Rigid Spherical Inclusion Research Articles

Implementation and numerical verification of a non-linear homogenization method applied to hyperelastic composites

We present a 3D implementation and verification of a micromechanical model dedicated to highly compressible hyperelastic composites having a random microstructure. The model is based on the second-order method proposed by Ponte Castañeda and Tiberio [P. Ponte Castañeda, E., Tiberio, J. Mech. Phys. Solids 48 (2000) 1389–1411]. After recalling the basic principles of this non-linear homogenization technique, we describe its application to composites made up of an hyperelastic matrix and deformable (or rigid) spherical inclusions. The inclusions can be either particles or voids. Computational issues related to the implementation of the model are also presented. In order to provide a rigorous verification of the model for a large range of material contrasts, unit cell computations are performed for reinforced or porous hyperelastic materials. It is shown that in the two cases, the predictions of the model are in a good agreement with the reference finite elements solutions. Finally, voids shape effects on the macroscopic behavior of porous hyperelastic materials are analyzed.

The reciprocal theorem and rigid spherical inclusion vis-à-vis certain point singularities

The elastic interaction of certain point singularities with a rigid spherical inclusion embedded into an otherwise infinite elastic medium is investigated. The particular singularities considered are point force, force-dipole (with and without moment), centre of dilatation and concentrated moment. In each case, simple, closed form expressions are deduced by application of Betti's reciprocal theorem for the net force and net torque acting on the inclusion.

Rotational microrheology of multilayered finite elastic media

This paper presents derivations of the torque-rotation angle relationship for a rigid spherical bead embedded in a composite medium made of n isotropic and linear elastic layers. Analytical solutions are provided for both compressible and incompressible solids, assuming no-slip conditions between the rigid spherical inclusion and its adjacent medium as well as between elastic layers. Thanks to these general formulas, we investigated the effect of finite size media on the torque-bead rotation response and derived the exact relationship linking apparent and intrinsic elastic moduli of the medium. Thus, this result can be applied to characterize precisely the mechanical heterogeneity and architecture of soft biomaterials, including cells, from real magnetocytometry experiments. This point is exemplified in the case of a bilayer medium which, interestingly, allows distinguishing cellular cortex from deep cytoskeleton. We found that rotational microrheology experiments are well suited to characterize locally the elastic properties of the layer in contact with the probe as soon as the layer thickness is larger than two-bead diameters.

A general procedure for solving boundary-value problems of elastostatics for a spherical geometry based on Love's approach

We develop a general procedure for solving the first and second fundamental problems of the theory of elasticity for cases where boundary conditions are prescribed on a spherical surface, using Love's general solution of the elastostatic equilibrium equations in terms of three scalar harmonic functions. It is shown that this general solution combined with a methodology by Brenner paves an elegant way to determine the three harmonic functions in terms of the boundary data. Thus, with this general scheme, solution of any such boundary-value problem is reducible to a routine exercise thereby providing some 'economy of effort'. Furthermore, we develop a similar general scheme for thermoelastic problems for cases when temperature type boundary conditions are prescribed on a spherical surface. We then illustrate the application of the procedure by solving a number of problems concerning rigid spherical inclusions and spherical cavities. In particular, apart from furnishing alternative solutions to the known problems, we demonstrate the use of this general procedure in solving the problem of interaction of a rigid spherical inclusion with a concentrated moment and that of a concentrated heat source situated at an arbitrary point outside the inclusion. We also derive closed-form expressions for the net force and the net torque acting on a rigid spherical inclusion embedded into an infinite elastic solid under an ambient displacement field characterized by an arbitrary-order polynomial in the Cartesian coordinates. To the best of our knowledge, these results are new.

Particle structuring under the effect of an uniaxial deformation in soft/hard nanocomposites

A model nanocomposite sample, made of rigid monodisperse spherical inclusions in a deformable matrix, was uniaxially stretched. The displacement field of the particles at the sample surface is analyzed using atomic force microscopy. It is shown that its 2D structure factor presents most of the characteristic features previously described from scattering experiments on similar materials. At the scale of the particles, distortions from affinity are observed. They can be explained by the radial interactions between neighboring inclusions, related to the mechanical confinement of the matrix between particles. At larger scales, remarkable alignments of particles are observed along a direction which is roughly perpendicular to the stretching direction. We show that this effect is found in other soft/hard nanocomposites. It may contribute to the mechanical properties of this class of heterogeneous materials.

