Analysis of uniaxial stress state of an infinite sheet with complex geometry: impact of elliptical holes and linear cracks on structural stability

In engineering design, structural defects such as cracks can lead to catastrophic failures in machines and equipment. This study investigates the uniaxial stress state of an infinite sheet weakened by two elliptical holes with linear cracks. Due to the complexity of the geometric configuration and the absence of known conformal mapping functions for such regions, this problem has not been addressed in previous research. We solve a plane elasticity problem for a complex geometric configuration featuring two elliptical holes with linear cuts using the theory of complex variables and conformal mapping functions. The solution involves solving a system of linear algebraic equations derived from the theory of complex variables and Kolosov-Muskhelishvili potentials. By expanding the functions φ(z) and ψ(z) into series, we obtain an analytical solution and provide numerical examples to illustrate key theoretical aspects. The coefficients of the analytical functions are determined, and well-known elasticity theory formulas are applied to compute stress components at characteristic points. This research presents a novel approach to solving this specific problem, as conformal mapping functions for such complex configurations have not been previously established.