Two-scale Homogenization Procedure Research Articles

Homogenization method for microscopic characterization of the composite magnetoelectric multiferroics

Tuning of magnetization or electrical polarization using external fields other than their corresponding conjugate fields (i.e., magnetic field for the former or electric field for the latter response) attracts renewed interest due to its potential for applications. The magnetoelectric effect in multiferroic 1–3 composite composed of alternating magnetic and ferroelectric layers operating in linear regime consequent to external biasing fields is simulated and analysed theoretically. Two-scale homogenization procedure to arrive at the equilibrium overall physical properties of magnetoelectric multiferroic composite is formulated using variational analysis. This procedure is extended to quantify the underlying local (microscopic) electric, magnetic and elastic fields and thereby compute local distribution of stresses and strains, electrical and magnetic potentials, the electric and magnetic fields as well as the equivalent von Mises stresses. The computational model is implemented by modifying the software POSTMAT (material postprocessing). Computed local stress/strain profiles and the von Mises stresses consequent to biasing electrical and magnetic fields provide insightful information related to the magnetostriction and the ensuing electrical and magnetic polarization. Average polarization and magnetization against magnetic and electric fields respectively are computed and found to be in reasonable limits of the experimental results on similar composite systems. The homogenization model covers multiferroics and its composites regardless of crystallographic symmetry (with the caveat of assuming an ideal and semi-coherent interface connecting the constituent phases) and offer computational efficiency besides unveiling the nature of the underlying microscopic field characteristics.

Strain localization in reduced-order asymptotic homogenization

A reduced-order asymptotic homogenization-based multiscale technique that can capture damage and inelastic effects in composite materials is proposed. This technique is based on a two-scale homogenization procedure where eigenstrain representation accounts for the inelastic response and the computational efforts are alleviated by a reduction-of-order technique. Macroscale stress is derived by calculating the influence tensors from analysis of a representative volume element. At microscale, the damage in the material is modeled using a framework based on continuum damage mechanics. To solve the problem of strain localization, a method of alteration of the stress–strain relation of microconstituents based on the dissipated fracture energy in a crack band is implemented. The issue of spurious postfailure artificial stiffness at macroscale is discussed and the effect of increasing the order to alleviate this problem is checked. Verification studies demonstrated that the proposed formulation predicts the macroscale response and also captures the damage- and plasticity-induced inelastic strains.

On models of double porosity poroelastic media

This paper focuses on the modelling of fluid-filled poroelastic double porosity media under quasi-static and dynamic regimes. The double porosity model is derived from a two-scale homogenization procedure, by considering a medium locally characterized by blocks of poroelastic Biot microporous matrix and a surrounding system of fluid-filled macropores or fractures. The derived double porosity description is a two-pressure field poroelastic model with memory and viscoelastic effects. These effects result from the ‘time-dependent’ interaction between the pressure fields in the two pore networks. It is shown that this homogenized double porosity behaviour arises when the characteristic time of consolidation in the microporous domain is of the same order of magnitude as the macroscopic characteristic time of transient regime. Conversely, single porosity behaviours occur when both timescales are clearly distinct. Moreover, it is established that the phenomenological approaches that postulate the coexistence of two pressure fields in ‘instantaneous’ interaction only describe media with two pore networks separated by an interface flow barrier. Hence, they fail at predicting and reproducing the behaviour of usual double porosity media. Finally, the results are illustrated for the case of stratified media.

Numerical homogenization for composite materials analysis. Comparison with other micro mechanical formulations

This work presents a two-scale homogenization procedure to analyze three dimension composite structures by FEM. The theory implemented is compared with other micro-structural formulations: micro models and serial–parallel mixing theory, in terms of result accuracy and computational cost. The comparison shows that for linear analysis, the homogenization is an excellent alternative to the other formulations. Its computational cost is substantially lower than the one required by the micro-model and it is able to capture several micro-structural phenomena that it is not automatically recorded by the serial–parallel methodology. It will also be shown that the extension of the proposed theory to the non-linear range stills represents a challenge. The major limitation is its computational cost because it requires to solve the sub scale at each gauss point and load step. However, the comparison shows that this cost is in terms of CPU time but not in terms of memory. Based on the results obtained, it can be concluded that the homogenization is an excellent alternative for simulation of materials with complex micro structures. The method is also very promising for non linear simulations, when coupled with a threshold criteria to decide whether it is necessary to analyze the RVE or not.

Determination of the effective conductive properties of composites with curved oscillating interfaces by a two-scale homogenization procedure

Rough surfaces and interfaces appear in many situations of practical interest in physics and mechanics of solids. When the interfaces between the constituent phases of composites are rough instead of being smooth as assumed usually, all well-known micromechanical schemes resorting to Eshelby’s tensor are no longer applicable to computing their effective properties. The present work proposes a two-scale homogenization procedure aiming at determining the effective thermal properties of a two-dimensional composite in which the curved interfaces between the constituent phases oscillate periodically and quickly. An asymptotic analysis method is first used to homogenize a rough interface zone as an equivalent interphase, and the effective thermal conductivity tensor of this interphase at the mesoscopic scale is exactly determined. Then, by applying two micromechanical schemes, closed-form expressions for the effective thermal properties of composites at the macroscopic scale are derived. Finally, the analytical results obtained are compared with the relevant bounds and with the corresponding numerical results provided by the finite element method. The two-scale homogenization procedure turns out to be accurate and efficient.

