The displacement initial-boundary value problem in bending of thermoelastic plates weakened by cracks (Q947564)

The authors consider the initial-boundary value problem in a vector form with the edge-of-the-crack Dirichlet boundary conditions for bending of a thermoelastic plate \[ \begin{aligned} &\mathbb{B}_0\partial_t^2\mathbf{U}(x,t)+\mathbb{B}_1\partial_t\mathbf{U}(x,t)+\mathbb{A}\mathbf{U}(x,t) =\mathbf{Q},\;(x,t)\in S\times (0,\infty),\\ &\mathbf{U}(x,0)=\partial_t\mathbf{U}(x,0)=0,\;x\in S,\\ &\mathbf{U}^+(x,t)=\mathbf{F}^+(x,t),\;\mathbf{U}^-(x,t)=\mathbf{F}^-(x,t),\;(x,t)\in \partial S_0\times (0,\infty),\end{aligned} \] \(\mathbf{U}=(\mathbf{u}^T,u_4)^T,\;\mathbf{u}=(u_1,u_2,u_3)^T,\;u_4(x,t)=\frac{12}{h^3}\int_{-h/2}^{h/2}x_3\theta(x,x_3,t)dx_3,\;x=(x_1,x_2),\;\) \( \mathbb{B}_0=\text{diag}\{\frac{\rho}{12}h^2,\frac{\rho}{12}h^2,\rho,0\}\;\) and the matrices \(\mathbb{B}_1,\;\mathbb{A}\) contain the first-order and the second-order partial derivatives with respect to \(x_1,\;x_2\) respectively. The crack is modeled by an open connected arc \(\partial S_0\subset \partial S.\) The unique solvability in spaces of distributions is proved by means of combination of the Laplace transformation and variational methods.