An Eulerian formulation for large deformations of anisotropic elastic and viscoelastic solids and viscous fluids

An Eulerian formulation has been developed for the constitutive response of a group of materials that includes anisotropic elastic and viscoelastic solids and viscous fluids. The material is considered to be a composite of an elastic solid and a viscous fluid. Evolution equations are proposed for a triad of vectors m _i that represent the stretches and orientations of material line elements in the solid component. Evolution equations for an orthonormal triad of vectors s _i are also proposed to characterize anisotropy of the fluid component. In particular, for an elastic solid it is shown that the material response is totally characterized by the functional form of the strain energy and by the current values of m _i , which are measurable in the current state of the material. Moreover, it is shown that the proposed Eulerian formulation removes unphysical arbitrariness of the choice of the reference configuration in the standard formulation of constitutive equations for anisotropic elastic solids.     

Homogénéisation d'un milieu élastique fortement hétérogèneHomogenization of a strongly heteregeneous elastic medium

We consider the homogenization of an elastostatic problem in a strongly heterogeneous periodic medium made of two connected components having comparable tensors of elastic moduli, separated by a third medium (soft layer), the thickness of which is of the same order ε than the basic periodicity cell, and such that its elastic moduli tensor becomes infinitely small following a rate ε^r, r>0. If r?2, we identify the homogenized problem. Otherwise, we have to assume moreover that there are no volume forces in the third medium. To cite this article: M. Mabrouk, A. Boughammoura, C. R. Mecanique 330 (2002) 543–548.     

Approximate fundamental solutions and wave fronts for general anisotropic materials

The time-dependent differential equations of elastodynamics for homogeneous solids with a general structure of anisotropy are considered in the paper. A new method of computation of the fundamental solution for these equations is proposed. This method consists of the following. Applying the Fourier transformation with respect to space variables to these equations, we obtain a system of second order ordinary differential equations whose coefficients depend on Fourier parameters. Using the matrix transformations and properties of the coefficients, the Fourier image of the fundamental solution is computed. Finally, the fundamental solution is calculated by the inverse Fourier transformation to the obtained Fourier transform. The implementation and justification of the suggested method have been made by computational experiments in MATLAB. These experiments confirm the robustness of the suggested method. The visualization of the displacement components in general homogeneous anisotropic solids by modern computer tools allows us to see and evaluate the dependence between the structure of solids and the behavior of the displacement field. Our method allows users to observe the elastic wave propagation, arising from pulse point forces of the form e^mδ(x)δ(t), in monoclinic, triclinic and other anisotropic solids. The visualization of displacement components gives knowledge about the form of fronts of elastic wave propagation in Sodium Thiosulfate with monoclinic and Copper Sulphate Pentahydrate with triclinic structures of anisotropy.     

Asymptotic analysis of nonlinear elastic platesAnalyse asymptotique des plaques élastiques non linéaires

We study in this Note the asymptotic analysis of nonlinear elastic plates with varying thickness. We suppose that the material moduli of the plates are anisotropic and nonhomogeneous, and the plates are submitted to body forces, to surfaces forces on the lower and upper faces and to pressure forces on the lateral boundary such that the displacements remains plane. To cite this article: D.A. Chacha, C. R. Mecanique 330 (2002) 581–586.     

计算微裂纹损伤材料有效模量的一种简单方法

给出了一种基于Taylor模型的有效介质方法。用以计算微裂纹相互作用对有效本构关系的影响,该方法假设每一个微裂纹位于一种有效介质之中,该有效介质的弹性模量由不考虑微裂纹相互作用的Taylor模型计算、和自洽方法相比,这种方法计算简单,而且结果更准确。     

Homogenization of an elastic medium reinforced by anisotropic fibersHomogénéisation d'un milieu élastique renforcé par des fibres anisotropes

We present results in this Note concerning a vector version in the framework of linearized elasticity (see A. Sili, Homogenization of an elastic medium reinforced by anisotropic fibers, in press), of our previous work in which we have studied the homogenization of a scalar nonlinear monotone problem posed on a fibered medium (see A. Sili, Homogenization of a nonlinear monotone problem in an anisotropic medium, in press). Here, we assume that parallel elastic anisotropic fibers, periodically distributed with a period of size ε in a cube $\text{Ω}$, are surrounded by a soft elastic material, the elasticity coefficients of this material being in the ratio ε² with those of the fibers. We prove that the homogenized problem is nonlocal and involves variables linked together with the anisotropy of the fibers. To cite this article: A. Sili, C. R. Mecanique 331 (2003).     

Extrema of Young’s modulus for cubic and transversely isotropic solids

For a homogeneous anisotropic and linearly elastic solid, the general expression of Young’s modulus E(n), embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which E(n) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Young’s modulus on direction n are given as well.     

