Hashin-Shtrikman Principles Research Articles

Hydrothermal behavior of pure PA 6 and homogenization of discontinuous long carbon fiber‐reinforced PA 6

AbstractPolymer‐based composites play a crucial role in lightweighting applications by providing resource‐efficient semi‐structural materials. Among these, polyamides, in particular polyamide 6 (PA 6), serve as widely used matrix materials in various technical applications, especially in fiber‐reinforced composites. The mechanical properties and performance of PA 6 are significantly affected by environmental factors such as temperature and humidity. Previous research has investigated the influence of hydrothermal conditions on the mechanical properties of pure PA 6. This study aims to extend the preceding investigations with additional experimental tests. As a preliminary work to later investigate the influence of the fibers on the hydrothermal behavior, discontinuous long carbon fiber‐reinforced PA 6 is considered for micromechanical modeling of the effective behavior. In this context, a mean‐field homogenization approach based on the Hashin–Shtrikman principle is used. By means of a variable reference stiffness, an approach specifically tailored to the material system under consideration can be applied. The simulation data are compared with data from experiments under tension load indicating good agreements for a specific choice of the reference material.

Dynamic homogenization of heterogeneous piezoelectric media: A polarization approach using infinite-body Green’s function

Dynamic homogenization theories are powerful tools for describing and understanding the behavior of heterogeneous media such as composites and metamaterials. However, a major challenge in the homogenization theory is determining Green’s function of these media, which makes it difficult to predict their effective constitutive relations, particularly for the finite-size and/or non-periodic media in real-world applications. In this paper, we present a formulation for finding the elastodynamic effective constitutive relations for general heterogeneous media, including finite-size and non-periodic ones, via a polarization approach based on the Hashin–Shtrikman principle along with Green’s identities. Our proposed formulation relies on the infinite-body Green’s function of a homogeneous reference medium, making it free from the difficulty of determining Green’s function even for the homogenization of finite-size and/or non-periodic media. Additionally, we demonstrate the universal applicability of this formulation for both random and deterministic heterogeneous media. This work contributes to a better understanding of the homogenization theory and the design of next-generation metamaterials that require the accurate prediction of effective material characteristics for dynamic wave manipulation under desired operating environments.

Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites

In this paper, we develop a computational approach based on variational principles combined with FFT technique to determine the effective elastic properties of composite materials. The unit cell problem of the composite is recast in a weak and integral form by making use of classic variational approaches, based on the strain and the stress potential, and the Hashin–Shtrikman variational principles. The problem is discretized with Fourier series and the stationary point is computed numerically by means of an iterative scheme. These algorithms use a representation of the local elastic tensor on the double grid (twice the size of the discretization) which is introduced by the integral formulation; it advantageously accounts for the exact geometry of the microstructure and allows the derivation of rigorous bounds of the effective elasticity coefficients. In the second part of the paper, we establish a hierarchy between the different FFT solutions: the strain, the stress based solutions coming from the classic variational formulations and the polarization based solution derived from the Hashin–Shtrikman principle, which depends on the choice of the elastic moduli of the reference material. It is proved that the strain and the stress based solutions deliver optimal bounds since they are always better than those obtained from the Hashin–Shtrikman variational principle when the same discretization is used. Alternatively, when the elastic moduli of the reference material is negative, the polarization based solution provides an estimate of the effective properties which is comprised between the strain based upper bound and the stress based lower bound.

Extended Hashin-Shtrikman Variational Principles

Internal parameters, eigenstrains, or eigenstresses, arise in functionally graded materials, which are typically present in particulate, layered, or rock bodies. These parameters may be realized in different ways, e.g., by prestressing, temperature changes, effects of wetting, swelling, they may also represent inelastic strains, etc. In order to clarify the use of eigenparameters (eigenstrains or eigenstresses) in physical description, the classical formulation of elasticity is presented, and the two most important Lagrange's and Castigliano's variational principles are formulated in the sequel. Then the classical Hashin-Shtrikman principles are recalled and the involvement of eigenparameters is studied in more detail.

Non-local constitutive response of a random laminate subjected to configuration-dependent body force

The non-local constitutive behaviour of an infinite composite laminate is analyzed. The laminate is treated as random and is subjected to a combination of deterministic and configuration-dependent body force. In this case, in addition to the effective non-local elastic operator, other non-local constitutive operators must be considered in order to define the mean response of the body. For a laminate subjected to forces that vary only in the direction of lamination, these operators are obtained explicitly. The Hashin-Shtrikman principles developed by Luciano and Willis (Luciano, R., Willis, J.R., 2001a. Bounds on non-local effective relations for random composites loaded by configuration-dependent body force, to appear in J. Mech. Phys. Solids), which provide bounds for the operators for general composites, are shown to generate exactly the two operators that define the stress, while giving only bounds for the remaining operator that appears in the expression for the total energy. The case of a two-phase laminate with the layers arranged periodically is presented as an example.

