A polyconvex phase‐field approach to fracture with application to finite‐deformation contact problems

Abstract

Variationally consistent phase-field methods allow for an efficient investigation of complex three-dimensional fracture problems (see [1,2]). However, formulations for large deformation problems often exhibit a lack of numerical stability for different loading scenarios. In the underlying contribution a novel formulation for finite strain polyconvex elasticity is adapted to phase-field fracture problems. In particular we introduce a new anisotropic split based on the principal invariants of the right Cauchy-Green strain tensor for a proper treatment of fracture within the polyconvex framework (see [4]). This polyconvex phase-field fracture formulation can be implemented in a straightforward manner and improves the numerical stability. Furthermore, a fourth order crack density functional is considered to improve accuracy and convergence. To account for the C 1 requirement the system is embedded in a sophisticated isogeometric framework with the ability of local refinement. Eventually, a variationally consistent Mortar contact algorithm is applied (see [3]) to handle contact boundaries. Initial boundary value problemThe continuum mechanical contact and phase-field fracture problem with bodies B i , i = 1, 2 is comprised of phase-field, bulk and contact contributions (cf. [5,6]). To regularize the crack zone, a fourth-order Cahn-Hilliard type differential equation is appliedTherein s ∈ {0, 1} is the phase-field parameter, where s = 0 denotes the broken and s = 1 the intact state of the bulk. The degradation of the strain energy is the key element for the fracture behavior of the model. A most simplest model relies on an isotropic degradation of the strain energy density function Ψ defined aswhere a > 0 is a modeling parameter of the degradation function g(s). However, the above model contradicts with physical observations since compressive and tensile states are equally degradated. For an anisotropic degradation linear and nonlinear models based on the principal strains have been developed (see [7] and [2], respectively). The former is restricted to linear problems whereas the latter formulation is not rank-one convex which leads to stability issues. To remedy this drawback we introduce a novel polyconvex formulation based on an extended kinematic set. In particular the deformation gradient, the cofactor and the Jacobian determinant are introducedsuitable for performing the fibre ( dx = F dX), area ( da = H dA) and volume maps ( dv = J dV ), respectively. In the above equations the tensor crossproduct has been introduced (see [4] and [8] for more details).With the introduction of the extended kinematic set a polyconvex formulation of the strain energy function can be found. Variation and linearisation are greatly simplified. In particular we avoid to differentiate the inverse of the deformation gradient. On this basis we introduce a new anisotropic tension-compression split based on the principal invariants of the right CauchyGreen tensor and the Jacobian determinant. The tensile parts (• + ) of the isochoric inv...