To understand the fracture mechanics of quasi-brittle materials including rock and concrete, it is necessary to study the tension softening relationship within fracture process zone (FPZ). At present, inverse analysis method has become one of the most used methods to calculate the tension softening constitutive (TSC). However, the existing inverse analysis methods still have some shortcomings, so it is necessary to propose a convenient method for calculating the TSC. In this study, three-point bending (3-pb) tests were conducted on heat-treated limestone beams based on digital image correlation (DIC) technique first. Subsequently, a method for calculating the TSC under monotonic loading was proposed, and the proposed method was confirmed using the test data of heat-treated limestone beams. Following that, the effects of the TSC on the mechanical and deformation responses were analyzed. Finally, the effects of cohesive element types (zero-thickness and solid cohesive elements) and analysis methods (standard and explicit analyses) on the simulation results were investigated. The results show that limestone undergoes brittle-ductile transition after high temperature, and the nonlinear deformation of limestone increases. The proposed method is workable and correct, and it is more convenient to promote and apply compared to the existing inverse analysis methods due to only the specimen size and the load versus displacement curve need to be known. All the parameters of the TSC (E, ft, w2, w3 and w4) affect the mechanical and deformation responses. The choice of the cohesive element types and analysis methods has basically no effect on the simulation results.
Liu et al. (2026) studied this question.