Elastic Spherical Inclusion Research Articles

Elastic coupled phase theory based on the Cosserat equations: Propagation of coherent waves

We develop a new coupled phase theory (CPT) in order to model the propagation of elastic coherent waves in homogeneous solid media containing randomly distributed spherical elastic inclusions. To this end, the Buyevich’s theory previously developed for fluid media (Y. Buyevich et I. Shchelchkova, Progress in Aerospace Sciences 18, 1978.) has been adapted and applied to the Cosserat equations. A key point is to introduce the Independent Scattering Approximation (ISA). We show that the coherent wavenumbers depend explicitly on the external force applied to spheres, the torque characterizing their rotation and the stresslet which is the result of the resistance of the particles to the straining motion. Numerical results are in good agreement with experimental data.

Mean field homogenization schemes using averages of stress, strain and strain energy density and applications

The average of stress (AS) and average of strain (AN) were often used in the development of conventional mean-field homogenization (MFH) schemes, but their correlation with the average of strain energy density (AE) has rarely been discussed. In this work, we examined the correlations of AE with AS and AN, and found that AE can be reduced to the product of AS or AN under uniform boundary stress or linear surface displacement associated with a uniform strain field. By introducing a reference matrix and using any two among AE, AS and AN, three MFH schemes were developed: NS (using AN and AS); NE (using AN and AE); and SE (using AS and AE). Following the concept of the Mori-Tanaka's (MT) method, three MFH schemes obtained above were reduced to MT-NS, MT-NE and MT-SE, respectively. Effective elastic properties of composites consisting of isotropic elastic matrices and spherical isotropic elastic inclusions were predicted using these schemes and compared with experimental results. The comparison revealed that MT-SE and MT-NS can reasonably replicate the experimental results in the case of moderate particle volume fractions and ratios between the Young's moduli of matrix and the reinforcement particles, demonstrating the validity of the MFH schemes. This work is significant because it not only clarifies the correlation of AE with AN and AS, but also provides more options for the development of MFH schemes, which can enrich the theory and methods for the analysis of the effective properties of composites.

STRUCTURE AND MECHANICAL CHARACTERISTICS OF COMPOSITES WITH HIGH DAMPING PROPERTIES

The damping characteristics of hybrid composite materials reinforced with elastic spherical and fibrous inclusions with viscoelastic coatings are studied. It is shown that in composites with particles morphology, the effect of significant increase of dissipation loss can be realized, andᵇ the effective composite properties loss can exceed the dissipation loss of viscoelastic coatings more than twenty times. The analytical estimations for optimal parameters of hybrid composite materials are proposed. The influence of possible imperfections in composite structure on their effective dissipation properties are analyzed.

3D crystal plasticity analyses on the role of hard/soft inclusions in the local slip formation

The present work aims to investigate the phenomenon of local plastic slips formation in the surrounding of inclusions by means of 3D finite element analyses from a mesoscale point of view. Isotropic elastic spherical Al2O3 and MnS inclusions are modelled inside a cylindrical ferritic body subjected to tensile loading conditions with the superimposition of hydrostatic pressure. A crystal plasticity constitutive model within the framework of finite strains is adopted for the description of the matrix together with an anisotropic elasticity tensor. Several crystallographic orientations are considered, in relation to the loading direction, to observe the stress distribution and the formation of the plastic deformation in the surrounding of the inclusion. Two modelling strategies of the matrix-defect interface bond are investigated.

Surface tension and a self-consistent theory of soft composite solids with elastic inclusions.

The importance of surface tension effects is being recognized in the context of soft composite solids, where they are found to significantly affect the mechanical properties, such as the elastic response to an external stress. It has recently been discovered that Eshelby's inclusion theory breaks down when the inclusion size approaches the elastocapillary length L≡γ/E, where γ is the inclusion/host surface tension and E is the host Young's modulus. Extending our recent results for liquid inclusions, here we model the elastic behavior of a non-dilute distribution of isotropic elastic spherical inclusions in a soft isotropic elastic matrix, subject to a prescribed infinitesimal far-field loading. Within our framework, the composite stiffness is uniquely determined by the elastocapillary length L, the spherical inclusion radius R, and the stiffness contrast parameter C, which is the ratio of the inclusion to the matrix stiffness. We compare the results with those from the case of liquid inclusions, and we derive an analytical expression for elastic cloaking of the composite by the inclusions. Remarkably, we find that the composite stiffness is influenced significantly by surface tension even for inclusions two orders of magnitude more stiff than the host matrix. Finally, we show how to simultaneously determine the surface tension and the inclusion stiffness using two independent constraints provided by global and local measurements.

