1. the insufficient knowledge of the dynamic behavior and crashworthiness of the materials involved,
2. the often imperfectly known boundary conditions,
3. the heterogenous state implied by damage and rupture localization.

Here, we will be focusing on the first aspect. The dynamic tests we are considering are dynamic compression tests on laminated plates based on Split Hopkinson Pressure Bar principle. These lead to imprecise measurements of forces and velocities. Thus, Problem (2) is encountered from the very beginning of the identification process. Then, the question which comes to mind is:

How can one formulate an identification problem in dynamics in such a way that the results would be accurate in spite of the great measurement uncertainties?

Our guiding principle, which was directly inspired by studies on model updating in vibration [1], is to verify exactly, during the identification process, the properties which are considered to be reliable. The uncertain quantities are then taken into account by minimizing a modified constitutive relation error [2]. In this first work [3], we are considering the simple problem of an elastic rod with redundant displacement and force conditions at both ends, denoted and , which are assumed to be uncertain. The identification of the Young's modulus is carried out in two steps: for a fixed , the ill-posed problem is reformulated as the minimization of:

(54.1)

under the conditions:

KA to          DA to

(54.2)

One can note that the boundary conditions and are related to the fields used in the minimization problem and, therefore, can be different from the measured values and . This minimization under constraints is performed by introducing an appropriate number of Lagrange multipliers, which leads to the simultaneous resolution of both a direct problem and an adjoint (time-retrograde) problem related to one another. The resolution methods will be discussed during the presentation. Subsequently, the identification of the best is carried out using the same functional, whose gradient can be calculated directly from the fields which are solutions to the first problem with fixed . A numerical example, in which the exact measurements were modified by up to % on the average, will be presented. The method presented here, which turns out to be particularly robust, and another method inspired by [4] will be compared.

Figure 54.1: Variation of the error with : (a) classical approach, (b) our method

Work is currently underway to extend this method to elastic damage behavior, including the possibility of damage localization.

1

P. Ladevèze and A. Chouaki, "Application of a posteriori error estimation for structural model updating", Inverse Problems, 15, 49-58, 1999. doi:10.1088/0266-5611/15/1/009

2

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

3

P. Feissel, O. Allix, "Identification à partir d'essais dynamiques bruités", Rapport contractuel d'avancement, 2001.

4

L. Rota, "An inverse approach for identification of dynamic constitutive equations", Int. Symposium on Inverse Problems, Ed A.A.Balkema, 1994.

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