search space.

Improved Hashin–Shtrikman Bounds for Elastic Moment Tensors and an Application

This paper is devoted to the derivation of trace bounds for elastic moment tensors. Starting from the integral equation formulation of the elastic moment tensor, we establish that its trace can be obtained as a sum of minimal energies. We then recover the so-called Hashin–Shtrikman bounds, and show that these bounds can be tightened for inclusions which have some local thickness. As an application, we show that the volume of the inclusion can be estimated by the elastic moment tensor.

Random polycrystals of grains containing cracks: Model of quasistatic elastic behavior for fractured systems

A model study on fractured systems was performed using a concept that treats isotropic cracked systems as ensembles of cracked grains by analogy to isotropic polycrystalline elastic media. The approach has two advantages: (a) Averaging performed is ensemble averaging, thus avoiding the criticism legitimately leveled at most effective medium theories of quasistatic elastic behavior for cracked media based on volume concentrations of inclusions. Since crack effects are largely independent of the volume they occupy in the composite, such a non-volume-based method offers an appealingly simple modeling alternative. (b) The second advantage is that both polycrystals and fractured media are stiffer than might otherwise be expected, due to natural bridging effects of the strong components. These same effects have also often been interpreted as crack-crack screening in high-crack-density fractured media, but there is no inherent conflict between these two interpretations of this phenomenon. Results of the study are somewhat mixed. The spread in elastic constants observed in a set of numerical experiments is found to be very comparable to the spread in values contained between the Reuss and Voigt bounds for the polycrystal model. Unfortunately, computed Hashin-Shtrikman bounds are much too tight to be in agreement with the numerical data, showing that polycrystals of cracked grains tend to violate some implicit assumptions of the Hashin-Shtrikman bounding approach. However, the self-consistent estimates obtained for the random polycrystal model are nevertheless very good estimators of the observed average behavior.

A unified approach to predict overall properties of composite materials

This paper presents a unified approach to predict the overall elastic properties of homogeneous and isotropic composite materials from the micromechanics point of view. Within the framework of the approach, the commonly used approaches such as dilute estimate, Voigt estimate (VE), Reuss estimate (RE), modified rule of mixture, self-consistent estimate (SCE), Mori-Tanaka estimate (MTE) and Wakashima-Tsukamoto estimate (WTE) are expressible in the same form solely with different parameters. Particularly, the Voigt bound (VB) and Reuss bound (RB), as well as the Hashin-Shtrikman bounds (HSBs), are easily obtained by properly choosing the homogeneous comparison material. Furthermore, in light of the present approach, tighter than Hashin-Shtrikman bounds are proposed that can also be regarded as two kinds of estimates.

Elastic properties of forsterite–enstatite compositesup to 3.0 GPa

Which rule of mixture is the best for predicting the overall elastic properties ofpolyphase rocks based on the elasticities and volume fractions of their constituents? In order toaddress this question, we sintered forsterite–enstatite polycrystalline aggregates with a variedforsterite volume fraction (0, 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0). Elastic properties of thesesynthesized composites were measured as a function of pressure up to 3.0 GPa in aliquid-medium piston cylinder apparatus using a high precision ultrasonic interferometrictechnique. The experimental data can be much better described by the shear-lag model than bythe commonly used simple models such as Voigt, Reuss and Hill averages, Hashin–Shtrikmanbounds, Ravichandran bounds, Halpin–Tsai equations, and Pauls calculations. We interpret thisas an indication that the elastic interaction and stress transfer between phases are neglected in allmodels except the shear-lag model. In particular, the present study suggests that olivine in themantle may be more abundant than previously inferred from the comparison between seismicEarth model velocities and calculated overall elastic properties of mantle mineral compositesaccording to the Hill average or Hashin–Shtrikman bounds.