Computational Methods for Debonding in Composites

32 R. Rolfes et al.
symmetric, which is the case for unit cells used here under direct loads, for reasons
of symmetry the deformations have to be symmetric as well. Therefore, in
case of direct load it has to be ensured that the boundaries remain plane. For most
boundaries a deformation can be given
u(0,y,z) =0, u(a,y,z)=δx
v(x,0,z) =0
w(x,y,0) =0
(2.2)
whereas other boundaries can deform but have to remain straight and orthogonal,
which can be realized over a linear equation
v(x,b,z) =const. = δy
w(x,y,c) =const. = δz
(2.3)
Due to symmetry of unit cell and load it is only necessary to model one quarter
of the micromechanical unit cell. Under shear load opposing boundaries of the unit
cell have to remain parallel to each other:
but move in opposing directions:
u(0,y,z) =u(a,y,z)
v(x,0,z) =v(x,b,z) (2.4)
w(x,y,0) =w(x,y,c)=0
u(x,0,z) =−u(x,b,z)=−δx
v(0,y,z) =−v(a,y,z)=−δy
(2.5)
The forces F shown in Fig. 2.3 are the integrals over the stresses of the boundary
and yield the homogenized stresses when divided by the corresponding boundary
area. Deformations δi convert into homogenized unit cell strains and Poisson’s
ratios.
2.2.2 Voxel Mesh
Conventional modelling leads to a number of irregular elements, in particular for a
mesomechanical unit cell of a textile composite, but also for the micromechanical
unit cell. In combination with the strain energy based regularization irregular elements
lead to a mesh-dependent solution, see [5], because the regularization requires
elements with an aspect ratio of unity. To avoid this drawback of irregular elements,
the unit cells shown here are meshed with voxel elements, meaning “volume pixel”.
They have an aspect ratio of one, hence the geometry can only be approximated
because the mesh is regular.