Uniqueness theorems for displacement fields with locally finite energy in linear elastostatics

Abstract

Uniquencess theorems are proved for the fundamental boundary value problems of linear elastostatics in bodies of arbitrary shape. The displacement fields are required to have finite strain energy in bounded portions of the bodies and satisfy the principle of virtual work. For bounded bodies, the total strain energy is finite and uniquencess is proved without additional hypotheses. In particular, no restrictions other than the energy condition are placed on the field singularities that may occur at sharp edges and corners. For unbounded bodies, uniqueness can be proved as in the bounded case if the total strain energy is finite. Sufficient conditions for this are shown to be the finiteness of the strain energy in bounded portions of the body together with the growth restriction % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWdraqaaiaabwhadaWg% aaWcbaGaaeyAaaqabaGccaGGOaGaaeiEaiaacMcacaqG1bWaaSbaaS% qaaiaabMgaaeqaaOGaaiikaiaabIhacaGGPaGaaeizaiaabIhacaqG% 9aGaaGimaiaacIcacaqGYbGaaiykaiaacYcacaqGYbGaeyOKH4Qaey% OhIukaleaacqGHPoWvdaWgaaadbaGaaeOCaiaacYcacqaH0oazaeqa% aaWcbeqdcqGHRiI8aaaa!5E73!\[\int_{\Omega _{{\text{r}},\delta } } {{\text{u}}_{\text{i}} ({\text{x}}){\text{u}}_{\text{i}} ({\text{x}}){\text{dx = }}0({\text{r}}),{\text{r}} \to \infty } \] on the displacement fieldu _i , where Ω_{r, δ} is the portion of the body that lies between concentric spheres with radiir andr+δ and δ>0.

This research was supported by the Air Force Office of Scientific Research. Reproduction in whole or part is permitted for any purpose of the United States Government.