We present a method of solution of a class of fracture problems in the theory of elasticity. The method can be applied to any problem reducible to Poisson's equation, e.g. heat conduction and mass diffusion in solids, theory of consolidation and the like. The novelty of the paper is that we address regions of layered composites with notches, or, in a particular case, with a crack. Within the framework of classical analysis, we apply Fourier and Mellin transforms, 'fit' them together, and reduce the problem to solving a singular integral equation with fixed singularities on a semi-axis. We show the existence and uniqueness of solutions of the equations under consideration, and justify the asymptotics necessary for applications. We show the practical usefulness of the method on the examples of an antiplane problem of fracture mechanics. From our solution, we are able to find the stress intensity factor in the case when a crack tip penetrates a layered composite consisting of 60 layers, and show the limits of applicability of the anisotropic model of such composites.

A model describing the evolving shape of a growing pile is considered, and is shown to be equivalent to an evolutionary quasi-variational inequality. If the support surface has no steep slopes, the inequality becomes a variational one. For this case existence and uniqueness of the solution are proved.

We consider two-dimensional and axially symmetric critical-state problems in type-II superconductivity, and show that these problems are equivalent to evolutionary quasi-variational inequalities. In a special case, where the inequalities become variational, the existence and uniqueness of the solution are proved.

The self-similar source solution of the Barenblatt equation for elasto-plastic filtration through porous rock is known to be of the second kind. We determine the behaviour of the associated anomalous exponent and the profile of the solution in the limit of large compressibility and small elastic recovery of the rock.

We consider the non-local problem

It is found that for the case of decreasing f then: (i) for

there is a unique steady state which is globally asymptotically stable; (ii) for

then the problem can be scaled so that

in which case: (a) for λ < 8 there is a unique steady state which is globally asymptotically stable; (b) for λ = 8 there is no steady state and u is unbounded; (c) for λ > 8 there is no steady state and u blows up for all x, −1 < x, < 1. Some formal asymptotic estimates for the local behaviour of u as it blows up are obtained.

The diffraction of time-harmonic waves in a nonlinear medium with periodic structure is studied in this paper. In particular second harmonic generation – an important phenomenon in nonlinear optics-is modelled. The model, derived from a general nonlinear system of Maxwell's equations, is shown to have a unique solution for all but a discrete number of frequencies. The problem is solved numerically by combining a method of finite elements and a fixed-point iteration scheme. Numerical experiments for some simple grating structures are presented and discussed.

A general method is presented to facilitate the solution of a class of polydisperse spray problems in which a cloud of droplets can be described using a sectional or group model. The procedure involves replacing the original coupled droplet sectional variables conservation equations by a set of uncoupled sectional equations for auxiliary variables. The form of these latter equations is identical to that of the single spray equation for a quasi-monodisperse spray, solutions of which are more readily attainable even for multidimensional spray problems. Thus, these ready-made solutions can be exploited directly for the auxiliary variables, from which solutions can then be constructed in a straightforward manner for the desired original sectional variables. Three illustrative examples for spray diffusion flames with different features of complexity highlight the potential applicability of the proposed method, and indicate the sensitivity of flame characteristics to initial spray conditions and in-spray related phenomena.

In this work exact upper bounds of the drag force are found for curved plates of fixed total arc-length in flow with a wake, which forms according to the Joukowsky-Roshko-Eppler model. The shapes of maximum drag are determined in an explicit analytical form and can be interpreted as those of ideal parachutes.

We consider the propagation of an exothermic reaction wave which converts liquid reactants into a solid product. Such reaction waves are observed, for example in addition polymerization, where there is a propagating localized polymerization zone, in which monomer is converted to polymer. We study uniformly propagating travelling waves and their linear stability. We show that, though this problem is similar to the problem of gaseous combustion, which exhibits a hydrodynamic instability, here there is no such hydrodynamic instability. This theoretical result is consistent with experimental observations in the case when the polymerization process occurs under high external pressure conditions.

It is well-known that solutions to the one-dimensional supercooled Stefan problem (SSP) may exhibit blow-up in finite time. If we consider (SSP) in a half-line with zero flux conditions at t = 0, blow-up occurs if there exists T < ∞ such that limt↑Ts(t) > 0 and lim inft↑T⋅(t) = – ∞,s(t) being the interface of the problem under consideration. In this paper, we derive the asymptotics of solutions and interfaces near blow-up. We shall also use these results to discuss the possible continuation of solutions beyond blow-up.

