Fracture in tension–compression-asymmetry solids via phase field modeling

Many nonlinear elastic materials exhibit an asymmetric response when loaded in tension versus compression. In this paper, we study the fracture of such nonlinearly elastic materials with tension–compression asymmetry by virtue of phase field modeling. An additive decomposition of strain energy is utilized and extended to account for the modulus difference between tension and compression. This strain energy decomposition is demonstrated for both a neo-Hookean model and an Odgen model. The decomposition is proven to be consistent with the basic requirements of thermodynamics and is important for fracture modeling under the stress states with both compression and tension. The implementation of the phase field model with both decomposed energy of neo-Hookean and Odgen model is given. The implemented model is capable of capturing the tension–compression asymmetry of nonlinear elastic solids. It also can model crack initiation and propagation efficiently especially when the material undergoes both tension and compression, demonstrated through several typical specimens with pre-set crack. The stress fields around the crack tip break the classical law of singularity for the elastic solids with tension–compressionsymmetry (σ∝r^−0.5), which is greatly influenced by the ratio of tensile modulus to compressive modulus. The hardening of the material can delay the fracture with more diffusive fracture process zone, demonstrated by the modified Odgen model. The proposed approach shows a great potential to predict the damage behavior of the materials with tension–compression asymmetry at finite strain.