Computational Methods for Debonding in Composites

15 Numerical Simulation of Fiber Orientation and Resulting Thermo-Elastic Behavior 309
Fig. 15.12 Conditions imposed on the plate for a static deformation test
modulus E22 and E33 are weak compared to E11, which means that an important
rigidity was added in the flow direction.
The plate is now subjected to a simple traction in the 1-direction, as shown in
Fig. 15.12. The plate is fixed at the other extremity, near the injection gate.
To solve the mechanical problem, the behaviour law is rewritten as following:
σ = C c : ε − (q + ξ ∇.u)1 (15.36)
where u denotes the displacement and q and ξ are a scalar numbers taken such as:
q + ξ ∇.u = 0 (15.37)
The mechanical problem to be solved is then:
�
∇. Cc �
: ε(u) − ξ (∇.u)1 − ∇q + fv = 0
∇.u + 1
q = 0
ξ
(15.38)
where fv are the body forces.
Equation 15.38 takes the advantage to deal with the isotropic case when the
material is incompressible. In fact, for an isotropic material, Eq. 15.38 is rewritten
as:
�
�
∇. 2µε(u)+(λ − ξ )∇.u1 − ∇q + fv = 0
∇.u + 1
q = 0
ξ
(15.39)
where λ and µ are the Lame coefficients, defined as a function of the Young
modulus and the Poisson ratio:
µ =
E
νE
; λ =
2(1 + ν) (1 − 2ν)(1 + ν)
(15.40)