A variational phase-field model for brittle fracture in polydisperse elastomer networks

Abstract

The distribution of chain lengths in elastomer networks occurs naturally during polymerization, and has a critical role in the mechanical behavior of these materials, including the damage and fracture response. A common underlying assumption for the majority of constitutive models for rubber elasticity is that all the chains admit the same length. Moreover, in the classical statistical mechanical model for elastomer networks, the changes in internal energy are generally assumed to be negligible in comparison to the changes in configurational entropy. In contrast, the fracture process in a elastomer network, as already demonstrated in the well-known Lake-Thomas model, is essentially internal energy dominated. In this paper, we formulate a phase-field model for brittle fracture in polydisperse elastomer networks extending and merging recent advances in the fields of (a) homogenization of elastomer networks and (b) the variational approach to brittle fracture, allowing for predictions of crack nucleation, initiation and propagation. The free energy of the continuum is obtained employing an eight chain network model, accounting for (a) internal energy contributions from the extension of molecular bonds and (b) arbitrary chain length distribution. The representative chain in the eight chain network model takes into account the distribution of chain lengths, following the recently developed equal force model. We employ a mixed displacement-pressure formulation for the discretization of the incompressible large deformation elastic problem that arises. The analytical solution for the crack nucleation problem of an incompressible hyperelastic bar under uniaxial loading is compared with the three-dimensional simulation result. Finally, we demonstrate through a representative numerical simulation the capability of the gradient damage model to simulate crack propagation in an incompressible elastomer network.