Crack Lip Contact Modeling Based on Lagrangian Multipliers with X-FEM

The eXtended Finite Element Method (X-FEM), developed intensively
in the past 15 years has become a competitive tool for the solution of problems
with evolving discontinuities and singul
arities. In the present study, we focus
on the application of X-FEM on frictionless contact problems in the context of
fracture mechanics. A promising approach
in the literature counting for this problem
consists in applying Lagrangian multipliers. Meanwhile, as pointed out in Ji and
Dolbow (Int J Numer Methods Eng 61:2508–2535, 2004), a naive choice for
Lagrangian multiplier space leads to oscillatory multipliers on the contact surface.
This oscillation results from a non-uniform
but mesh-dependent inf-sup condition.
In this work, we adapt the algorithm proposed in Béchet et al. (Int J Numer Methods
Eng 78:931–954, 2009) on crack lip contact by di
scretizing the displacement field
with both scalar and vector tip enrichment functions (Chevaugeon et al., Int J
Multiscale Comput Eng 11:
597–631, 2013). The influence of the tip enrichment
functions on the stability of the formulation is addressed. We show evidences
that the vector enrichment functions can improve the conditioning of the problem
without jeopardizing the simulation
accuracy in the presence of contact.