A unified periodical boundary conditions for representative volume elements of composites and applications

Xia, Zihui; Zhang, Yunfa; Ellyin, F.

doi:10.1016/s0020-7683(03)00024-6

Cited by 783 publications

(339 citation statements)

References 22 publications

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“…Pecullan et al [229] also concluded that the effective bulk moduli obtained under linear DBCs for very high matrix to inclusion stiffness contrast ratio may not satisfy the Hashin-Shtrikman upper bound. Xia et al [234] reported that the homogeneous boundary conditions, when applied on periodic microstructures, "are not only overconstrained, but may also violate the boundary traction periodicity conditions" under loading types with shear components. Hazanov and Huet [235], Hazanov and Amieur [236], and Pahr and Zysset [237] proposed uniform mixedtype boundary conditions that consider applying constant traction boundary conditions on some parts of the boundary and linear displacement boundary conditions to the other parts such that the apparent elasticity tensor for this boundary condition lies between the apparent tensors obtained with homogeneous boundary conditions.…”

Section: Computational Homogenizationmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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Section: Computational Homogenizationmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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“…In terms of the unit cell studied here, the PBCs and minimization of mesh mismatches were achieved through increasing the number of unit cells analysed in a single simulation, while merging mismatched nodes on contacting faces. According to our previous work [31], seven independent boundary conditions (BCs) in the form of uniform displacements were specified to obtain the material properties of fibre tows, as shown in Fig. 3.…”

Section: Mesh Generation and Boundary Conditionsmentioning

confidence: 99%

Strength prediction for bi-axial braided composites by a multi-scale modelling approach

Wang

Zhong²,

Adaikalaraj³

et al. 2016

J Mater Sci

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“…Rodriguez and Cabeza did not report finite element results for a hexagonal array, which are supplemented by using the commercial software ANSYS and are listed in Table 2. The detailed implementation procedure refers to the one for calculating the effective mechanical properties [17] and the thermoelectroelastic properties [20]. From Tables 1 and 2 (Fig.…”

Section: Quantitative Analysismentioning

confidence: 99%

“…Thus, a corresponding periodic boundary condition should be prescribed in the analysis based on a unit cell model. Xia et al [17] and Jiang et al [18] pointed out that the effective moduli/conductivities obtained under periodic boundary conditions are bounded by those obtained under homogeneous displacement/temperature boundary conditions and homogeneous traction/flux boundary conditions, hence are more reasonable. This paper is devoted to presenting a new complex variable method for two-dimensional steady-state heat conduction problems of cylindrically orthotropic fibers.…”

Section: Introductionmentioning

confidence: 98%

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A complex variable solution of two-dimensional heat conduction of composites reinforced with periodic arrays of cylindrically orthotropic fibers

Peng

Jiang

Song

2010

Computational Materials Science

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a b s t r a c tProblems of two-dimensional steady-state heat conduction for composites with doubly periodic arrays of cylindrically orthotropic fibers are dealt with. A new complex variable method is presented by introducing an appropriate coordinate transformation to convert the governing differential equation into a harmonic one, and the eigenfunction expansions of the field variables in a unit cell are derived. Then by using a generalized variational functional which absorbs the periodicity condition, an eigenfunction expansion-variational method based on a unit cell is developed to solve such problems. A convergence analysis and a comparison with finite element calculations are conducted to demonstrate the correctness and efficiency of the present method. A discussion is made about the effects of the cylindrical orthotropy of the fiber and the existence of the isotropic core in the fiber on the effective conductivity of the composite. An engineering equivalent parameter, which reflects the overall influence of the thermal conductivities of the matrix and fibers as well as the interfacial characteristic on the effective thermal conductivity of the composite, is found. It is shown that the present first-order approximation of the effective thermal conductivity of the composite can be written in a unified formula for different microstructural characteristics and possesses a good engineering accuracy.

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A unified periodical boundary conditions for representative volume elements of composites and applications

Cited by 783 publications

References 22 publications

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Strength prediction for bi-axial braided composites by a multi-scale modelling approach

A complex variable solution of two-dimensional heat conduction of composites reinforced with periodic arrays of cylindrically orthotropic fibers

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