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On the Iterlayer Contact Conditions in Multilayer Thin Body Theory and Some Issues of Splitting Initial-Boundary Value Problems

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Abstract

The interlayer contact conditions for various connections of adjacent layers of a multilayer body are written out. The formulations of initial-boundary value problems are discussed. Tensors-operators of cofactors to the tensor-operator of the equations of motion of the elasticity theory in the displacements of an isotropic homogeneous material and to the stress-operator are obtained, which allow us to decompose the equations and boundary conditions, respectively. From the decomposed equations of the classical theory of elasticity, the corresponding decomposed equations of the static (quasistatic) problem of the theory of prismatic bodies of constant thickness in displacements are obtained. From them the equations in the moments of unknown vector-functions with respect to any systems of orthogonal polynomials are derived. As a special case, the system of equations of the fifth approximation in moments with respect to the system of Legendre polynomials is obtained. Moreover, these equations are derived both without taking into account the boundary conditions on the front surfaces, and with these conditions. The system of equations splits into two systems. One of them is a system with respect to the moments of even orders of the unknown vector-function, and the other one is a system with respect to moments of odd orders of the same functions. Moreover, the matrix differential operators of these systems have a triangular form and their determinants are different from zero. Hence, these systems are consistent. Based on the constructed operator of cofactors to the operator of one of these systems it is decomposed and for each moment of the unknown vector-function we get an elliptic equation of high order (the order of the system depends on the order of approximation), whose characteristic roots are easily found.

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Funding

This work was published with the financial support of the Shota Rustaveli National Science Foundation (project no. FR-21-3926).

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Authors and Affiliations

  1. Lomonosov Moscow State University, 119991, Moscow, Russia

    M. Nikabadze

  2. Bauman Moscow State Technical University, 105005, Moscow, Russia

    M. Nikabadze

  3. International Research Center on Mathematics and Mechanics of Complex Systems, Universita degli Studi dell’Aquila, L’Aquila, Italy

    A. Ulukhanyan

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  1. M. Nikabadze
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  2. A. Ulukhanyan
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Correspondence to M. Nikabadze or A. Ulukhanyan.

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(Submitted by A. M. Elizarov)

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Nikabadze, M., Ulukhanyan, A. On the Iterlayer Contact Conditions in Multilayer Thin Body Theory and Some Issues of Splitting Initial-Boundary Value Problems. Lobachevskii J Math 43, 1945–1961 (2022). https://doi.org/10.1134/S1995080222100304

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