Isao Saiki

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.

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.

Cyclic shear force–slip behavior of studs under alternating and pulsating load condition

Abstract The objective of this study is to investigate the maximum strength and the fatigue strength of studs subjected to the alternating load as well as the pulsating load. First, in order to conduct a series of static and fatigue test of studs under both the load conditions, we devise a new specimen to which the alternating shear force can be applied easily by general fatigue testing machines. Employing the devised specimens, the shear force–slip behavior, the maximum strength and the fatigue strength of studs under the alternating load condition are investigated and are compared with those under the pulsating load condition.

FLOWER PATTERNS APPEARING ON A HONEYCOMB STRUCTURE AND THEIR BIFURCATION MECHANISM

Illuminative deformation patterns of a honeycomb structure are presented. A representative volume element of a honeycomb structure consisting of 2 × 2 hexagonal cells is modeled to be a -equivariant system. The bifurcation mechanism and an exhaustive list of possible bifurcated patterns are obtained by group-theoretic bifurcation theory. A flower mode of the honeycomb is shown to have the same symmetry as the so-called anti-hexagon in the Rayleigh–Benard convection. A numerical bifurcation analysis is conducted on an elastic in-plane honeycomb structure consisting of 2×2 cells to produce beautiful wallpapers of bifurcating deformation patterns and, in turn, to highlight the achievement of the paper. New deformation patterns of a honeycomb structure have been found and classified in a systematic manner. Knowledge of the symmetries of the bifurcating solutions has turned out to be vital in the successful numerical tracing of the bifurcated paths. This paper paves the way for the introduction of the results hitherto obtained for flow patterns in fluid dynamics into the study of patterns on materials.

GROUP-THEORETIC BIFURCATION MECHANISM OF PATTERN FORMATION IN THREE-DIMENSIONAL UNIFORM MATERIALS

An underlying mathematical mechanism for formation of periodic geometric patterns in uniform materials is investigated. Symmetry of a rectangular parallelepiped domain with periodic boundaries is modeled as an equivariance to a group O(2) × O(2) × O(2). The standard group-theoretic approach is used to investigate possible patterns of this domain that emerge through direct and some secondary bifurcations. This investigation clarifies the mechanism of successive symmetry-breaking bifurcation, which entails a variety of geometrical patterns in three-dimensional uniform materials. In particular, a few characteristic geometric patterns, such as oblique layer, column and diamond patterns, are identified and classified. Pattern simulations are conducted on geometrical patterns of joints in a calcite and folds in a stratum to reinforce pertinence of the pattern formation mechanism. Images of three-dimensional patterns of joints and folds are expanded into the triple Fourier series, and transient processes of bifurcation are reconstructed to arrive at possible courses of successive bifurcation. Qualitative information from this approach can offer insight into transient courses of deformation, which have been overlooked up to now.

Image simulation of uniform materials subjected to recursive bifurcation

Abstract Deformations of uniform materials are well known to display characteristic geometrical patterns such as en echelon cracks. A systematic procedure for the image simulation of the progress of deformation patterns of uniform materials is proposed here by highlighting recursive symmetry-breaking bifurcation as the fundamental mechanism to generate patterns. We here focus on a rectangular domain with periodic boundaries. That is, to better express the local uniformity at the sacrifice of the consistency with the boundary conditions, we employ the infinite-periodic-domain approximation which assumes that the domain is periodically extended in the x - and y -directions, respectively. Since real material properties manifest itself sufficiently away from the boundaries and usually form some characteristic patterns, the use of periodic boundaries is essential in the simulation of true material properties. Rules of the recursive bifurcation, which are expressed in terms of a hierarchy of subgroups labeling the symmetries of deformation patterns, are constructed by extending the pre-existing group-theoretic studies for this domain. The use of periodic boundaries has led to the emergence of the subgroups labeling stripe and echelon symmetries that disappear if these boundaries are not used. These rules of bifurcation are interpreted in terms of the double Fourier series to prepare for the image analysis of deformations in a rectangular domain. The use of the Fourier series has physical necessity in that the direct bifurcation modes of uniform domains are always harmonic and that periodic properties are better expressed in the frequency domain. Mode interference with high frequencies after bifurcation is advanced as the mechanism of localization of deformations. The computational analysis on a rectangular domain (plate) with periodic boundaries at four sides is conducted to present a numerical example of echelon-mode formation through recursive (cascade) bifurcation. The procedure for image simulation is applied to a few uniform materials, including: kaolin and steel specimens. The intensity of the digital images of the deformation patterns of these specimens in the frequency domain is successfully classified with the use of the rules of recursive bifurcation. As a result of these, the transient process of deformations, which was not discernible by the mere visual observations and was less understood so far, is identified based on a firm theoretical basis. The recursive bifurcation has thus been acknowledged to be the underlying mechanism of pattern formation of uniform materials.

NONLINEAR SIMULATION METHOD FOR VEHICLE-BRIDGE VIBRATIONAL PROBLEM

橋梁上の車両の走行性や鉄筋コンクリート床版のひび割れ性状などの使用性能の確認の必要性から, 構造物を構成する部材の一部が非線形の復元力特性を有する車両―橋梁系振動問題を解くことが必要となってきている.そこで本研究では, 構造物を構成する部材の一部が非線形の復元力特性を有する車両―橋梁系振動問題のシミュレーション解法を定式化した. そして, そのシミュレーション解法を用いた, 連続合成桁橋の鉄筋コンクリート床版のひび割れやずれ止めに作用する水平せん断力の動的応答性状の解析例などを示し, そのシミュレーション解法の有用性を確認した.