The effective elastic moduli of columnar composites made of cylindrically anisotropic phases with rough interfaces

The present work aims to determine the effective elastic moduli of a composite having a columnar microstructure and made of two cylindrically anisotropic phases perfectly bonded at their interface oscillating quickly and periodically along the circular circumferential direction. To achieve this objective, a two-scale homogenization method is elaborated. First, the micro-to-meso upscaling is carried out by applying an asymptotic analysis, and the zone in which the interface oscillates is correspondingly homogenized as an equivalent interphase whose elastic properties are analytically and exactly determined. Second, the meso-to-macro upscaling is accomplished by using the composite cylinder assemblage model, and closed-form solutions are derived for the effective elastic moduli of the composite. Two important cases in which rough interfaces exhibit comb and saw-tooth profiles are studied in detail. The analytical results given by the two-scale homogenization procedure are shown to agree well with the numerical ones provided by the finite element method and to verify the universal relations existing between the effective elastic moduli of a two-phase columnar composite.

Appearance of the nonlinearity from the nonlocality in diffusion through multiscale fractured porous media

We shall consider diffusion or single-phase flow in a multiscale porous medium which represents an infinite set of self-similar double-porosity media. At each scale, the medium consists of a highly permeable network of connected channels and low-permeable blocks. The characteristic scale of heterogeneity is e at the highest level of hierarchy, wherein e is a small parameter. The ratio between the channel and block permeability at each scale is e 2. The process analyzed is described using a diffusion equation with an oscillating multiscale diffusion parameter. The macroscale behavior is of interest. The transition to the macroscale is performed by means of the two-scale homogenization procedure. One step of averaging at each level of hierarchy leads to the appearance of the memory terms in the averaged equation. The successive averaging steps lead to progressive memory accumulation, so at each step of averaging, the macroscale model changes its type, and even the result of the second step is unknown a priori. The objective was to determine the macroscopic limit model for the infinite number of scales. By the method of induction, we obtained the macroscale model for an arbitrary number of scales and its limit for the infinite hierarchy. The limit model represents the system of two equations with memory terms. The kernel of the memory operator is the solution of a nonlinear integro-differential equation. Its solution is obtained through Laplace transform.

Adaptive Two-Scale Nonlinear Homogenization

An adaptive two-scale homogenization procedure combining a reduced order and a direct mathematical homogenization method is developed. A simple strain-based indicator is used to guide the construction of a nearly optimal model consisting of either a direct or a reduced order unit cell model. Consistent interscale transfer operators are developed to map the information between the two homogenization models. Verification and validation studies are conducted on a composites plate with a center crack, metal-polyurea sandwich plate, and fabric composites circular tube.

Derivation of the localization and homogenization conditions for electro-mechanically coupled problems

The challenges in high-performance mechatronics and micromechatronics are the modeling, analysis and design of electro-mechanical systems. Piezoceramics are the prototype of materials which exhibit characteristic electro-mechanical coupling effects used in a wide range of industrial applications. In complex designs of piezoceramic devices concentrations of stresses and electric field occur, which may lead to nonlinear electro-mechanical responses, typical for the large signal range. The challenge in the field of ferroelectric ceramics is the modeling of the complicated interactions between electrical and mechanical quantities on the macroscale, which are caused by switching processes on the microscale. In this paper we derive the basic equations for a possible direct two-scale homogenization procedure of electro-mechanically coupled boundary value problems.

Multiscale modeling of oriented thermoplastic elastomers with lamellar morphology

Thermoplastic elastomers (TPEs) are block copolymers made up of “hard” (glassy or crystalline) and “soft” (rubbery) blocks that self-organize into “domain” structures at a length scale of a few tens of nanometers. Under typical processing conditions, TPEs also develop a “polydomain” structure at the micron level that is similar to that of metal polycrystals. Therefore, from a continuum point of view, TPEs may be regarded as materials with heterogeneities at two different length scales. In this work, we propose a constitutive model for highly oriented, near-single-crystal TPEs with lamellar domain morphology. Based on small-angle X-ray scattering (SAXS) and transmission electron microscopy (TEM) observations, we consider such materials to have a granular microstructure where the grains are made up of the same, perfect, lamellar structure (single crystal) with slightly different lamination directions (crystal orientations). Having identified the underlying morphology, the overall finite-deformation response of these materials is determined by means of a two-scale homogenization procedure. Interestingly, the model predictions indicate that the evolution of microstructure—especially the rotation of the layers—has a very significant, but subtle effect on the overall properties of near-single-crystal TPEs. In particular, for certain loading conditions—namely, for those with sufficiently large compressive deformations applied in the direction of the lamellae within the individual grains—the model becomes macroscopically unstable (i.e., it loses strong ellipticity). By keeping track of the evolution of the underlying microstructure, we find that such instabilities can be related to the development of “chevron” patterns.