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Abstract The crack band model, which was shown to provide a superior computational representation of fracture of quasibrittle materials (in this journal, May 2022), still suffers from three limitations: (1) The material damage is forced to be uniform across a one-element wide band because of unrestricted strain localization instability; (2) the width of the fracture process zone is fixed as the width of a single element; and (3) cracks inclined to rectangular mesh lines are represented by a rough zig-zag damage band. Presented is a generalization that overcomes all three, by enforcing a variable multi-element width of the crack band front controlled by a material characteristic length l0. This is achieved by introducing a homogenized localization energy density that increases, after a certain threshold, as a function of an invariant of the third-order tensor of second gradient of the displacement vector, called the sprain tensorη, representing (in isotropic materials) the magnitude of its Laplacian (not expressible as a strain-gradient tensor). The continuum free energy density must be augmented by additional sprain energy Φ(l0η), which affects only the postpeak softening damage. In finite element discretization, the localization resistance is effected by applying triplets of self-equilibrated in-plane nodal forces, which follow as partial derivatives of Φ(l0η). The force triplets enforce a variable multi-element crack band width. The damage distribution across the fracture process zone is non-uniform but smoothed. The standard boundary conditions of the finite element method apply. Numerical simulations document that the crack band propagates through regular rectangular meshes with virtually no directional bias.
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An algorithm for the simulation of curvilinear plane‐strain and axisymmetric hydraulic fractures with lag using the universal meshes
Summary We present an algorithm to simulate curvilinear hydraulic fractures in plane strain and axisymmetry. We restrict our attention to sharp fractures propagating in an isotropic, linear elastic medium and driven by the injection of a laminar, Newtonian fluid governed by lubrication theory, and we require the existence of a finite lag region between the fluid front and the crack tip. The key novelty of our approach is in how we discretize the evolving crack and fluid domains: we utilize universal meshes (UMs), a technique to create conforming triangulations of a problem domain by only perturbing nodes of a universal background mesh in the vicinity of the boundary. In this way, we construct meshes, which conform to the crack and to the fluid front. This allows us to build standard piecewise linear finite element spaces and to monolithically solve the quasistatic hydraulic fracture problem for the displacement field in the rock and the pressure in the fluid. We demonstrate the performance of our algorithms through three examples: a convergence study in plane strain, a comparison with experiments in axisymmetry, and a novel case of a fracture in a narrow pay zone.
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- Award ID(s):
- 1662452
- PAR ID:
- 10453432
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- International Journal for Numerical and Analytical Methods in Geomechanics
- Volume:
- 43
- Issue:
- 6
- ISSN:
- 0363-9061
- Page Range / eLocation ID:
- p. 1251-1278
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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