Topological sensitivity for 3D elastodynamic and acoustic inverse scattering in the time domain

Building on previous work for 3D inverse scattering in the frequency domain, this article develops the concept of topological derivative for 3D elastic and acoustic-wave imaging of media of arbitrary geometry using data in the time domain. The topological derivative, which quantifies the sensitivity of the cost functional associated with the inverse scattering problem due to the creation at a specified location of an infinitesimal hole (for the elastodynamic case) or rigid inclusion (for the acoustic case), is found to be expressed in terms of the time convolution of the free field and a supplementary adjoint field. The derivation of the topological derivative follows the generic pattern proposed in previous studies, which is transposable to a variety of other physical problems. A numerical example, where the featured cost function is defined in terms of synthetic data arising from the scattering of plane acoustic waves by a rigid spherical inclusion, illustrates the utility of the topological derivative concept for defect identification using time-varying data.

The Green functions of an elastic medium surrounding a rigid spherical inclusion

A homogeneous isotropic elastic medium surrounds and adheres to a rigid inclusion of spherical shape and otherwise fills the whole of space. A point force is located at an arbitrary one of its points. We determine the associated elastic field explicitly for four fundamental cases, in order to identify the Green tensor functions which are needed for the construction of the more complex elastic fields of other singularities, or of an internal fracture. The first case has a fixed inclusion, with no permitted translation or rotation, the second has an unconstrained one, not subject to any external force or couple, the third has an untranslated one which is not subject to any external torque about its centre, and the fourth has an unrotated inclusion which is subject only to an external couple. When the elastic medium is incompressible (with its Poisson's ratio equal to one-half), we are able to verify for the first case that, in keeping with the analogy noted by Rayleigh, our Green function is coincident in mathematical terms with the one given long ago by Oseen, for slow viscous flow outside a fixed spherical boundary.

Force-displacement relationships for spherical inclusions in finite elastic media

Here we present derivations of the force-displacement relationship for a rigid spherical inclusion embedded in homogeneous, isotropic, linear solids. Formulas are given for both the case of perfect interfacial bonding (no slip) and the case of a sliding contact (slip) between the medium and the inclusion. The formulas are applicable to both compressible and incompressible solids and are applicable for elastic media of finite extent. The results allow the determination of the elastic modulus of the medium from force-displacement measurements on the inclusion provided Poisson’s ratio is known. We find that, when the size of the medium is much larger than that of the inclusion, the inclusion displacement in response to an applied force is 50% larger when slip is present.

1401 2個の球体剛体介在物をもつ弾性体の熱応力(S18-1 介在物・空孔・き裂,S18 不規則性,不均一性構造・材料の数理モデルとその応用)

This paper deals with the thermal stresses of an infinite elastic solid with two rigid spherical inclusions under a uniform temperature field. A body force distribution method of solution is presented for the problem by applying Green's functions for axisymmetric problems of elasticity. The problem is formulated by an integral equation with unknown body forces distributed. The distance between the inclusions is changed by their displacements. Stress distributions around the inclusion and the displacement are shown by numerical calculations.

Effective properties for single size, rigid spherical inclusions in an elastic matrix

The main problem is that of determining the effective moduli for a compressible isotropic elastic medium containing single size, rigid, spherical inclusions at non-dilute concentrations. A solution is synthesized from available rigorous elasticity results that have been found under asymptotic conditions. Preliminary to obtaining this result for the compressible medium case, results are first found for the incompressible case over a range of rigid particle size distributions. All results extend from the dilute condition up through the full packing limit, which depends upon the size distribution.

The Nonaxisymmetric Contact Thermoelastic Problem for a Half-Space with a Motionless Rigid Spherical Inclusion

Based on the generalized Fourier method used simultaneously in the spherical and cylindrical systems of coordinates, we suggested an analytical method for solving the contact problem of thermoelasticity for an elastic half‐space with a rigid spherical inclusion. The problem is reduced to an infinite system of linear algebraic equations with the Fredholm operator provided that the boundary surfaces do not intersect. An approximate solution of the system in the form of a series with respect to a small parameter is obtained. The numerical analysis of the problem is presented.

