Mathematical model of the stress-strain state of multilayered structures with different elastic properties

The article examines the features of stress-strain state of structures made of two and multilayer elements. The relevance of use of multi-layer load-bearing structures in rapidly constructed protective structures in Ukraine under conditions of possible shock-explosive and fire damage is justified.The method of interpolating trigonometric Lagrange polynomials in mixed plane problems of theelasticity theory is described. In the problems of elasticity theoryfor the regionunder conditions of planar setting, two of the four boundary conditions are known. It is proposed permissive system of integral equations relative to an unknown pair of boundary conditions on each side of the area under consideration, based on the solution of theorem on reciprocity of work and reciprocity of displacements for a plane under the influence of a unit force.Solutions for each of the sides of region are built by developing the Multopa-Kalandia collocation method. The boundary conditions are the result of solving the system of integral equations. Features on the contour, namely points of application of unit forces, boundary breaks, etc., are taken into account by additional functions that are introduced into the kernels of integral expressions in the form of coefficients of the required boundary conditions.To represent the cumbersome functions of movements and stresses of multilayer structures in a compact form, the apparatus of the theory of functions of a complex variable is used, which leads to compact expressions that are convenient for programming. To obtain more accurate solutions and reduce the duration of computer programs, all ordinary and singular integrals given in the work are calculated analytically, that is, the system of integral equations to be solved is reduced to a system of linear algebraic equations with unknown density functions at the interpolation points.The proposed mathematical model provides the universality inherent in general numerical methods and allows to study the general stress-strain state of multilayer structures and local effects in zones of junction of layers with different elastic properties.