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Abstract <p>Within the framework of the generalized Melan–Buffler model for a thin elastic inclusion, the problem of determining the stress state of an elastic infinite plate reinforced with such an inclusion is considered. The thin inclusion consists of a thin rectangular part in profile, which at its end sections is symmetrically coupled with parts of an ellipse; the semi-minor axis of the ellipse is much smaller than its semi-major axis. Solving the problem is reduced to solving the Prandtl integro-differential equation (IDE), which is reduced to an infinite system of linear algebraic equations (ISLAE) using the mathematical method of Chebyshev polynomials. The regularity of this ISLAE has been proven. The well-known numerical-analytical method for solving singular integral equations (SIE), based on Gauss quadrature formulas for calculating integrals, is applied to the governing Prandtl IDE. The main characteristics of the problem such as tangential and normal contact stresses on the edges of the inclusion, axial forces in the sections of the inclusion are represented by explicit formulas. Special cases are considered and numerical analysis is carried out.</p>

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Application of a Novel Elastic Thin Inclusion Model to the Stress State Problem of an Infinite Elastic Plate Reinforced with an Inclusion

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • S. M. Mkhitaryan,
  • E. G. Kanetsyan,
  • M. S. Mkrtchyan

摘要

Abstract

Within the framework of the generalized Melan–Buffler model for a thin elastic inclusion, the problem of determining the stress state of an elastic infinite plate reinforced with such an inclusion is considered. The thin inclusion consists of a thin rectangular part in profile, which at its end sections is symmetrically coupled with parts of an ellipse; the semi-minor axis of the ellipse is much smaller than its semi-major axis. Solving the problem is reduced to solving the Prandtl integro-differential equation (IDE), which is reduced to an infinite system of linear algebraic equations (ISLAE) using the mathematical method of Chebyshev polynomials. The regularity of this ISLAE has been proven. The well-known numerical-analytical method for solving singular integral equations (SIE), based on Gauss quadrature formulas for calculating integrals, is applied to the governing Prandtl IDE. The main characteristics of the problem such as tangential and normal contact stresses on the edges of the inclusion, axial forces in the sections of the inclusion are represented by explicit formulas. Special cases are considered and numerical analysis is carried out.