Simulation of Hertzian cone cracks using a phase field description for fracture

Abstract

The present contribution focuses on fracture caused by indentation loading on the surface of a brittle solid. Its theoretical prediction is a challenging task due to the fact that crack nucleation is not geometrically induced, but is caused by the stress concentration in the contact near-field. The application of the phase field model requires constitutive assumptions to ensure a tension-compression asymmetric material response and prevent damage in compressed regions. This is achieved at the cost of giving up the variational concept of brittle fracture. We simulate the indentation of a cylindrical flat-ended punch on brittle materials like silicate glass. In order to reduce the numerical effort, we exploit axisymmetric conditions for the finite element formulation. After crack initiation stable propagation of a cone crack can be observed in good agreement with experiments. Indentation fractureThe occurrence of cone cracks under indentation loading has already been described by Hertz [1] in 1881: When a cylindrical stiff punch is pressed against the surface of a brittle elastic solid a circular crack is spontaneously initiated around the contact area. Under further loading this initial ring crack grows into a cone, the so called Hertzian cone crack. The growth at this stage is stable since the radius of the cone extends with the evolving crack. A particular challenge is to compute the crack initiation from the defect-free surface. The initial crack starts some distance outside of the indenter, as known from experiments [2]. Previous studies (e.g. [3]) modeled pre-existing ring cracks in order to investigate the further progress of the cone crack. The spontaneous formation of the ring crack is a non-standard problem of fracture mechanics which requires to take strength σ c and toughness G c into account, e.g. [4]. Thus the phase field approach appears to be well suited to handle such a problem since it also comprises both parameters. Furthermore, the method does not require any assumptions concerning the unknown location of the crack initiation or crack path. Governing equations of the phase field approachPhase field models have already proven their excellent ability to reproduce situations with complex crack patterns including initiation and the determination of unknown crack paths. For the smeared description of a crack in a linear elastic body Ω, with mass density ρ and external boundary ∂Ω = ∂Ω t ∪ ∂Ω u , an additional scalar field S(x, t) is introduced to describe the state of the material between the fully broken state S = 0 and undamaged material S = 1. Thereby the crack surface energy, including Griffith's critical energy release rate G c , is approximated by a smooth function with a small regularization parameter ℓ which is related to the width of the transition zone between both states. In order to obtain the coupled Euler-Lagrange equations for dynamic brittle fracture we employ Hamilton's principle. With the choice of a quadratic degradation function, which describes the loss of stiffness in damaged ...