Real Potentials of Elasticity Theory

Consider the elastostatics equations for a region D, finite or infinite: 1.1% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq % GHciITcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOa % IyRaamiEamaaBaaaleaacaWGQbaabeaaaaGccqGH9aqpcaaIWaGaai % ilaiaaywW7caWGPbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaa % iodaaaa!484E!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\frac{{\partial {\sigma _{ij}}}}{{\partial {x_j}}} = 0,\quad i = 1,2,3$$, where σ _ij are components of a stress tensor in the global co-ordinates which are assumed to be Cartesian. Here we use Einstein’s summation rule. The components of the stress tensor are connected to the components of the strain tensor by Hooke’s law: 1.2% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS % baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadogadaWgaaWcbaGa % amyAaiaadQgacaWGRbGaamiBaaqabaGccqaH1oqzdaWgaaWcbaGaam % 4AaiaadYgaaeqaaOGaaiilaiaaywW7caWGPbGaaiilaiaadQgacqGH % 9aqpcaaIXaGaaiilaiaaikdacaGGSaGaaG4maaaa!4CCF!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\sigma _{ij}} = {c_{ijkl}}{\varepsilon _{kl}},\quad i,j = 1,2,3$$, where c _ijkl = c _ijlk = c _klij are elastic constants; ε _kl are the components of the strain tensor: 1.3% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS % baaSqaaiaadUgacaWGSbaabeaakiabg2da9maalaaabaGaaGymaaqa % aiaaikdaaaWaaeWaaeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWcba % Gaam4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadYgaaeqa % aaaakiabgUcaRmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGSb % aabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaam4AaaqabaaaaaGc % caGLOaGaayzkaaGaaiilaaaa!4D9A!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\varepsilon _{kl}} = \frac{1}{2}\left( {\frac{{\partial {u_k}}}{{\partial {x_l}}} + \frac{{\partial {u_l}}}{{\partial {x_k}}}} \right),$$, u _k are the components of the displacement vector u. Substitution of (1.3) into (1.2) and the result into (1.1) leads to the complete system of elasticity theory in terms of displacements: 1.4% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa % aaleaacaWGPbGaamOAaiaadUgacaWGSbaabeaakmaalaaabaGaeyOa % Iy7aaWbaaSqabeaacaaIYaaaaOGaamyDamaaBaaaleaacaWGRbaabe % aaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGccqGHciIT % caWG4bWaaSbaaSqaaiaadYgaaeqaaaaakiabg2da9iaaicdacaGGUa % aaaa!48DD!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${c_{ijkl}}\frac{{{\partial ^2}{u_k}}}{{\partial {x_j}\partial {x_l}}} = 0.$$.