1. Introduction#

1.1. Why use X- FEM?#

The X- FEM [1] method is a simple extension of the finite element method. It allows a mesh independent of the geometry of the problem. Borders, holes, cracks become entities that can be inserted, moved, propagated, without having to modify the mesh. A simple and unique mesh thus replaces several complex meshes. To represent a discontinuity or a singularity within finite elements, the base of form functions is enriched, using the properties of the unit partition. In cracking, the displacement (or temperature) discontinuity due to the crack is introduced by a generalized Heaviside function and the addition of asymptotic fields at the crack end improves the precision in elastic fracture mechanics. In addition, the « level sets » method is particularly practical for representing 3D cracks and effective for the propagation phase, the main idea being to consider the interface as the iso-zero of a distance function.

You can use X- FEM to:

  • represent a crack (discontinuity of movements or temperature),

  • represent an interface between two disjoint solids (discontinuity of movements or temperature),

  • represent a hole or a void (underthickness for example),

  • represent the interface between two materials (stress discontinuities).

Representation of a crack

In fracture mechanics, a crack has two lips and a crack bottom. The lips are initially confused, otherwise it is called a cut. The field of movement (or temperature) is discontinuous across the lips of the crack, and the stress field is singular at the bottom of the crack.

In Code_Aster, it is possible to define a crack (see the implementation in § 18) using two level set functions and to have it propagated.

Representation of an interface between two disjoint solids

The two solids are separated by an interface (examples: a geological fault between two layers, the crushing of a piece of land on a fixed structure, two thermally insulated solids). The displacement or temperature field is discontinuous across the interface. The materials on either side of the interface may be different, but have the same law of behavior. Contact is possibly defined on the interface.

This functionality is possible in*Code_Aster*, by defining the interface with a level set function (see the implementation in § 4).

Representation of a hole or a vacuum

This is a particular case of the previous case where one of the solids is empty. For this, no contact conditions are defined on the interface. If no load is applied to the solid corresponding to the vacuum (except the blocking of rigid modes) it is not involved in the calculation, and plays the role of « vacuum ».

This functionality is possible in Code_Aster, by defining the interface between matter and vacuum by a level set function (see the implementation in § 5).

Representation of an interface in a bimaterial

In a bi-material, the two materials are « glued », the movement across the interface between the two materials is continuous, but the stresses are discontinuous.

This feature is not possible in*Code_Aster*.

In the rest of this document, the general term « crack » will be used, which will refer to either a crack or an interface.

1.2. Specificity of a calculation with X- FEM#

Compared to a classical calculation, some steps are specific:

  • creation of the mesh: the mesh does not have a crack,

  • definition of the crack: since the crack is not contained in the mesh, it must be defined in another way,

  • modification of the model: some elements must be enriched in order to represent a discontinuity of movement or temperature through the crack and the singularity of stress at the bottom of the crack,

  • visualization post-processing: in order to visualize the opening of the crack or the discontinuity of the temperature field, it is necessary to create a visualization mesh and the associated result fields.