• Journal of the Korean Mathematical Society
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• v.46 no.5
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• pp.1041-1069
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• 2009

We study the periodic homogenization of the non-stationary Stokes equations. The fundamental homogenization theorem and corrector theorem are proved under a very general assumption on the viscosity coefficients and data. The proofs are based on a weak formulation suitable for an application of classical Tartar's method of oscillating test functions. Such a weak formulation is derived by adapting an argument in Teman's book [Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1984].

• Coupled systems mechanics
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• v.7 no.1
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• pp.61-77
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• 2018

This paper establish a high concentration ratio approximation of linear elastic properties of materials with periodic microstructure with cubic inclusions. The approximation is derived using first few terms of power series expansion of the solution of the equivalent eigenstrain problem with a homogeneous eigenstrain approximation. Viability of the approximation at high concentration ratios is proved by comparison with a numerical solution of the homogenization problem. To this end some theoretical result of symmetry properties of the homogenization problem are given. Using these results efficient numerical computation on a reduced computational domain is presented.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.22 no.11
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• pp.1603-1609
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• 1998

Numerical solutions of Stokes flow through periodic arrays of cylinders were sought using Darcy's law and homogenization theory. Drag and lift forces of each cylinder were computed for various attack angles and pitch-to-diameter ratios. It was found that drag force decreased as principal pressure gradient direction deviated from array direction and that drag force increased exponentially as pitch-to-diameter ratio approached unity. Similar tendency was found in lift force except that lift force increased and then decreased in quadratic manner as attack angle varied.

• Structural Engineering and Mechanics
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• v.72 no.6
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• pp.713-722
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• 2019

Micromechanics is a technique for the analysis of composites or heterogeneous materials which focuses on the components of the intended structure. Each one of the components can exhibit isotropic behavior, but the microstructure characteristics of the heterogeneous material result in the anisotropic behavior of the structure. In this research, the general mechanical properties of a 3D anisotropic and heterogeneous Representative Volume Element (RVE), have been determined by applying periodic boundary conditions (PBCs), using the Asymptotic Homogenization Theory (AHT) and strain energy. In order to use the homogenization theory and apply the periodic boundary conditions, the ABAQUS scripting interface (ASI) has been used along with the Python programming language. The results have been compared with those of the Homogeneous Boundary Conditions method, which leads to an overestimation of the effective mechanical properties. According to the results, applying homogenous boundary conditions results in a 33% and 13% increase in the shear moduli G₂₃ and G₁₂, respectively. In polymeric composites, the fibers have linear and brittle behavior, while the resin exhibits a non-linear behavior. Therefore, the nonlinear effects of resin on the mechanical properties of the composite material is studied using a user-defined subroutine in Fortran (USDFLD). The non-linear shear stress-strain behavior of unidirectional composite laminates has been obtained. Results indicate that at arbitrary constant stress as 80 MPa in-plane shear modulus, G₁₂, experienced a 47%, 41% and 31% reduction at the fiber volume fraction of 30%, 50% and 70%, compared to the linear assumption. The results of this study are in good agreement with the analytical and experimental results available in the literature.

• Nuclear Engineering and Technology
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• v.44 no.1
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• pp.43-52
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• 2012

To compute a permeability coefficient along a rough fracture that takes into account the fracture geometry, this study performed detailed measurements of fracture roughness using a confocal laser scanning microscope, a quantitative analysis of roughness using a spectral analysis, and a homogenization analysis to calculate the permeability coefficient on the microand macro-scale. The homogenization analysis is a type of perturbation theory that characterizes the behavior of microscopically inhomogeneous material with a periodic boundary condition in the microstructure. Therefore, it is possible to analyze accurate permeability characteristics that are represented by the local effect of the facture geometry. The Cpermeability coefficients that are calculated using the homogenization analysis for each rough fracture model exhibit an irregular distribution and do not follow the relationship of the cubic law. This distribution suggests that the permeability characteristics strongly depend on the geometric conditions of the fractures, such as the roughness and the aperture variation. The homogenization analysis may allow us to produce more accurate results than are possible with the preexisting equations for calculating permeability.

