An integral-transform-based phase-field formulation with an analytic mapping of the Park–Paulino–Roesler (PPR) cohesive law

An unsolved problem in cohesive fracture phase field modelling is how to establish a clear analytical mapping relationship between a prescribed cohesive law and its corresponding intrinsic energy density. This paper proposes an integral-transform-based phase-field formulation and its numerical implementation scheme, which provides an explicit analytical mapping relationship for the Park-Paulino-Roesler (PPR) cohesive law. The proposed phase-field formulation has been reconstructed within a thermodynamically consistent framework. Within this framework, a unified analytical construction of the intrinsic energy density associated with the PPR traction–separation law is derived. This construction provides a direct analytical mapping between the phase-field representation and the underlying PPR cohesive model, without relying on fitting or interpolation procedures. The BFGS global algorithm with unified stiffness correction is further developed within an implicit finite element framework for the numerical implementation of the model. To address the inherent endpoint singularities in the numerical realization of integral-transform phase-field models, a strategy based on the Maclaurin expansion is proposed to eliminate these singularities. Representative numerical examples demonstrate accurate reproduction of the target cohesive behavior, robust crack propagation predictions, and insensitivity to both mesh size and phase-field length scale. The proposed approach therefore offers a systematic route for incorporating shape-controllable PPR softening behavior into an integral-transform phase-field framework, while avoiding repeated fitting or interpolation procedures for different cohesive responses and the singularities inherent in the original model. Another important contribution is that it provides a clear and feasible approach for the numerical implementation of the integral-transform-based phase-field model.