Computational Methods for Debonding in Composites

270 M. Kästner et al.
including problems related to distorted element shapes and poor numerical condition
of the system of equations to be solved.
In order to improve flexibility and efficiency of the homogenization technique
an alternative modelling strategy is examined. For this purpose X-FEM – whose
major benefits such as regular meshes and reduced meshing effort while retaining all
advantages of the classical finite element approach have been demonstrated recently
[1, 6, 9, 12, 13, 16] – is applied to modelling of composite materials here.
13.2.1 Fundamentals
Based on the partition of unity technique [11] X-FEM offers the possibility to model
arbitrary discontinuities using regular finite element meshes that do not need to
match interfaces, a fact that is very advantageous in modelling of cracks. Various
extensions include enrichment functions for the crack tip [19], the application of
higher order elements [15], geometrically nonlinear formulations [3], as well as the
simulation of 3D [4, 5, 18] and bi-material cracks [17].
Refering to textile-reinforced composites the indepence of mesh geometry and
internal discontinuities implies that element boundaries do not conform to a material
interface. Instead, the mechanical behaviour which is characterized by a discontinuity
of strain perpendicular to the interface ∂G (Fig. 13.6)
�
˜ε1 j
�
˜εαβ
� � �
= ˜ε j1 �= 0
�
= 0
i, j = 1,2,3; α, β = 2,3 (13.37)
with ˜εij = cikc jlεkl; cij = ˜ei · e j; ˜e1 = n; [a]=a + − a − is represented by a local
enrichment of the FE displacement approximation within the element
Fig. 13.6 Interface with strain
discontinuity
uX-FEM = ∑Niui +
i
� �� �
∑ Nja jF
j∈E
� �� �
FEM Enrichment
(13.38)