V.I. Kushch

Microstresses and effective elastic moduli of a solid reinforced by periodically distributed spheroidal particles

Abstract The particle composite consisting of a continuous matrix with the spheroidal particles arranged in several triply periodic arrays is considered. The problem on macroscopically uniform stressed state of this composite is solved accurately. The essence of the method used is the representation of a displacement vector by a series of triply periodic partial vectorial solutions of Lames equation written in a spheroidal basis. Exact satisfaction of the interfacial boundary conditions reduces the primary boundary-value problem to an infinite set of linear algebraic equations. By solving it numerically the displacements, strains and stresses at an arbitrary point of composite can be determined with any desirable accuracy. Analytical averaging of the strain and stress tensors gives the exact expressions for all components of effective elasticity tensor of composite considered. The influence on stress concentration and effective moduli of the structural parameters of composite is investigated and the comparison is made with known approximate solutions

Finite-Weber-number motion of bubbles through a nearly inviscid liquid

A method is described for computing the motion of bubbles through a liquid under conditions of large Reynolds and finite Weber numbers. Ellipsoidal harmonics are used to approximate the shapes of the bubbles and the flow induced by the bubbles, and a method of summing flows induced by groups of bubbles, using a fast multipole expansion technique is employed so that the computational cost increases only linearly with the number of bubbles. Several problems involving one, two and many bubbles are examined using the method. In particular, it is shown that two bubbles moving towards each other in an impurity-free, inviscid liquid touch each other in a finite time. Conditions for the bubbles to bounce in the presence of non-hydrodynamic forces and the time for bounce when these conditions are satisfied are determined. The added mass and viscous drag coefficients and aspect ratio of bubbles are determined as a function of bubble volume fraction and Weber number

Interacting cracks and inclusions in a solid by multipole expansion method

Abstract An accurate series solution is obtained of the elastic problem for a solid containing penny-shaped cracks and spheroidal inclusions of cavities. The method of solution is based on the general solution procedure developed by Kushch [(1996) Elastic equilibrium of a solid containing a finite number of aligned inclusions. International Journal of Solids and Structures33, 1175–1189] and consists in representation of the displacement vector by a series of the vectorial partial solutions of Lames equation, written in a spheroidal basis. By using the addition theorems for these partial solutions the primary boundary-value problem is reduced to an infinite set of linear algebraic equations. An asymptotic analysis of the problem is performed and the series expansion of the opening-mode stress intensity factor is obtained. Numerical analysis of model problems is performed and some results demonstrating the effect on the stress intensity factor of the pair interactions in crack-crack, crack-cavity and crack-inclusion geometries are presented.

Elastic equilibrium of a medium containing a finite number of aligned spheroidal inclusions

Abstract The strict solution in series is obtained of the elasticity theory problem for an unbounded domain containing some aligned spheroidal inhomogeneities under uniform far field loads. The essence of the method used is the representation of the displacement field in a multiply connected domain as a sum of general solutions for corresponding single connected domains. Each term of this sum, in turn, is expanded into series on vectorial partial solutions of Lames equation in a local spheroidal basis. In order to satisfy exactly all interfacial boundary conditions, the re-expansion formulae (addition theorems) for external partial solutions are used. As a result, the primary boundary value problem of elasticity theory is reduced to an infinite set of linear algebraic equations. The convergence rate of the proposed solution procedure is evaluated numerically. Some numerical results demonstrating the influence on stress distribution of material properties, spatial position of inclusions and external load are presented.

Stress intensity factor and effective stiffness of a solid containing aligned penny-shaped cracks

Abstract The stress state and effective elastic moduli of an isotropic solid containing equally oriented penny-shaped cracks are evaluated accurately. The geometric model of a cracked body is a spatially periodic medium whose unit cell contains a number of arbitrarily placed aligned circular cracks. A rigorous analytical solution of the boundary-value problem of the elasticity theory has been obtained using the technique of triply periodic solutions of the Lame equation. By exact satisfaction of the boundary conditions on the cracks’ surfaces, the primary problem is reduced to solving an infinite set of linear algebraic equations. An asymptotic analysis of the stress field has been performed and the exact formulae for the stress intensity factor (SIF) and effective elasticity tensor are obtained. The numerical results are presented demonstrating the effect of the crack density parameter and arrangement type on SIF and overall elastic response of a solid and comparison is made with known approximate theories.

