Hashin-Shtrikman Type Research Articles

The variational principle for probabilistic measure and Hashin–Shtrikman bounds

The paper is a review of Hashin–Shtrikman type bounds for effective moduli of conductivity and elasticity of polycrystals and composites written from the perspective of the variational principle for probabilistic measure. The results for such bounds are rederived in probabilistic terms. Remarkably, in probabilistic terms the Hashin–Shtrikman approach gets especially simple form. Besides, a clear distinction arises between the basic assumption, the choice of the trial field, and the simplifying assumptions, like geometrical isotropy, physical isotropy, texture isotropy, etc. We filled out several gaps. First, we derive an integral equation to be solved to get the bounds when the simplifying assumptions do not hold. Second, we extend the bounds for polycrystals with the cubic symmetry of crystallites to all thermodynamically possible crystallites; previously such bounds were found for crystallites with special elastic properties. One practical outcome considered is the derivation of approximate formulae for the temperature dependence of effective elastic moduli. Third, for crystallites with non-cubic symmetries, we formulated algebraic variational problems to be solved numerically to obtain the bounds, and solved these problems for several materials.

On the optimality of the variational linear comparison bounds for porous viscoplastic materials

This paper is concerned with the optimality of the variational bounds of the Hashin-Shtrikman type (VHS) for nonlinear composites, first obtained by Ponte Castañeda (1991a) by means of the corresponding HS bounds for suitably optimized linear comparison composites (LCCs). For simplicity, this problem is addressed in the context of porous viscoplastic materials with incompressible, isotropic matrix phase and two-dimensional microstructures, subjected to plane-strain loading conditions. Although special, this case—exhibiting infinite heterogeneity contrast and compressible macroscopic response—is expected to be fully representative of more general three-dimensional porous materials, as well as more general two-phase, well-ordered composites. Thus, it is shown that the VHS bounds, which were originally derived for the class of nonlinear composites with statistically isotropic microstructures, are in fact attained over the larger class of microstructures with anisotropic nonlinear response but isotropic linear response. By appealing to an exact variational representation for the effective potential of finite-rank nonlinear laminates, it is shown that there exist certain values of the applied macroscopic stress for which the finite-rank laminate (closed-cell porous) microstructures attaining the linear HS bounds also attain the nonlinear VHS bound. Explicit results are obtained in the ideally plastic limit for the yield surface of the finite-rank laminates attaining the VHS bound. In particular, the results of the paper highlight the fact that bounds for nonlinear composites are much more sensitive to microstructural details than bounds for linear composites.

Efficient two–scale simulations of engineering structures using the Hashin–Shtrikman type finite element method

An efficient FE2-like numerical homogenization approach for nonlinear microstructures is proposed using the Hashin–Shtrikman type finite element (HSFE) method to solve the microscale problem. The latter combines the small computational effort of Hashin–Shtrikman type homogenization approaches with the accuracy of full-field FE-solutions. The key point is a reduced order method based on a Hashin–Shtrikman type variational formulation combined with data-clustering, which is based on offline FE-simulations of microstructures (snapshots). The presented microscopic model has significantly less microscopic degrees of freedom in comparison to the classic FE2-method. The number of stress computations within the microstructure is highly reduced. The tangent operator, which incorporates the coupling between the microscopic and macroscopic scale, is derived analytically. Different numerical examples are investigated, where a nonlinear RVE is attached to each integration point of a macrostructure. A comparison to full-field FE-simulations shows that the macro-response and local fields are well captured by the HSFE method.

Transmission and reflection of waves at an interface between ordinary material and metamaterial

