High-rank nonlinear sequentially laminated composites and their possible tendency towards isotropic behavior

deBotton, Gal; Hariton, I.

doi:10.1016/s0022-5096(02)00049-2

Cited by 31 publications

(35 citation statements)

References 27 publications

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“…A key point in this procedure is that the length scale of the embedded laminate is assumed to be much smaller than the length scale of the embedding laminate. This assumption allows to regard the rank-ðM À 1Þ laminate in the rank-M laminate as a homogeneous phase, so that available expressions for the effective potential of simple laminates (e.g., deBotton and Ponte Castañeda, 1992) can be used at each step of the process to obtain an exact expression for the effective potential of the rank-M sequential laminate (e.g., Ponte deBotton and Hariton, 2002). From this construction process, it follows that the microstructure of these sequential laminates can be regarded as random and particulate, with phase 1 playing the role of the (continuous) matrix phase embedding the (discontinuous) porous phase.…”

Section: Porous Materials With Sequentially Laminated Microstructuresmentioning

confidence: 99%

“…The effective stress potential of the resulting rank-M porous laminate can be shown to be (deBotton and Hariton, 2002;Idiart, 2006) e U M ð rÞ ¼ min…”

Section: Porous Materials With Sequentially Laminated Microstructuresmentioning

confidence: 99%

“…In order to check the accuracy of the new model, the resulting estimates will be compared with exact results for isotropic porous materials with a special class of microstructures known as ''sequential laminates" (Ponte Castañeda, 1992;deBotton and Hariton, 2002;Idiart, 2007). As will become clear below, sequential laminates are particularly suitable for assessing the accuracy of ''linear comparison" methods like the one considered in this work.…”

Section: Introductionmentioning

confidence: 99%

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A homogenization-based constitutive model for isotropic viscoplastic porous media

Danas

Idiart

Castañeda

2008

International Journal of Solids and Structures

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International audienceAn approximate model based on the "second-order" nonlinear homogenization method is proposed to estimate the effective behavior of isotropic, viscoplastic, porous materials. The model is constructed in such a way that it reproduces exactly the behavior of a "composite-sphere assemblage" in the limit of hydrostatic loadings, and therefore coincides with the hydrostatic limit of Gurson's criterion in the special case of ideal plasticity. As a consequence, the new model improves on earlier homogenization estimates, which have been found to be quite accurate for low triaxialities but overly stiff for sufficiently high triaxialities and nonlinearities. Additionally, the estimates delivered by the model exhibit a dependence on the third invariant of the macroscopic stress tensor, which has a nontrivial effect on the effective response of the material at moderate triaxialities. The proposed model is compared with exact results obtained for a special class of porous materials with sequentially laminated microstructures. The agreement is found to be quite good for the entire range of stress triaxialities, and all values of the porosity and nonlinearity considered

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Section: Porous Materials With Sequentially Laminated Microstructuresmentioning

confidence: 99%

“…The effective stress potential of the resulting rank-M porous laminate can be shown to be (deBotton and Hariton, 2002;Idiart, 2006) e U M ð rÞ ¼ min…”

Section: Porous Materials With Sequentially Laminated Microstructuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A homogenization-based constitutive model for isotropic viscoplastic porous media

Danas

Idiart

Castañeda

2008

International Journal of Solids and Structures

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“…[141] and [142]. Later, deBotton and Hariton [143] and deBotton [144] obtained a general expression for the behavior of incompressible sequentially laminated composites in small deformation and finite elasticity and compared their results with Hashin-Shtrikman bounds and proposed estimates of Ponte Castañeda [136]. In passing, we mention that an important application of homogenization is to predict the behavior of fiber-reinforced materials reported in Refs.…”

Section: Introductionmentioning

confidence: 92%

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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“…In this example, the NEXP1 technique is used. In relation to the composite under investigation, there are exact results (LAM) for nonlinear laminates characterized by power-law models of deBotton and Hariton [44] and the variational (VAR) and second-order (SO) estimates of Idiart and Ponte Castañeda [45]. For the SO estimates, the notation SO(W) and SO(U) correspond to the stress and strain formulations, respectively.…”

Section: D Isotropic Incompressible Compositementioning

confidence: 99%

Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials

Yvonnet

González

2009

Computer Methods in Applied Mechanics and Engineering

120

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a b s t r a c tThe homogenization of nonlinear heterogeneous materials is much more difficult than the homogenization of linear ones. This is mainly due to the fact that the general form of the homogenized behavior of nonlinear heterogeneous materials is unknown. At the same time, the prevailing numerical methods, such as concurrent methods, require extensive computational efforts. A simple numerical approach is proposed to compute the effective behavior of nonlinearly elastic heterogeneous materials at small strains. The proposed numerical approach comprises three steps. At the first step, a representative volume element (RVE) for a given nonlinear heterogeneous material is defined, and a loading space consisting of all the boundary conditions to be imposed on the RVE is discretized into a sufficiently large number of points called nodes. At the second step, the boundary condition corresponding to each node is prescribed on the surface of the RVE, and the resulting nonlinear boundary value problem, is solved by the finite element method (FEM) so as to determine the effective response of the heterogeneous material to the loading associated to each node of the loading space. At the third step, the nodal effective responses are interpolated via appropriate interpolation functions, so that the effective strain-energy, stress-strain relation and tangent stiffness tensor of the nonlinear heterogeneous material are provided in a numerically explicit way. This leads to a non-concurrent nonlinear multiscale approach to the computation of structures made of nonlinearly heterogeneous materials. The first version of the proposed approach uses multidimensional cubic splines to interpolate effective nodal responses while the second version of the proposed approach takes advantage of an outer product decomposition of multidimensional data into rank-one tensors to interpolate effective nodal responses and avoid high-rank data. These two versions of the proposed approach are applied to a few examples where nonlinear composites whose phases are characterized by the power-law model are involved. The numerical results given by our approach are compared with available analytical estimates, exact results and full FEM or concurrent multilevel FEM solutions.

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High-rank nonlinear sequentially laminated composites and their possible tendency towards isotropic behavior

Cited by 31 publications

References 27 publications

A homogenization-based constitutive model for isotropic viscoplastic porous media

A homogenization-based constitutive model for isotropic viscoplastic porous media

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

Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials

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