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Numerical multi-scale modeling and analysis of granular media based on the homogenization theory
Published in Y. Kishino, Powders and Grains 2001, 2020
K. Kaneko, Y. Kishino, T. Kyoya, K. Terada
In this study, a multi-scale numerical model is developed for investigating the mechanical behavior of granular media based on the mathematical homogenization theory (Sanchez-Palencia 1980). The model is described in a quasi-static boundary value problem with friction-contact behavior of particulates and then formulated in terms of two distinct scales; i.e. micro- and macro-scales. A representative volume element (RVE) is defined to realize the particulate nature in a micro-scale and is regarded as a periodic assembly of elastic particulates. On the other hand, the macroscopic field variables are simply obtained as the volume average of the corresponding microscopic ones over the RVE. The derived two-scale boundary value problem allows us to analyze the micro-scale behavior by a granular element model (Kishino 1989), in which spring and friction devices connect rigid particulates with each other, while the macroscopic problem is solved by the continuum-based FEM. A simulation of bi-axial compression tests on a plane specimen demonstrates the capability of this two-scale numerical model and illustrates the applicability to geotechnical problems.
Failure surface of granular materials predicted by global-local computations
Published in Masayuki Hyodo, Hidekazu Murata, Yukio Nakata, Geomechanics and Geotechnics of Particulate Media, 2017
K. Kaneko, R. Aizawa, K. Kumagai, S. Tsutsumi, Y. Kishino
One of the authors developed a two-dimensional global-local numerical method (Kaneko et al. 2003) based on mathematical homogenization theory (Lions 1981; Allaire 1992) to simulate the boundary value problems that faithfully reflect the micromechanical deformation characteristics of granular medium. We have also extended the analysis method to the plane strain global-local problem that uses threedimensional particle aggregates in the micro-scale problem (Kaneko et al. 2005). The mathematical homogenization theory divides structures composed of aggregates of grains into an equivalent homogenous medium and microscopic structures called representative volume elements (RVE), and describes them by boundary value problems in two scales such as macro- and micro-scales. The global-local method combines these macro- and micro-scale problems to solve them simultaneously and it enables structural analysis that faithfully reflects the microscopic deformation characteristics of a granular medium allocated to the micro-scales. In the global-local method of granular medium, the macro-scale problem solved by the finite element method (FEM), and the microscale problem by the granular element method (GEM) (Kishino et al. 2001;Kishino 2002). In the global-local method based on the mathematical homogenization theory, the role of the constitutive equation in FEM is performed by micro-scale analysis by the GEM with periodic boundary control. In other words, according to the mathematical homogenization theory, micro-scale analysis under a periodic boundary condition provides a macroscopic constitutive relationship.
Homogenization
Published in Georgios A. Drosopoulos, Georgios E. Stavroulakis, Nonlinear Mechanics for Composite Heterogeneous Structures, 2022
Georgios A. Drosopoulos, Georgios E. Stavroulakis
Homogenization takes advantage of these periodic microstructural patterns by considering a representative sample of the material, called Representative Volume Element or RVE. The RVE includes all the constituent materials, e.g. fibres and the matrix or masonry blocks and the mortar. Homogenization methods are then applied to the RVE in order to derive the effective material properties which will be adopted in the structural scale. It is noted that quite often in analytical or numerical homogenization, the term unit cell is also used to express this representative microstructural sample. An example of a periodic microstructure is schematically presented in figure 6.1.