Generalized Hashin–Shtrikman variational principle for boundary-value problem of linear and non-linear heterogeneous body

When the in-situ measurement of the effective properties is difficult, a ground or a crust of large dimensions is modeled as a body with probabilistically varying materials. As the variance of heterogeneity is large, conventional analysis methods require enormous numerical computation. As an alternative, this paper proposes the generalized Hashin–Shtrikman variational principle which provides upper and lower bounds for the expectation of the behavior of such a probabilistically varying body. The bounds are obtained by analyzing two fictitious bodies which are rigorously defined when probabilistic distributions of material properties are given. The generalized Hashin–Shtrikman principle can be applied to non-linear initial boundary-value problems. The fault formation process in surface ground layers is solved as an illustrative example. The surface layers are modeled as a probabilistically varying elasto-plastic body, and it is shown that the upper and lower bounds for the expectation actually bound the average behavior which is computed by the Monte-Carlo simulation. Discussions are made on these numerical results.

On methods for bounding the overall properties of periodic piezoelectric fibrous composites

The homogenization problem of linearly piezoelectric fibrous composites with a periodic microstructure is formulated by adopting the free- and complementary-energy minimum principles, as well as the Hashin Shtrikman principles. The auxiliary electroelastic problems involved in the Hashin-Shtrikman principles are solved by a transformation from the space domain to the Fourier domain. Approximations of those variational principles, suitable for numerical computations, are built up by adopting the Fourier-series method and the finite-element method. In this way, rigorous upper and lower bounds on the overall material moduli are obtained. An example is presented, in order to show the effectiveness of the proposed approximation techniques and to compare them.

Variational bounds for the overall properties of piezoelectric composites

In this paper, variational bounds for the overall properties of periodic heterogeneous media with nonlinear and nonlocal piezoelectric constitutive relationships are obtained. First, elementary bounds are developed by extending to piezoelectric materials the well-known Voigt and Reuss bounds. Then, by generalizing the two Hashin-Shtrikman principles, eight new variational principles are derived and applied to obtain bounds. In fact, the variational principles developed are based on auxiliary electroelastic equilibrium problems, which can be solved by a transformation from the space domain to the Fourier domain. As an application, fibre-reinforced linear piezoelectric composites are considered, and expressions of upper and lower bounds for the overall properties of these composites are developed in closed form.

Upper and lower bounds for the overall properties of a nonlinear composite dielectric. II. Periodic microgeometry

The study of the effective properties of a nonlinear composite dielectric begun in the companion to this paper is continued in the context of a simple cubic array of inclusions embedded in a matrix. Use of a nonlinear comparison material in the nonlinear Hashin-Shtrikman variational principles permits the generation of both upper and lower bounds for any composite. It is straightforward to apply the new approach to the simple composite considered here as the trial electric field which is substituted into the Hashin-Shtrikman variational principle is finite everywhere. The results obtained indicate the likely improvement over the simple classical bounds that can be obtained using the Hashin-Shtrikman principles.

Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions

This paper is concerned with two-dimensional, linearly elastic, composite materials made by mixing two isotropic components. For given volume fractions and average strain, we establish explicit optimal upper and lower bounds on the effective energy quadratic form. There are two different approaches to this problem, one based on the “Hashin-Shtrikman variational principle” and the other on the “translation method". We implement both. The Hashin-Shtrikman principle applies only when the component materials are “well-ordered", i.e., when the smaller shear and bulk moduli belong to the same material. The translation method, however, requires no such hypothesis. As a consequence, our optimal bounds are valid even when the component materials are not well-ordered. Analogous results have previously been obtained by Gibianski and Cherkaev in the context of the plate equation.

Variational principles for dynamic problems for inhomogeneous elastic media

Some new variational principles for elastodynamic problems, which reduce to the Hashin-Shtrikman principle in the static limit, are presented. They are deduced from dynamical analogues of the classical energy principles of elastostatics by performing appropriate rearrangements and then neglecting certain terms that are quadratic in a measure of the error associated with the approximating trial fields. One of the new principles is then employed to develop equations that provide an approximate description of waves in a random composite. These equations make optimal use of the pair correlations of the composite, in the sense that better approximation (relative to the variational principle) would unavoidably involve correlations of higher order.