Phase‐field approach to fracture for finite‐deformation contact problems

Abstract

The present contribution deals with a variationally consistent Mortar contact algorithm applied to a phase-field fracture approach for finite deformations, see [4]. A phase-field approach to fracture allows for the numerical simulation of complex fracture patterns for three dimensional problems, extended recently to finite deformations (see [2] for more details). In a nutshell, the phase-field approach relies on a regularization of the sharp (fracture-) interface. In order to improve the accuracy, a fourth-order Cahn-Hilliard phase-field equation is considered, requiring global C 1 continuity (see [1]), which will be dealt with using an isogeometrical analysis (IGA) framework. Additionally, a newly developed hierarchical refinement scheme is applied to resolve for local physical phenomena e.g. the contact zone (see [3] for more details). The Mortar method is a modern and very accurate numerical method to implement contact boundaries. This approach can be extended in a straightforward manner to transient phase-field fracture problems. The performance of the proposed methods will be examined in a representative numerical example. Governing equationsThe continuum mechanical problem with bodies B (i) 0 in the reference configuration (i = 1, 2) is comprised of phase-field, bulk and contact contributions. To regularize the crack zone, a fourth-order partial-differential-equation is applied for the crack density functional. Postulating that crack growth is subject to tensile state we provide an anisotropic split for which we utilize an eigendecompostion of the deformation gradient. For the ease of exposition we restrict our considerations to frictionless contact problems. With the displacement ϕ (i) and the phase-field parameter s (i) as primary unknowns we obtain the initial boundary value problem (IBVP) for the considered time interval I as follows(1)Equations (1) 1 and (1) 2 denote the balance of linear momentum and the balance equation for the phase-field, respectively. B (i) and ρ (i) 0 denote the the prescribed body force and the reference density, respectively. G (i) c and l (i) denote the critical local fracture energy density and the length scale parameter, respectively. Equations (1) 3 -(1) 4 are the Dirichlet and Neumann boundary conditions with prescribed displacementφ and tractionT (i) , respectively. Additionally, N (i) denotes the unit outward normal to the Neumann boundary Γ (i) n . Moreover, equations (1) 6 -(1) 7 denote the boundary conditions for the phasefield. Equations (1) 8 are the initial conditions for both, the displacement as well as the phase-field. The IBVP is supplemented by the constitutive relations for the second Piola-Kirchhoff stress tensor and the driving force for the phase-fieldwheree denotes the elastic part of a strain energy density (for more informations see [4]). For the contact description we need to enforce the well-known Karush-Kuhn-Tucker conditions given by equations (1) 5 , comprised of the impenetrability condition, the condition which only permits compressive traction...

Discrete and Phase Field Methods for Linear Elastic Fracture Mechanics: A Comparative Study and State-of-the-Art Review

“…It should also be noted that the use of 4 th -order model leads to a more accurate approximation of stresses, which in turn facilitates higher rates of crack growth. More applications of higher-order phase-field models can be found in [259][260][261].…”

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

“…The 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 applied…”