Kenjiro Terada

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.

A class of general algorithms for multi-scale analyses of heterogeneous media

Abstract A class of computational algorithms for multi-scale analyses is developed in this paper. The two-scale modeling scheme for the analysis of heterogeneous media with fine periodic microstructures is generalized by using relevant variational statements. Instead of the method of two-scale asymptotic expansion, the mathematical results on the generalized convergence are utilized in the two-scale variational descriptions. Accordingly, the global–local type computational schemes can be unified in association with the homogenization procedure for general nonlinear problems. After formulating the problem in linear elastostatics, that with local contact condition and the elastoplastic problem, we present representative numerical examples along with the computational algorithm consistent with our two-scale modeling strategy as well as some direct approaches.

Characterization of the mechanical behaviors of solid-fluid mixture by the homogenization method

The mechanical behaviors of a solid-fluid mixture are characterized by using the homogenization method which is based on the method of asymptotic expansions. According to the choice of the so-called effective parameters, the formal derivation yields two distinct systems of well-known macromechanical governing equations; one for poroelasticity and the other for viscoelasticity. The homogenized equations representing the asymptotic behaviors entail the locally defined field equations and the geometry of a repeating unit. In addition to the identities of both formulations with ones in classical mechanics, the formulation enables the evaluation of actual mechanical responses of microstructures. This distinctive feature of the homogenization method is called the localization, which must be a key capability that provides a bridge between micromechanics and macromechanics. Thus, the present developments and several numerical simulations will provide insight into a variety of engineering problems in regard to solid-fluid coupled systems.

Two-scale kinematics and linearization for simultaneous two-scale analysis of periodic heterogeneous solids at finite strain

Abstract We introduce the notion of two-scale kinematics and the procedure of two-scale linearization , which are indispensable to the simultaneous two-scale analysis method for the mechanical behavior of periodic heterogeneous solids at finite strain. These are accomplished by formulating the two-scale boundary value problem in both material and spatial descriptions with reference to the two-scale modeling strategy developed in [Comput. Methods Appl. Mech. Engrg. 190 (40–41) (2001) 5427] that utilized the convergence results of mathematical homogenization. The formulation brings the intimate relationship between micro- and macro-scale kinematics in describing the micro–macro coupling behavior inherent in heterogeneous media. It is also shown that the two-scale linearization necessitates the strict consistency with the micro-scale equilibrated state and naturally invites the tangential homogenization process for both material and spatial descriptions. Several numerical examples of simultaneous two-scale computations are presented to illustrate the two-scale nature of the deformation of a heterogeneous solid at finite strain.

Global–local analysis of granular media in quasi-static equilibrium

Abstract A method of global–local analysis is developed for quasi-static equilibrium problems for granular media. The two-scale modeling based on mathematical homogenization theory enables us to formulate two separate boundary value problems in terms of macro- and microscales. The macroscale problem governs the equilibrium of a global structure composed of granular assemblies, while the microscale one is posed for the particulate nature of a local structure with the friction-contact mechanism between particles. The local structure is identified with a periodic representative volume element, or equivalently, a unit cell, over which averaging is performed. The mechanical behavior of unit cells is analyzed by a discrete numerical model, in which spring and friction devices connect rigid particles, whereas the continuum-based finite element method is used for the macroscopic one. Representative numerical examples are presented to demonstrate the capability of the proposed two-scale analysis method for granular materials.

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.

Microstructure-based Evaluation of the Influence of Woven Architecture on Permeability by Asymptotic Homogenization Theory

A microstructure-based computational approach is taken to predict the permeability tensor of woven fabric composites, which is a key parameter in resin transfer moulding (RTM) simulation of polymer-matrix composites. An asymptotic homogenization theory is employed to evaluate the permeability from both macro- and microscopic standpoints with the help of the finite-element method (FEM). This theory allows us to study the relation between microscopic woven architecture and macroscopic permeability based on the method of two-scale asymptotic expansions. While the fluid velocity is introduced for Stokes flow microscopically, the macroscopic one is for seepage flow with the Darcys law. The latter can be characterized for arbitrary configurations of unit microstructures that are analyzed for the former under the assumption of the periodicity. After discussing the validity of this approach, we present a typical numerical example to discuss the permeability characteristics of plain weave fabrics undergoing shear deformation in the preforming in comparison with the undeformed one. Another notable feature of the proposed method is that the correlation between the macroscopic behaviors and the microscopic ones can be analyzed, which is important to analyze and/or design the RTM process. Hence, it is also demonstrated that the microscopic velocity field evaluated with macroscopic pressure gradient provides important information about the flow in RTM processes.

Imperfection sensitive variation of critical loads at hilltop bifurcation point

Variation of critical loads due to initial imperfections at the hilltop bifurcation point is described by elastic stability theory. We derive a system of bifurcation equations for a potential system expressing local behavior at this bifurcation point, which is a double critical point occurring as a coincidence of a simple pitchfork bifurcation point and a limit point. The piecewise linear law of imperfection sensitivity of critical loads in Thompson and Schorrock [J. Mech. Phys. Solids 23 (1975) 21] is revised by extending initial imperfections to be considered in the bifurcation equations. Based on this sensitivity law, a procedure to determine the most influential (worst or optimum) initial imperfection is formulated. As the most essential development of this paper, under the assumption that initial imperfections are subject to a multi-variate normal distribution, we derive the probability density function of critical loads that follows a Weibull-like distribution. The validity of theoretical developments is assessed through its application to elastic truss structures.

Recursive bifurcation of tensile steel specimens

Failure modes of steel specimens subjected to uniaxial tension are investigated. These modes are well known to display complex geometrical characteristics of deformation accompanied by the plastic instability behavior. As an underlying mechanism of such complexity, we here focus on the recursive occurrence of bifurcations. In the theory, the rule of recursive bifurcation of a rectangular parallelepiped domain is obtained by the group-theoretic bifurcation theory so as to exhaust all the mathematically possible courses of bifurcation. In the experiment, we examine the representative failure modes with reference to the rules to identify actual courses of recursive bifurcation. Three-dimensional finite element analysis of a thin specimen is conducted to observe the recursive bifurcation, in which diffuse necking is formed by the direct bifurcation and the single shear band by the secondary bifurcation. The recursive bifurcation has thus been identified as the mechanism to create the complex failure modes.

Bifurcation mechanism underlying echelon-mode formation

Abstract This paper presents a theory on the underlying mathematical mechanism of the echelon mode (a series of parallel short wrinkles that looks like a flight of stairs or wild geese arranged in formation) which has been observed ubiquitously with uniform materials, but which has long denied successful numerical simulations. It is shown by means of the group-theoretic bifurcation theory that the echelon mode formation can be explained as a recursive (secondary, tertiary, …) symmetry-breaking bifurcation if O(2) × O(2) is chosen as the underlying symmetry to model the local uniformity of materials. This implies, for example, that the use of periodic boundaries is essential to successfully realize the oblique stripe patterns and the subsequent echelon mode formation in numerical simulations. In fact, a recursive bifurcation analysis of a rectangular domain with periodic boundaries subject to uniform uniaxial compression yields various kinds of patterns, such as diamond, stripe and echelon modes, which are often observed for materials under shear.