Summary We consider a three-phase composite in which the internal circular inhomogeneity obeying Neuber’s nonlinear stress-strain law is bonded to an infinite linear isotropic elastic matrix via a middle linear isotropic elastic annular coating when the matrix is subjected to uniform remote anti-plane stresses. A neutral circular inhomogeneity that does not disturb the prescribed uniform stress field in the matrix is identified by numerically solving the resulting two coupled nonlinear equations via iteration or equivalently by solving a single sextic equation to arrive at the constant effective strain within the inhomogeneity and the ratio of the shear modulus of the matrix to that of the coating. The upper bound of the shear modulus ratio is the classical Hashin–Shtrikman formula while the lower bound is the classical Hashin–Shtrikman formula for a cavity. A neutral coated nonlinear spherical inhomogeneity in conductivity obeying Neuber’s law is also designed.

Summary This paper is concerned with the study of a circular inclusion in an infinite elastic matrix under the combined action of a point force applied at a finite point of the matrix and far away stresses acting in the matrix. The interface between the inclusion and the matrix resists stretching and bending, and may have a prescribed surface prestress. The interface is modeled using the Steigmann-Ogden model of surface elasticity. The problem is solved using the Somigliana identities connecting the stresses and the displacements in the matrix and the inclusion, and the Fourier series expansions of the stresses and displacements on the interface. The solution obtained in this paper can be viewed as Green’s function for a nanosized elastic circular inclusion in an infinite elastic matrix. The obtained Green’s function can serve as a basis solution for numerical studies, such as, for instance, boundary integral equations of elasticity. Parametric studies and comparisons with the known results are given in the paper.

Summary The effect of the Coriolis force is demonstrated for chiral continuum models describing waves in the equatorial region and the polar regions on a rotating sphere. Novel asymptotic features of equatorial waves are presented in this paper. We show that the shape of a ridge of a polar vortex can be approximated by the governing equations of a gyropendulum. Theoretical deductions are accompanied by illustrative examples.

Summary This article investigates shear flow in a Hele-Shaw cell, driven by varying horizontal buoyancy forces resulting from a horizontal density gradient induced by a scalar field. By employing asymptotic methods and taking the dependence of density and transport coefficients on the scalar field into account, effective two-dimensional hydrodynamic equations coupled with the scalar conservation equation are derived. These equations determine an effective diffusion coefficient for the scalar field accounting for shear-induced diffusion, and an effective shear-induced buoyancy force that modifies the classical Darcy’s law. The derived equations provide a foundation for future research into various problems involving scalar transport in horizontal Hele-Shaw cells.

Summary The present paper introduces the notion of chiral gravitational elastic waves and explores their connections to equatorial and planetary waves. The analysis of the gravity-induced waveforms in gyroscopic systems composed of gyropendulums provides important insights into the dynamics of waves in the vicinity of the equatorial belt. We show that the direction of motion of the chiral waveforms can be controlled by choosing the orientation of the spinners. The presence of gravity is shown to affect the stop band frequencies for such structures, providing an additional control parameter for the chiral waveguides. The theoretical work is accompanied by illustrative examples.

Summary Waves in a thermo-visco-elastic medium interact with a bounded penetrable obstacle of arbitrary shape. The T-matrix for such a scattering problem connects the expansion of the incident wave in terms of regular vector spherical wavefunctions to the expansion of the scattered waves in terms of outgoing vector spherical wavefunctions. It is shown that the T-matrix is symmetric, and an algorithm for its calculation is presented.

Summary In this article, we solve exactly an initial boundary value problem (IBVP) posed for Burgers’ equation on the quarter plane. Asymptotic behaviors of the solution of Burgers’ equation in different regions of the quarter plane are obtained from the exact solution. The special solutions of the Burgers’ equation, such as traveling wave solution and stationary solution, describe the large time asymptotic behaviors of the solutions of the IBVP. The important contribution here is that we give a clear picture of the competition of different terms in the exact solution and the dominance of different terms leading to different asymptotic solutions. The asymptotic expansions of the exact solution of the IBVP studied here may help us in constructing asymptotic solutions of generalized Burgers’ equations that are not explicitly solvable.

Abstract The evolution of solitary waves governed by perturbations of the Korteweg-de Vries (KdV) equation is considered, focussing in particular on the Burgers-Korteweg-de Vries (BKdV) equation. Using matched asymptotic expansions the structure of the wave is determined for all timescales. A tail appears behind the main waveform, the structure of which is determined in the form of a convolution integral. Numerical results are presented using a pseudospectral scheme but modified so that linear terms are incorporated into an integrating factor. All details of the asymptotic structure of the waveform are validated by numerical results. Comparisons are made with earlier asymptotic analyses of decaying solitary waves.

Summary A steady state plane problem of an inhomogeneous half-plane subjected to a load running along the boundary at subsonic speed is analyzed. The Lame coefficients and the density of the half-plane are assumed to be power functions of depth. The model is different from the standard static model have been used in contact mechanics since the Sixties and originated from the 1964 Rostovtsev exact solution of the Flamant problem of a power-law graded half-plane. To solve the governing dynamic equations with variable coefficients written in terms of the displacements, we propose a method that, by means of the Fourier and Mellin transforms, maps the model problem into a Carleman boundary value problem for two meromorphic functions in a strip with two shifts or, equivalently, to a system of two difference equations of the second order with variable coefficients. By partial factorization the Carleman problem is recast as a system of four singular integral equations on a segment with a fixed singularity and highly oscillating coefficients. A numerical method for its solution is proposed and tested. Numerical results for the displacement and stress fields are presented and discussed.

Summary We study the design of an imperfectly bonded neutral isotropic elastic ellipsoidal inhomogeneity that does not disturb the prescribed uniform normal stresses in an isotropic elastic matrix. The imperfect inhomogeneity-matrix interface is modeled by a spring-type imperfect interface characterized by a single imperfect interface function. The same degree of interface imperfection is realized in both the normal and tangential directions. The two loading parameters and the imperfect interface function are determined by solving a resulting cubic equation for each one of the three types of elastic constants of the composite. All three roots of the cubic equation are permissible provided that the Young’s modulus (for the first two types) or the Poisson’s ratio (for the third type) of the inhomogeneity is higher or lower than that corresponding to the matrix. The design of a neutral superellipsoidal or paraboloidal elastic inhomogeneity with another kind of spring-type interface that does not sustain shear traction under a prescribed uniform hydrostatic stress field in the matrix is also achieved.