Stress concentrations in the particulate composite with transversely isotropic phases

Abstract The accurate series solution have been obtained of the elasticity theory problem for a transversely isotropic solid containing a finite or infinite periodic array of anisotropic spherical inclusions. The method of solution has been developed based on the multipole expansion technique. The basic idea of method consists in expansion the displacement vector into a series over the set of vectorial functions satisfying the governing equations of elastic equilibrium. The re-expansion formulae derived for these functions provide exact satisfaction of the interfacial boundary conditions. As a result, the primary spatial boundary-value problem is reduced to an infinite set of linear algebraic equations. The method has been applied systematically to solve for three models of composite, namely a single inclusion, a finite array of inclusions and an infinite periodic array of inclusions, respectively, embedded in a transversely isotropic solid. The numerical results are presented demonstrating that elastic properties mismatch, anisotropy degree, orientation of the anisotropy axes and interactions between the inclusions can produce significant local stress concentration and, thus, affect greatly the overall elastic behavior of composite.

A kinetic theory for particulate systems with bimodal and anisotropic velocity fluctuations

Observations of bubbles rising near a wall under conditions of large Reynolds and small Weber numbers have indicated that the velocity component of the bubbles parallel to the wall is significantly reduced upon collision with a wall. To understand the effect of such bubble-wall collisions on the flow of bubbly liquids bounded by walls, a model is developed and examined in detail by numerical simulations and theory. The model is a system of bubbles in which the velocity of the bubbles parallel to the wall is significantly reduced upon collision with the channel wall while the bubbles in the bulk are acted upon by gravity and linear drag forces. The inertial forces are accounted for by modeling the bubbles as rigid particles with mass equal to the virtual mass of the bubbles. The standard kinetic theory for granular materials modified to account for the viscous and gravity forces and supplemented with boundary conditions derived assuming an isotropic Maxwellian velocity distribution is inadequate for descri...

Evaluation of some approximate estimates for the effective tetragonal elastic moduli of two-phase fiber-reinforced composites

This paper examines three sets of approximate formulae for the overall tetragonal effective elastic properties of two-phase fiber-reinforced unidirectional composites with isotropic phases. The fibers are of circular cross-sections and periodically distributed in a matrix in a square pattern. The formulae by Kantor and Bergman, Luciano and Barbero, and estimates based on non-interacting Maxwell’s type approximations are rewritten in unified notations. The latter approximations coincide with the most of well-known estimates of the effective medium theories (composite cylinder model, generalized self-consistent model and the Mori–Tanaka method), as well as with one of the Hashin–Shtrikman variational bounds. The approximate estimates are compared with the exact periodic solutions to determine the range of their applicability. The simplest and most accurate formulae are identified and suggested as a set of approximate expressions for accurate estimates of the effective elastic properties of composite materials with a square symmetry.

On convergence of the generalized Maxwell scheme: conductivity of composites containing cubic arrays of spherical particles

The Maxwell concept of equivalent inhomogeneity generalized to account for the interactions between the particles in the cluster and combined with recently reported results on the polarizability of a cube is used to evaluate the effective conductivities of the materials reinforced by cubic arrays of spherical particles. New numerical results demonstrate that the estimates of the effective properties based on the generalized Maxwell scheme with the equivalent inhomogeneity of cubic shape converge to the accurate periodic benchmark solutions, unlike spherical shape-based estimates.

Local fields and effective conductivity tensor of ellipsoidal particle composite with anisotropic constituents

An accurate semi-analytical solution of the conductivity problem for a composite with anisotropic matrix and arbitrarily oriented anisotropic ellipsoidal inhomogeneities has been obtained. The developed approach combines the superposition principle with the multipole expansion of perturbation fields of inhomogeneities in terms of ellipsoidal harmonics and reduces the boundary value problem to an infinite system of linear algebraic equations for the induced multipole moments of inhomogeneities. A complete full-field solution is obtained for the multi-particle models comprising inhomogeneities of diverse shape, size, orientation and properties which enables an adequate account for the microstructure parameters. The solution is valid for the general-type anisotropy of constituents and arbitrary orientation of the orthotropy axes. The effective conductivity tensor of the particulate composite with anisotropic constituents is evaluated in the framework of the generalized Maxwell homogenization scheme. Application of the developed method to composites with imperfect ellipsoidal interfaces is straightforward. Their incorporation yields probably the most general model of a composite that may be considered in the framework of analytical approach.