Computational Methods for Debonding in Composites

230 J.A. Oliveira et al.
χ jk
i (0,y2,y3) =χ jk
i (y0 1 ,y2,y3)
χ jk
i (y1,0,y3) =χ jk
i (y1,y 0 2 ,y3)
χ
and (11.23)
jk
i (y1,y2,0) =χ jk
i (y1,y2,y 0 3 )
In order to prevent rigid body motion, displacements and rotations of an arbitrary
point of the unit-cell must be locked. In this work, this restriction is created acting
only on the translation degrees of freedom of one of the vertices of the RUC. Rigid
body motion is avoided by the periodicity constraints, which, as shown in relations
11.23, force the restriction to act equally over all other vertices.
11.3.3 Homogenised Elasticity Matrix D h
The homogenised elasticity matrix D h is obtained from Eq. (11.17), resulting in
D h =
ne
∑
k=1
Y k
Y Dk (I − B k χχχ k ) (11.24)
where Y k is the volume of element k, Y the total geometry volume and I the identity
matrix. Note that if χχχ = 0, this equation becomes the classical volume average of
the elastic properties of the microscale elements.
11.4 Numerical Procedures
The numerical tools developed by the authors are based on the finite element
method (FEM) [13, 15, 25] and use tetrahedral and hexahedral finite elements. The
main focus of this section is on the auxiliary algorithms for RUC generation and
periodicity boundary condition management.
11.4.1 The Main Program
The finite element code is developed for structural linear elastic computations [13].
It solves 3-D problems using tetrahedral or hexahedral (linear or quadratic) finite
elements.
Since the FEM analyses considered in this work often lead to sparse coefficient
matrices, using the complete matrix on the numerical calculations greatly reduces
the efficiency of the matrix operations involved. This may become critical with the
increase of the size of the problem, as the number of zero coefficients tends to
increase, leading to a waste of computational effort. This limitation can be overcome