Stress-Strain State Of a Polygon With Cracks Caused By Thermal Stresses
analytic functions, elliptical holes, stress intensity factor, conformal mapping functions, stress components.
AuthorsAbstractWhen addressing engineering design challenges, engineers frequently encounter various structural defects—including cracks—that can lead to the failure of machinery and equipment. This study examines a regular polygon affected by linear cracks caused by thermal stresses. Since conformal mapping functions for such complex domains containing linear cracks were previously unknown, this problem had not been addressed in earlier studies. In the present paper, a plane elasticity problem involving a complex geometric configuration—specifically, a domain weakened by two openings with rectilinear cuts—is solved using the theory of functions of a complex variable and conformal mapping functions. The problem was addressed by solving a system of linear algebraic equations utilizing the theory of complex variables and Kolosov-Muskhelishvili potentials 7,8; the system was solved by expanding the functions φ(z) and ψ(z) into series. An analytical solution was obtained, and subsequent studies will present numerical examples illustrating key aspects of the theory. Solving these linear algebraic equations yielded the coefficients of the analytic functions; subsequently, established formulas from the theory of elasticity were used to determine the stress components at characteristic points of the cross-sections. This problem is being solved for the first time, as no conformal mapping functions had previously been found for such shapes—specifically, for the complex geometric configuration of the body in question.
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