Keywords: sandwich composite, viscoelasticity, elasto-plasticity, anisotropy, homogenisation.

In the present paper, the modelling of a sandwich composite is presented within the framework of the continuum mechanical theory. The investigated composite sandwich is composed of three layers, namely two top panels made of an aluminium alloy and one polymeric core reinforced with carbon fibers. In a first part, the material modelling of the separate layers is presented [1,2]. Whereas the metal panels have an elasto-plastic material behaviour, the polymeric core matrix is viscoelastic [3,4,5], and the carbon fibers are supposed to be purely elastic. As a consequence, the polymer matrix reinforced with one family of fibers will be simulated as a transverse isotropic material.

First, a three-dimensional model of the plate consisting of the individual layers is applied for a cyclic traction test and for a cyclic flexion test. Whereas the results for the simulation of a single material are quite easy to analyse, the ones for the composite made of (only) three layers are much more complex. The curves showing the stress as function of the displacement demonstrate a complicated behaviour where the influence of the plasticity and the effects of the viscoelasticity are difficult to differentiate. Moreover, the stress distribution as function of the layer's thickness demonstrates a discontinuous and nonlinear curve. The nonlinearity of the material behaviour (especially plasticity) forbids the use of a classical plate theory. Because of the complexity of the results and of the large computation time required for the three-dimensional model, a homogenisation method is preferred.

In a last part, two-scale modelling is presented. A numerical homogenisation is of great interest in the field of modelling complex material at a finite strain state. The principle of a numerical homogenisation for a plate material was first presented by Landervik and Larsson [6,7]. The principle of a homogenisation is to project the deformations of the macroscale into the microscale, in order to take into account the heterogeneities of the microscale. In this scale, a micro boundary problem will be numerically solved within the representative volume element (RVE). In case of a hybrid laminate, assumptions have to be made for the definition of the RVE, since it must include the anisotropy of the material. As a matter of fact, the RVE has to include the whole thickness of the plate and can only vary in the other two directions. Because of the nonlinearity of the deformations observed within the RVE, a second order homogenisation has to be used. The stress resultant calculated over the microscale will be in a last step projected back into the macroscale, with help of an extension of the Hill condition adapted to a second-order homogenisation. Further developments on the shell kinematics, the investigation of a RVE and the projection back to the macroscale will be presented in a forthcoming work.

1

P. Haupt, "Continuum Mechanics and Theory of Materials", Springer Verlag, Second Edition, 177-249, 435-501, 2002. doi:10.1115/1.1451084

2

G.A. Holzapfel, "Nonlinear Solid Mechanics, A continuum Approach for engineering", Wiley, 205-251, 2000.

3

M. Johlitz, H. Steeb, S. Diebels, A. Chatzouridou, J. Batal, W. Possart, "Experimental and theoretical investigation of nonlinear viscoelastic polyurethane systems", J Mater Sci, 42, 9894-9904, 2007. doi:10.1007/s10853-006-1479-4

4

K. Sedlan, "Viskoelastisches Materialverhalten von Elastomerwerkstoffen: Experimentelle Untersuchung und Modellbildung, Dissertation", PhD, Kassel University/Universität Gesamthochschule Kassel, 60-120, 2000.

5

A. Lion, "Thermodynamik von Elastomeren", Berichte des Instituts für Mechanik der Universität Kassel (Bericht 1/2000), ISBN 3-89792-023-9.

6

M. Landervik, R. Larsson, "Multiscale Homogenization and shell theory for modeling thin porous layers", Chalmers Reproservice, 2008.

7

M. Landervik, R. Larsson, "A higher-order stress-resultant shell formulation based on multiscale homogenization", Chalmers Reproservice, 2008.

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