Enhanced isogeometric analysis of fracturing in solid and porous media

EXtended Finite Element Method (XFEM) decouples the crack path from the discretisation, a major advantage compared to interface elements for simulating arbitrary fracturing. Also, avoiding the costly option of remeshing, XFEM enables propagating discontinuities to be modelled on the original mesh layout. XFEM, however, has been replaced by other techniques, such as meshless methods, to provide higher-order continuity across element boundaries. This is required to remove stress jumps at element boundaries, to improve stress estimates particularly in the presence of a discontinuity, and to locally satisfy mass conservation of the fluid part needed at element boundaries of a porous medium. With the advent of IsoGeometric Analysis (IGA), spline bases have been proposed to provide $C^\mathbbm{p}$-continuity ($\mathbbm{p}>0$) across element boundaries, which also have been incorporated in XFEM, so-called eXtended IsoGeometric Analysis (XIGA), to replace the underlying $C^0$-continuity of the Lagrangian interpolations. Within the realm of Linear Elastic Fracture Mechanics (LEFM), XIGA has been well addressed, but not so far cohesive zone models. The interelement sharing of control points in splines removes the Kronecker-Delta property available for Lagrange interpolation underlying the customary finite element method. These features complicate elementwise enrichment of individual control points, as well as the compatibility enforcement. The latter is of utmost importance as a possible source of error, which has not been well addressed for XIGA despite the full investigation carried out in the original contribution combining IGA and XFEM. Shifting technique can localise the effect of the Heaviside function, although, unlike XFEM, it stretches to the blending elements. This is similar to the effect of blending technique to enforce compatibility in XFEM. An XIGA formulation is proposed for Non-Uniform Rational B-splines (NURBS) to account for small strain and geometrically nonlinear cohesive fractures. Compatibility enforcement is realised of shifting and blending techniques perpendicular and parallel to the discontinuity, respectively. The latter enables removing the undesired effect of the discontinuity ahead of the crack tip, which is absent in recent XIGA contributions. To provide a standard finite element data structure B{\'e}zier extraction is utilised. In a further development, XIGA is augmented with the Local Maximum Entropy (LME), a meshless technique with superior features to elevate the accuracy of the enhanced section, which is coined X-IGALME. The results indicate a guaranteed stability of the solution for X-IGALME, where XIGA and XFEM fail. Unlike elementwise approaches, the singularity-free characteristic of X-IGALME facilitates a single prescription for enrichment in any condition, including void level set. Finally, XIGA for a progressively fracturing porous medium is investigated, representing a successful fracture simulation in fluid saturated porous media using a 2PDOF (two pressure degrees of freedom) model.