Abstract: This study focuses on developing and applying the Boundary Integral Equation Method (BIEM) to model nonlinear deformation and fracture processes in locally homogeneous media, including three-dimensional structures and their complex-shaped elements. This work is relevant because there is a need to investigate the volumetric aspects of these processes and their impact on the load-bearing capacity, durability, and structural integrity of materials, especially when defects are present. Method: The method uses a direct approach with single- and double-layer potentials to represent the boundary values of displacements and forces. Singular integrals are computed using adaptive schemes and specified accuracy, employing Gaussian quadrature formulas. This method involves forming discrete subdomains to enable analysis of complex geometries with local singularities and elements of contrasting sizes. Quadratic Lagrangian boundary elements describe domain boundaries and solutions, and volume elements account for physical nonlinearity. The method of successive linearization is combined with a model of hardening elastoplastic media. Results: The application of the discussed method demonstrated high accuracy in simulating deformation and fracture in locally homogeneous 3D media, as confirmed by comparisons with nonlinear benchmark solutions and experimental data. It solved practical problems, such as analyzing nonlinear stress and strain fields in a two-layer strip with a surface crack and in the toroidal chamber of a tokamak- type fusion device under complex thermomechanical loading. Discussion: The results underscore the substantial potential of the BIEM in reliably analyzing the behavior of nonlinear deformable media under mechanical, thermal, and electromagnetic fields. This method allows for the precise evaluation of limit states, service life prediction, and material selection optimization. Limitations are associated with computational complexity for highly nonlinear problems and the need for experimental validation under specific loading conditions. Conclusion: The developed BIEM is a robust tool for modeling nonlinear deformation and fracture processes in inhomogeneous media. Its application to practical problems involving composite materials and fusion device components confirms its versatility, accuracy, and computational efficiency. Future research will involve integrating multiphysics models and utilizing machine learning for parameter optimization.
V. A. Petushkov (Thu,) studied this question.