On the 1D homogeneous and localized solutions of variational phase field method for pressurized fracture

While 1D analytical solutions of the variational phase-field fracture method have been extensively reported and provide critical insights into crack nucleation and damage evolution criteria, their counterparts for pressurized fractures have received scant attention. This work presents a systematic study of a family of variational phase-field models for pressurized fracture. We derive both homogeneous and localized analytical solutions for a 1D bar under uniaxial tension. We find that in these models, the material’s strength depends on two key parameters: the internal length scale l₀ and the crack surface pressure p. We show how to modulate l₀ to correctly capture the material strength, similar to classical methods. A key finding is that, unlike in classical models, solely modulating the internal length scale l₀ to capture material strength becomes insufficient under non-zero Poisson’s ratio. By testing different models, we identify one (the AT1-I2 model) that is not affected by this issue. All analytical findings and the identified model’s efficacy are validated through two-dimensional finite element simulations.