Junction problem for Euler-Bernoulli and Timoshenko elastic inclusions in elastic bodies (Q2821877)

The authors study an elasticity problem for a 2D elastic body filling in a domain \(\Omega \) with Lipschitz boundary. This domain contains a line Euler-Bernoulli inclusion \(\gamma _{b}=(0,1)\times \{0\}\) and a line Timoshenko elastic inclusion \(\gamma _{t}=(1,2)\times \{0\}\). The authors consider the energy \(\pi (u,v,w,\phi )=\frac{1}{2}\int_{\Omega \setminus \overline{\gamma _{b}\cup \gamma _{t}\cup \{(1,0)}\}\sigma (u)\varepsilon (u)-\int_{\Omega _{\gamma }}fu+\frac{1}{2}\int_{\gamma _{b}}v_{xx}^{2}+\frac{ 1}{2}\int_{\gamma _{b}}\cup \gamma _{t}\cup \{(1,0)\}}w_{x}^{2}+\frac{1}{2} \int_{\gamma _{t}}\{\varphi _{x}^{2}+(v_{x}+\varphi )^{2}\}\), where \( \varepsilon \) is the linearized strain tensor, \(\sigma \) is the stress tensor which is linked to \(\varepsilon \) through a fourth-order tensor which satisfies the usual symmetry, continuity and coercivity properties, and \( f\in L^{2}(\Omega )^{2}\). Using classical arguments, the authors first prove that this energy has a unique minimizer on the space \( W=\{(u,v,w,\varphi )\in H_{0}^{1}(\Omega )^{2}\times H^{1}(\Omega )\cap H^{2}(\gamma _{b})\times H^{1}(\Omega )\times H^{1}(\gamma _{t})\); \(v=u\cdot \nu \), \(w=u\cdot \tau \) on \(\gamma _{b}\cup \gamma _{t}\cup \{(1,0)\}\), \( v_{x}(1-)+\varphi (1+)=0\}\) and they write the equilibrium problem associated to this energy. They also prove the equivalence between these two problems and they write the junction conditions which occur at the extremities of the line inclusions. The authors then suppose that the Euler-Bernoulli inclusion is delaminated and they write the associated problems (energy and equilibrium) for which they prove an equivalence result. They then introduce a rigidity parameter \(\lambda >0\) in this last equilibrium problem and, proving uniform estimates on the corresponding solution, they compute the limit of this solution when \(\lambda \rightarrow \infty \). Finally, they compute the limit of the solution when \(\lambda \rightarrow 0\).