Computational Methods for Debonding in Composites

48 R. Rolfes et al.
A dev is the deviator of the structural tensor A, A is the constant bending tensor
and B is the first derivative of the linear terms in σ of the quadratic yield locus. This
enables us to state the yield function Eq. (2.37) in the more general form:
2.3.2.3 Parameter Identification
f = 1
σ : A : σ + B : σ − 1 (2.40)
2
To determine the five material parameters α1, α2, α3, α32 and α4 of the yield function,
material simulations of the micromechanical unit cell are used. As already
explained, in fiber direction yielding is not assumed, because the strength in the preferred
direction is determined by the strength of the fibers. So the parameter α4 is set
to zero and there remain four parameters to be determined by means of four virtual
material tests done with the micro model. The material tests and there representation
in stress space are:
1. Simple shear in the plane perpendicular to the fiber (transverse shear)
σ = devσ = σ pind ⎡ ⎤ ⎡ ⎤
0 y⊥⊥ 0
0
⎢ ⎥ ⎢ ⎥
= ⎣ y⊥⊥ 0 0⎦,
a = ⎣ 0 ⎦
0 0 0
1
I1 =(y⊥⊥) 2 , I2 = 0, I3 = 0, I4 = 0
� f = α1 (y⊥⊥) 2 − 1 = 0
α1 := 1/(y⊥⊥) 2 (2.41)
2. Simple shear in the fiber plane (in-plane shear)
σ = devσ = σ pind ⎡ ⎤
0 y⊥� 0
⎢ ⎥
= ⎣ y⊥� 0 0⎦,
⎡ ⎤
1
⎢ ⎥
a = ⎣ 0 ⎦
0 0 0
0
I1 = 0, I2 =(y ⊥�) 2 , I3 = 0, I4 = 0
� f = α2 (y ⊥�) 2 − 1 = 0
α2 := 1/(y ⊥�) 2 (2.42)
3. Uniaxial tension and uniaxial compression perpendicular to the fiber
⎡ ⎤
00 0
⎢ ⎥
σ = ⎣ 00 0⎦,
⎡ ⎤
1
⎢ ⎥
a = ⎣ 0 ⎦
00y⊥
0