Fracture Modeling of Nearly Incompressible Hyperelastic Materials: A Phase‐Field Approach via Simo–Taylor–Pister Three‐Field Variation

Abstract

Incompressibility is commonly observed in the majority of hyperelastic materials, which are fundamental to a wide range of technical applications owing to their distinctive properties, particularly their flexibility, stretchability, and resilience. However, the incompressibility constraint poses a significant challenge in modeling such materials, further complicating efforts to accurately describe or reproduce crack propagation through numerical simulations. Mixed element formulations, for example, the so‐called Q1P0 approach, are extensively employed to mitigate locking issues associated with incompressibility at finite deformations. In this study, the Q1P0 approach is formulated via the Simo–Taylor–Pister (STP) three‐field formulation, which emanates from the Hu‐Washizu variational principle, resulting in an equivalent single‐field displacement formulation. This framework is subsequently coupled with the phase‐field method to predict finite strain fracture of nearly incompressible hyperelastic materials. To the best of the authors' knowledge, this represents the first successful coupling of the phase‐field approach with the STP‐based Q1P0 formulation. Through a consistent derivation and condensation of the pressure field, the novel four‐field formulation is reduced to a pure displacement‐phase‐field approach. This reduction constitutes a major advantage of the proposed formulation, as it preserves the size of the global system without introducing additional degrees of freedom. To alleviate the innate conflict between the incompressibility constraint and diffuse crack opening, a tailored phase‐field degradation function is integrated into the volumetric contribution, accelerating pressure release and thereby facilitating crack opening. Accordingly, the incompressibility constraint is relaxed in damaged material while remaining strictly enforced in intact material. The proposed formulation offers two more benefits: (i) it provides enhanced generality and applicability with respect to the choice of the volumetric strain‐energy function, and (ii) it results in a physically consistent material response in both the nearly incompressible and the compressible regimes. Subsequently, several numerical examples are presented to illustrate the characteristics of the proposed formulation.