Making use of Symmetries in the Elastic Inverse Homogenization Problem

MENDEZ, CARLOS GUSTAVO; JUAN MANUEL PODESTÁ; SEBASTIAN TORO; ALFREDO HUESPE; OLIVER, XAVIER

New York

Congreso; 13th World Congress in Computational Mechanics (WCCM); 2019

World Congress in Computational Mechanics (WCCM)

Itis a known fact that even the highest symmetry (isotropic) of theelastic tensor can be achieved through topology design using a unitcell with arbitrary shape whose material distribution does notpresent any symmetry (think about a polycrystal or an amorphousmaterial). However,it is also well known that an adequate choice of the unit cell andthe symmetries imposed in the design process can significantlyfacilitate the finding of certain classes of composites (likeVigdergauz microstructures or new ones proposed by Sigmund [1]).Inthis work, we make a comprehensive analysis of the connection betweenthe symmetry of the material distribution in the microstructure andthe properties of the resulting elastic tensor.Consideringperiodic structures, we analyze all the possible Bravais lattices andall the plane (wallpaper) groups in order to study the way in whichthe symmetries of these patterns are reflected in the homogenizedelastic tensor. For the unit cell we adopt Wigner-Seitz cells, whichare primitive cells that preserve all the symmetries of the subjacentBravais lattice and simplify the implementation of plane groups.Givenan arbitrary elastic tensor, we propose a procedure for the inversehomogenization that allow us to choose the most convenient shape forthe unit cell and to select the symmetries to be imposed thatguarantee (during the whole optimization process) that thehomogenized elastic tensor will have the same symmetry of the tensorto be designed. Concerning the design, several well establishedtools were used, such as algorithms for the rotation of the tensor totheir material axes [2] and topology optimization methods based onSIMP [1] and topological derivative [3].Someexamples regarding the search of new classes of extreme materials areshown, where it can be seen how different composites classes emergedepending on the enforced symmetries. [1]O. Sigmund(2000). Anew Class of Extremal Composites. Journalof the Mechanics and Physics of Solids,48(2),397-428.[2]N. Auffray and P. Ropars (2016). Invariant-based reconstruction ofbidimensional elasticity tensors. InternationalJournal of Solids and Structures, 87,183-193.[3]S. Amstutz et al. (2010). Topological derivative for multi-scalelinear elasticity models applied to the synthesis of microstructures.International Journal for NumericalMethods in Engineering, 84(6), 733-756.p { margin-bottom: 0.1in; direction: ltr; line-height: 120%; text-align: left; }a:link { color: rgb(0, 0, 255); }