A Finite Difference Method for 3D Fracture Characteristics of Hard-Thick Strata Considering Elastic–Plastic Supports and Strain-Softening

Mid-Thick Plate Theory oversimplifies boundary conditions and ignores key attributes like material plasticity and strain-softening behavior. In response to these challenges, this study introduces a refined method aimed at calculating 3D fracturing characteristics in hard-thick layers, especially those considering an elastic–plastic coal and rock mass. Then, the methodology is applied to the 8202 Working Face of the Tashan Coal Mine. Finite difference methods are employed to derive equations governing stress components, boundary conditions, and failure criteria. In addition, MATLAB is utilized to calculate the stresses of the K3 key layer, subsequently allowing for an assessment of the pattern and size of its first breaks. The following conclusions can be obtained: (1) The methodology integrates traditional boundary conditions, like clamping and simple supports, but augments them with elastic–plastic supports. (2) A comprehensive constitutive model is established that describes the full compression behavior of the elastic–plastic coal–rock mass, replete with equations for stiffness calculations. (3) The initial fracture position of the thick-hard key stratum K3 is advanced by 10.1 m in the region ahead of the goaf, with 7.4 m on either side. (4) The precision of the methodology is validated after comparisons between the fracture dimensions and patterns of on-site measurements and medium-thick plates under conditions of both four-sided fixed support and four-sided simple support.