The equation

is a possible dimensionless version of a model for the configuration of a very long strut resting on a nonlinear elastic foundation with axial loading P. By seeking to establish the existence of homoclinic orbits connecting the zero equilibrium of (*), now regarded as defining a four dimensional dynamical system, to itself one is pursuing the so-called ‘dynamical phase-space analogy’ for the spatial configuration suggested by the form of the equation. The existence of homoclinic solutions is then interpreted as indicating the presence of spatially localized buckling of the deformed strut.

A simplified version of Reissner's theory of thin shells of revolution suffering small strains but arbitrarily large deflections and rotations reduces, when specialized to axi-symmetric deformations of annular membranes under a vertical surface load, to a nonlinear ordinary differential equation which is free of Poisson'ratio. Within this framework the questions of existence and non-existence of non-negative solutions of the associated stress and displacement boundary value problem are brought to a final answer. Progress in this direction was made in an earlier study (Grabmüller & Pirner 1987). In this paper a continuous monotone curve is constructed which effects a subdivision of the respective ranges of boundary data into complementary domains of existence and non-existence of strictly positive solutions

This paper is concerned with the mathematical study of a nonlinear system modelling an irreversible phase change problem. Uniqueness of the solution is proved using the accretivity of the system in (L1)2. Expressing one of the two unknowns as an explicit functional of the other reduces the system to a single nonlinear evolution equation and ultimately leads to an existence theorem.

In this paper the existence and uniqueness of the solution of a nonlinear system modelling some irreversible phase changes is established.

This note deals with a new model of elastic–perfectly plastic materials in which the yield stress is regarded as a threshold above which plastic flow occurs rather than a constraint which cannot be violated. This modelling change allows us to treat a number of signalling and impact problems not solvable within the classic framework of elastic–perfectly plastic materials.

We discuss the self-similar solutions of the second kind associated with the propagation of turbulent bursts in a fluid at rest. Such solutions involve an eigenvalue parameter μ, which cannot be determined from dimensional analysis. Existence and uniqueness are established and the dependence of μ on a physical parameter λ in the problem is studied: estimates are obtained and the asymptotic behaviour as λ → ∞ is established.

We describe a specific measurement process that works well in practice for locating steel reinforcement bars in concrete. For the case that the total volume of these bars is small, we derive an approximate linear, but sufficiently accurate, mathematical model for which we can prove a uniqueness result.

In this paper we investigate the mathematical model of the equilibrium of a finite volume in ℝn (n = 1,2, 3) of a two-phase continuous medium, under the assumption that each pure phase is an isotropic elastic solid. The main results in this paper are:

(i) the existence and uniqueness of a solution of this mathematical model;

(ii) a discussion of the stress-strain law associated with the free energy of this two-phase continuous medium, which is multiple-valued due to the non-smoothness of the Gibbs potential (complementary energy);

(iii) a description of the structure of solutions in plane strain.

The diffraction of a plane sound wave in a fluid by an adjacent elastic solid containing a surface flaw is analysed using ray techniques. By solving the eikonal equation with suitable boundary data, the pattern of the rays leaving the boundary and propagating into the fluid and solid respectively is established, with the corresponding amplitudes being furnished by the appropriate system of transport equations. For the acoustic and elastic cylindrical bulk waves that emanate from the flaw itself, the amplitude directivities cannot be found from this ray analysis alone.

Low permeability formations are often hydrofractured to increase the production rate of oil and gas. This process creates a thin, but highly permeable, fracture which provides an easy path for oil and gas to flow through the reservoir to the borehole. Here we examine the payoff of hydrofracturing by determining the increased production rate of a hydrofractured well. We find explicit formulas for the steady production rate in the three regimes of small, intermediate, and large (dimensionless) fracture conductivities. Previously, only the formula for the large fracture conductivity case was known.

We assume that Darcy flow pertains throughout the reservoir. Then, the steady fluid flow through the reservoir is governed by Laplace's equation with a second-order boundary condition along the fracture. We first analyze this boundary value problem for the case of small fracture conductivities. An explicit formula for the production rate is obtained for this case, essentially by combining singular perturbation methods with spectral methods in a function space which places the second-order boundary condition on the same footing as Laplace's equation. Next, we re-cast Laplace's equation as a variational principle which has the second-order boundary condition as its natural boundary condition. This allows us to use simple trial functions to derive accurate estimates of the production rate in the intermediate conductivity case. Then, an asymptotic analysis is used to find the production rate for the large fracture conductivity case. Finally, the asymptotic and variationally-derived production rate formulasare compared to exact values of the production rate, which have been obtained numerically.

It may be feasible to create more than a single fracture about a borehole. So we also develop similar asymptotic and variational formulas for the production rate of a well with multiple fractures.

This paper deals with the motion of a fluid in a closed loop under the effect of natural convection and a given external heat flux. More precisely, we show that the stationary solutions of a system describing the intermediate asymptotics of the previous problem are structurally linearly unstable.

A concentrated capacity problem is posed for the heat equation in a multidimensional domain. In the concentrated capacity (i.e. in a portion of the boundary of the domain) a change of phase takes place, and a Stefan-like problem is posed. This scheme has been introduced in the literature as the formal limiting case of a certain class of diffusion problems.

Our main result is a theorem of continuous dependence of the solution on the data. It is also used to prove the existence of the solution (in a weak sense), assuming only integrability of the data. The solution is found as the limit of the solutions of the approximating problems.