A computational homogenization approach for limit analysis of heterogeneous materials

Abstract

model together with coupled kinematic and static theorems as well as homogenization theory. Based on finite elements and primal-dual interior point optimization algorithm, the assessment of the macroscopic strength criterion and stability analysis of soils reinforced by stone columns were performed in [9][10][11][12][13]. For periodic plates, a theoretical study on the overall homogenized Love-Kirchhoff strength domain of a rigid perfectly plastic multi-layered plate was reported by [14,15], and numerical determination of the macroscopic bending strength criterion was presented in [16,17]. By means of homogenization techniques and kinematic limit analysis in conjunction with non-linear programming, the plastic limit loads and failure modes of periodic composites governed by ellipsoid yield criteria can be determined [18][19][20][21][22][23]. Using the elastic stresses of the periodic microstructure, a static direct method in combination with homogenization was proposed in [24][25][26][27][28] for two and threedimensional limit analysis of periodic metal-matrix composites. In the method, the strong form of the equilibrium equations was transformed into the so-called weak form and to be satisfied locally in an average sense using approximated virtual displacement fields. Based on linear matching method, limit analysis of reinforced concrete structures has also been studied in [29][30][31][32][33]. Practical multi-scale modeling and homogenization can be found in [34].Direct approaches for limit analysis of structures and materials lead to a problem of constrained optimization involving either linear or nonlinear programming, and hence, the development of efficient optimization algorithms to enable solutions to be obtained is one of the major research directions in the field. From a mathematical point of view, linear programming is very attractive and has been widely applied to limit analysis problems [35][36][37]. Although powerful software based on interior point algorithms is available for the resolution of the resulting optimization problem, a large number of constraints generated in the linearization process would be needed in order to provide accurate solutions, thereby increasing the computational cost. Attempts have been made to solve problems involving exact convex yield functions using nonlinear programming algorithms [38][39][40][41][42][43]. In recent years, thanks to the development of such an efficient primal-dual interior point algorithm, implemented in a commercial optimization software Mosek [44], limit analysis problems have gained increasing attention. In fact, most commonly used yield criteria can be cast in the form of conic or semi-definite constraints, and optimization problems involving such constraints can be solved efficiently by such an algorithm, evidenced by recent works [11-13, 16, 17, 45-50].The objective of the present paper is to develop a novel computational homogenization procedure for kinematic limit analysis of periodic materials. The total velocity fields at microscopic level ar...

Limit analysis of microstructures based on homogenization theory and the element-free Galerkin method

“…It can be observed that proposed procedure can prove the highly accurate solutions with low computational cost in comparison with approaches presented in [12,16]. The number of variables in present procedure is less than those used in [16], while the numerical result is approximate. Moreover, with the use of cone-based algorithm the resultant optimization problem in present study can be solved rapidly, the CPU-time required for handling the problem is much less than those in [12] using iterative algorithm.…”

“…Tab. 1 shows present numerical solutions compared with those in [12,16]. It can be observed that proposed procedure can prove the highly accurate solutions with low computational cost in comparison with approaches presented in [12,16].…”

“…Two perforated material models including a rectangular hole of size L 1 ⇥ L 2 and a circle hole of radius R at center are considered. The problems have been investigated using kinematic formulation in [11,12,16] and quasi-static formulation in [10]. The RVE is subjected to an orthogonal macroscopic stress (⌃ 11 , ⌃ 22 ) in plane (x 1 , x 2 ) as shown in The effect of microscopic hole on the overall strength of perforated materials is investigated.…”

“…In those works, both of isotropic and anisotropic materials obeying the von Mises or elliptic yield criterion were considered. In order to improve the computational aspect, a numerical procedure based on combination of kinematic limit analysis and homogenization theory for periodic materials proposed in [16]. However, these studies only focused on the application of finite element method.…”

Statistical Analyses of the Strengths of Particulate Reinforced Metal Matrix Composites (PRMMCs) Subjected to Multiple Tensile and Shear Stresses

“…Yu [22,23], V.L. Canh et al [24]. Strengths of PRMMC materials are not only affected by the material microstructure, but also by the type of loading they are exerted to.…”