Brittle fracture propagation in rocks is a complex process due to significant grain‐scale heterogeneity and evolving stress states under dynamic loading conditions. In this work, we use digital image correlation and linear elastic fracture mechanics to make instantaneous measurements of the opening (mode I) and in plane shear (mode II) components of the stress intensity field during dynamic mixed mode crack initiation and propagation in crystalline and granular rocks. Both rock types display some similar fracture behaviors as observed in engineered materials, including rate dependent fracture initiation toughness and a direct relationship between propagation toughness and crack velocity; however, measured propagation toughness is higher than quasi‐static values at crack velocities well below the branching velocity in both rocks. Additionally, due to grain scale controls on the fracture process, mixed mode crack propagation is fundamentally different between these two rock types. Mixed mode propagation is energetically more favorable than pure opening mode propagation in sandstone, while the opposite is true in granite. Furthermore, following initiation, propagation in granite occurs so as to minimize the mode II contribution, irrespective of the initiation conditions, while fractures in sandstone maintain a non‐negligible mode II contribution during propagation across the sample.
This work presents a stabilized formulation for phase‐field fracture of hyperelastic materials near the limit of incompressibility. At this limit, traditional mixed displacement and pressure formulations must satisfy the inf‐sup condition for solution stability. The mixed formulation coupled with the damage field can lead to an inhibition of crack opening as volumetric changes are severely penalized effectively creating a pressure‐bubble. To overcome this bottleneck, we utilize a mixed formulation with a perturbed Lagrangian formulation which enforces the incompressibility constraint in the undamaged material and reduces the pressure effect in the damaged material. A mesh‐dependent stabilization technique based on the residuals of the Euler–Lagrange equations multiplied with a differential operator acting on the weight space is used, allowing for linear interpolation of all field variables of the elastic subproblem. This formulation was validated with three examples at finite deformations: a plane‐stress pure‐shear test, a two‐dimensional geometry in plane‐stress, and a three‐dimensional notched sample. In the last example, we incorporate a hybrid formulation with an additive strain energy decomposition to account for different behaviors in tension and compression. The results show close agreement with analytical solutions for crack tip opening displacements and performs well at the limit of incompressibility.
- Award ID(s):
- 2038057
- Publication Date:
- NSF-PAR ID:
- 10371084
- Journal Name:
- International Journal for Numerical Methods in Engineering
- Volume:
- 123
- Issue:
- 19
- Page Range or eLocation-ID:
- p. 4655-4673
- ISSN:
- 0029-5981
- Publisher:
- Wiley Blackwell (John Wiley & Sons)
- Sponsoring Org:
- National Science Foundation
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