Maximum principles and pointwise error estimates for torsion of shells of revolution

Abstract

This paper is concerned with the question of assessing the quality of approximate thin shell solutions for the problem of axisymmetric torsion of elastic shells of revolution, an issue previously considered by Ho and Knowles [1]. In both works, the objective is to obtain pointwise estimates, based on the threedimensional theory of elasticity, for the errors involved in using an approximate shell theory.The mathematical problem is that of obtaining explicit pointwise gradient estimates for the solutions of homogeneous and nonhomogeneous second-order uniformly elliptic equations, an issue of interest beyond the present context. In contrast to arguments using energy inequalities, here we apply a technique based on maximum principles for such equations.The particular examples of cylindrical, spherical and conical shells of revolution are discussed in detail. In the former two cases, it is shown that the shell theory solution is in fact an exact elasticity solution in the Saint-Venant sense. Asymptotic error estimates for general shells of revolution are also given. RI~SUMIEL'6tude qi-dessous traite de la qualit6 de l'approximation des coques minces appliquEe aux probl~mes de torsion asym6trique des coques 61astiques de r6volution, probl6me d6j~ consid6r6 par Ho et Knowles [1].L'objectif des deux 6tudes est d'6valuer en chaque point, par la th6orie de l'61asticit6 en trois dimensions, les erreurs cons6quentes h l'utilisation de la th6orie de l'approximation des coques.Le probl~me math6matique consiste h obtenir une estimation explicite en chaque point des gradients pour les solutions d'6quations elliptiques uniformes du deuxi~me ordre, homog~nes et non-homog6nes, probl~me d'un inter& qui d6passe le pr6sent contexte. Nous utilisons ici une technique bas6e sur le principe du maximum pour de telles 6quations, par contraste avec les raissonnements utilisant des in6galit6s 6nerg6tiques.Les exemples particuliers des coques de r6volution cylindriques, sph6riques et coniques sont discut6s en d6tail. Dans les deux premiers cas, nous montrons que la solution par la th6orie des coques est en fait une solution exacte dans le sens de Saint-Venant. Des estimations asymptotiques des erreurs dans le cas de coques de r6volution plus g6n6rales sont 6galement donn6es.

The effect of nonlinearity on a principle of Saint-Venant type

“…In order to arrive at the appropriate boundary condition at x~ = ~, we first observe that, if f(x~) were constant for 0--< x2--< h, the function u(xl, xz) = kx~ (4.4) would satisfy the differential equation (3.2), the free surface conditions (4.1), and the end condition (4.2), provided the constant k is chosen so that 13 See p. 203 of [11]. In order to arrive at the appropriate boundary condition at x~ = ~, we first observe that, if f(x~) were constant for 0--< x2--< h, the function u(xl, xz) = kx~ (4.4) would satisfy the differential equation (3.2), the free surface conditions (4.1), and the end condition (4.2), provided the constant k is chosen so that 13 See p. 203 of [11].…”

“…Bearing (5.1), (5.4) in mind, we find that the linearized version of the differential equation ( 20 v(p, q) is found for hardening and for softening materials in §7. Similar techniques are employed in [12], [13] to obtain explicit spatial decay results for gradients of solutions of some linear second-order problems. Similar techniques are employed in [12], [13] to obtain explicit spatial decay results for gradients of solutions of some linear second-order problems.…”

Maximum principles and bounds on stress concentration factors in the torsion of grooved shafts of revolution

“…On On which is sufficient for obtaining a lower bound for ~'max-A more elaborate generalization of Theorem 2 would consist in establishing a version of the theorem in a weak formulation, involving less stringent smoothness hypotheses on ~b, thereby facilitating the construction of suitable comparison functions. Such weak maximum principles were employed successfully in elasticity problems in [18,19] for example. We defer these considerations at the present time.…”