Nonstationary two-dimensional filtration in a porous medium is considered, whereby part of the medium is saturated, another part is unsaturated but wet, and the remaining part is dry. The saturated/unsaturated and unsaturated/dry interfaces are free boundaries. It is shown that there exists a unique solution, and that the saturation function is continuous in the wet portion of the medium; this implies that the two interfaces are separated. Under some monotonicity-type conditions on the initial and boundary data it is shown that the free boundaries are continuous.

In the Calculus of Variations, several notions of convexity have emerged, corresponding to different properties of the functionals to be minimized. The relations between these various notions are not yet fully understood. In this context, we present a numerical study of quasiconvexity for some functions of the type f(ξ) = g(|ξ|2, det ξ), where Ξ is a 2×2-matrix. The corresponding global optimization problems are solved using a simulated annealing-like algorithm. The computations strongly indicate that the considered functions are quasiconvex if and only if they are rank-one convex. The relation to Morrey's conjecture, various applications and implementation problems are discussed.

In this paper we investigate the movement of free boundaries in the two-dimensional Hele-Shaw problem. By means of the construction of special solutions of self-similar type we can describe the evolution of free boundary corners in terms of the angle at the corner. In particular, we prove that, in the injection case, while obtuse-angled corners move and smooth out instantaneously, acute-angled corners persist until a (finite) waiting time at which, at least for the special solutions, they suddenly jump into an obtuse angle, precisely the supplement of the original one. The critical values of the angle π and π/2 are also considered.

We treat the mathematical properties of the one-parameter version of the Mróz model for plastic flow. We present continuity results and an energy inequality for the hardening rule, and discuss different versions of the flow rule regarding their relation to the basic laws of thermodynamics.

An analysis is made of the damping of sound and structural vibrations by vorticity production in the apertures of a bias flow, perforated elastic plate. Unsteady motion causes vorticity to be generated at the aperture edges; the vorticity and its energy are swept away by the bias flow and result in a net loss of acoustic and vibrational energy. In this paper we investigate the interaction of an arbitrary fluid-structure disturbance with a small circular aperture in the presence of a high Reynolds number, low Mach number bias flow. By considering the limit in which the aperture is small compared to the length scale of the impinging disturbance, it is shown that the effect of the interaction can be represented by a concentrated source in the plate bending wave equation consisting of a delta function and two of its axisymmetric derivatives. A generalized bending wave equation is then formulated for a plate perforated with an homogeneous distribution of small, bias flow circular apertures. This equation is used to predict the attenuation of sound and resonant bending waves by vorticity production. Acoustic damping is found to be significant provided the fluid loading is sufficiently small for the plate to be regarded as rigid (e.g. for an aluminium plate in air when the frequency is not too small). On the other hand, a bending wave is effectively damped only when the fluid loading is large enough for the wave to produce a substantial pressure drop across the plate; when this occurs the predicted attenuations are comparable with those usually achieved by the application of elastomeric damping materials. Numerical predictions are presented for steel and aluminium plates in air and water.

In this paper we construct formal large-time solutions of a model equation, with initial data possessing bounded support, describing transport of a reacting and decaying contaminant in a porous medium. This we do in one, two and three space dimensions where, depending on the reaction model used, the solution may or may not have bounded support for all time. In the former case, working with what we call the reduced equation, we prove convergence, in one space dimension, to an outer limit as t → ∞. The outer solution has to be supplemented by inner solutions valid near the edges of the support. These inner solutions take the form of decaying travelling waves which we analyse using phase plane methods. Using the travelling waves as sub- and super-solutions, we establish the large-time behaviour of the interfaces which we refine using asymptotic matching. These ideas can be formally extended to higher space dimensions where we deduce the shape of the support of the large-time profiles which turns out to be ellipsoidal. For reaction models where the support is unbounded we prove convergence of the solution of the reduced equation to a travelling and decaying fundamental solution of the linear heat equation with convection and absorption. Finally, we indicate how the results for the reduced equations can be formally embedded in an asymptotic analysis of the original model equation.