Keywords: shock waves, shell theory, viscoplasticity, material damage.

The motivation for the present paper was the need for a material model that is able to predict the damage evolution during the dynamic response of structures, i.e. the onset of damage, the gradual degradation and final failure of structures, under impulsive loading conditions. The focus is on a critical review of the ductile damage law of Lemaitre [3], which had been originally derived for strain rate independent materials with intention to apply it for strain rate dependent material behaviour. The adaptation for impulsive loading conditions of strain rate dependent material is supported by results of uniaxial tension tests, consisting of loading and unloading cycles at different strain rates. These experimental tests have been carried out to investigate the effect of different strain rates on the damage initiation and evolution.

In literature a wide range of structural models, constitutive equations and experimental methods for investigating the deformations of impulsively loaded structures is available. However, the mechanical models are very diverse and can, therefore, lead to different results in numerical simulations. For this reason, validations by means of experiments are necessary, providing a possiblity to assess the applied structural and material models.

Special investigations on impulsively loaded stuctures are described e.g. by Cristescu [1], Stronge and Yu [8]. In these studies loadings and measurements of the resulting structural deformations are typically performed during short time periods of several microseconds. Common methods for applying impulses on structural specimens are impact, explosion and shock wave-loadings. A summary of short time measurement techniques for recording deformation and load histories in impulsive loading situations is given by Knauss [2]. In this context, the shock tube technique provides a distinct advantage over other experimental methods applying impulsive loads on structures. This advantage consists in the fact that the shock waves can be made nearly ideally plane, if the shock tube is only long enough. Thus, the loading history for a plane specimen can be measured next to the the examined structure. In the contrary, explosions typically lead to spherical shock waves and consequently to very complex pressure distributions. These loading histories can hardly be modelled accurately enough. An alternative method for subjecting structures to impulsive loads are impact experiments which cause concentrations of stresses and strains in the surrounding of the acting force. Because of its advantages the shock tube technique is used in the present investigation.

In our previous studies [6,7] viscoplastic laws were validated by means of comparisons between measured and simulated deflections of shock wave-loaded circular metal plates. Two different shock tubes were used for subjecting initially flat plates to shock waves. By applying short time measurement techniques, the midpoint displacement and the pressure acting on the plate specimen were measured during the impulse duration. It was shown that the applied non-linear first-order shear deformation shell theory of Schmidt and Reddy [5] in combination with the viscoplastic law of Lemaitre-Chaboche [4] is appropriate to predict the measured deformation history.

Here, the validation study is extended by including material damage in the experiments and in the simulations. For this purpose an experimental set-up is required by means of which dynamic plastic ductile damage can be initiated and its evolution can be traced. Subjecting aluminium plates alternately to cyclic shock waves leads to accumulated plastic strains associated with material damage and failure of the structure. Due to the inelastic deformations, a snap-through occurs during the impulsive loading cycles. For FE-simulations the damage initiation, damage progession and final failure of the structure are taken into account by means of an adaptation of the damage law of Lemaitre [3] for strain rate dependent material. Special consideration is paid to the proper treatment of dynamic viscoplastic problems by Lemaitre-type damage laws which concerns especially the treatment of the overstresses in this framework.

1

N. Cristescu, "Dynamic plasticity", North-Holland Publishing Company, Amsterdam, 1967.

2

W.G. Knauss, "Perspectives in experimental solid mechanics", Int. J. Solids Structures, 37, 251-266, 2000. doi:10.1016/S0020-7683(99)00092-X

3

J. Lemaitre, "A continous damage mechanics model for ductile fracture", J. Eng. Mat. Tech., 107, 83-89, 1985.

4

J. Lemaitre, J.L. Chaboche, "Mechanics of Solid Materials", Cambridge University Press, 1994.

5

R. Schmidt, J.N. Reddy, "A refined small strain moderate rotation theory of elastic anisotropic shells", ASME J. Appl. Mech., 55, 611-617, 1988.

6

M. Stoffel, R. Schmidt, D. Weichert, "Shock wave-loaded plates", Int. J. Solids Structures, 38, 7659-7680, 2001. doi:10.1016/S0020-7683(01)00038-5

7

M. Stoffel, "Evolution of plastic zones in dynamically loaded plates using different elastic-viscoplastic laws", Int. J. Solids Structures, 41, 6813-6830, 2004. doi:10.1016/j.ijsolstr.2004.05.060

8

W.J. Stronge, T.X. Yu, "Dynamic Models for Structural Plasticity", Springer-Verlag, Berlin-Heidelberg-New York, 1993.

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