Keywords: topology optimization, micropolar materials, chirality.

The topology optimization problem for linearly elastic micropolar solids is applied to the optimal design of three-dimensional solids made of isotropic micropolar materials. The constituent materials are supposed to generally lack centrosymmetry, which means that force stresses and microcurvatures are coupled, and so are couple stresses and micropolar strains. Non-centrosymmetric materials are also called chiral materials. Chirality is the lack of planes of symmetry in material elements, so that the mirror image of the element is not identical to the element itself. Macroscopically, this is the case of composites with helical or screw-shaped inclusions [1]; particular honeycombs can also be modeled as chiral materials at the macroscale [2].

The maximum global stiffness is taken as the objective function, and the material density as the design variable. According to the solid isotropic material with penalisation (SIMP) model [3], the constitutive tensors are assumed to be smooth functions of the density. Optimal material distributions are obtained for several significant cases, including beams subjected to axial force and bending.

The differences with respect to the optimal configurations obtained with classical Cauchy materials and centrosymmetric materials are pointed out and discussed. Similar to micropolar centrosymmetric solids, the optimum material localizes into curved beam-shaped regions [4], in contrast to the well-known truss-like solutions that arise in the optimal topological design of Cauchy solids [5]. The inherent material non-symmetry is matched by non-symmetric optimal geometries, as the material, in the optimal configuration, must resist the bending and torsional deformation of the structure.

1

R. Lakes, "Elastic and viscoelastic behavior of chiral materials", Int. J. Mech. Sci., 43(7), 1579-1589, 2001. doi:10.1016/S0020-7403(00)00100-4

2

D. Prall, R.S. Lakes, "Properties of a chiral honeycomb with a Poisson's ratio of -1", Int. J. Mech. Sci., 39(3), 305-314, 1997. doi:10.1016/S0020-7403(96)00025-2

3

M.Ph. Bendsøe, "Optimal shape design as a material distribution problem", Struct. Optim., 1(4), 193-202, 1989. doi:10.1007/BF01650949

4

M. Rovati, D. Veber, "Optimal topologies for micropolar solids", Struct. Multidisc. Optim., 33(1), 47-50, 2007. doi:10.1007/s00158-006-0031-0

5

M.Ph. Bendsøe, O. Sigmund, "Topology Optimization - Theory, Methods and Applications", Springer, Berlin, 2003.

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