## Application to damage

There are two main applications in this section, either the identification of damage as it grows or the identification of the effect of pre-existing damage (in the spirit of non destructive testing).

### Identification of a damage law

This was studied on unidirectional composites where the highly non-linear in-plane shear behaviour can be described by a damage law. The parameterization of this damage law can follow the now rather standard model proposed by Ladevèze et al. (see reference below). Therefore, one can write the shear stress as the function of the shear strain either as a polynomial of the shear strain or using a dmagae threshold. In the first case, the equations remain linear and the four in-plane orthotropic stiffness components plus the parameters driving the damage polynomial can be identified using a test like the unnotched Iosipescu test (see Chalal et al. 2004). However, experimentally, a law with a damage threshold seems more adapted and the system then becomes non-linear and has to be solved by minimizing a cost function. Results can be found in Chalal et al. 2006 below.
It should be pointed out that a group at the Polytechnic University della Marche in Ancona, Italy, is trying to identify a damage law in metals from combined fringe projection and digital image correlation, using the Virtual Fields Method (see Rossi et al. 2010).

### References on identification of a damage law

- Ladevèze P., Allix O., Deü J.-F., Lévêque D., A mesomodel for
localisation and damage computation in laminates. Computational Methods in
Applied Mechanical Engineering, vol. 183, pp. 105–22, 2000.
- Chalal H., Meraghni F., Pierron F., Grédiac M.,
Direct identification of the damage behaviour of composite materials using the virtual fields method,
*Composites Part A: Applied Science and Manufacturing*, vol. 35, n° 7-8, pp. 841-848, 2004.
- Chalal H., Avril S., Pierron F., Meraghni F.,
Experimental identification of a nonlinear model for composites using the grid technique coupled to the virtual fields method,
*Composites Part A: Applied Science and Manufacturing*, vol. 37, n° 2, pp. 315-325, 2006.
- Rossi M., Chiappini G., Sasso M., Characterization of Aluminum Alloys Using a 3D Full Field Measurement, Annual SEM conference (Society for Experimental Mechanics), 7-10 June 2010 in Indianapolis, IN, USA.

### Identification of a stiffness reduction map in impacted composites plates

This application considers pre-existing damage in impcated composite plates. This damage leads to a local reduction of stiffness in the plate. It many techniques exist to locate the damage and evaluate its size (ultrasounds, infrared thermography, shearography etc.), it is much more difficult to quantitatively assess the stiffness reduction caused by this damage. It should be noted that a similar approach could be used for any phenomenon leading to a heterogeneous stiffness map (manufacturing defects, for instance).
The idea is to the plates in bending at very low strains in order not to grow the damage (NDT procedure). The bending stiffness matrix will be written as a function of the virgin material stiffness matrix times a stiffness reduction function. This function can be discrete (piecewise) or continuous (polynomial for instance), depending on the stiffness contrast and size of the damage. This topic was the subject of the PhD thesis of Dr Jin-Hwan Kim that can be downloaded on the link given in the references below. This work, in close collaboration with Professor Michael R. Wisnom at the University of Bristol (UK), is being pursued through the PhD of Mr Cédric Devivier (to be completed in 2011).

### References on identification of a stiffness reduction map

- Kim J.-H,
Identification of the local stiffness reduction of damaged composite plates using the virtual fields method, PhD thesis, Arts et Métiers ParisTech, France, 2008.
- Kim J.-H., Pierron F., Wisnom M., Syed-Muhamad K.,
Identification of the local stiffness reduction of a damaged composite plate using the virtual fields method,
*Composites Part A: Applied Science and Manufacturing*, vol. 38, n° 9, pp. 2065-2075, 2007.
- Kim J.H., Pierron F., Wisnom M., Avril S.,
Local stiffness reduction in impacted composite plates from full-field measurements,
*Composites Part A: Applied Science and Manufacturing*, vol. 40, n° 12, pp. 1961-1974, 2009.