Guy Bonnet

Basic singular solutions for a poroelastic medium in the dynamic range

The basic singular solution to the problems of dynamic poroelasticity is produced in analogy with thermoelasticity. The solution, given for a harmonic excitation, is valid for all distances and the whole frequency range by analytical expressions.

Green’s functions in an infinite transversely isotropic saturated poroelastic medium

The purpose of this article is to study Green’s functions for an infinite transversely isotropic saturated poroelastic medium. In the first part, equations of motions of such a medium are recalled according to Biot’s theory and completed by homogenization techniques, for harmonic displacements. The second part gives Green’s functions for such an infinite medium in the 3‐D Fourier‐transformed geometrical space. In the third part, a 1‐D inverse analytical Fourier transformation, followed by an inverse Hankel transformation, leads to an analytical expression of Green’s functions for harmonic motions. Some numerical methods to compute final Green’s functions in time space are listed.

A computational procedure for predicting the long term residual settlement of a platform induced by repeated traffic loading

Abstract A general structural analysis approach is developed in the present paper, allowing the evaluation of the residual settlement of a platform induced by repeated traffic loading. It notably relies upon the formulation of a cyclic constitutive law, which describes the progressive accumulation of irreversible (permanent) deformations locally exhibited by the different underlying granular materials when subjected to long term stress cycling generated by the traffic loading. This constitutive law is incorporated into a step-by-step numerical scheme where two kinds of elastic calculations are implemented: the first one concerns the determination of the so called reference stress cycles, while the second one is aimed at calculating the residual displacement and stress fields of the platform derived from the integration of the permanent non elastic deformations. The whole procedure is illustrated on the simplified model of a moving strip-load acting upon a homogeneous half-space, adopting a cyclic constitutive law formulated for a particular unbound granular material used in road pavements.

Eshelby's tensor fields and effective conductivity of composites made of anisotropic phases with Kapitza's interface thermal resistance

Eshelbys results and formalism for an elastic circular or spherical inhomogeneity embedded in an elastic infinite matrix are extended to the thermal conduction phenomenon with a Kapitza interface thermal resistance between matrix and inclusions. Closed-form expressions are derived for the generalized Eshelby interior and exterior conduction tensor fields and localization tensor fields in the case where the matrix and inclusion phases have the most general anisotropy. Unlike the relevant results in elasticity, the generalized Eshelby conduction tensor fields and localization tensor fields inside circular and spherical inhomogeneities are shown to remain uniform even in the presence of Kapitzas interface thermal resistance. With the help of these results, the size-dependent overall thermal conduction properties of composites are estimated by using the dilute, Mori–Tanaka, self-consistent and generalized self-consistent models. The analytical estimates are finally compared with numerical results delivered by the finite element method. The approach elaborated and results provided by the present work are directly applicable to other physically analogous transport phenomena, such as electric conduction, dielectrics, magnetism, diffusion and flow in porous media, and to the mathematically identical phenomenon of anti-plane elasticity.

Shear Correction Factors for Functionally Graded Plates

The Reissner-Mindlin plate model for calculation of functionally graded materials has been proposed in literature by using shear correction coefficient of homogeneous model. However, this use is a priori not appropriate for the gradient material. Identification of the transverse shear factors is thus investigated in this paper. The transverse shear stresses are derived by using energy considerations from the expression of membrane stresses. Using the obtained transverse shear factor, a numerical analysis is performed on a simply supported FG square plate whose elastic properties are isotropic at each point and vary through the thickness according to a power law distribution. The numerical results of a static analysis are compared with available solutions from previous studies.

Linear elastic properties derivation from microstructures representative of transport parameters.

It is shown that three-dimensional periodic unit cells (3D PUC) representative of transport parameters involved in the description of long wavelength acoustic wave propagation and dissipation through real foam samples may also be used as a standpoint to estimate their macroscopic linear elastic properties. Application of the model yields quantitative agreement between numerical homogenization results, available literature data, and experiments. Key contributions of this work include recognizing the importance of membranes and properties of the base material for the physics of elasticity. The results of this paper demonstrate that a 3D PUC may be used to understand and predict not only the sound absorbing properties of porous materials but also their transmission loss, which is critical for sound insulation problems.

Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

In this paper, the derivation of irreducible bases for a class of isotropic 2nth-order tensors having particular ‘minor symmetries’ is presented. The methodology used for obtaining these bases consists of extending the concept of deviatoric and spherical parts, commonly used for second-order tensors, to the case of an nth-order tensor. It is shown that these bases are useful for effecting the classical tensorial operations and especially the inversion of a 2nth-order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed-form expression of the strain field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.

New closed-form thermoelastostatic Green function and Poisson-type integral formula for a quarter-plane

A new Greens function and a new Poisson-type integral formula for a boundary value problem (BVP) in thermoelastostatics for a quarter-plane subject by mixed homogeneous mechanical boundary conditions are derived in this paper. The thermoelastic displacements are generated by a heat source, applied in the inner points of the quarter-plane and by temperature, prescribed on its boundary semi-straight-lines. All results, obtained in terms of elementary functions, are formulated in a special theorem. The first difficulty to obtain these results is in deriving the functions of influence of a unit concentrated force onto elastic volume dilatation @Q^(^k^). The second difficulty is in calculating a volume integral of the product of function @Q^(^k^) and Greens function G in heat conduction. A closed-form solution for a particular BVP of thermoelastostatics for a quarter-plane also is included. Using the proposed approach, it is possible to extend the obtained results not only for any canonical Cartesian domain, but also for any canonical orthogonal one.

Closed-form solutions for the effective conductivity of two-phase periodic composites with spherical inclusions

In this paper, we use approximate solutions of Nemat-Nasser et al. to estimate the effective conductivity of two-phase periodic composites with non-overlapping spherical inclusions. Systems with different inclusion distributions are considered: cubic lattice distributions (simple cubic, body-centred cubic and face-centred cubic) and random distributions. The effective conductivities of the former are obtained in closed form and compared with exact solutions from the fast Fourier transform-based methods. For systems containing randomly distributed spherical inclusions, the solutions are shown to be directly related to the static structure factor, and we obtain its analytical expression in the infinite-volume limit.

Low frequency locally resonant metamaterials containing composite inclusions

One main feature of metamaterials is the occurrence of a negative dynamic mass density that is produced when an inner local resonance is present. The inner resonance can be obtained in composite materials containing composite inclusions. For suitable ratios of the physical properties of the constituting materials, the composite inclusions act as spring-mass systems. The scaling of physical properties leading to such an inner resonance and the associated effective dynamic properties of materials containing composite inclusions are briefly recalled. The resonance frequencies and dynamic mass densities are obtained in a closed form for materials containing cylindrical composite fibers or spherical composite inclusions, after solving the related boundary value elasticity problems.