The propagation of waves though bulk composite material (or metamaterial) is to a large extent understood, but the problem of how waves are transmitted and reflected at any interface between two different composites has hardly been considered, particularly if the composites have random microgeometry. The present work makes a contribution towards the solution of this problem by studying a configuration of particular simplicity. A randomly inhomogeneous composite which occupies the half-space x2 > 0 is bonded at x2=0 to a half-space x2 < 0 of a homogeneous medium, and an acoustic wave is incident on the interface from below. The composite has two constituents which have the same elastic modulus but different densities. A variational approximation of Hashin–Shtrikman type yields an explicit formula for the Fourier/Laplace transform of its effective density and a corresponding dispersion relation for the plane mean waves that the composite can support. When its two-point correlation is chosen as h(r)=e−r/a, the dispersion relation admits two distinct mean acoustic waves, and the only directly available conditions at the interface x2=0 are continuity of displacement and traction, which are insufficient in number to fix the two transmission coefficients and one reflection coefficient. It is possible, however, to apply the variational approach directly to an infinite medium comprising the composite in x2 > 0 and the homogeneous medium in x2 < 0, leading to a problem of Wiener–Hopf type. When the homogeneous half-space has the same properties as one of the constituents of the composite, the Wiener–Hopf problem involves only a single unknown “polarization” supported in x2 > 0 and is solved explicitly.

An efficient reduced computational method for nonlinear homogenization problems: the Hashin–Shtrikman type Finite Element method (HSFE)

AbstractThis contribution is dedicated to a numerically efficient homogenization method for nonlinear microstructures (HSFE), which combines the small computational effort of the Hashin‐Shtrikman type approach with the accuracy of full‐field finite element solutions. The key point of the method is a model order reduction scheme based on data‐clustering, which leads to a significant reduction of computational time. The derivation of the analytical tangent operator realises the coupling between the microscopic and macroscopic scale. The presented approach is tested in the context of small strains by a numerical example, where a nonlinear RVE is attached to each integration point of a 2D macrostructure. A comparison to the FE‐simulations shows that the global‐ and local‐responses are well captured by the HSFE method.

Two-scale simulation of the hot stamping process based on a Hashin–Shtrikman type mean field model

The hot stamping process of sheet metals into a w-shape profile is considered in experiments and numerical simulation using the FEM tool ABAQUS. The influence of the processing parameters of the hot stamping process on the final shape and the residual stress state are investigated. Tensile tests of thermomechanically treated specimens serve as a starting point for the modeling of the thermomechanical behavior of the steel grade 22MnB5. Based on the thermo-micromechanical (TMM) model for small deformations suggested by Neumann and Böhlke (2016), the updated Lagrangian formulation is used to render the finite deformation process.The model captures in a two-scale framework the temperature dependent elasto-plastic effect of the different phases, the phase transformation effect, and the phase transformation accompanying transformation strain occurring during the quenching of the austenitized sheet metal. Additionally, a model for the transformation induced plasticity (TRIP) effect in the context of the two-scale approach is suggested. The homogenization and localization of the thermomechanical properties is performed with a HashinöShtrikman (HS) type homogenization scheme. In this work, a reduced formulation for isotropic elastic behavior of the different phases is found reducing the amount of numerical costs. Furthermore, a self-consistent scheme based on the HS type method is introduced.Finally, the suggested TMM model is validated by considering the time-temperature-transformation- (TTT) and continuous-cooling-time-curves (CCT), dilatation tests under external load, and the hot stamping of w-shape profiles. The TMM model is compared to both experimental results and a phenomenological reference model usually used for the simulation of hot stamping processes.It is found that the suggested TMM model fits the final shape and the residual stress state of hot stamped parts better than the phenomenological reference model.

Effective field hypothesis in Hashin–Shtrikman bounds estimations for effective moduli of composites with noncanonical shape of inclusions

One consider linear elastic composite materials (CMs), which consist of a homogeneous matrix containing a statistically homogeneous random set of identical aligned inhomogeneous inclusions of non-canonical (i.e. nonellipsoidal) shape. Estimates of the Hashin-Shtrikman type are developed by extremizing of the classical variational functional involving a general integral equation (GIE) connecting the strain fields in the point being considered with the strain fields in the surrounding points. An improvement of the approach proposed is based on the use of a piece wise constant approximation of the trial effective field rather than the trial polarisation tensor that, in its turn, allows one to take into account any inhomogeneity of the strain distributions inside the inclusions. Due to a sensitivity of the proposed approach to the inhomogeneity of these strain distributions, a final Hashin-Shtrikman bounds representation demonstrates more significant dependence on the shape of noncanonical inclusions than the corresponding classical bounds (that is shown by the concrete numerical examples).

