Saint-Venant's principle on unbounded regions

Abstract

SynopsisWe consider an anisotropic non-homogeneous linear elastic material in equilibrium and occupying an open region with non-compact boundary. In both the linearised and classical linear theories the asymptotic behaviour of the solution is determined and a clear relationship established with Saint-Venant's principle on such regions. Although the treatment is discussed with special reference to elasticity, it is equally applicable to general systems of elliptic differential equations, and thus reveals a relationship with the classical theorems of Phragmen-Lindelof and Liouville.

Spatial Behaviour in Thermoelastostatic Cylinders of Indefinitely Increasing Cross-Section

“…Such studies, clearly related to Saint-Venant's principle, are significant for investigations into boundary effects and similar phenomena. In elasticity, both growth and decay estimates have been derived for nonprismatic and prismatic finite and semi-infinite cylinders (e.g., [1,2]), and for cone-like regions (e.g., [5]) unbounded in a given direction. The method of derivation, however, does not readily extend to thermoelasticity, although the cylinder has been successfully treated by separate weighting of the mechanical and thermal energies [6].…”

Spatial stability in linear thermoelasticity

“…Moreover, subject to conditions (5.7) and (5.8), the relations (5.4) and (5.9) generate the upper bound 13) which is of later use. The next task is to obtain an upper bound for the absolute value of H(x 3 ) in terms of H (x 3 ).…”

“…For other types of unbounded bodies, Poincaré's inequality is replaced by Wirtinger's inequality in the construction of a differential inequality with respect to measures taken over spherical or other suitable curvilinear surfaces. An application to isothermal elasticity presented in [13] demonstrates that decay is characteristically algebraic and not exponential, but whether such behaviour generally occurs in corresponding thermoelastic problems awaits clarification. Exact solutions, however, derived in [15] to certain problems in isotropic thermoelasticity confirm that for these particular problems the decay is algebraic.…”

On Saint-Venant's principle in a poroelastic arch-like region

“…For unbounded wedge-shaped domains, the decay is shown to be of power law type rather than exponential. Issues related to those just described have been established in linear elasticity for cone-like domains by Knops et al [27]. This paper considers a curvilinear strip in the form of an arch-like region R described in the polar coordinates r and by R : a<r<b, 0< < , where a, b, and (<2 ) are prescribed positive constants.…”