Phase-field models for brittle and cohesive fracture

In this paper we first recapitulate some basic notions of brittle and cohesive fracture models, as well as the phase-field approximation to fracture. Next, a critical assessment is made of the sensitivity of the phase-field approach to brittle fracture, in particular the degradation function, and the use of monolithic versus partitioned solution schemes. The last part of the paper makes extensions to a recently developed phase-field model for cohesive fracture, in particular for propagating cracks. Using some simple examples the current state of the cohesive phase-field model is shown

The cohesive band model: A cohesive surface formulation with stress triaxiality

In the cohesive surface model cohesive tractions are transmitted across a two-dimensional surface, which is embedded in a three-dimensional continuum. The relevant kinematic quantities are the local crack opening displacement and the crack sliding displacement, but there is no kinematic quantity that represents the stretching of the fracture plane. As a consequence, in-plane stresses are absent, and fracture phenomena as splitting cracks in concrete and masonry, or crazing in polymers, which are governed by stress triaxiality, cannot be represented properly. In this paper we extend the cohesive surface model to include in-plane kinematic quantities. Since the full strain tensor is now available, a three-dimensional stress state can be computed in a straightforward manner. The cohesive band model is regarded as a subgrid scale fracture model, which has a small, yet finite thickness at the subgrid scale, but can be considered as having a zero thickness in the discretisation method that is used at the macroscopic scale. The standard cohesive surface formulation is obtained when the cohesive band width goes to zero. In principle, any discretisation method that can capture a discontinuity can be used, but partition-of-unity based finite element methods and isogeometric finite element analysis seem to have an advantage since they can naturally incorporate the continuum mechanics. When using interface finite elements, traction oscillations that can occur prior to the opening of a cohesive crack, persist for the cohesive band model. Example calculations show that Poisson contraction influences the results, since there is a coupling between the crack opening and the in-plane normal strain in the cohesive band. This coupling holds promise for capturing a variety of fracture phenomena, such as delamination buckling and splitting cracks, that are difficult, if not impossible, to describe within a conventional cohesive surface model. © 2013 Springer Science+Business Media Dordrecht

Modelling inter- and transgranular fracture in piezoelectric polycrystals

A cohesive zone finite element model for quasi-static fracture of piezoelectric polycrystals is proposed. Interface elements are used to model both inter- and transgranular fracture. Electromechanical constitutive relations are derived by enhancing commonly used mechanical traction-opening laws with relations for a parallel plate capacitor. Numerical simulations demonstrate that the proposed model correctly mimics several experimentally observed phenomena. Most importantly the switch from mainly intergranular to mainly transgranular fracture with increasing grain size is modelled correctly. The model also correctly mimics the influence of an electric field on the ultimate load

A discussion on gradient damage and phase-field models for brittle fracture

\u3cp\u3eGradient-enhanced damage models find their roots in damage mechanics, which is a smeared approach from the onset, and gradients were added to restore well-posedness beyond a critical strain level. The phase-field approach to brittle fracture departs from a discontinuous description of failure, where the distribution function is regularised, which also leads to the inclusion of spatial gradients. Herein, we will consider both approaches, and discuss their similarities and differences.\u3c/p\u3

Discrete fracture analysis using locally refined T-splines

\u3cp\u3eLocally refined T-splines are used to model discrete crack propagation without a predefined interface. The crack is introduced by meshline insertions in the locally refined T-mesh, which yields discontinuous basis functions. To implement the method in existing finite element programs, Bézier extraction is used. A detailed description is given as to how the crack path is inserted and how the domain is reparameterized after insertion. The versatility and accuracy of the approach to model discrete crack propagation without the crack path being predefined is demonstrated by two examples, namely, an L-shaped beam and a single-edge notched beam. When the crack approaches the physical boundaries, limitations to reparameterization arise, as will be discussed at the hand of a double-edge notched specimen.\u3c/p\u3

Isogeometric analysis for modelling of failure in advanced composite materials

Isogeometric analysis (IGA) has recently received much attention in the computational mechanics community. The basic idea is to use splines as the basis functions for finite-element calculations. This enables the integration of computer-aided design and numerical analysis and allows for an exact representation of complex, curved geometries. Another feature of isogeometric basis functions, their higher-order continuity, is even more important for the development of shell and continuum shell elements to analyse structural stability and damage in thin-walled composite structures. The higher-order shape functions can be used to implement relatively straightforward but powerful shell elements. In addition, these shape functions contribute to a better representation of stresses in continuum elements. Finally, interfaces and delaminations can be modelled by reducing the order of the isogeometric shape functions by knot-insertion. In this chapter, we will give an overview of the recent developments in IGA for shell and continuum shell formulations

Finite element simulation of pressure-loaded phase-field fractures

\u3cp\u3eA non-standard aspect of phase-field fracture formulations for pressurized cracks is the application of the pressure loading, due to the fact that a direct notion of the fracture surfaces is absent. In this work we study the possibility to apply the pressure loading through a traction boundary condition on a contour of the phase field. Computationally this requires application of a surface-extraction algorithm to obtain a parametrization of the loading boundary. When the phase-field value of the loading contour is chosen adequately, the recovered loading contour resembles that of the sharp fracture problem. The computational scheme used to construct the immersed loading boundary is leveraged to propose a hybrid model. In this hybrid model the solid domain (outside the loading contour) is unaffected by the phase field, while a standard phase-field formulation is used in the fluid domain (inside the loading contour). We present a detailed study and comparison of the Γ-convergence behavior and mesh convergence behavior of both models using a one-dimensional model problem. The extension of these results to multiple dimensions is also considered.\u3c/p\u3

A phase-field description of dynamic brittle fracture

In contrast to discrete descriptions of fracture, phase-field descriptions do not require numerical tracking of discontinuities in the displacement field. This greatly reduces implementation complexity. In this work, we extend a phase-field model for quasi-static brittle fracture to the dynamic case. We introduce a phase-field approximation to the Lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase-field. We study the behavior of the model in one dimension and show how it influences material properties. For the temporal discretization of the equations of motion, we present both a monolithic and staggered time integration scheme. We study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments. We also introduce a local adaptive refinement strategy and study its performance in the context of locally refined T-splines. We show that the combination of the phase-field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions. (C) 2012 Elsevier B.V. All rights reserve