Model order reduction of nonlinear homogenization problems using a Hashin–Shtrikman type finite element method

This work presents a computational nonlinear homogenization approach, the starting point of which is a model order reduction method based on data-clustering. To this end, the micromechanical data from numerical experiments (snapshots) is analyzed in order to identify characteristic microstructural deformation patterns. These describe how the macroscopic strain typically localizes within the microstructure. The outcome of the procedure is a subdivision of the microstructure into a set of clusters of material points. Within each cluster the strain is then approximated as being constant.The mechanical problem is formulated in terms of a three-field Hashin–Shtrikman type variational formulation which is based on the introduction of a linear-elastic reference medium. After discretization, most of the global unknowns can be eliminated via static condensation leaving the piecewise constant cluster strains as the primary unknowns. The resulting homogenization scheme includes, as special cases, the finite element method as well as Hashin–Shtrikman and Talbot–Willis type homogenization approaches with phase-wise constant trial fields (as well as related bounds). The limit case ’finite element method’ allows to transfer knowledge from finite element technology and thus provides new strategies for the choice of the stiffness of the reference material. The method is applied to several nonlinear microstructures with different inclusion volume fractions and varying degree of anisotropy. The results are shown to be in good agreement with full-field FE-simulations. Furthermore, the method is used to compute a refined upper bound of the Talbot–Willis type (compared to phase-wise constant trial fields), which converges to the finite element solution with increasingly refined discretization.

Bounds for nonlinear composite conductors via the translation method

Hashin–Shtrikman type bounds are proposed for nonlinear isotropic composite conductors in two dimensions. Those bounds are obtained by combining the translation method with the idea of embedding the original two-dimensional problem in an extended problem of dimension 6. Invariance properties allow the evaluation of the bounds to be dramatically simplified. Explicit results are obtained for the problem of dielectric breakdown. Numerical results are given for two-phase composites governed by power-law energy functions. The obtained bounds are shown to improve on the linear comparison bounds of the Hashin–Shtrikman type that are delivered by the Talbot-Willis (1985) approach and the Ponte Castañeda (1991) variational method.

Mechanism based mean-field modeling of the work-hardening behavior of dual-phase steels

Low-alloyed dual-phase (DP) steels exhibit good mechanical properties due to their composite-like microstructure of strong martensitic inclusions embedded in a ductile ferritic-matrix. This work presents an efficient two-scale approach to model the work-hardening of DP steels based on physically-motivated work-hardening contributions from geometrically necessary dislocations. The resulting mean-field model of Hashin-Shtrikman type comprises a minimal amount of free parameters and incorporates two main physical aspects motivated from presented experimental data: First, the constitutive equations for ferrite incorporate the averaged microstructural morphology (obtained by electronic backscatter diffraction) in terms of grain size and martensite coverage. Second, a direct interaction between ferrite and martensite phases is achieved via kinematic-hardening which models the long-range stresses experimentally observed in tensile tests of the bulk material. The presented approach is able to both reproduce the measured DP600 tensile curve and predict the distinctly different work-hardening rates observed for high-strength DP steels.

Estimates for the overall linear properties of pointwise heterogeneous solids with application to elasto-viscoplasticity

New estimates are derived for the overall properties of linear solids with pointwise heterogeneous local properties. The derivation relies on the use of ‘comparison solids’ which, unlike comparison solids considered previously, are themselves pointwise heterogeneous. The estimates are then exploited within an incremental homogenization scheme to determine the overall response of multiphase elasto-viscoplastic solids under arbitrary loading histories. By way of example, the scheme is applied to incompressible Maxwellian solids with power-law plastic dissipation; particularly simple estimates of the Hashin–Shtrikman type are obtained. Predictions are confronted with full-field simulations for particulate composites under cyclic and rotating loading conditions. Good agreement is found for all cases considered. In particular, elasto-plastic transitions, tension-compression asymmetries (Bauschinger effect) and stress-path distortions induced by material heterogeneity are all well-captured, thus improving significantly on commonly used elastic-plastic decoupled schemes.

