A time-dependent obstacle problem in linearised elasticity
The author deals with a unilateral hyperbolic problem for an elastic body moving against a plane obstacle. It has the following variational formulation: \begin{align*} &\text{Find}\ \boldsymbol{u}\equiv (u_i):(0,T)\to\boldsymbol{V}(\varOmega) \ \text{s.t.}\\& \boldsymbol{u}\in L^\infty(0,T;\boldsymbol{U}(\varOmega)),\ \dot{\boldsymbol{u}}\in L^\infty(0,T;\boldsymbol{L}^2(\varOmega)),\ \ddot{\boldsymbol{u}}\in \mathcal{M}([0,T];\boldsymbol{L}^2(\varOmega)), \\ & 2\rho\langle\langle\text{d}\ddot{\boldsymbol{u}}(t),\boldsymbol{v}(t)-\boldsymbol{u}(t)\rangle\rangle_{\boldsymbol{L}^2(\varOmega))}+\int_0^T \int_\varOmega A^{ijk\ell}e_{k||\ell}(\boldsymbol{u}(t))e_{i||j}(\boldsymbol{v}(t)-\boldsymbol{u}(t))\,\text{d}x\text{d}t\ge\\ & \int_0^T \int_\varOmega f^i(t)(v_i(t)-u_i(t))\,\text{d}x\text{d}t\ \forall \boldsymbol{v}\in \mathcal{D}(0,T;\boldsymbol{V}(\varOmega))\ \text{s.t.} \ \boldsymbol{v}(t)\in \boldsymbol{U}(\varOmega)\ \text{for a.e.}\ t\in (0,T)\\ &\text{and}\quad \boldsymbol{u}(0)=\boldsymbol{u}_0,\ \dot{\boldsymbol{u}}(0)=\boldsymbol{u}_1. \end{align*} The existence theorem is verified using the penalization method and rather strong assumption on the uniform estimate of penalized solutions.
scientific article; zbMATH DE number 7187775
scientific article