A homogenization-enriched viscodamage model for cement-based material creep

We develop in this study a viscodamage model for cementitious materials based on the concept of pseudo-strains introduced by Schapery [1] coupled with the damage model of Mazars [2]. The pseudo-strains approach allows for reformulation of the initial viscoelastic problem as an equivalent elastic one by using a simple correspondence principle. The viscoelastic model results from a homogenization procedure in which the Mori–Tanaka scheme is applied in the Laplace–Carson space for estimating the effective linear viscoelastic bulk and shear moduli of the material. A composite made of a linear viscoelastic matrix described by a generalized Maxwell model and containing distributed spherical elastic inclusions and pores is adopted as a simplified representation of the material under consideration. One interesting feature of our approach is that exact analytical expressions for both bulk and shear moduli in the time space are derived in simple cases where a limited number of Maxwell chains are involved to describe the matrix behavior. The evolutions of the damage variable are governed by an equivalent pseudo-strain calculated from the pseudo-strains. The viscodamage model is applied to the simulation of cement paste basic creep tests available in the literature. After a proper identification of the parameters, an analysis of the results obtained with moderate creep loading reveals that the total strains evolutions are due mainly to the viscous characteristics of the material and to a lesser extent to the damage growth. The effects of the pore volume fraction of the material and of high load-levels leading to the specimen rupture on the creep response are further investigated and discussed.

Non-linear extension of FFT-based methods accelerated by conjugate gradients to evaluate the mechanical behavior of composite materials

FFT-based methods are used to solve the problem of a heterogeneous unit-cell submitted to periodic boundary conditions, which is of a great interest in the context of numerical homogenization. Recently (in 2010), Brisard and Zeman proposed simultaneously to use Conjugate Gradient based solvers in order to improve the convergence properties (when compared to the basic scheme, proposed initially in 1994). The purpose of the paper is to extend this idea to the case of non-linear behaviors. The proposed method is based on a Newton–Raphson algorithm and can be applied to various kinds of behaviors (time dependant or independent, with or without internal variables) through a conventional integration procedure as used in finite element codes. It must be pointed out that this approach is fundamentally different from the traditional FFT-based approaches which rely on a fixed-point algorithm (e.g. basic scheme, Eyre and Milton accelerated scheme, Augmented Lagrangian scheme, etc.). The method is compared to the basic scheme on the basis of a simple application (a linear elastic spherical inclusion within a non-linear elastic matrix): a low sensitivity to the reference material and an improved efficiency, for a soft or a stiff inclusion, are observed. At first proposed for a prescribed macroscopic strain, the method is then extended to mixed loadings.

A generalized Vlasov-Jones foundation model for micromechanics studies of syntactic foams

Syntactic foams are lightweight particulate composites that are synthesized by dispersing thin hollow inclusions in a matrix material. A spherical elastic inclusion embedded in a dissimilar medium is often used as a unit cell model to study syntactic foams’ micromechanics. The analysis of such unit cell problem is generally conducted by modeling both the inclusion and the matrix using three-dimensional linear elasticity. In this paper, we take a different approach by developing a generalized Vlasov-Jones foundation model for the matrix and describing the inclusion using Donnell shell theory. This framework is specialized to uniaxial tensile loading of a spherical matrix in the presence of two equal interfacial cracks introduced to model inclusion–matrix debonding. A variational method is used to formulate a nonlinear boundary value problem for the shell and foundation displacements and an iterative solution procedure based on the Ritz method is presented. For a polymer matrix–glass inclusion system, a parametric study is performed to investigate the effect of debonding extent, inclusion volume fraction, and shell thickness on debonding energetics and effective elastic modulus. The proposed model is validated using finite element results and experimental data on Young’s modulus of epoxy-glass syntactic foams.