Hashin–Shtrikman type mean field model for the two-scale simulation of the thermomechanical processing of steel

Thermomechanical treatment of steel plays an important role in production, e.g., hot stamping. To capture the complex polymorphic material behavior of steel under thermal and mechanical loads, the usual approach in literature is to use a macroscopic phenomenological ansatz. In contrast, the aim of this work is to establish a more physically based two-scale model taking the material behavior of the different phases on the micro scale into account. The scale transition from macro scale to micro scale is performed by a nonlinear localization method of Hashin–Shtrikman type, which is based on a phase-wise constant stress polarization. The thermomechanical constitutive law is determined by both the phase transformation on micro scale and the temperature-dependent mechanical behavior of each phase. The mechanical behavior is based on a thermodynamical approach, i.e. introduction of a Helmholtz free energy for each phase and use of potential relations gained from the Clausius–Duhem (CD) inequality. The temperature-affected diffusionless evolution of the microstructure is modeled by both a Koistinen–Marburger (KM) consistent rate law and a suggested nonlinear extension of the KM model. The suggested transformation model leads to a better agreement with experimental results compared to the usually used approaches. The Johnson–Mehl–Avrami–Kolmogorov (JMAK) model is chosen to describe the diffusion driven phase transformation. After a parameter identification of the steel 42CrMo4 and a validation of the thermomechanical coupling, the influence of the latent heat, the transformation strain occurring during phase transformation, and the additional parameter arising in the homogenization scheme on the macroscopic behavior is investigated.

Microstructure based prediction and homogenization of the strain hardening behavior of dual-phase steel

The mechanical behavior of automotive dual-phase steel (DP) is modeled by two different approaches: with a full-field representative volume element (RVE) and with a mean-field model. In the first part of this work, the full-field RVE is constituted by a crystal plasticity-based ferrite matrix with von Mises-type martensite inclusions. To isolate the martensite influence, the full-field DP results were compared to a full-field comparison RVE. In the comparison RVE, all martensite inclusions were replaced by a phase that exhibits the average ferrite behavior. A higher relative martensite grain boundary coverage facilitates an increased average dislocation density after quenching. However, for uniaxial deformations above ∼10%, the grain size-dependent relation reverses and exhibits slowed-down hardening. In the second part, we incorporate the main findings from the full-field simulations into a nonlinear mean-field model of Hashin–Shtrikman type. The dislocation density production parameter and the saturated dislocation density are modeled based on grain size and martensite coverage. The comparison of both approaches shows good agreement for both the overall and constituent averaged behavior.

Variational formulation of the equivalent eigenstrain method with an application to a problem with radial eigenstrains

In this paper a variational formulation of the equivalent eigenstrain method is established. A functional of the Hashin–Shtrikman type is proposed such that the solution of the equivalent eigenstrain equation is a unique minimizer of the functional. Moreover, it is also shown that the equivalent eigenstrain equation is the Euler–Lagrange equation of the potential energy of the inclusions. An approximate solution of the equivalent eigenstrain equation is then found as a minimizer of the functional on a finite dimensional span of basic eigenstrains. Special attention is paid to possible symmetries of the problem. The variational formulation is illustrated by determination of effective linear elastic properties. In particular, material with a simple cubic microstructure is considered in detail. A solution for the polynomial radial basic eigenstrains approximation is found. In particular, for the homogeneous eigenstrain approximation, the effective moduli are derived in an exact closed form.

Nonlinear Homogenization of Microstructures in Steel with Temperature‐Controlled Phase Transformation

AbstractFor the description of the phase transformation effect TRIP (transformation induced plasticity), a thermo‐micro‐mechanical model is introduced. The scale transition is realized by a Hashin‐Shtrikman type homogenization scheme. The macroscopical behavior is purely based on micro‐mechanical, thermo‐mechanical constitutive models. The model takes into account the thermal strains arising during cooling, and the Greenwood‐Johnson effect. The Greenwood‐Johnson effect describes the TRIP strain developing during the growth of a product phase in the parent phase. The TRIP strain is determined by a micromechanical model gained by an extended Leblond ansatz. For the diffusion driven phase transformation, the JMAK model is incorporated into the homogenization scheme. The diffusionless transformation is described by a rate law of Koistinen‐Marburger type. (© 2013 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Macroscopic diffusivity in concrete determined by computational homogenization

