Keywords: composite materials, delamination, fracture mechanics, numerical analysis.

This paper presents an implicit algorithm to simulate delamination growth in structures made of composite materials and submitted to monotone as well as fatigue loading. In both cases, the main feature of the algorithm is to generate a non-local relation between the delamination driving force or local energy release rate and the delamination front displacement.

In the case of monotone loading and for stable growth, the method consists in minimising the total energy of the structure with respect to a delamination front displacement [1]. This energy is defined as the sum of the mechanical energy and of the Griffith's fracture energy in such a way that the characterisation of the minimum gives the well-known Griffith's criterion in a weak form. To solve the minimisation problem, the two first derivatives of the energy with respect to the front displacement are needed. Their analytical expressions are obtained using the method of Destuynder and Djaoua [2]. The numerical approximation is then made approaching the front by -splines and the front displacement by the product of -splines along the front with a bell-shaped function in the plane perpendicular to the front so that the unknowns are the splines co-ordinates of the front displacement. Let the front location and the loading be given, the new front location is obtained performing the following steps:

• The solution of the problem of elasticity is first computed.
• Then the two first derivatives of both the mechanical energy and the fracture energy are computed.
• The front displacement is obtained performing one iteration of the Newton's method.
• Finally, the front is moved and the part of the structure near the delamination is meshed again.

These steps are repeated until the convergence is reached.

In the case of fatigue loading, the same approach is used for one-dimensional cracks, defining the fracture energy in such a way that the characterisation of the minimum of the total energy gives the evolution law of interest, here the Paris' law [3]. An extension to delamination growth can be made writing the Paris' law in terms of the delaminated area increase rate in place of the front displacement increase rate. Unfortunately, a lot of work is required to validate such an evolution law and to identify the related parameters. In this paper, an implicit algorithm is obtained defining a weak form of the classical Paris' law and solving the resulting problem by the Newton's method.

For monotone loading, the growth stability is studied looking at the spectrum of the second derivative of the mechanical energy:

• If all the eigenvalues are positive, the growth is stable.
• If all the eigenvalues are negative, the growth is unstable.

Numerical computations made starting with an artificial delamination showed that the spectrum could have negatives eigenvalues though the growth is stable. For such cases, it is expected that the initial front shape was not adapted to the loading. Furthermore, it was observed that the number of negative eigenvalues was decreasing as the front shape approached the converged front shape.

For fatigue loading, the problem of growth stability remains as the second derivative of the mechanical energy is required. The numerical experiments showed that, if the spectrum of the total energy was positive, the algorithm converged, otherwise it diverged. In this case, an explicit algorithm is better. However, when convergence is reached, the number of iterations of the algorithm was revealed insensitive to the value of the number of cycles increment so that large increments can be made.

1

Y. Ousset, "Numerical simulation of delamination growth in layered composite plates", European Journal of Mechanics A/Solids, 18, 291-312, 1999. doi:10.1016/S0997-7538(99)80017-5

2

P. Destuynder, M. Djaoua, "Sur une interprétation mathématique de l'intégrale de Rice en théorie de la rupture fragile", Mathematical Methods in Applied Science, 3, 70-87, 1981. doi:10.1002/mma.1670030106

3

P. Paris, F. Erdogan, "A critical analysis of crack propagation laws", Journal of Basic Engineering, 85, 528-534, 1963.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)