Dynamic contact between a spherical inclusion and a matrix upon incidence of an elastic wave

UDC 539.3 Using the boundary element method, we have studied the dynamic displacements and stresses in an infinite elastic matrix with a spherical elastic inclusion, caused by the propagation of an elastic wave. The original problem has been reduced to a system of boundary integral equations for the contact displacements and tractions on the interface between the inclusion and matrix. Based on the numerical solution of these equations, we have analyzed the influence of the direction of wave propagation and frequency on the important physical parameters, depending on the elastic characteristics of composite constituents.

A frequency-domain BIEM combining DBIEs and TBIEs for 3-D crack-inclusion interaction analysis

The influence of a spherical elastic inclusion on a penny-shaped crack embedded in an infinite elastic matrix subjected to a time-harmonic crack-face or incident wave loading is investigated. A boundary integral equation method (BIEM) combining displacement boundary integral equations (DBIEs) on the matrix-inclusion interface and traction boundary integral equations (TBIEs) on the crack-surface is developed and applied for the numerical solution of the corresponding 3-D elastodynamic problem in the frequency domain. The singularity subtraction and mapping techniques in conjunction with a collocation scheme are implemented for the regularization and the discretization of the BIEs by taking into account the local structure of the solution at the crack-front. As numerical examples, the interaction of an elastic inclusion and a neighboring penny-shaped crack subjected to a tensile crack-surface loading or an incident plane longitudinal wave loading is investigated. The effects of the inclusion are assessed by the analysis of mixed-mode dynamic stress intensity factors (DSIFs) in dependence on the wave number, the material combination of the matrix and the inclusion, and the crack-inclusion orientation, size and distance.

Onset of residual stress fields near solitary spherical inclusions in a perfectly elastoplastic medium

When computing residual stresses in deformable solids, one has to use the theory of elastoplastic solids, because the final level and distribution of residual stresses is determined exactly by the accumulated reversible strains. In turn, to compute the elastic strains, one needs to determine the displacement field. The problem of determining displacements in statically determinate problems of the theory of perfect elastoplastic solids was considered for the first time in [1, 2]. The techniques proposed there permitted solving the problem of finding the residual stresses near a cylindrical cavity in a perfectly elastoplastic medium [3]. It was shown that secondary plastic flow [4] may arise in the unloading processes, which significantly redistributes the final residual stresses. In the present paper, we consider the loading and unloading problems for a ball with a rigid or elastic spherical inclusion. We study the onset of secondary plastic flow under unloading and compute the residual stresses. Thus, we model the onset of the residual stress field near a more rigid inhomogeneity. The case of a softer inhomogeneity was essentially considered in [3], where the onset of the residual stress field near a continuity flaw was studied.

Dynamic mechanical response of elastic spherical inclusions to impulsive acoustic radiation force excitation

Acoustic radiation force impulse imaging has been used clinically to study the dynamic response of lesions relative to their background material to focused, impulsive acoustic radiation force excitations through the generation of dynamic displacement field images. Dynamic displacement data are typically displayed as a set of parametric images, including displacement immediately after excitation, maximum displacement, time to peak displacement, and recovery time from peak displacement. To date, however, no definitive trends have been established between these parametric images and the tissues' mechanical properties. This work demonstrates that displacement magnitude, time to peak displacement, and recovery time are all inversely related to the Young's modulus in homogeneous elastic media. Experimentally, pulse repetition frequency during displacement tracking limits stiffness resolution using the time to peak displacement parameter. The excitation pulse duration also impacts the time to peak parameter, with longer pulses reducing the inertial effects present during impulsive excitations. Material density affects tissue dynamics, but is not expected to play a significant role in biological tissues. The presence of an elastic spherical inclusion in the imaged medium significantly alters the tissue dynamics in response to impulsive, focused acoustic radiation force excitations. Times to peak displacement for excitations within and outside an elastic inclusion are still indicative of local material stiffness; however, recovery times are altered due to the reflection and transmission of shear waves at the inclusion boundaries. These shear wave interactions cause stiffer inclusions to appear to be displaced longer than the more compliant background material. The magnitude of shear waves reflected at elastic lesion boundaries is dependent on the stiffness contrast between the inclusion and the background material, and the stiffness and size of the inclusion dictate when shear wave reflections within the lesion will interfere with one another. Jitter and bias associated with the ultrasonic displacement tracking also impact the estimation of a tissue's dynamic response to acoustic radiation force excitation.

