EXTENSION OF AN ENERGY THEOREM OF BARENBLATT'S TO ELASTODYNAMICS

In some ixoblems of the mechanics of continuous media one encctufcers the situ-aticm that the soltfdcn of the differenelal equations of the problem, satisfy-ing initial and botmdary ccmditicms, is determined only to within one or nDre arbitrary paL~=ters. In order to determine the appropriate value of these parameters for a given suc/h ~...-oblem ore u~l ly makes use of subsiai~vy con-ditdnns such as the finiteness of velocities, stresses, etc. For elastostatic problems, Barenblatt [i] has given a theorem to establish these additional cmmditicms which is an extensicm of the principle of minimum potentlal energy. ~hese conditions L...-ovide the equations to be a p ~ to the ~ value problem in c~er to obtain a cumplete solution to the problem. Barenblatt illustrates the use of his theorem to rederive such o~litlc~s for a contact ~oblem and a cra~k problem; these results previously had been established by re~,i~ing stresses to remain finite. In the present note we generalize Barenblatt's result to co~er a class of ela-s t i c problems. The problems included are those for which the system is in a s t ~ state in the sense that the to1-~l k/netic energy, the total strain energy and the w~rk dome by the external forces are independent of time. Extemsicm of Barenblatt's ~KEem to the S State Elasto- "c Case Ore%sider a dynanic, linear, elastic systan which is in a stead ~ state in the sense that the total k/netic energy, the total ~train energy and the total w3rk done by the external forces are finite constants*. Note that if a sys-ten is in a S t~y state acucrdirg to the us~l de f in i t i cm.,ni%ich re~,ires that the velocity of ~ point of the ~ be t ime-~t, then the system surface tractions and the body forces at a ~aterial point are assumed to be functions of its position only. Scientific Communicatio