Muneo Hori

Simulation of the multi-scale convergence in computational homogenization approaches

Although the asymptotic homogenization is known to explicitly predict the thermo-mechanical behaviors of an overall structure as well as the microstructures, the current developments in engineering fields introduce some kinds of approximation about the microstructural geometry. In order for the homogenization method for periodic media to apply for general heterogeneous ones, the problems arising from mathematical modeling are examined in the framework of representative volume element (RVE) analyses. Here, the notion of homogenization convergence allows us to eliminate the geometrical periodicity requirement when the size of RVE is sufficiently large. Then the numerical studies in this paper realize the multi-scale nature of the convergence of overall material properties as the unit cell size is increased. In addition to such dependency of the macroscopic field variables on the selected size of unit cells, the convergence nature of microscopic stress values is also studied quantitatively via the computational homogenization method. Similar discussions are made for the elastoplastic mechanical responses in both macro- and microscopic levels. In these multi-scale numerical analyses, the specific effects of the microstructural morphology are reflected by using the digital image-based (DIB) finite element (FE) modeling technique which enables the construction of accurate microstructural models.

On two micromechanics theories for determining micro–macro relations in heterogeneous solids

The average-field theory and the homogenization theory are briefly reviewed and compared. These theories are often used to determine the effective moduli of heterogeneous materials from their microscopic structure in such a manner that boundary-value problems for the macroscopic response can be formulated. While these two theories are based on different modeling concepts, it is shown that they can yield essentially the same effective moduli and boundary-value problems. A hybrid micromechanics theory is proposed in view of this correspondence. This theory leads to a more accurate computation of the effective moduli, and applies to a broader class of microstructural models. Hence, the resulting macroscopic boundary-value problem gives better estimates of the macroscopic response of the material. In particular, the hybrid theory can account for the effects of the macrostrain gradient on the macrostress in a natural manner.

Bounds and estimates of overall moduli of composites with periodic microstructure

Abstract Bounds on the overall moduli of a broad class of composites with periodic microstructure are obtained by generalized Hashin—Shtrikman variational principles. Piecewise constant eigenstrains and eigenstresses are used, and exact, computable bounds are developed. The formulation is valid for composites comprised of an anisotropic (or isotropic) matrix with an arbitrary number of periodically distributed anisotropic (or isotropic) inhomogeneities. Examples of two-phase particulate, whisker, and fiber-reinforced composites are considered for illustration. Finally, an estimate of the overall moduli, based on the selection of the effective medium as the reference material, is proposed for periodic microstructure.

Solids with periodically distributed cracks

Abstract A systematic method is presented for estimating the overall properties of solids with periodically distributed cracks. In view of the periodicity, the displacement, strain and stress fields of the cracked solid can be expressed in Fourier series. Elastic solids with periodically distributed flat voids are considered first. The results for cracks are then obtained by letting the thickness aspect ratio of the void approach zero. This limiting process is performed with care. The only approximation involved is the distribution of the homogenization eigenstrains, which is assumed to be piecewise constant. The estimate of overall elastic moduli, crack opening displacements and stress intensity factors eventually reduces to the calculation of several infinite series. The formulation is valid for elliptic as well as two-dimensional line (slit-like) cracks and cracks with arbitrary shapes. It fully includes the interaction effects.

Appropriate number of unit cells in a representative volume element for micro-structural bifurcation encountered in a multi-scale modeling

The paper proposes a method to determine the number of unit cells (basic structural elements) to be employed for a representative volume element (RVE) of the multi-scale modeling for a solid with periodic micro-structures undergoing bifurcation. Main difficulties for the multi-scale modeling implementing instability are twofold: loss of convexity of the total potential energy that should be homogenized and determination of a pertinent RVE that contains multiple unit cells. In order to resolve these difficulties, variational formulation is achieved with the help of Γ-convergence theory within the framework of non-convex homogenization method, while the number of unit cells in an RVE is determined by the block-diagonalization method of group-theoretic bifurcation theory. The latter method enables us to identify the most critical bifurcation mode among possible bifurcation patterns for an assembly of arbitrary number of periodic micro-structures. Thus, the appropriate number of unit cells to be employed in the RVE can be determined in a systematic manner. Representative numerical examples for a cellular solid show the feasibility of the proposed method and illustrate material instability at a macroscopic point due to geometrical instability in a micro-scale.

