Computational Methods for Debonding in Composites

15 Numerical Simulation of Fiber Orientation and Resulting Thermo-Elastic Behavior 295
the Space-Time Discontinuous Galerkin method, applied by Redjeb et al. [36],
and the experimental ones, on an industrial test case. Once the part solidifies, we
consider an anisotropic elastic behaviour, described in Sect. 15.3. The goal is to
determine the effective elastic properties of the composite, for a given orientation
state (which is supposed not to be modified after the filling step of the injection
cycle). As proposed in the literature, our approach proceeds in two steps: first, using
homogenisation techniques, the unidirectional properties are determined; secondly,
the Advani and Tucker model [1] is used to compute the anisotropic mechanical
properties as a function of the unidirectional ones and of the second order orientation
tensor. In Sect. 15.4, an industrial example is considered to show the feasibility
of this methodology for the prediction of the thermo-elastic properties in the solid
state.
15.2 Modelling Flow-Induced Fiber Orientation
15.2.1 Evolution Equation of Fiber Orientation
For a single fiber, the orientation can be classically described by a unit vector p
which indicates the direction of the fiber axis (Fig. 15.1):
The evolution of p for a single fiber in a Newtonian fluid was given by Jeffery
[25]:
∂ p
� � � �
+ v.∇p = Ω.p + λ ˙ε p − ˙ε : p ⊗ p p
(15.1)
∂t
where v is the local velocity of the fluid, Ω =(∇v− ∇vt )/2 is the rotation tensor,
˙ε =(∇v + ∇vt )/2 is the strain rate tensor and λ is a function of the fiber aspect
ratio β :
λ = β 2 − 1
β 2 l
; β = (15.2)
+ 1 D
In this expression, l is the length of the fiber and D its diameter, supposed constants.
For a set of fibers, it is hard to follow each fiber but it is feasible to consider
the orientation distribution, given by a continuous function ψ(p,t), which represents
Fig. 15.1 Definition of the
vector p which characterizes
the orientation of a single
fiber [1]