Keywords: identification, inverse problem, corrupted measurements, transient dynamic, viscoplasticity, constitutive relation error, LATIN method.

The motivation of this study is the identification of the parameters of a rate dependent damage model [1] from crash tests which are characterized by highly scattered measurements. In this case, contrary to other experimental situations, there is no a priori information available on the nature or on the level of the perturbation, which is why we were a priori unable to rely on a Tikhonov regularization approach [3,4] or on a Kalman filter approach [5,6]. Therefore, a specific method has been studied. It is based on the concept of modified constitutive relation error (CRE) [7] which has been proven to be effective in the case of model updating in vibrations [8]. Its principle consists in splitting the experimental relations as well as the theoretical ones into two groups: the reliable relations and the less reliable relations. The verification of the reliable ones are enforced throughout the identification process, whereas the less reliable ones are relaxed and verified globally, at best, throughout the identification strategy by the minimization of the sum of a modeling error term and an experimental error term.

Since the identification of the above damage model appeared to be too complex to be studied at first, the method was first developed in the elastic case. In [2], the method exhibits a very robust "behavior" with respect to the corruption of the measurements. The present work deals with the extension of this method to the case of visco-plasticity.

In the case of visco-plastic material, the reliable quantities appear to be the initial conditions, the dynamic equilibrium equation and the state law. The model distance is chosen, among other possibilities, as the quadratic distance to the evolution law. This choice allows the minimization of a quadratic function under non-linear constraints. The use of the adjoint method leads us to solve a non linear wave propagation problem with both initial and final conditions. The large time increment (LATIN) method [9] is therefore applied in order to linearize this problem over the whole time domain. Thus, the problem is treated by an iterative procedure, where at each iteration we solve successively:

• a system of non-linear equations which is local in space variable,
• a problem of minimizing a quadratic function under linear constraints, which is global in space variable.

The resolution of the second one is done by using the method developed for the case of elasticity [2].

This identification strategy has been applied in the case of a visco-plastic model with isotropic hardening. The results obtained show that the iterative resolution strategy based on the LATIN method converges with different measurements and different initial guesses of material parameters; and the identification method appears to be robust with respect to perturbations on the measurements up to 40%.

The next step will be to apply the methodology to the damage case.

1

O. Allix and J.F. Deü. Delay-damage modeling for fracture prediction of laminated composites under dynamic loading, Engineering Transactions, Vol. 45, 29-46, 1997.

2

O. Allix, P.Feissel and H.M. Nguyen. Identification strategy in the presence of corrupted measurements, Engineering Computations, Vol 22, 5/6, 487-504, 2005. doi:10.1108/02644400510602989

3

A.N. Tikhonov and V.Y. Arsenin. Solutions of Ill-Posed Problems, Winston and Sons, 1977.

4

S. Andrieux and F. Voldoire. Stress identification in steam generator tubes from profile measurements, Nuclear Engineering and Design, Vol.158, 2-3, 417-427, 1995. doi:10.1016/0029-5493(95)01048-M

5

R.E. Kalman. A new approach to linear filtering and prediction problems, Trans. ASME J. Basic Engineering, Vol.82, 35-45, 1960.

6

A. Corigliano and S. Mariani. The extended Kalman filter for model identification in impact dynamics, Complas VII, E. Oñate and D.R.J. Owen, Barcelone, 2003.

7

P. Ladevèze. A modeling error estimator for dynamical structural model updating, Advances in Adaptive Computational Methods in Mechanics, Elsevier, 1998. doi:10.1016/S0922-5382(98)80009-3

8

A. Deraemaeker, P. Ladevèze and Ph. Leconte. Reduced bases for model updating in structural dynamics based on constitutive relation error, Computer Methods in Applied Mechanics and Engineering, Vol. 191, 21-22, 2427-2444, 2002. doi:10.1016/S0045-7825(01)00421-2

9

P. Ladevèze. Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation, Springer Verlag, 1999.

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