Plastic behavior of composites and porous media under isotropic stress

Chu, Tingxiang; Hashin, Zvi

doi:10.1016/0020-7225(71)90029-2

Cited by 36 publications

(25 citation statements)

References 4 publications

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“…To ease the numerical upscaling, simplifying assumptions have to be introduced (e.g. on plasticity [24][25][26]); however, the resulting macroscopic descriptions are quite sensitive to them [15]. Among contributions via incremental schemes, let us mention those of Wieckowski [27] and Michel [28] on inelastic composites, of Doghri and Ouaar [29] on cyclic plasticity of metals, of Luizar-Obregon et al [30] on non-linear coupling in porous media, of Kristenssen and Ahadi [7] on coarse and clayed soils, and of [11] in elasto-plasticity with damage for geomaterials.…”

Section: Generalities On Homogenization Schemes Of Non-linear Compositesmentioning

confidence: 99%

Compressibility and permeability of sand–kaolin mixtures. Experiments versus non‐linear homogenization schemes

Boutin

Kacprzak

Doanh

2010

Num Anal Meth Geomechanics

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SUMMARYThis study deals with the behaviour of mixtures of sand and saturated kaolin paste considered as composite materials made of permeable and deformable (with non-linear behaviour) matrix (the kaolin paste) with rigid and impervious inclusions (the sand grains). Oedometric and permeability tests highlight the key role of the state of the clay paste, and show the existence of a threshold of sand grain concentration above which a structuring effect influences both compressibility and permeability. At the light of these experiments two homogenization schemes (with simplifying assumptions to make the problem manageable) are considered to model these two parameters. Qualitative and quantitative comparisons with experimental data point out their respective domain of interest and limitations: a tangent homogenization scheme is shown to be sufficient to describe the macroscopic properties for dilute sand concentration; above the concentration threshold, the structuring effect is captured by the new homogenization scheme developed in this paper.

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Section: Generalities On Homogenization Schemes Of Non-linear Compositesmentioning

confidence: 99%

Compressibility and permeability of sand–kaolin mixtures. Experiments versus non‐linear homogenization schemes

Boutin

Kacprzak

Doanh

2010

Num Anal Meth Geomechanics

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“…Within this approximation, classical micromechanical schemes, such as the secant method, have been applied (see Chu andHashin, 1971, Berveiller andZaoui, 1979, among others), as well as more recent homogenization techniques for nonlinear composites with one potential (see for instance Li and Ponte Castañeda, 1993, Suquet, 1997Bardella, 2003and Gonzalez et al, 2004. These recent homogenization methods are based on the existence of a single potential and make use of the variational characterization of the local fields and of the effective potential.…”

Section: Introductionmentioning

confidence: 99%

On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles

Lahellec

Suquet

2007

Journal of the Mechanics and Physics of Solids

158

167

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A new method for determining the overall behavior of composite materials comprised of nonlinear inelastic constituents is presented. Upon use of an implicit timediscretization scheme, the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). In this first part of the study the variational technique of Ponte Castañeda is used to approximate the nonuniform eigenstrains by piecewise uniform eigenstrains and to linearize the nonlinear thermoelastic problem. The resulting problem is amenable to simpler calculations and analytical results for appropriate microstructures can be obtained. The accuracy of the proposed scheme is assessed by comparison of the method with exact results.

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“…In Fig. 3 we plot the potential along the strain path defined by 11 been used. The NEXP2 solution is obtained by choosing an accuracy d ¼ 10 À6 leading to R ¼ 85.…”

Section: Preliminary Numerical Testsmentioning

confidence: 99%

“…These methods, including the ones proposed by Willis [2], Dvorak [3], Qiu and Weng [4], Ponte Castañeda [5], Hu [6], Milton and Serkov [7], can be viewed as extending to the nonlinear case of some wellestablished techniques available for estimating or bounding the effective behavior of linear heterogeneous materials (see, e.g., Nemat-Nasser [8], Torquato [9] and Milton [10]). At the same time, a few exact results have been obtained for nonlinear heterogeneous materials presenting simple microstructures and undergoing particular loadings [11][12][13][14][15] of nonlinear heterogeneous materials are of both theoretical and practical importance. However, due to the difficulties inherent in analytically solving nonlinear homogenization problems, all of them have been obtained under rather restrictive assumptions and are not sufficient for the computation of structures consisting of nonlinear heterogeneous materials of complex microstructure and subjected to arbitrary macroscopic loadings.…”

Section: Introductionmentioning

confidence: 98%

Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials

Yvonnet

González

2009

Computer Methods in Applied Mechanics and Engineering

114

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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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Plastic behavior of composites and porous media under isotropic stress

Cited by 36 publications

References 4 publications

Compressibility and permeability of sand–kaolin mixtures. Experiments versus non‐linear homogenization schemes

Compressibility and permeability of sand–kaolin mixtures. Experiments versus non‐linear homogenization schemes

On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles

Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials

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