A phase-field model for fractures in incompressible solids

TL;DR

This work develops a phase-field fracture model for incompressible solids by introducing a mixed displacement-pressure formulation to mitigate volume-locking as $\nu\to0.5$. The discrete system includes $u$, $p$, $\varphi$, and a Lagrange multiplier $\tau$ for crack irreversibility, solved monolithically with a Newton method, and stabilized by Taylor-Hood $Q_2$-$Q_1$ elements for $u$ and $p$, with $\varphi$ and $\tau$ discretized appropriately. Numerical tests on a single-edge notched shear and an L-shaped panel compare FE orders, mesh refinements, and Poisson ratio variations, revealing that element choice strongly affects the L-shaped panel results and that incompressible effects increase pre-crack resistance and pressure contributions. The study demonstrates the viability of the proposed mixed formulation for rubber-like, nearly incompressible materials and highlights directions for future work such as error estimation and adaptive refinement to improve robustness and efficiency.

Abstract

Within this work, we develop a phase-field description for simulating fractures in incompressible materials. Standard formulations are subject to volume-locking when the solid is (nearly) incompressible. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor-Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on two numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson's ratio approaching the incompressible limit, are presented.

Figures (14)

• Figure 1: Conforming quadrilateral Stokes-elements of the type $Q_2 Q_1$: $Q_2$ for the displacement variable $u$ (the filled blue and the empty red bullets) and $Q_1$ for the scalar-valued pressure variable $p$ (empty red bullets).
• Figure 2: Geometry and boundary conditions of the single edge notched shear test. On the left and right side, the boundary condition in $y$-direction is $u_y = 0 mm$ and traction-free in $x$-direction. On the bottom boundary it is determined $u_x = u_y = 0 mm.$ On the top boundary, it holds $u_y = 0 mm$ and in $x$-direction a time-dependent non-homogeneous Dirichlet condition: $u_x = t \cdot 1 mm/s.$
• Figure 3: Geometry and boundary conditions of the L-shaped panel test. The lower left boundary is fixed with $u_x = u_y = 0 mm.$ In the right, marked corner, a special cyclic displacement condition for $u_y$ is given, defined in (\ref{['uy']}) and depicted in Figure \ref{['CyclicLoading']}.
• Figure 4: The cyclic loading history on $\Gamma_{u_y}$.
• Figure 5: Load-displacement curves for the single edge notched shear test with $5$ steps of uniform refinement for the original implementation with $Q_2 Q_1$ and $Q_1 Q_1$ elements in comparison to the new model $Q_2 Q_1 Q_1 Q_1^*$.

Theorems & Definitions (6)

• Remark 3.1
• Remark 3.3
• Remark 3.4
• Proposition 3.1: inf-sup condition for saddle point problems with penalty
• Lemma 3.5
• Proposition 4.1: Stable Taylor-Hood elements