APPLICATION OF GREEN’S THEOREM IN SOLID ELECTROSTATIC AND STRESS FIELDS

The physical quantity in a given region of static electric and stress fields is combined with their system energy, as functional. When their variables are equal to zero, these systems are located in an energy valley or local minima, and, therefore, are stable. Important results can be derived by applying Greens` theorem such as; for instance, Poison’s equation can be applied with its boundary conditions in static electric fields. After division of an arbitrary area of the field using FEM, the potentials on the nodes can be determined by calculating the variable of functional. Thus, Van der Puaw ` s equation1] is deduced on the boundary of a square sample and a cross sample by using the variable principle2]. Initially, Van der Puaw `s equation was deduced in arbitrarily shaped samples using conformal transformation1]. In addition, it is rare to use the variable principle based on Green`s theorem for mechanical system. There have been extensive discussions spanning a century3-11] among mechanical scientists and engineers on stress distribution in thick rings loaded by opposing forces along the diameter. However, these solutions are complicated. Using the variable principle, the sesolutions11] can be simplified with clear physical means. Nevertheless, the challenge of stress analysis in rings loaded by multiple-fold symmetrical forces is daunting. Consequently, associated literature is scare. Therefore, when setting variables of a functional for normal strain energy to zero, the energy valley condition will be realized, resulting in a restrain for normal stress distribution within several periods along the boundary. Finally， practical real stress solutions are obtained, after combining with high energy solutions11] for Airy equation12]. These practical real stress solutions not only fulfill the energy valley theorem, but are also consisting with photo-elasticity13] results