SUMMARYEffective moisture and chloride ion diffusivity coefficients for concrete are determined by computational homogenization, where concrete is modeled on the mesoscale as a heterogenous three‐phase composite material. By imposing moisture and chloride ion gradients on a representative volume element, effective macroscale properties are obtained through finite element analysis. A parametric study of the effects of the ballast content was carried out. The numerical results correspond well with an estimate of the Hashin–Shtrikman type available in the literature. The computational homogenization strategy proposed here also includes the interfacial transition zone, and its influence on the effective diffusivity coefficients is assessed. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Bounds on strong field magneto-transport in three-dimensional composites

This paper deals with bounds satisfied by the effective non-symmetric conductivity of three-dimensional composites in the presence of a strong magnetic field. On the one hand, it is shown that for general composites the antisymmetric part of the effective conductivity cannot be bounded solely in terms of the antisymmetric part of the local conductivity, contrary to the columnar case studied by Briane and Milton [SIAM J. Appl. Math. 70(8), 3272–3286 (2010)10.1137/100798090]. Thus a suitable rank-two laminate, the conductivity of which has a bounded antisymmetric part together with a high-contrast symmetric part, may generate an arbitrarily large antisymmetric part of the effective conductivity. On the other hand, bounds are provided which show that the antisymmetric part of the effective conductivity must go to zero if the upper bound on the antisymmetric part of the local conductivity goes to zero, and the symmetric part of the local conductivity remains bounded below and above. Elementary bounds on the effective moduli are derived assuming the local conductivity and the effective conductivity have transverse isotropy in the plane orthogonal to the magnetic field. New Hashin-Shtrikman type bounds for two-phase three-dimensional composites with a non-symmetric conductivity are provided under geometric isotropy of the microstructure. The derivation of the bounds is based on a particular variational principle symmetrizing the problem, and the use of Y-tensors involving the averages of the fields in each phase.

Minimum variational principles for time-harmonic waves in a dissipative medium and associated variational principles of Hashin–Shtrikman type

Minimization variational principles for linear elastodynamic, acoustic or electromagnetic time-harmonic waves in dissipative media were obtained by Milton et al. (Milton et al. 2009 Proc. R. Soc. A 465 , 367–396 ( doi:10.1098/rspa.2008.0195 )), generalizing the quasistatic variational principles of Cherkaev and Gibiansky (Cherkaev &amp; Gibiansky 1994 J. Math. Phys. 35 , 127–145 ( doi:10.1063/1.530782 )). Here, a further generalization is made to allow for a much wider variety of boundary conditions, and in particular Dirichlet and Neumann boundary conditions. In addition minimization or maximization principles of the Hashin–Shtrikman type, incorporating ‘polarization fields’, are developed. The corresponding principles for static problems have found substantial use in bounding the effective static properties of composite materials. The new dynamical principles offer the prospect of developing bounds on the effective dynamic properties of such materials.

A non-convex lower bound on the effective energy of polycrystalline shape memory alloys

This paper addresses the estimation of the effective free energy of polycrystalline shape memory alloys, in the framework of nonlinear elasticity and infinitesimal strains. The translation method is combined with a Hashin–Shtrikman type variational formulation to provide rigorous lower bounds on the effective free energy. Those bounds incorporate both intra-grain compatibility conditions (resulting in a non-convex bound) and inter-grain constraints (by taking one- and two-point statistics into account). Some examples are given to compare the results obtained with other bounds from the literature.

Second-order theory for nonlinear composites and application to isotropic constituents

New prescriptions are proposed for the ‘reference’ fields in the context of the ‘second-order’ nonlinear homogenization method [P. Ponte Castañeda, Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—Theory, J. Mech. Phys. Solids 50 (2002) 737–757], and are used to generate estimates for the effective behavior and first moments of the local fields in nonlinear composites. The new prescriptions yield simple, analytical expressions not only for the effective potentials, but also for the macroscopic stress-strain relation, as well as for the phase averages of the strain and stress fields. For illustrative purposes, ‘second-order’ estimates of the Hashin–Shtrikman type are provided for two-phase, transversely-isotropic composites with power-law phases, and are compared with exact results available for power-law, multiple-rank, sequential laminates. The agreement is found to be quite good for all ranges of nonlinearities and inclusion concentrations considered. To cite this article: M.I. Idiart et al., C. R. Mecanique 334 (2006).