A practical method is given that can determine whether or not it is possible for all or part of a problem in crystal elasticity to be expressed in explicit rational or radical terms when the characteristic sextic polynomial is known to be unsolvable. It involves the determination of the group to which an elastic quantity (displacement, stress, strain, energy density) belongs under permutations of the roots of the sextic polynomial. If this is not the symmetric group then a rational or radical expression is impossible. It is applied to the three-dimensional problem of a point force (the Green's function) in a cubic crystal. It is proved that it is impossible that the Green's function could be given by any radical expression that is valid for all elastic constants and directions in the crystal, as the existence of such an expression would lead to the solution of a proven unsolvable polynomial. It follows that the rational expression given by Dikici (1986) for the Green's function is not just incorrect, but that it is impossible that there could be any such rational expression. The same impossibility is found to apply to the displacements, stresses and strains of a straight dislocation in a cubic crystal. The application to other elasticity problems is discussed.

An experimental study using an analogue electronic model of the equation x‴ + x′ = є sin x, modified by the addition of small terms ax″ and βx with 0 < β < α ≫ є shows that these dissipative terms have a profound effect on the solutions for large time. If ∈ is not too large, experimental solutions tend to a simple periodic form, unlike the case α=β = 0. The existence of this limiting periodic form suggests the possibility of a simple analytic treatment using the method of harmonic balance, and this treatment leads to excellent agreement with the experimental results for a wide range of initial conditions and values of the parameters. The approach towards attracting limiting periodic solutions is analysed by using the method of multiple scales.

We apply the method of matched asymptotic expansions to link the outgoing wave solution at infinity of the differential equation y″(x)+(lgr;+єxn)y(x) = 0, x∈(0, ∞),λ∈Cє>0, n∈N across the turning point x = (—λ/є)1/n nearest to the positive real axis to a linear homogeneous boundary condition at the origin. The equation with n = 1 models the leakage of energy from the core of a bent fibre optic waveguide, the rate of leakage corresponding to Im λ, which was shown to be exponentially small like O(exp[— 1/є]) by Paris & Wood (1989). The extension n = 2 by Brazel et al. (1990) obtained Im λ = O(exp[— 1/є1/2]). Both these papers involve delicate analysis of the asymptotics of special functions near to Stokes' lines. When n > 2 no special functions are available, and completely different methods must be employed to obtain the result Im λ = O(exp[— 1/є1/n]).

A mathematical analysis is carried out for the Cahn–Hilliard equation where the free energy takes the form of a double well potential function with infinite walls. Existence and uniqueness are proved for a weak formulation of the problem which possesses a Lyapunov functional. Regularity results are presented for the weak formulation, and consideration is given to the asymptotic behaviour as the time becomes infinite. An investigation of the associated stationary problem is undertaken proving the existence of a nontrivial stationary solution and further regularity results for any stationary solution. Stationary solutions are constructed in one and two dimensions; a formula for the number of stationary solutions in one dimension is derived. It is then natural to study the asymptotic behaviour as the phenomenological parameter λ→0, the main result being that the interface between the two phases has minimal area.

Following Krasnoselskii & Pokrovskii (1983) we express the constitutive law for the Prandtl–Reuss elastoplastic model in terms of a hysteresis operator, and introduce the vector Ishlinskii model. We investigate some properties (continuity, energy inequalities, dependence on spatial variables) of these operators.