Backscattering from spherical elastic inclusions and accuracy of the Kirchhoff approximation for curved interfaces

The Kirchhoff (or tangent plane) approximation, derived from the theoretically complete Kirchhoff—Helmholtz integral representation for the seismic wavefield, has been used extensively for the analysis of seismic-wave scattering from irregular interfaces; however, the accuracy of this method for curved interfaces has not been rigorously established. This paper describes an efficient Kirchhoff algorithm to simulate scattered waves from an arbitrarily curved interface in an elastic medium. Synthetic seismograms computed using this algorithm are compared with exact synthetics computed using analytical formulae for scattering of plane P waves by a spherical elastic inclusion. A windowing technique is used to remove strong internal reverberations from the analytical solution. Although the Kirchhoff method tends to underestimate the total scattering intensity, the accuracy of the approximation improves with increasing value of the wavenumber-radius product, kR. The arrival times and pulse shapes of primary reflections from the sphere are well approximated using the Kirchhoff approach regardless of curvature of the scattering surface, but the amplitudes are significantly underestimated for kR≤ 5. The results of this work provide some new guidelines to assess the accuracy of Kirchhoff-synthetic seismograms for curved interfaces.

Analytical Derivation of Cosserat Moduli via Homogenization of Heterogeneous Elastic Materials

Abstract Why do experiments detect Cosserat-elastic effects for porous, but not for stiff-particle-reinforced, materials? Does homogenization of a heterogeneous Cauchy-elastic material lead to micropolar (Cosserat) effects, and if so, is this true for every type of heterogeneity? Can homogenization determine micropolar elastic constants? If so, is the homogeneous (effective) Cosserat material determined in this way a more accurate representation of composite material response than the usual effective Cauchy material? Direct answers to these questions are provided in this paper for both two- (2D) and three-dimensional (3D) deformations, wherein we derive closed-form formulae for Cosserat moduli via homogenization of a dilute suspension of elastic spherical inclusions in 3D (and circular cylindrical inclusions in 2D) embedded in an isotropic elastic matrix. It is shown that the characteristic length for a homogeneous Cosserat material that best mimics the heterogeneous Cauchy material can be derived (resulting in surprisingly simple formulae) when the inclusions are less stiff than the matrix, but when these are equal to or stiffer than the matrix, Cosserat effects are shown to be excluded. These analytical results explain published experimental findings, correct, resolve and extend prior contradictory theoretical (mainly numerical and limited to two-dimensional deformations) investigations, and provide both a general methodology and specific results for determination of simple higher-order homogeneous effective materials that more accurately represent heterogeneous material response under general loading conditions. In particular, it is shown that no standard (Cauchy) homogenized material can accurately represent the response of a heterogeneous material subjected to a uniform plus linearly varying applied traction, while a homogenized Cosserat material can do so (when inclusions are less stiff than the matrix).

10402 球状弾性介在物をもつ弾性円柱の引張り(材料(1))

This paper deals with a stress concentration problem of an elastic circular solid cylinder with an elastic spherical inclusion under tension. Effects of Poisson's ratio, especially effects of negative Poisson's ratio, on the stress concentrations are investigated. Lakes has shown that there are materials with negative Poisson's ratios. A method of solution is presented for the problem by applying Green's functions for axisymmetric body force problems of infinite circular solid cylinder and fundamental solutions for axisymmetric problems of elasticity.

A Boussinesq's Problem of an Elastic Half Space with an Elastic Spherical Inclusion

The problem of determing of stresses and displacements of an elastic half space subjected to a concentrated force acting at a point of the plane surface is called Boussinesq's problem. This paper deals with Boussinesq's problems of an elastic half space with an elastic spherical inclusion. We use a body force distribution method for the stress concentration problem. For this purpose, we apply the Green's functions for axisymmetric body force problems of an elastic half space and the fundamental solution for axisymmetric body force problems of elasticity. The problem is formulated by an integral equation with unknown functions corresponding to the intensities of distributed body forces. Stress distributions around the inclusion are shown by numerical calculations and influences of elastic moduli on the stresses are also investigated.

