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The paper deals with the three-dimensional problem in linear isotropic elasticity for a coated half-space. The coating is modelled via the effective boundary conditions on the surface of the substrate initially established on the basis of an ad hoc approach and justified in the paper at a long-wave limit. An explicit model is derived for the surface wave using the perturbation technique, along with the theory of harmonic functions and Radon transform. The model consists of three-dimensional 'quasi-static' elliptic equations over the interior subject to the boundary conditions on the surface which involve relations expressing wave potentials through each other as well as a two-dimensional hyperbolic equation singularly perturbed by a pseudo-differential (or integro-differential) operator. The latter equation governs dispersive surface wave propagation, whereas the elliptic equations describe spatial decay of displacements and stresses. As an illustration, the dynamic response is calculated for impulse and moving surface loads. The explicit analytical solutions obtained for these cases may be used for the non-destructive testing of the thickness of the coating and the elastic moduli of the substrate.

In the literature, it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods. Recently, it has been shown that this conjecture is true. One might guess that when compressibility is taken into account, such a wave, which is both a solitary wave and a shock wave, can still arise. One of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description. It is di¯cult to directly tackle the fully nonlinear rod equations. Here, by using a non-dimensionalization process and the reductive perturbation technique, we derive a new type of nonlinear dispersive equation as the model equation. We then focus on the travelling-wave solutions of this new equation. As a result, we obtain a system of ordinary di¬erential equations. An important feature of this system is that there is a vertical singular line in the phase plane, which leads to the appearance of shock waves. By considering the equilibrium points and their relative positions to the singular line, we are able to determine all qualitatively di¬erent phase planes. Those paths in phase planes which represent physically acceptable solutions are discussed one by one. It turns out that there is a variety of travelling waves, including solitary shock waves, solitary waves, periodic shock waves, etc. Analytical expressions for all these waves are obtained. A new phenomenon is also found: a solitary wave can suddenly change into a periodic wave (with nite period). In dynamical systems, this represents a homoclinic orbit suddenly changing into a closed orbit. To the authors' knowledge, such a bifurcation has not been found in any other dynamical systems.

The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we define M-eigenvalues for an elasticity tensor. The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the elasticity tensor is positive. If the strong ellipticity condition holds, then the elasticity tensor is rank-one positive definite. The elasticity tensor is rank-one positive definite if and only if the smallest Z-eigenvalue of the elasticity tensor is positive. A Z-eigenvalue of the elasticity tensor is an M-eigenvalue but not vice versa. If the elasticity tensor is second-order positive definite, then the strong ellipticity condition holds. The converse conclusion is not right. Computational methods for finding M-eigenvalues are presented.

This paper derives a finite-strain plate theory consistent with the principle of stationary threedimensional potential energy under general loadings with a fourth-order error. Starting from the three-dimensional nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the three-dimensional field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a two-dimensional virtual work principle. An alternative approach based on a two-dimensional truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a twodimensional energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Compared with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of loadings, applicability to finite-strain problems and no involvement of non-physical quantities.