By Bob Svendsen (auth.), Bernd Markert (eds.)

Modern computational options, resembling the Finite aspect technique, have, on account that their improvement a number of a long time in the past, effectively exploited continuum theories for varied functions in technological know-how and know-how. even though ordinary continuum equipment dependent upon the Cauchy-Boltzmann continuum are nonetheless of serious value and are universal, it more and more looks that fabric homes stemming from microstructural phenomena need to be thought of. this can be quite real for inhomogeneous load and deformation states, the place lower-scale measurement results start to impact the macroscopic fabric reaction; anything general continuum theories fail to account for. Following this concept, it really is obtrusive that ordinary continuum mechanics should be augmented to seize lower-scale structural and compositional phenomena, and to make this data available to macroscopic numerical simulations.

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2 and 3 shows that there are several possible orientations of the honeycomb structure with respect to the shear direction, cf. Teko˘glu and Onck [50]. What can be expected is a dependency of the mechanical properties from the hexagons’ orientation. For that reason another configuration in which the hexagons are rotated by 90◦ is examined too, cf. Fig. 3. Note that Gibson and Ashby [26] u1 = u2 = 0, ϕ = 0 Fig. 2 Honeycomb micromodel (config. 1) with complete set of boundary conditions for shear test.

The constitutive equations define T and M as functions of the Cosserat strain ε¯ and the curvature κ¯ , cf. de Borst [6]: T = 2μ ε¯ sym + 2μC ε¯ skw + λ (ε : I) I , M = 2μC (lC )2 κ¯ . (5) (6) Therein λ and μ are the Lamé constants known from Hooke’s law describing the symmetric stress tensor for elastic material behavior in the classical theory. The further parameters μC and lC are the already mentioned additional Cosserat parameters representing an additional stiffness as well as some kind of internal length.

Fig. 5. Fig. 5 Unit cells for the honeycomb structure with different orientations. To receive reference values for the nodal displacements and rotations the values at the shear sample’s top and bottom are included as well as average values of the nodal displacements and rotations per unit cell. As already mentioned a unit cell consists of three second order Timoshenko beams with three nodes each. This means the average displacement and rotation, respectively, for each unit cell found in the shear samples regardless from it’s orientation is represented by the mean of 7 nodal displacements and rotations each.

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