Elastic plastic shakedown of 3D periodic heterogeneous media: a direct numerical approach

A numerical direct method is presented for analyzing shakedown of three-dimensional heterogeneously periodic and elastic-perfectly plastic media. The main objective is to predict the long-term behavior of such media submitted to variable loads by coupling the periodic homogenization and shakedown theories. The macroscopic domains of external loads—the macroscopic strains or stresses—are obtained by solving the shakedown problem on a unit cell, representative of the heterogeneities. This coupled problem leads to a nonlinear constrained optimization problem. The developed method is tested on a layered material, on a perforated sheet, and on a material composed of spherical elastic inclusions. The results are compared with those of analytical and numerical analyses.

836 球状弾性体介在物をもつ半無限弾性体の Boussinesq の問題

The problem of determing of stresses and displacements of an elastic half space subjected to a concentrated force acting at a point of the plane surface is called Boussinesq's problem. This paper deals with Boussinesq's problems of an elastic half space with an elastic spherical inclusion. We use a body force distribution method for the stress concentration problem. For this purpose, we apply the Green's functions for axisymmetric body force problems of an elastic half space and the fundamental solution for axisymmetric body force problems of elasticity. The problem is formulated by an integral equation with unknown functions corresponding to the intensities of distributed body forces. Stress distributions around the inclusion are shown by numerical calculations and influences of elastic moduli on the stresses are also investigated.

A micromechanical damage model for effective elastoplastic behavior of partially debonded ductile matrix composites

A micromechanical damage model considering progressive partial debonding is presented to investigate the effective elastoplastic-damage behavior of partially debonded particle reinforced ductile matrix composites (PRDMCs). The effective, evolutionary elastoplastic-damage responses of three-phase composites, consisting of perfectly bonded spherical particles, partially debonded particles and a ductile matrix, are micromechanically derived on the basis of the ensemble-volume averaging procedure and the first-order effects of eigenstrains. The effects of random dispersion of particles are accommodated. Further, the evolutionary partial debonding mechanism is governed by the internal stresses of spherical particles and the statistical behavior of the interfacial strength. Specifically, following Zhao and Weng (1996), a partially debonded elastic spherical isotropic inclusion is replaced by an equivalent, transversely isotropic yet perfectly bonded elastic spherical inclusion. The Weibull's probabilistic function is employed to describe the varying probability of progressive partial particle debonding. The proposed effective yield criterion, together with the assumed overall associative plastic flow rule and the hardening law, forms the analytical framework for the estimation of the effective elastoplastic-damage behavior of ductile matrix composites. Finally, the present predictions are compared with the predictions based on Ju and Lee's (2000) complete particle debonding model, other existing numerical predictions, and available experimental data. It is observed that the effects of partially debonded particles on the stress–strain responses are significant when the damage evolution becomes rapid.

Interfacial debonding of a spherical inclusion embedded in an infinite medium under remote stress

In the study of strength of particle reinforced composites, it is important to understand the energy release rate due to interfacial debonding between the particle and the matrix which is induced by manufacturing imperfection. This paper is aimed at the investigation of the critical condition for growth of the interfacial debonding and the corresponding volume increase due to void formation. The model used in the study is an isotropic elastic spherical inclusion embedded in an infinite isotropic elastic matrix under remote stress. Initial axisymmetrical interfacial debondings are assumed to exist in the vicinity of poles of the spherical inclusion. Axisymmetrical deformations of the matrix and the inclusion are analyzed based on the theory of three-dimensional elasticity in spherical coordinates. In order to avoid oscillatory stress singularity at the interfacial debonding front between two dissimilar materials, a condition of free slipping without friction at the interface is imposed. A Fredholm integral equation of the first kind is formulated based on the continuity conditions in the normal components of stress and displacement at the contact interface. The kernel function of the integral equation is expressed in terms of an infinite series of Legendre functions. Two types of remote stresses are considered in this study. The first type is the remote tension in the axial direction of the spherical inclusion and the second type is the remote compression in the transverse direction with respect to the axis of the spherical inclusion. Energy release rate is determined according to the rate of change of work done by remote stresses. In this paper, energy release rate and volume of the deformed void due to debonding are computed for any given size of initial interfacial debonding.