Modeling Discontinuities
Material failure in solid mechanics, particularly in geomaterials such as soils, concrete, rocks,
etc., is characterized, at the macroscopic level, by the appearance of locus of damage, that
exhibit jumps (discontinuities) in the displacement field across them. These discontinuities, depending on the context, are termed, cracks, shear bands or fractures.
In recent years finite elements with discontinuities have gained increasing interest in modelling material failure, due to
their specific ability to provide, unlike standard finite elements, specific kinematics to capture strong discontinuities. They
essentially consist of enriching the (continuous) displacement modes of the standard finite elements, with additional
(discontinuous) displacements. The
discontinuity path is placed inside the elements irrespective of the size and specific orientation of them. Then, typical drawbacks
of standard finite elements in modelling displacement discontinuities, like spurious mesh size and mesh bias dependences,
can be effectively removed. In addition, unlike with standard elements, mesh refinement isn't necessary to
capture those discontinuities, and the simulation can be done with relatively coarse meshes. By using that technology, in
conjunction with some additional refinements, realistic simulations of multiple strong discontinuities propagating in three dimensional
bodies can be achieved, with small computers, in reasonable computational times.
As for the enriching technique, two broad families can be distinguished in terms of the support of the enriching discontinuous
displacement modes:
- Elemental enrichment: This method deals with the simulation of strain localization phenomena through the Strong Discontinuity Approach (SDA) developed by Oliver et al [1996] by enriching at elemental level the finite elements of the mesh.
The theory is based on EAS (Enchanced Assumed Strains Framework) and developed in the ambit of Continuum Mechanics using an Isotropic Continuum Damage model and its variations, which serves to simulate materials like concrete, ceramics, rocks and ice.
- Nodal enrichment: The support of each mode is the one of a given nodal shape function i.e.: those elements
surrounding a specific node. Most of the formulations of this family, available in the literature, have
been developed in the context of the Partition of Unity Method (PUM), under the name of X-FEM method.
X-FEM
The eXtended Finite Element Method (X-FEM) is a numerical method for modeling strong (displacement) as well as
weak (strain) discontinuities within a standard finite element framework. In the X-FEM, special functions are added to the finite element approximation using the framework of Partition of Unity. For crack modeling in isotropic linear
elasticity, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are used to account
for the crack. This enables the domain to be modeled by finite elements without explicitly meshing the crack
surfaces, and hence quasi-static crack propagation simulations can be carried out without remeshing.
Since its introduction in 1999, the X-FEM method has enjoyed a considerable level of success and popularity from researchers in the computational and applied mechanics communities.
The X-FEM grew out of research into meshfree methods by the computational mechanics group at Northwestern University directed by Ted Belytschko.
The basic philosophy of the X-FEM is that features of interest in a problem, for example crack surfaces, phase boundaries, and fluid-structure interfaces, can be represented independently of the finite element mesh. As a result, simulating the evolution of these features is greatly facilitated. The finite element mesh need not explicitly "fit" these features with the X-FEM, circumventing the need to re-mesh in many cases and facilitating adaptivity in others.
The basic ideas behind the method are easy to understand. Most finite element approximations to bulk fields (e.g. displacement, temperature) can be expressed as a linear combination of nodal shape functions. These shape functions are only able to represent discontinuities in the bulk fields if the mesh is constructed in a particular way. For example, the classical approach to representing the jump in displacement field across a crack front is to explicitly mesh both crack faces. With the X-FEM, the classical mesh need only overlap the geometry of the crack front and does not need to be carefully aligned with it. The linear combination is then augmented with enrichment functions that capture the jump in displacement field across the crack. Crack growth can in turn be simulated through the identification of additional enriched nodes and a new construction for the enrichment function, a process that is typically much simpler than re-meshing.
References:
- Belytschko T., Black T. [1999] "Elastic crack growth in finite elements with minimal remeshing", International Journal for Numerical Methods in Engineering, Volume 45, Issue 5, 1999, pp. 601-620
- Oiver et al. [1996] "Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1: Fundamentals. International", Journal for Numerical Methods in Engineering, Volume 39, Issue 21, 1996, pp. 3575-3600