Stability and bifurcation analysis in the plane compression test with gradient damage models

In this paper, we present an analytical solution for the stability and bifurcation analyses in 2D uniaxial compression tests. We established a relation between the value of the fracture angle resulting from the uniaxial compression test and the stability and bifurcation criteria. Through this study, a minimization of a Rayleigh ratio is introduced. This result serves as a practical tool to identify the stable state of the studied case from its fracture angle. This work is based on the variational approach to fracture, i.e., the phase field models that are characterized by material softening and instabilities. The standard phase-field model without unilateral effects fails to distinguish between tensile and compressive fractures, leading to non-physical patterns. Tension–compression split models can be an alternative, introducing a fracture angle due to the presence of shear modes. Numerical simulations of a 2D beam under compression were conducted using COMSOL Multiphysics. The numerical results are in agreement with the analytical ones, verifying the condition imposed on the fracture angle corresponding to stable states and the possibility of bifurcation.