Powell–Sabin B-splines for smeared and discrete approaches to fracture in quasi-brittle materials

Abstract

Non-Uniform Rational B-splines (NURBS) and T-splines can have some drawbacks when modelling damage and fracture. The use of Powell–Sabin B-splines, which are based on triangles, can by-pass these drawbacks. Herein, smeared as well as discrete approaches to fracture in quasi-brittle materials using Powell–Sabin B-splines are considered.For the smeared formulation, an implicit fourth-order gradient damage model is adopted. Since quadratic Powell–Sabin B-splines employ C1-continuous basis functions throughout the domain, they are well-suited for solving the fourth order partial differential equation that emerges in this higher order damage model. Moreover, they can be generated from an arbitrary triangulation without user intervention. Since Powell–Sabin B-splines are generated from a classical triangulation, they are not necessarily boundary-fitting and in that case they are not isogeometric in the strict sense.For discrete fracture approaches, the degree of continuity of T-splines is reduced to C0 at the crack tip. Hence, stresses need to be evaluated and weighted at the integration points in the vicinity of the crack tip in order to decide when the critical stress is reached. In practice, stress fields are highly irregular around crack tips. Furthermore, aligning a T-spline mesh with the new crack segment can be difficult. Powell–Sabin B-splines also remedy these drawbacks as they are C1-continuous at the crack tip and stresses can be directly computed, which vastly increases the accuracy and simplifies the implementation. Moreover, re-meshing is more straightforward using Powell–Sabin B-splines. A current limitation is that, in three dimensions, there is no procedure (yet) for constructing Powell–Sabin B-splines on arbitrary tetrahedral meshes.