Plasticity of a two-phase composite with partially debonded inclusions

The effective elastoplastic behavior of a two-phase composite consisting of partially debonded elastic inclusions and a ductile matrix is investigated by a homogenization method. The method drew information from a recent study by the authors on the effective elastic moduli of the said composite and from an energy approach suggested by Qui and Weng, J. Appl. Mech., 59, 261 [1992] to address the homogenized plastic state of the heterogeneously deformed ductile matrix. Two types of partial debonding configuration are considered; the first is on the top and bottom of the aligned oblate inclusions and the other is on the lateral surface of the prolate ones, with special reference to spherical inclusions for both types of debonding. The transversely isotropic elastoplastic properties of the partially debonded composite are found to be highly dependent upon the debonding mode and the volume concentration and shape of inclusions. A damage mechanics based on Weibull's statistical function is also proposed to study the progressive partial debonding of the initially bonded composite under pure tension and under biaxial tension, respectively, for these two types of partial debonding. It is found that the interfacial strength, particle concentration, inclusion shape and debonding mode all play significant role in the overall response of the heterogeneous system during the progressive debonding process.     

Chargements axisymétriques d'un bicouche transversalement isotropeAxisymmetric loading on a transversely isotropic two-layered medium

An exact three-dimensional analysis is developped for an axisymmetric loading acting on the surface of a semi-infinite medium composed by two transversely isotropic materials. The loading is assumed to be parallel to the elastic symmetry axis of the upper layer. The solutions of a concentrated force and a uniform loading distributed on a circle are obtained by exact integral expressions. The numerical results are performed to show the anisotropic effect with isovalue curves of stress. To cite this article: C. Ruimy, M. Dahan, C. R. Mecanique 330 (2002) 469–473.     

Induced damage by ellipsoidal cracks in an anisotropic medium

This paper is concerned with an estimation of the effective elastic properties of an anisotropic body permeated by ellipsoidal cracks following the Eshelbys method. The classical integral expression of the P-tensor, the symmetrised derivative of the Greens tensor, is given for a 3D defined crack embedded in an anisotropic medium. The numerical evaluation of the P-tensor is validated with several limiting cases of simplified geometry cracks. The interest of a 3D representation of the cracks is shown with several applications: influence of the ellipticity and crack thickness aspect ratios, growing cracks. To conclude, a comparison between theoretical results and experimental data is done for the load-induced change of the compliance tensor for a damaged composite material.     

Approximate yield criteria for anisotropic metals with prolate or oblate voidsCritères macroscopiques pour des métaux plastiques anisotropes contenant des cavités non sphériques

Following the study of Gologanu et al. (1997) which has extended the well-known approach of Gurson (1975), we propose approximate yield criteria for anisotropic plastic voided metals containing non spherical cavities. The plastic anisotropy of the matrix is described by means of Hill's quadratic criterion. The procedure to establish the closed form expression of approximate macroscopic criteria, in which void shape and plastic anisotropic effects are included, is detailed. The new criteria allow us to recover existing results in the cases of spherical and cylindrical voids in an Hill type plastic matrix. Moreover, they agree with previous criteria for non spherical voids in an isotropic plastic matrix. Finally, for validation purposes, we provide, in the general case of non spherical cavities in the anisotropic matrix, a comparison with the numerical exact two field criteria. To cite this article: V. Monchiet et al., C. R. Mecanique 334 (2006).     

On the thermal anisotropy affecting transfers in a multilayer porous mediumSur l'effet de l'anisotropie thermique sur les transferts dans un milieu poreux multicouches

An analytical model for studying double-diffusive natural convection within a multilayer anisotropic porous medium is developed and validated with respect to a direct numerical silmulation. The studied domain is composed of two horizontal porous layers where the lower one is thermally anisotropic and is submitted to a uniform horizontal heat flux and a vertical mass flux. The assumption of parallel flow is validated and the effect of anisotropy on dynamic transitions is investigated. To cite this article: R. Bennacer et al., C. R. Mecanique 332 (2004).     

Méthode particulaire anisotrope pour des écoulements de fluide visqueuxAnisotropic particle method for viscous flows

The anisotropic particle method has been extended to the case of viscous flows. The moment transport equation is modified to account for viscous effects. The diffusion term has been evaluated by using the PSE method and the particle moments. The modified transport equation includes geometrical moments for which a specific transport equation has been introduced. The study of the evolution of two corotating vortices allowed the comparison of the anisotropic particle method with the usual particle method. To cite this article: A. Beaudoin et al., C. R. Mecanique 332 (2004).     

Boundary element elastic stress analysis of 3D generally anisotropic solids using fundamental solutions based on Fourier series

The authors have very recently proposed an efficient, accurate alternative scheme to numerically evaluate etc. Green’s function, U(x), and its derivatives for three-dimensional, general anisotropic elasticity. These quantities are necessary items in the formulation of the boundary element method (BEM). The scheme is based on the double Fourier series representation of the explicit, exact, algebraic solution derived by Ting and Lee (1997) [Ting, T.C.T., Lee, V.G., 1997. The three-dimensional elastostic Green’s function for general anisotropic linear elastic solid. Q. J. Mech. Appl. Math. 50, 407–426] expressed in terms of Stroh’s eigenvalues. By taking advantage of some its characteristics, the formulations in this double Fourier series approach are revised and several of the analytical expressions are re-arranged in the present study. This results in significantly fewer terms to be summed in the series thereby improving further the efficiency for evaluating the Green’s function and its derivatives. These revised Fourier series representations of U(x) and its derivatives are employed in a BEM formulation for three-dimensional linear elastostatics. Some numerical examples are presented to demonstrate the veracity of the implementation and its applicability to the elastic stress analysis of generally anisotropic solids. The results are compared with known solutions in the literature where possible, and with those obtained using the commercial finite element code ANSYS. Excellent agreement is obtained in all cases.     