Double-Inclusion Model and Overall Moduli of Multi-Phase Composites

The double inclusion model consists of an ellipsoidal inclusion of arbitrary elasticity, containing another ellipsoidal heterogeneity of arbitrary elasticity, size, and orientation, which are embedded in an infinitely extended homogenous domain of yet another arbitrary elasticity. Average field quantities for the double inclusion are obtained analytically, and used to estimate the overall moduli of two phase composites. The technique includes the self consistent and other related methods as special cases. Furthermore, exact bounds for the overall moduli are obtained on the basis of the double inclusion model

Physics-based urban earthquake simulation enhanced by 10.7 BlnDOF × 30 K time-step unstructured FE non-linear seismic wave simulation

With the aim of dramatically improving the reliability of urban earthquake response analyses, we developed an unstructured 3-D finite-element-based MPI-OpenMP hybrid seismic wave amplification simulation code, GAMERA. On the K computer, GAMERA was able to achieve a size-up efficiency of 87.1% up to the full K computer. Next, we applied GAMERA to a physics-based urban earthquake response analysis for Tokyo. Using 294,912 CPU cores of the K computer for 11 h, 32 min, we analyzed the 3-D non-linear ground motion of a 10.7 BlnDOF problem with 30 K time steps. Finally, we analyzed the stochastic response of 13,275 building structures in the domain considering uncertainty in structural parameters using 3 h, 56 min of 80,000 CPU cores of the K computer. Although a large amount of computer resources is needed presently, such analyses can change the quality of disaster estimations and are expected to become standard in the future.

Earthquake Motion Simulation with Multiscale Finite-Element Analysis on Hybrid Grid

The prediction of strong ground motion with high resolution is a challenging task. In this article, the authors propose a multiscale analysis based on a singular perturbation and a new finite-element method with a hybrid of structured and unstructured elements, for a full 3D numerical simulation of earthquake wave propagation from fault to surface, including soft surface layers. The multiscale analysis refines a solution of lower resolution by considering effects of ground structures on wave propagation, and the hybrid grid radically reduces the amount of numerical computation. Several numerical experiments are carried out to show the validity and usefulness of the finite-element method with the hybrid grids, by comparing the results with those obtained using existing methods. An exemplary problem of wave propagation is solved using the proposed method. The potential usefulness of the proposed method is discussed, with particular attention paid to the accuracy of computing strong ground motion with higher spatial resolution.

Universal Bounds for Overall Properties of Linear and Nonlinear Heterogeneous Solids

For a sample of a general heterogeneous nonlinearly elastic material, it is shown that, among all consistent boundary data which yield the same overall average strain (stress), the strain (stress) field produced by uniform boundary tractions (linear boundary displacements), renders the elastic strain (complementary strain) energy an absolute minimum. Similar results are obtained when the material of the composite is viscoplastic. Based on these results, universal bounds are presented for the overall elastic parameters of a general, possibly finite-sized, sample of heterogeneous materials with arbitrary microstructures, subjected to any consistent boundary data with a common prescribed average strain or stress. Statistical homogeneity and isotropy are neither required nor excluded. Based on these general results, computable bounds are developed for the overall stress and strain (strain-rate) potentials of solids of any shape and inhomogeneity, subjected to any set of consistent boundary data. The bounds can be improved by incorporating additional material and geometric data specific to the given finite heterogeneous solid. Any numerical (finite-element or boundary-element) or analytical solution method can be used to analyze any subregion under uniform boundary tractions or linear boundary displacements, and the results can be incorporated into the procedure outlined here, leading to exact bounds. These bounds are not based on the equivalent homogenized reference solid (discussed in Sections 3 and 4). They may remain finite even when cavities or rigid inclusions are present. Complementary to the above-mentioned results, for linear cases, eigenstrains and eigenstresses are used to homogenize the solid, and general exact bounds are developed. In the absence of statistical homogeneity, the only requirement is that the overall shape of the sample be either parallelepipedic (rectangular or oblique) or ellipsoidal, though the size and relative dimensions of the sample are arbitrary. Then, exact analytically computable, improvable bounds are developed for the overall moduli and compliances, without any further assumptions or approximations. Bounds for two elastic parameters are shown to be independent of the number of inhomogeneity phases, and their sizes, shapes, or distribution. These bounds are the same for both parallelepipedic and ellipsoidal overall sample geometries, as well as for the statistically homogeneous and isotropic distribution of inhomogeneities. These bounds are therefore universal. The same formalism is used to develop universal bounds for the overall non-mechanical (such as thermal, diffusional, or electrostatic) properties of heterogeneous materials.

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

Abstract 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.