A phase‐field model of brittle fracture for an isogeometric Reissner‐Mindlin shell formulation

Abstract

One of the biggest challenges in fracture modeling is the correct physical description of the fracture processes. Over the past decade, the well‐established phase‐field modeling framework has gained a lot of attention, due to its ability to predict the fracture phenomena adequately. It has shown very promising results for a three dimensional solid formulation and has been extended to an isogeometric Kirchhoff‐Love formulation in order to describe brittle fracture in plates and shells, see [1]. However, the Kirchhoff‐Love theory only describes shells in the thin regime. For thick shells, the deformation behavior and subsequently the fracture behavior additionally depends on the transverse shear strains which are not present in the Kirchhoff‐Love approach. In this work, the phase‐field fracture framework is extended to an isogeometric Reissner‐Mindlin shell formulation [2]. In this approach, the shell is described using the midsurface and a director vector field for the thickness direction. Subsequently, the phase‐field is also defined only on the midsurface. In order to distinguish the cracking behavior in tension and compression, as proposed in [1], the spectral decomposition of the strain for the tension‐compression split is done on the total strain, which varies through the thickness. The proposed method is tested for several numerical examples and compared to a three dimensional solid formulation and the Kirchhoff‐Love shell formulation in order to confirm its accuracy and efficiency.

Phase‐field modeling of brittle fracture along the thickness direction of plates and shells

“…In order to couple both of these fields-now remarkably referring to different geometric entities-with each other, R = {X ∈ V | Θ 3 = 0} has to hold to enable the translation between each material point of the shell midsurface and the corresponding material point of the solid. Given (33) and (34), the following coupling quantities can be identified:…”

“…• The phase-field variable c ( Θ 3 ) involved in the degradation function g is needed for the computation of the stress resultants when solving for the displacement field (33).…”

“…The set of coupled partial differential equations given by the strong forms in (33) and (34) are solved using NURBS-based IGA since it allows a rotationless FE formulation. 49 For that purpose, the weak forms with appropriate test functions-here indicated by (⋅)-are introduced: the weak form of the phase-field evolution equation of the solid part is given by…”

“…While the first two aforementioned contributions for phase‐field shell models do not consider this decomposition correctly with the first only splitting the membrane part and the second performing no split at all, Kiendl et al 32 in 2016 demonstrated for KL shells that a split of the complete strain tensor involving both membrane as well as bending action needs to be performed at various points over the shell thickness. Apart from various splits into tensile and compressive terms being presented, next to KL shell kinematics, the phase‐field approach to fracture has also been combined with higher‐parametric shell models such as the Reissner ‐ Mindlin (RM) shell theory, see for instance References 33,34. Further approaches to deal with the thinness of the structure in the context of phase‐field modeling exist, such as a formulation with a mixed interpolation of tensorial components (MITC)4+ RM degenerated shell 35 and a solid‐shell approach 36,37 .…”