A comparative study of micromechanics models to evaluate effective coefficients of 1-3 piezoelectric composite
Published in Mechanics of Advanced Materials and Structures, 2023
Sanjeev Kumar Singh, Saroja Kanta Panda
For carrying out finite element study of a large-scale or macroscopic structure, a unit cell-based approach [12, 13, 15] is adopted. A representative volume element (RVE) or a unit cell that captures the main features of the underlying microstructure is created. A homogenization technique is adopted to find a globally homogeneous medium equivalent to the original composite where the strain energy stored in both systems remains conserved. A numerical method based on FEM has been developed to carry quasi static analysis of periodic 1-3 piezocomposites. In general, this approach involves the following five major steps.An appropriate unit cell containing sufficient information of both fiber and matrix phases is identified for a specified fiber volume fraction.The unit cell is subjected to mechanical and electrical loading constraints under the designated boundary conditions.The resultant stress and electric field components are measured which develop when loads and boundary conditions are applied to the unit cell.A homogenized coupled response is captured by invoking an appropriate averaging technique.The effective coefficients of the piezocomposites are calculated with the help of mathematical relations shown in Eqn. (1). These constitutive equations relate measured stresses and electric displacements to imposed strain and electric fields.
Bounds on size-dependent behaviour of composites
Published in Philosophical Magazine, 2018
S. Saeb, P. Steinmann, A. Javili
In order to predict the response of a heterogeneous material, several multi-scale techniques have been developed in the past. Reviews of the different multi-scale approaches can be found in [33–36]. Among these techniques, first-order computational homogenisation method or more specifically direct micro-to-macro transition method has become the most popular technique. An extensive body of literature is devoted to study this technique, among which we refer to [37–52]. The main assumption of homogenisation is that the microstructure of the heterogeneous material is far smaller than the characteristic length of the macrostructure. This separation of scales allows to view the problem as two coupled subproblems at the macro- and micro-scale. It is assumed that the constitutive response of the microstructure is known and, in an average sense, results in the effective response of the macrostructure. Usually, both the macro-problem and the micro-problem are discretised and solved using the finite element method [53–56]. The statistically similar sample of the microstructure is commonly referred to as representative volume element (RVE). For further details and the identification of RVEs, see [57–59]. The major limitation associated with classical first-order computational homogenisation is that it lacks a physical length scale and, thus, fails to account for the size-dependent behaviour of the material response commonly referred to as size effect. It has been recently shown that including interfaces at the micro-scale introduces a physically interpretable length scale[8,60] and agrees well with atomistic simulations [61], see also [62,63]. Another methodology to introduce a physical length scale at the micro-scale is to employ second-order computational homogenisation developed in [64] where higher order gradients are incorporated in the material response.
Optimal grading of TPMS-based lattice structures with transversely isotropic elastic bulk properties
Published in Engineering Optimization, 2021
In a standard topology optimization formulation, solid isotropic material with penalization (SIMP) or rational approximation of material properties (RAMP) models are set such that the intermediate density variables are penalized in order to obtain an optimal solution containing only zeros and ones. In the framework presented in this article, the density variable represents the volume fraction of the TPMS-based lattice structure of interest. Therefore, material interpolation laws representing these lattices are derived by numerical homogenization. From a fundamental point of view, the scale of the periodic structure should be much smaller than the macro scale. This could of course be questioned here. However, the homogenization method utilized in this work is frequently applied in order to find material interpolation laws for TPMS-based lattices on a meso scale and these laws compare well with experiment. A recent work on numerical homogenization of the gyroid structure compared with experiment was performed by Castro et al. (2019). The articles by Chen et al. (2019) and Lu et al. (2019) are other recent works on this topic. An article explaining how to determine material properties using numerical homogenization was given by Andreassen and Andreasen (2014). The articles by Hollister and Kikuchi (1992) and Sun and Vaidya (1996) are examples of early work on numerical homogenization. The term homogenization was introduced even earlier by Suquet (1987) meaning how to calculate effective material properties for an equivalent homogenous structure, which is typically done for a representative volume element (RVE), a concept introduced many years before by Hill (1963). Of course, homogenization also plays an important role for topology optimization and dates back to the original article on topology optimization by Bendsøe and Kikuchi (1988), where the optimal designs were the result of homogenization. A recent article on homogenization-based topology optimization generating optimal micro-structures was presented by Groen, Wu, and Sigmund (2019).