Computational Methods for Debonding in Composites

6 Study of Delamination with in Composites 127
material is completely damaged, d = 1. The effective stresses, shown in Eq. (6.22),
correspond to the stresses that would be obtained in the material if it is not
damaged:
σσσ 0 = C0 : εεε (6.23)
And the real stress tensor can be obtained, from the strain state of the material,
by coupling Eqs. (6.22) and (6.23).
6.2.3.2 Damage Criterion
σσσ =(1 − d)σσσ0 =(1 − d)C0 : εεε (6.24)
The damage criterion predicts when the damage process begins to take place in
the material, this is: when the material elastic behaviour is lost as a result of the
degradation. This criterion is used to represent different material behaviours, and it
depends on the stress tensor and on the value of the internal variables found in the
material point under study. Its expression is:
F(σσσ 0, q)= f (σσσ 0) − c(d) ≤ 0 with q ≡{d} (6.25)
where c(d) is a function defining the damage limit value and f (σσσ 0) defines the damage
surface. Damage will start the first time that f (σσσ 0) value is equal or larger than
c(d). Usually, instead of working with Eq. (6.25), the damage criterion is converted
to an equivalent one by using a scalar function G which is positive and with its
derivative positive, monotonously increasing and invertible:
F ∗ (σσσ 0,d)=G[ f (σσσ 0)] − G[c(d)] ≤ 0 (6.26)
The damage surface to be used with this formulation can be any of the existing in
literature (i.e. Von-Mises, Mohr-Coulomb, etc.) and its election will depend on the
material to be modeled. In the present work, the damage surface that will be used
for matrix material corresponds to the one defined by norm of principal stresses,
weighted according to the compression–tension proportion found in the principal
stress tensor. This damage surface can be written as:
f (σσσ 0)=ρ ·�σσσ I� (6.27)
being σσσ I the principal stress tensor and ρ the compression–tension weight function,
defined as:
ρ = r0N +(1 − r0) (6.28)
where N = σ max
t is the material ratio between the maximum accepted compressive
strength/tensile strength. And r0 is a scalar function that defines the relation
c /σ max