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.     

Equations of generalized thermoelasticity of a Cosserat medium in stresses

Thermoelasticity equations in stresses are derived in this paper for a Cosserat medium taking into account the finiteness of the heat propagation velocity. A theorem is proved on the uniqueness of the solution for one of the obtained systems of such equations.Notation u displacement vector - small rotation vector - absolute temperature - ₀ initial temperature of the medium - relative deviation of the temperature from the initial value - , , , , , ,, m constants characterizing the mechanical or thermophysical properties of the medium - density - I dynamic characteristic of the medium reaction during rotation - k heat conduction coefficient - ₀ a constant characterizing the velocity of heat propagation - X external volume force vector - Y external volume moment vector - w density of the heat liberation sources distributed in the medium - E unit tensor - T force stress tensor - M moment stress tensor - nonsymmetric strain tensor - bending-torsion tensor - s entropy referred to unit volume - V volume occupied by the body - surface bounding the body - (T)_ki, (M)_ki components of the tensorsT andM - q thermal flux vector Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 482–488, March, 1981.     

Dyadic Green's function of a sphere with an eccentric spherical inclusion

An exact, analytical solution to the problem of point-source radiation in the presence of a sphere with an eccentric spherical inclusion has been obtained by combined use of the dyadic Green's function formalism and the indirect mode-matching technique. The end result of the analysis is a set of linear equations for the vector wave amplitudes of the electric Green's dyad. The point source can be anywhere, even within the aforesaid nonspherical body, and there is no restriction with regard to the electrical properties in any part of space. Several checks confirm that this solution obeys the energy conservation and reciprocity principles. Numerical results are presented for an electric Hertz dipole radiating from within an acrylic sphere, which contains an eccentric spherical cavity.     

Scattering of a pulsed wave by a sphere with an eccentric spherical inclusion

A dielectric sphere with an eccentric spherical dielectric inclusion and an incident amplitude-modulated plane electromagnetic wave constitute an exterior radiation problem, which is solved in this paper. A solution is obtained by combined use of the Fourier transform and the indirect-mode-matching method. The analysis yields a set of linear equations for the wave amplitudes of the frequency-domain expansion of the electric-field intensity within and outside the externally spherical inhomogeneous body; that set is solved by truncation and matrix inversion. The shape of the backscattered pulse in the time domain is determined by application of the inverse fast Fourier transform. Numerical results are shown for a pulse backscattered by an acrylic sphere that contains an eccentric spherical cavity. The effects of cavity position and size on pulse spreading and delay are discussed.     

Temperature conditions of the interaction of a medium with a thin inclusion

A mathematical model of a thin linear inclusion (layer) with a heat-liberating or thermally insulated surface is proposed for the calculation of the temperature in arbitrary bodies.     

Thermal stress-focusing effect in a spherical Zirconia inclusion with dynamically transforming strains

Some composite materials, such as Zirconia-toughened ceramics, are remarkable materials which have high strength, a high elastic modulus, and an improved toughness, etc. These good qualities are made possible through the stress-induced phase transformation of composite particles, which is accompanied by an impact cooling. When a spherical inclusion in an infinite elastic domain is suddenly subjected to an instantaneous phase transformation, stress waves occur at the surface of a spherical inclusion at the moment thermal impact is applied. The wave may accumulate at the center and show stress-focusing effects, even though the initial stress may be relatively small. This paper analyzes the thermal stress-focusing effect caused by the instantaneous anisotropic phase transformations in the spherical Zirconia inclusion. By use of ray theory, the numerical results give a clear indication of the mechanism of stress-focusing in an inclusion embedded in an infinite elastic medium.     

Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion

Abstract

The theory of interaction between an arbitrary electromagnetic shaped beam and a sphere with an eccentrically located spherical inclusion is presented. This theory is built as a synthesis between two available theories (i) the generalized Lorenz-Mie theory for a homogeneous sphere (illuminated by an arbitrary shaped beam) and (ii) the theory of interaction between a plane wave and an eccentrically stratified dielectric sphere.     

Transient heat conduction in a medium with multiple spherical cavities

This paper considers a transient heat conduction problem for an infinite medium with multiple non‐overlapping spherical cavities. Suddenly applied, steady Dirichlet‐, Neumann‐ or Robin‐type boundary conditions are assumed. The approach is based on the use of the general solution to the problem of a single cavity and superposition. Application of the Laplace transform and the so‐called addition theorem results in a semi‐analytical transformed solution for the case of multiple cavities. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. A large‐time asymptotic series for the temperature is obtained. The limiting case of infinitely large time results in the solution for the corresponding steady‐state problem. Several numerical examples that demonstrate the accuracy and the efficiency of the method are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Molecular heat exchange of a solid spherical particle with a gaseous medium

The process of molecular heat exchange of a motionless rather large solid spherical aerosol particle with the surrounding medium has been mathematically simulated at significant variations of the temperature in its vicinity. The obtained formulas make it possible to find directly the temperature distribution in the vicinity of the particle and the value of the molecular heat flow conducted from the particle surface taking into account the temperature jump and the temperature dependence of the thermal conductivity coefficient. The analysis of the theoretical results has shown that the increase in the surface temperature of the particle leads to a monotonic increase in the jump temperature of the gas near its surface. In the case of a rather large particle, this can lead to a strong decrease in the value of the molecular heat flow conducted from its surface.     