• Journal of the Korean Society for Precision Engineering
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• v.22 no.2
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• pp.79-88
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• 2005

Evaluating the effective properties of materials containing various types of in-homogeneities is an important issue in the analysis of structures composed of those materials. A simple and effective method for the purpose is to impose the periodic displacement boundary conditions on the finite element model of a unit cell. Their theoretical background is explained based on the purely kinematical relations in the regularly spaced in-homogeneity problems, and the strategies to implement them into the analysis and to evaluate the homogenized material constants are introduced. The creep behavior of a thin sheet with square arrayed rectangular voids is characterized, where the orthotropy is induced by the presence of the voids. The homogenization method is validated through the comparison of the analysis of detailed model with that of the simplified one with the effective parameters.

• Aerospace Engineering and Technology
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• v.12 no.2
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• pp.48-56
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• 2013

In this paper, the equivalent elastic properties of fiber reinforced plastic laminar are investigated using various homogenization schemes. Although there are several methods for predicting the equivalent elastic properties such as analytical formula or semi-empirical formula, most of them have some limitations or are not much accurate when handling new composite material consisting of various fiber, matrix and fiber-volume fraction ratio. To resolve the issues, computational homogenization scheme is adopted with a representative volume element (RVE) comprised of a set of finite elements. Finally, the equivalent elastic properties are obtained by applying periodic boundary conditions. The obtained results are compared with those by the existing methods and test results. Also its effect on structural analysis results of the composite satellite panel is investigated.

• Structural Engineering and Mechanics
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• v.45 no.4
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• pp.439-454
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• 2013

Hollow cylindrical tubes with a prismatic sandwich lining designed to replace the solid cross-sections are studied in this paper. The sections are divided by a number of revolving periodic unit cells and three topologies of unit cells (Square, Triangle and Kagome) are proposed. Some types of multiple-topology designed materials are also studied. The feasibility and accuracy of a homogenization method for obtaining the equivalent parameters are investigated. As the curved elements of a unit cell are represented by straight elements in the method and the ratios of the lengths of the curved elements to the lengths of the straight elements vary with the changing number of unit cells, some errors may be introduced. The frequencies of the first five modes and responses of the complete and equivalent models under an internal static pressure and an internal step pressure are compared for investigating the scope of applications of the method. The lower bounds and upper bounds of the number of Square, Triangular and Kagome cells in the sections are obtained. It is shown that treating the multiple-topology designed materials as a separate-layer structure is more accurate than treating the structure as a whole.

• Steel and Composite Structures
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• v.42 no.4
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• pp.489-499
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• 2022

This paper proposes a numerical optimization method to design the mesoscale architecture of textile composite for simultaneously enhancing mechanical and thermal properties, which compete with each other making it difficult to design intuitively. The base cell of the periodic warp and fill yarn system is served as the design space, and optimal fibre yarn geometries are found by solving the optimization problem through the proposed method. With the help of homogenization method, analytical formulae for the effective material properties as functions of the geometry parameters of plain-woven textile composites were derived, and they are used to form the inverse homogenization method to establish the design problem. These modules are then put together to form a multiobjective optimization problem, which is formulated in such a way that the optimal design depends on the weight factors predetermined by the user based on the stiffness and thermal terms in the objective function. Numerical examples illustrate that the developed method can achieve reasonable designs in terms of fibre yarn paths and geometries.

• Journal of Power System Engineering
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• v.19 no.3
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• pp.55-62
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• 2015

This paper discusses a new model for investigating the micro-mechanical behavior of beam-like structures composed of various elastic moduli and complex geometries varying through the cross-sectional directions and also periodically-repeated along the axial directions. The original three-dimensional problem is first formulated in an unified and compact intrinsic form using the concept of decomposition of the rotation tensor. Taking advantage of two smallness of the cross-sectional dimension-to-length parameter and the micro-to-macro heterogeneity and performing homogenization along dimensional reduction simultaneously, the variational asymptotic method is used to rigorously construct an effective zeroth-order beam model, which is similar a generalized Timoshenko one (the first-order shear deformation model) capable of capturing the transverse shear deformations, but still carries out the zeroth-order approximation which can maximize simplicity and promote efficiency. Two examples available in literature are used to demonstrate the consistence and efficiency of this new model, especially for the structures, in which the effects of transverse shear deformations are significant.