## Application to elastoplasticity

The application of the virtual fields method to elasto-plastic constitutive equations is rather recent. The first efforts started in 2003. The main difference
from elasticity is that the stresses cannot be written as an explicit function of the measured strains. Indeed, the measurements are total strains and it
is therefore necessary to separate elastic from plastic strains. The principle of the method is therefore as follows:

- identification of the elastic stiffnesses in the first stages of the test, when plasticity has not yet initiated;
- choose a first set of plastic parameters to initiate the process
- use these parameters to separate plastic and elastic strains
- calculate the stresses from the calculated elastic strains and the identified elastic parameters
- compute a cost function which is the difference between the virtual work of the internal and external forces. It is possible to add as many terms
as selected virtual fields to enrich the cost function
- minimize the cost funtions by iterating on the plastic parameters

Although also iterative, this procedure is much faster than finite element model updating since the calculation of the cost function does not require
a complete finite element calculation but simply an extraction of elastic from total strains. The identification typically runs in a couple of minutes.

The following slideshow explains how the method works.

The VFM in elasto-plasticity
### References

- Grédiac M., Pierron F.,
Applying the Virtual Fields Method to the identification of elasto-plastic constitutive parameters,
*International Journal of Plasticity*, vol. 22, pp. 602-627, 2006.
- Pannier Y., Avril S., Rotinat R., Pierron F.,
Identification of elasto-plastic constitutive parameters from statically undetermined tests using the virtual fields method,
*Experimental Mechanics*, vol. 46, n° 6, pp. 735-755, 2006.
- Avril S., Pierron F., Pannier Y., Rotinat R.,
Stress reconstruction and constitutive parameter identification in elastoplasticity using measurements of deformation fields,
*Experimental Mechanics*, 2007. Accepted.