Conservation properties for plane deformations of isotropic and anisotropic linearly elastic strips (Q1330767)

The authors have established two conservation properties of the stresses generated in an homogeneous anisotropic elastic strip occupying the space \(-\ell< x_1< \ell\), \(0< x_2<h\). The stress free conditions on the ends \(x_2=0\) and \(x_2=h\) are \(\tau_{22}= \tau_{12}=0\) for \(-\ell< x_1< \ell\). It is further assumed that the end conditions at \(x_1= \pm\ell\) are such that if the stress conditions at \(x_2=0\) and \(x_2=h\) are expressed in terms of the Airy stress function \(\varphi\), then, integrated with respect to \(x_1\), they reduce to \(\varphi (x_1, 0)= \varphi(x_1, h)= \varphi_{12} (x_1, 0)= \varphi_{12} (x_1, h)=0\). This actually is the case when the loads are self-equilibrated. Two conservation properties under the above conditions are \(\int^h_0[\beta_{11} \varphi^2_{,22}- \beta_{22} \varphi^2_{,11}- 2(\beta_{12}+ \beta_{66}) \varphi^2_{,12}+ 2\beta_{22} \varphi_{,1} \varphi_{,111}+ 4\beta_{26} \varphi_{,12} \varphi_{,11} ]dx_2= c_3\) and \(\int^h_0 [\beta_{11} \varphi^2_{,221}+ (2\beta_{12}+ \beta_{66}) \varphi^2_{,112}+ 2\beta_{11} \varphi_{,112} \varphi_{,222}- 4\beta_{16} \varphi_{,112} \varphi_{,122} ]dx_2= c_4\). These results can also be expressed in terms of stresses.