Detachment of an elastic matrix from a rigid spherical inclusion

An approximate theoretical treatment is given for detachment of an elastomer from a rigid spherical inclusion by a tensile stress applied to the elastomeric matrix. The inclusion is assumed to have an initially-debonded patch on its surface and the conditions for growth of the patch are derived from fracture energy considerations. Catastrophic debonding is predicted to occur at a critical applied stress when the initial debond is small. The strain energy dissipated as a result of this detachment, and hence the mechanical hysteresis, are also evaluated. When a reasonable value is adopted for Young's modulus E of the elastomeric matrix, it is found that detachment from small inclusions, of less than about 0.1 mm in diameter, will not occur, even when the level of adhesion is relatively low. Instead, rupture of the matrix near the inclusion becomes the preferred mode of failure at an applied stress given approximately by E/2. For still smaller inclusions, of less than about 1 m in diameter, rupture of the matrix becomes increasingly difficult, due to the increasing importance of a surface energy term. These considerations account for the general features of reinforcement of elastomers. Small-particle fillers become effectively bonded to the matrix, whereas larger inclusions induce fracture near them, or become detached from the matrix, at applied stress that can be calculated from the particle diameter, the strength of adhesion, and the elasticity of the matrix material.     

Steady-state temperature distribution in an inhomogeneous medium with local inclusions

We present a modification of the method of image regions [G. I. Marchuk, Methods of Numerical Mathematics, Springer-Verlag [1975)] to solve the boundary-value problem for the steady-state temperature distribution in an irregular multiply connected region.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 1, pp. 158–163, July, 1981.     

The elastic interaction between a screw dislocation and a spherical inclusion

Using an extension of a classical technique due to Kelvin, the general solution for the interaction between a spherical inclusion in an infinite medium and any self-equilibrated source of stress outside the inclusion is derived in the form of infinite series involving spherical surface harmonics. This solution is then applied to the special case of the interaction between a screw dislocation and a spherical inclusion. The main features of the interaction are illustrated by examining the Peach-Koehler force per unit length on the dislocation. Expressions for the net force and interaction energy are also found, and it is seen that truncation of these various expansions leads to expressions which agree with existing long-range interaction expressions. The long-range prediction of the net force and energy is seen to be in good agreement with the exact results when the dislocation is sufficiently far from the inclusion. Finally a comparison is made between the spherical inclusion-screw dislocation interaction problem and its two dimensional analogue involving a cylindrical inclusion, and it is shown that there is not good agreement between the distribution of force on the dislocation for these two problems.     

Propagation of diffuse light in a turbid medium with multiple spherical inhomogeneities

We develop a fast and accurate solver for the forward problem of diffusion tomography in the case of several spherical inhomogeneities. The approach allows one to take into account multiple scattering of diffuse waves between different inhomogeneities. Theoretical results are illustrated with numerical examples; excellent numerical convergence and efficiency are demonstrated. The method is generalized for the case of additional planar diffuse-nondiffuse interfaces and is therefore applicable to the half-space and slab imaging geometries.     

Eshelby tensors for a spherical inclusion in microelongated elastic fields

Eshelby tensors are found for a spherical inclusion in a microelongated elastic field. Here, a special micromorphic model is introduced to describe the damaged material which defines the damage as the formation and the growth of microcracks and microvoids occurred in the material at the microstructural level. To determine the new material coefficients of the model, an analogy is established between the damaged body and the composite materials and then Mori–Tanaka homogenization technique is considered to obtain overall material moduli. Following this idea, the determination of the Eshelby tensors which establish the relation between the strains of the matrix material and of the inclusion becomes the first task. Introducing the concept of eigenstrain and microeigenstrain, the general constitutive theory is given for a homogeneous isotropic centrosymmetric microelongated media with defects. Then by the use of Green’s functions, micro and macro elastic fields are presented for the case of spherical inclusions embedded in an infinite microelongated material. Thus, the Eshelby tensors are obtained for a microelongated elastic field with a spherical inclusion and it is also shown that the classical Eshelby tensors can be obtained as a limit case of the microelongation.     

Three-dimensional dynamic stress analysis on a penny-shaped crack interacting with a suddenly transformed spherical inclusion

In this paper, the interaction between a penny-shaped crack and a near-by suddenly transformed spherical inclusion in 3-dimensional solid is investigated to assess the dynamic effect of the transformation. To simplify the solution procedure, the current problem is divided into two sub-problems by using the superposition principle. A time domain boundary integral equation method (BIEM) is adopted to evaluate the stress and displacement fields. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. The impact effect of the spherical inclusion when it experiences a pure dilatational eigenstrain on the penny-shaped crack is studied. The relationship between the relative location of the inclusion and its impact effect on the time history of the Mode I crack intensity factor is discussed in detail.     

Steady-state thermoelastic state of a heterogeneous plane with a slit along an arc of the circumference of a round inclusion

An analysis was carried out of the steady-state thermoelastic state of a plane containing a dissimilar round inclusion for the case when along a part of the interface there is a break in the continuity of mechanical and thermal contact due to the presence of a thermally insulated crack. Stress distribution in the vicinity of the crack tips was investigated.