Computational Methods for Debonding in Composites

6 Study of Delamination with in Composites 121
stresses are zero is similar, as it will be proved in this work, to the performance
shown by a delaminated material.
To obtain this structural behaviour, the matrix material has to loose its strength for
a certain stress state. This lost of strength must be permanent in order to simulate the
real crack produced by delamination in the material. This is achieved with a damage
formulation based on the fracture energy of the material.
A detailed description of the different formulations required to simulate the
delamination process: the S/P mixing theory and the damage formulation used in
matrix material, are described in the following section. Afterwards, to prove the
validity of the scope used to simulate the delamination phenomenon, as well as the
ability of the formulations proposed to simulate it, this work compares the results
obtained from the experimental test made to obtain the mode II fracture energy of a
composite with the results obtained from a numerical simulation of the same model.
The experimental test is the End Notch Flexure (ENF) test defined by the European
Structural Integrity Society (ESIS). The agreement between experimental and
numerical results will prove the ability of the S/P mixing theory, together with the
damage formulation used, to simulate delamination processes.
6.2 Formulation
The formulation shown in this work to simulate the delamination effect in composite
materials made of laminates of fibre reinforced polymers is the following
one: The composite behaviour is obtained from its constituent materials with the
serial/parallel mixing theory developed by Rastellini [13]. This theory is described
in Sect. 6.2.1. To obtain a good convergence ratio in the process and, in most cases,
to be able to obtain convergence, it is necessary to use a tangent constitutive tensor.
This tensor can be obtained analytically for some constitutive equations but not
for the damage formulation used in the presented simulation. To solve this problem,
the tangent constitutive tensor is obtained performing a numerical derivation
with a perturbation method. This methodology is exposed in Sect. 6.2.2. Finally, the
delamination process is obtained as a lost of strength and stiffness in matrix material.
This effect is characterized using a damage formulation, which is described in
Sect. 6.2.3.
6.2.1 Serial/Parallel Mixing Theory
The serial/parallel mixing theory considers that in a certain direction (or directions)
the compounding materials behave in parallel, while their behaviour is serial in the
remaining directions. For this reason it is necessary to define, and split, the serial and
parallel parts of the strain and stress tensors. This is done with two complementary
fourth order projector tensors, one corresponding to the serial direction (PS)andthe