Computational Methods for Debonding in Composites

300 H. Miled et al.
temperature is kept constant to 130 ◦ C. The temperature balance equation is:
ρCp
� ∂T
∂t
+ v.∇T
�
= div(k∇T )+ ˙w (15.15)
ρ is the volume density, Cp is specific heat, k is the thermal conductivity and ˙w
is the viscous dissipation. These properties are considered not dependant from the
induced flow anisotropy.
For short fiber reinforced materials, the flow becomes viscoelastic and the stress
tensor is expressed as a function of the fourth order orientation tensor [5]:
σ = 2η � ˙¯ε,T ��
� ��
˙ε + Np a : ˙ε − p1 (15.16)
where η(˙¯ε,T ) is the temperature-dependent viscosity due to both the polymer
matrix and fibers. The induced anisotropy is represented by the parameter Np which
depends on the fiber concentration and on the fiber aspect ratio, and is difficult to be
predicted. Thus, its contribution to the material’s rheology is not taken into account
in this work. As a consequence, the problem is considered governed by the Navier
and Stokes equations:
�
∂v
ρ
∂t +(v.∇)v
�
� �
= −∇p + ∇. 2η ˙ε + f
(15.17)
∇.v = 0
p is the polymer’s pressure and f are the body forces (such as gravity). We
suppose that the viscosity is given by the Carreau-Yasuda law [9, 43]:
� �
η(˙¯ε,T )=η0(T ) 1 + η0(T ) ˙¯ε
τs
�α
� m−1
α
(15.18)
where α and m are constant parameters, and η0(T ) represents the temperature
dependency, following the Arrhenius law:
� � ��
1 1
η0 (T)=η0 (Tref).exp β − (15.19)
T Tref
The velocity-pressure mechanical problem is solved using the mixed finite element
method with the P1+/P1 element and the thermal problem using a classical
Galerkin formulation. Both problems are weakly coupled: at each time increment,
the temperature-dependant viscosity used is the one computed at time t − ∆t. Furthermore,
the position of the flow front is advected using a Level set technique [4].
Parameters used in Eqs. 15.15, 15.18 and 15.19 are given in Table 15.1.