A mathematical model of a contact interaction between a plate and rigid stamp is derived taking into account physical and design details. The plate is considered to have a crack, that changes its form. The problem of the contact is evaluated based on the theory of variational inequalities. The shape of the stamp is assumed to be perpendicular to the plate surface and the Poisson’s ratio is between 0 and 0.5. Analytical formulation of the study consists of transformation equation, boundary conditions and integral equation. The result is used in maximization and minimization problems for choosing extremal shape of the vertical break in the plate.
This study presents mathematical model of the internal waves and examines wave propagation in a two-layer fluid flow. Elements of the functional-analytical approach are used to develop the model. A flat unsteady motion of a two-layer liquid under a cover over a flat bottom is considered. The fluid is assumed to be ideal and incompressible. Internal waves are caused by external pressure application to the interface between the layers, oscillation of the flat bottom and disturbances in the flow. The power of the wave source is characterized by dimensionless parameter ε. The problem is formulated, and its solution is based on asymptotic analysis for 0<ε<1.
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