Decaying states of plane strain in a semi-infinite strip and boundary conditions for plate theory
چکیده
Friedrichs and Dressler and Gol’denveiser and Kolos have independently shown that the classical plate theory of Kirchhoff is the leading term of the outer expansion solution (in a small thickness parameter) for the linear elasto-statics of thin, flat, isotropic bodies. As expected, neither this leading term nor the full outer solution alone is able to satisfy arbitrarily prescribed edge conditions. On the other hand, the inner solution, which is significant only near the edge, is determined by a sequence of boundary value problems which are very difficult to solve, nearly as difficult as the original problem. For stress edge-data, St. Venant’s principle may be invoked to generate a set of stress boundary conditions for the classical plate theory as well as for some higher order terms in the outer expansion without any reference to the inner solution. Attempts in the literature to derive the corresponding boundary conditions for displacement edge-data have not been successful. With the help of the Betti-Rayleigh reciprocity theorem, we have derived the correct set of boundary conditions for classical and higher order plate theories with arbitrary edge-data. In this paper, we work out these conditions for an infinite plate strip with edgewise uniform data. We show that the conditions for individual terms in the outer expansion may be summed to give a simple set of appropriate boundary conditions for the full outer solution at the mid -plane. The boundary conditions obtained for the semi-infinite plate case a r e rigorously correct and the result for the stress data case rigorously justifies the application of St. Venant’s principle. Applications of the displacement boundary conditions obtained are illustrated by two simple problems: (i) The shearing of an infinitely long rectangular block, and (ii) A clamped infinite plate strip under uniform face pressure. Abklingende ebene Dehnungszustände in einem halb-unendlichen Streifen und Randbedingungen fur die Plattentheorie (Zusammenfassung) Friedrichs und Dressler sowie Gol’denveiser und Kolos haben unabhängig voneinander gezeigt, dass das erste Glied der durch äussere Entwicklung (nach einem kleinen Dickeparamter) gewonnenen Lösung für die lineare Elastostatik dünner ebener, isotroper Körper zur klassischen Kirchhoffschen Plattentheorie führt. W i e erwartet k a n n weder dieses erste Glied noch d i e vollständige aussere Lösung allein willkürlich vorgegebene Randwerte erfüllen. Andererseits ist die innere Lösung die nur in Randnähe von Bedeutung ist, durch eine Folge von Randwertproblemen bestimmt, die sehr schwer zu lösen sind, nahezu ebenso schwer wie + The research is partly supported by NSERC Operating Grant No. A9259 and, in the case of the second author, also by a UBC Killam Senior Fellowship.