球状剛体介在物をもつ弾性円柱の引張り : 負のポアソン比をもつ材料(S13-1 き裂・介在物問題,S13 数理弾性力学応用の最前線)

球状剛体介在物をもつ弾性円柱の引張り : 負のポアソン比をもつ材料(S13-1 き裂・介在物問題,S13 数理弾性力学応用の最前線)

球状剛体介在物をもつ弾性円柱の表面圧力による応力集中(OS3-4 応力集中、最適設計他,OS3 先端機能性構造材料の力学的挙動に関する数理解析とマルチスケールアナリシスへの進展)

This paper deals with a stress concentration problem of an elastic circular solid cylinder with a rigid spherical inclusion under uniform surface pressures. A method of solution is presented for the problem by using Green's functions for axisymmetric body force problems of an elastic solid cylinder. Stress distributions around the inclusion are investigated.

618 球状剛体介在物をもつ半無限弾性体の Boussinesq の問題 : 続, すべり境界の場合

This paper deals with the problem of determing of stresses and displacements of an elastic elastic half space with a rigid spherical inclusion in case of sliding interface. It is assumed that the half space is subjected to a concentrated normal force at a point on the plane surface of the half space. A method of solution is presented for the problem by using the Green's functions for axisymmetric body force problems of an elastic half space. Stresses and displacements around the inclusion are shown by numerical calculations and compared with the perfect bonding case. Influnces of Poisson's ratio on stresses and displacements are shown.

502 球状剛体介在物をもつ半無限弾性体の Boussinesq の問題

This paper deals with the problem of the determing of the stress and the displacement of an elastic half space with a rigid spherical inclusion. It is assumed that the half space is subjected to a concentrated normal force at a point on the plane surface of the half space. A method of solution is presented for the problem by using the Green's functions for axisymmetric body force problems of an elastic half space. The problem is formulated by an integral equation with unknown functions of the intensities of distributed body forces. Stresses and displacements around the inclusion of the half space are shown by numerical calculations.

Physical Basis for Sound Absorbing Materials Using a Medium with a Complex Density

The paper gives the results a theoretical and experimental study of human made composite media in the form of a rubber–like material with rigid compact spherical or cylindrical inclusions. It is shown that in contrast to rubber–like materials with voids, which can be described by a complex compressibility, materials with solid compact inclusions can be described by a complex density. On the basis of a material with inclusions of spherical shape, the example is given of the synthesis of a wideband absorber whose properties are practically independent of the static pressure.

Effective moduli of particulate solids: Lubrication approximation method

To efficiently calculate the effective properties of a composite, which consists of rigid spherical inclusions not necessarily of the same sizes in a homogeneous isotropic elastic matrix, a method based on the lubrication forces between neighbouring particles has been developed. The method is used to evaluate the effective Lame moduli and the Poisson's ratio of the composite, for the particles in random configurations and in cubic lattices. A good agreement with experimental results given by Smith (1975) for particles in random configurations is observed, and also the numerical results on the effective moduli agree well with the results given by Nunan & Keller (1984) for particles in cubic lattices.

Point singularities and the singularity method in classical elastodynamics

The object of the article is to elucidate the application of the singularity method in classical elastodynamics. To this end, solutions for higher–order point singularities, e.g. time–harmonic concentrated moment, centre of dilatation, centre of rotation and force tensor, are derived from the influence tensor by a method usually employed in the theory of electrostatics for the study of potential of a system of charged particles. Solutions of some elastodynamic problems concerning spherical cavities and rigid spherical inclusions are then derived by combining these point singularities.

838 部分はく離のある球状剛体介在物を有する無限弾性体の熱応力

This paper deals with the thermal stress problems of an elastic solid with a rigid spherical inclusion under a uniform temperature change. It is assumed that there are partially debonded interfaces between the surfaces of the inclusion and the matrix. We use a body force distribution method for the stress concentration problem. For this purpose, we apply the fundamental solution for axisymmetric body force problems of elasticity. The thermal stress fields are expressed by the elastic field due to temperature and the elastic filed due to body forces. Stress distributions around the inclusion are shown, and compared with the results for an elastic solid with a partially debonded rigid sphericaal inclusion under tension.

Effective moduli of particulate solids: The completed double layer boundary element method with lubrication approximation

This paper reports the numerical results of the effective moduli of a composite consisting of identical rigid spherical inclusions in a homogeneous isotropic elastic medium. The effective moduli are calculated for particles in simple, body-centered, face-centered cubic lattices, and in a random configuration. The numerical technique used is the Completed Double Layer Boundary Element Method (CDLBEM) augmented by a short-range lubrication approximation. The results agree well with previous numerical results obtained by a completely different technique. Agreement with experimental results for particles in random configurations is also observed.