In this work, features of surface-to-surface contact discretization are combined with the extended finite element method to handle contact problems along virtual surfaces in 2D. We focus on the incorporating of geometrical changes, which result from wear process and are taken into account by mobile virtual interfaces.

1. Frictional Contact and Wear Along Virtual Interfaces:
Coupling the Mortar Method with the X-FEM
Basava Raju Akula1,2
, Julien Vignollet2
, Vladislav A. Yastrebov1
1
MINES ParisTech, PSL Research University, Center des Mat´eriaux, CNRS UMR 7633, BP 87, 91003
Evry, France
E-mail: basava-raju.akula@mines-paristech.fr, vladislav.yastrebov@mines-paristech.fr
2
Safran Tech, Safran Group, 78772 Magny-les-Hameaux, France
E-mail: basava-raju.akula@safrangroup.com, julien.vignollet@safrangroup.com
Keywords: mortar method, extended finite element method, contact, wear.
Interface phenomena like contact, friction and wear are complex both with regard to their mathematical
description and numerical treatment. Their localized nature lays a strong emphasis on the kind of
interface discretization scheme employed. Ensuring stability and appropriate patch-test performance of
these schemes are necessary ingredients to the overall accuracy and robustness of the contact treatment.
A relative motion between contacting bodies can lead to material removal on rubbing surfaces, affecting
the contact pressure and thus a change in global system response. Numerical simulation of this wear, can
involve re-meshing procedures to capture the shape changes at the interface. This re-meshing generally
requires the mapping of history variables onto the new mesh to incorporate load history dependent
material behavior.
The extended finite element method (X-FEM) presents an attractive alternative technique to address
wear specific issues. In this framework, surface changes can be incorporated as evolving internal dis-
continuities. Complementing this feature with a stable and accurate mortar discretization scheme [1]
would result in a simplified numerical recipe to treat frictional contacts involving wear effects which
would not require re-meshing or complex field mapping.
In this work, features of mortar methods in the context of contact [2, 3], and of the X-FEM in the
context of void/inclusion [4] treatment are exploited to formulate a generalized framework in 2D. As
illustrated in Fig. 1, we propose to use the virtual interface Γv, to describe the geometrical changes
resulting from wear. The virtual interface Γv acts as a slave/master surface of domain Ω1 getting in
contact with the contact surface of domain Ω2. Surface Γv coincides with the unworn surface Γc1 ⊂ ∂Ω1
at the beginning of the simulation. Upon loading, the material of Ω1 is worn-out and the surface Γv
propagates into the bulk to capture the surface evolution. This approach includes: (i) the level set
method (LSM) to describe the location of worn-surface, (ii) X-FEM Heaviside enrichment (topological)
within elements intersected by the worn-surface, and (iii) the mortar method to model contact along the
interface made of Γv and Γc2. We use a monolithic augmented-Lagrangian formulation [5] to resolve
the contact inequality constraints for both the normal and tangential directions.
The mortar interface virtual work (δWmortar
), is evaluated by setting the virtual boundary Γv as
the non-mortar side and Γc2 as the mortar side. The gap function g(x, ˜x) determines here the integral
gap measure, which depends on actualized positions of the boundary Γc2 and of the nodal positions of
the blending elements intersected by Γv. Tilded notations denote quantities of the domain Ω1 subject to
wear. The contribution of the domain Ω1 (δWint
1 ) to the global virtual work of the system is evaluated
using the X-FEM enrichment (1). The worn out volume Ωw evolves with time and is evaluated based
on the sliding distance and the local contact pressure. As illustrated in Fig. 1, the body subject to wear
is modeled using the X-FEM framework, i.e. the internal virtual work is integrated in a classical way in
1

2. elements non-intersected by the virtual boundary and only partially integrated on the unworn volume of
the blending elements.
δWint
1 =
˜Ω=Ω1Ωw
˜σ : δ˜ε d˜Ω, δWmortar
=
Γc2
δ



λ(x) g(x, ˜x) +
ε
2
g2
(x, ˜x) dΓ, λ + εg ≤ 0
−
1
2ε
λ(x)2
dΓ, λ + εg > 0.
(1)
where ε is the augmentation parameter. The displacement test functions for both domains are chosen
from appropriate functional spaces and have to satisfy Dirichlet boundary conditions.
Enriched nodes
Standard nodes
Standard element
Blending element
Removed element
Figure 1: Material removal due to wear at the interface, modeled using a virtual interface Γv. Ωw is the
worn-out volume where the internal virtual work is not integrated.
The proposed method consolidates diverse and well established fields of the mortar domain decom-
position methods and the X-FEM, working together to simplify and provide a general framework to
treat efficiently interface phenomenon like contact and wear. In perspective the method will be taken
forward into the realm of parallel computing, thus enabling to solve large and complex problems in a
flexible and accurate way.
References
[1] El-Abbasi, N., Bathe, K.J., “Stability and patch test performance of contact discretizations and a
new solution algorithm”, Computers & Structure, 79 (16), 1473-1486 (2001).
[2] Wriggers, P., Fischer, K.A., “Frictionless 2D contact formulations for finite deformations based on
the mortar method”, Computational Mechanics, 36 (3), 226-244 (2005).
[3] Glitterle, M., Popp, A., Gee, M.W., Wall, W.A., “Finite deformation frictional mortar contact using
a semi-smooth Newton method with consistent linearization”, International Journal for Numerical
Methods in Engineering, 84 (5), 543-571 (2010).
[4] Sukumar, N., Belytschko, T., “Arbitrary branched and intersecting cracks with the extended finite
element method”, International Journal for Numerical Methods in Engineering, 48, 1741-1760
(2000).
[5] Alart, P., Curnier, A., “A mixed formulation for frictional contact problems prone to Newton like
solution methods”, Computer methods in applied mechanics and engineering, 92 (3), 353-375
(1991).
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