An alternative approach to the heat equation

The technique of conformal transformations of the metric, widely used in general relativity and in cosmology, is applied to the analysis of heat conduction in an anisotropic medium, in which the thermal conductivity is described by a tensor instead of a scalar. The anisotropic medium is reduced to an effective equivalent one, which is isotropic. The simplification is achieved for a particular conformal factor of the transformation, uniquely determined under physically reasonable assumptions on the thermal conductivity, density, and specific heat of the medium. Another application consists in the formal elimination of source or sink terms from the heat equation by using a suitable conformal transformation.

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Effects of radially varying moduli on stress distribution of nonhomogeneous anisotropic cylindrical bodies

In this study, the stress distribution in a nonhomogeneous anisotropic cylindrical body is investigated. Using equilibrium equations, Hooke’s law and strain–displacement relations, a system of equations is obtained in cylindrical coordinates in terms of stress potentials where elastic properties change in radial direction. Young’s and shear moduli are expressed as power functions of r and Poisson’s ratios are kept constant. Closed-form solutions for stress potentials and stress distribution are obtained for an axisymmetric, orthotropic cylinder. Results are checked with FE results. A pressurized thick walled cylinder example is studied in details. Stresses in radial, tangential and axial directions and Von Mises stresses are plotted for different powers of r.     

A method of plasticity for general aligned spheroidal void or fiber-reinforced composites

Based on the general concept of the secant moduli method, together with a new way of evaluating the average matrix effective stress originally proposed by Qiu and Weng (“A Theory of Plasticity for Porous Materials and Particle-Reinforced Composites”, ASME J. Appl. Mech. (1992), 59, 261.), a method for nonlinear effective properties of general aligned fiber or void composites is proposed. The method is capable of predicting composite (especially for porous materials) yielding under a hydrostatic load. Compared to the Tandon and Weng (“A Theory of Particle-Reinforced Plasticity,” ASME J. Appl. Mech. (1988), 55, 126.), model the proposed method always gives softer prediction in the uniaxial tension. The proposed method will predict the same nonlinear stress and strain relation as the Ponte Castaneda (“The Effective Mechanical Properties of Nonlinear Isotropic Composite,” J. Mech. Phys. Solids (1991), 39, 45.) variational model if the same estimates or bounds for the linear comparison composite are adopted.     

Dispersion and Dispersivity Tensors in Saturated Porous Media with Uniaxial Symmetry

The coefficient of dispersion, D _ij , and the dispersivity, a _ijkl , appear in the expression for the flux of a solute in saturated flow through porous media. We present a detailed analysis of these tensors in an axially symmetric porous medium, e.g., a stratified porous medium, with alternating layers, and show that in such a medium, the dispersivity is governed by six independent moduli. We present also the constraints that have to be satisfied by these moduli. We also show that at least two independent experiments are required in order to obtain the values of these coefficients for any three-dimensional porous medium domain.     

Hydrodynamic formation of stable anisotropic structures in Couette flow of dilute suspensionFormation hydrodynamique pour structures anisotropiques stables dans un flux de Couette d'une suspension diluée

The problem of rotary motion of rigid axially symmetric elongated particles in the Couette flow of dilute suspension with anisotropic carrier fluid is solved. It is shown that the stable stationary solutions of the dynamical set of ordinary differential equations describing the particles rotary motion are possible in the case of forming the stationary anisotropy in the carrier fluid of the suspension. It allows us to detect the stationary orientation of suspended particles and formation of stable anisotropic liquid-crystalline structures in the considered suspension under the action of hydrodynamic forces. The study of rheological properties of such a structured suspension shows that it behaves as a viscoelastic quasi-Newtonian anisotropic liquid medium. To cite this article: E.Yu. Taran et al., C. R. Mecanique 332 (2004).     

Surface responses induced by point load or uniform traction moving steadily on an anisotropic half-plane

Surface responses induced by point load or uniform traction moving steadily with subsonic speed on an anisotropic half-plane boundary are investigated. It is found that the effects of the material constant on surface displacements are through matrices L^?1(v) and S(v)L^?1(v), while those on surface stress components are through matrices Ω(v) and Γ(v). Explicit expressions for the elements of these four matrices are expressed in terms of elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with symmetry plane at x₁ = 0, x₂ = 0 and x₃ = 0, and the case for orthotropic materials are all deduced. Results for isotropic material may be recovered from present results. For monoclinic materials with a plane of symmetry at x₃ = 0, two of the elements of matrix Ω(v) are found to be independent of subsonic speed.