Asymptotic modeling of contact interaction between a system of rigidly combined punches and an elastic fundament (Q2714006)

The author studies the contact of an elastic layer with a system of ideal rigidly paraboloid punches. With boundary value conditions of the one-side contact, the problem is stated as follows: NEWLINE\[NEWLINE \begin{gathered} L(\nabla_x)\mathbf u(\varepsilon;\mathbf x) \equiv - \mu\nabla_x\cdot\nabla_x \mathbf u(\varepsilon;\mathbf x) - \frac{\mu}{1 - 2\nu}\nabla_x\nabla_x\cdot \mathbf u(\varepsilon;\mathbf x) = 0, \quad\mathbf x\in \mathbb R^2\times (0,H); \\ \sigma_{31}(\mathbf u;\mathbf x',0) = \sigma_{32}(\mathbf u;\mathbf x',0) = 0,\quad \mathbf x' = (x_1,x_2)\in \mathbb R^2; \quad \sigma_{33}(\mathbf u;\mathbf x',0) = 0,\quad \mathbf x'\notin \Gamma(\varepsilon); \\ u_3(\varepsilon;\mathbf x',0)\geq \delta_j - \Phi^j(y_1^j,y_2^j),\quad \sigma_{33}(\mathbf u;\mathbf x',0) \leq 0, \\ [u_3(\varepsilon;\mathbf x',0) - \delta_j + \Phi^j(y_1^j,y_2^j)] \sigma_{33}(\mathbf u;\mathbf x',0) = 0, \quad \mathbf x'\in \omega_j(\varepsilon)\quad (j = 1,\dots, N); \\ \mathbf u(\varepsilon;\mathbf x', H) = 0,\quad \mathbf x'\in\mathbb R^2. \end{gathered} NEWLINE\]NEWLINE Here \(L(\nabla_x)\) is the Lamé operator; \(\sigma_{3i}(\mathbf u)\) are the components of the stress tensor that corresponds to the vector of displacements \(\mathbf u = (u_1,u_2,u_3)\); \(\Gamma(\varepsilon)\) is a set containing the so-called spots of contact.NEWLINENEWLINENEWLINEUsing the method of matching the asymptotic expansions, the author constructs the formal asymptotic of a solution to the above-mentioned problem. Next, the author obtains a variational inequality for the resulting nonlinear problem and proves an existence and uniqueness theorem.