Computational Methods for Debonding in Composites

6 Study of Delamination with in Composites 129
10
0
-10
-20
-30
-0.15 -0.1 -0.05 0 0.05
(a) (b)
10
0
-10
-20
-30
-0.15 -0.1 -0.05 0.05
Fig. 6.1 Stress–strain graph obtained with the damage formulation. In (a) is represented a material
with linear softening while (b) shows a material with exponential softening
Exponential softening: This function was first proposed by Oliver et al. [9], to
obtain an exponential softening in the material. The expression of the damage
parameter is:
d = G[ f (σσσ 0)] = 1 − τττ0
f (σσσ 0) e
A
⎛
⎝1− f (σσσ 0)
τττ 0
⎞
⎠
(6.34)
The parameter A depends of the fracture energy of the material. Its expression is
defined in the following section. And, the value τττ 0 , corresponds to the limit elastic
stress that can be found in the material. When using the damage surface based on the
norm of the principal stresses, differentiating between the compression and tension
states, the limit stress to be defined is the one corresponding to the compression
case, σ max
c .
Figure 6.1 a shows the stress-strain relation obtained for a material with a linear
softening; and Fig. 6.1b shows the evolution of the same material when an exponential
softening is applied to it. In both cases the relation defined between the
compression strength and the tension strength is: N = 2.
6.2.3.4 Parameter A
The parameter A, appearing in Eqs. (6.33) and (6.34), is obtained from the dissipation
equation of the material, considering an uniaxial process under a monotonous
increasing load. The parameter deduction can be obtained from [11] and their
expression is,
Linear softening: A = − 1 (τττ
2
0 ) 2
gcC0
Exponential softening: A =+
1
gcC0
(τττ 0 1
−
) 2 2
(6.35)