R. J. Knops

Asymptotic behaviour of solutions to semi-linear elliptic equations on the half-cylinder

Various, but closely related, forms of a Phragmén-Lindelöf principle are established for certain classes of semi-linear elliptic equations (including systems) defined on a half-cylinder. For example, it is shown that under either homogeneous Dirichlet or Neumann lateral boundary conditions, the smooth solution either fails to exist globally, or when it does exist globally it must tend asymptotically to zero with increasingly large distance along the cylinder from the base. For uniformly elliptic equations, the decay to zero is at least exponential. The method employed relies upon cross-sectional estimates and is additionally applicable to both the finite and whole cylinder. In the latter case, global existence fails for all non-trivial solutions.

The effect of a variation in the elastic moduli on Saint-Venant's principle for a half-cylinder

A semi-infinite prismatic cylinder composed of a linear anisotropic classical elastic material is in equilibrium under zero body force and either zero displacement or zero traction on the lateral boundary. The elastic moduli become perturbed. Under suitable conditions on the base load decay estimates are derived for the difference between corresponding quantities in the unperturbed and perturbed bodies. The amplitude in each estimate involves a multiplicative factor that tends to zero as the perturbation tends to zero. The analysis, based upon a first-order differential inequality, introduces apparently new modifications of Korns inequalities of the first and second kind.

On Saint-Venant’s Principle for Elasto-Plastic Bodies

This paper concerns Zanaboni’s version of Saint-Venant’s principle, which states that an elongated body in equilibrium subject to a self-equilibrated load on a small part of its smooth but otherwise arbitrary surface, possesses a stored energy that in regions of the body remote from the load surface decreases with increasing distance from the load surface. We here prove this formulation of Saint-Venant’s principle for elastic-plastic bodies. The present proof, which for linear elasticity considerably simplifies that developed by Zanaboni, depends crucially upon the principle of minimum strain energy to obtain a fundamental inequality that leads to the required result. Differential inequalities are not involved. The conclusion is not restricted to cylinders but is valid for plastic bodies of general geometries. Although no conditions are imposed on the plastic theories discussed here, counter-examples indicate that in certain circumstances the fundamental inequality, and hence the result, may be valid only for restricted data that includes the body’s minimum length.

Uniqueness and continuous data dependence in the elastic cylinder

Selected continuous data dependence is established for the linearised elastic cylinder under relaxed positive-definite and homogeneity conditions on elasticities in the superposed linear deformation and for constrained sets of the solution. The proofs are based on those in [Math. Proc. Cambridge Philos. Soc. 103 (1988) 535] and employ a Lagrange identity obtained from Bettis reciprocal theorem. The treatment is intended only to be indicative of the method and is restricted to dependence on body force, certain base data, and the superposed elasticities in the half-length of the finite cylinder. Estimates are also presented for the semi-infinite cylinder under alternative boundary conditions. Uniqueness of the solution is also established.

Continuous dependence theorems in the theory of linear elastic materials with microstructure

Abstract It is shown that the solution to the equations of a linear micropolar elastic solid (see Eringen [1]), in an exterior domain in R3, depends continuously on initial and boundary data, body forces and material coefficients. The linear dipolar elastic theory of Green and Rivlin [2] is also briefly examined.

Zanaboni's treatment of Saint-Venant's principle

Zanabonis version of Saint-Venants principle, which concerns an elastic body subjected to self-equilibrated loads distributed over a part of its surface, states that the stored energy tends to zero in regions increasingly remote from the load surface. Unlike other formulations, Zanabonis version applies to bodies of arbitrary shape. Here, unnecessarily complicated aspects of the proof are simplified and rendered mathematically precise by appeal to a variant minimum principle. For cylindrically shaped bodies, a new decay estimate is derived that supplements Zanabonis contributions and confirms that his approach provides a viable alternative to other studies of Saint-Venants principle. In conclusion, various generalizations of the original results are briefly discussed, including an extension to nonlinear elasticity.

Uniqueness and Complementary Energy in Nonlinear Elastostatics

Global uniqueness of the smooth stress and deformation to within the usual rigid-body translation and rotation is established in the null traction boundary value problem of nonlinear homogeneous elasticity on a n-dimensional star-shaped region. A complementary energy is postulated to be a function of the Biot stress and to be para-convex and rank-(n-1) convex, conditions analogous to quasi-convexity and rank-(n-2) of the stored energy function. Uniqueness follows immediately from an identity involving the complementary energy and the Piola-Kirchhoff stress. The interrelationship is discussed between the two conditions imposed on the complementary energy, and between these conditions and those known for uniqueness in the linear elastic traction boundary value problem.

Elements of Elastic Stability Theory

General notions of stability are probably well understood nowadays, especially the refinements encountered in applications to continuum mechanics and elastic theory. Movchan (1960 a,b) was the first to extend Liapunov’s original approach to continuous systems but difficulties encountered for nonlinear elasticity, considered in these lectures, in part account for the continuing popularity of other methods for investigating stability properties. These include The static energy criterion, The adjacent equilibrium method, Spectral analysis, Singular perturbations.

Alternative spatial behaviour in the incompressible linear elastic prismatic constrained cylinder

Differential inequalities are derived for two cross-sectional energy fluxes. Integration establishes exponential growth and decay estimates for the cross-sectional mean square displacement, displacement gradient and pressure. The conclusions relate to Saint--Venants principle for an incompressible elastic cylinder and more generally to the Phragmén--Lindelöf principle and Liouvilles theorem. Other contributions to the literature are briefly noted.

Filon's construct for dislocations and related topics

A unified account is presented of the relationships in plane homogeneous isotropic linear elasticity between dislocations, thermoelasticity, inclusions, and the variation of Poissons ratio. While basic principles are emphasised, there is also indicated how solutions in one theory may be used to generate solutions in another. The connexion between dislocations and the variation of Poissons ratio, and between dislocations and thermoelasticity are due respectively to Filon and Mushkelishvili.