Diploma thesis – Holger Mitschke

Abstract
This thesis addresses finite deformations of plane, periodic, symmetric skeletal structures, consisting of rigid bars linked at joints where they can pivot freely. In particular skeletal structures based on tessellations of the plane by polygons are considered. Deformations are computed by a numerical approach that explicitly maintains some or all symmetries of the tessellation. This method is applied to search for auxetic mechanisms in catalogues of tessellations. Auxetic mechanisms are deformations with a negative Poisson’s ratio, i.e. they expand in the transversal directions when stretched in the longitudinal direction. In contrast a positive Poisson’s ratio stands for the opposite reaction of an imposed elongation in one direction, namely a contraction in the perpendicular direction. A deformation of a periodic skeletal structure is studied by imposing an elongation in a given direction, and determining all joint positions as the solution of the system of bar equations, stating that all bar lengths remain unchanged. A skeletal structure is rigid if this disturbed system has no solution; it has a unique deformation, if there is a 1-dimensional solution space. Often the solution space is multidimensional. A numerical solution of this system of quadratic equations is obtained using the Newton- Raphson method. The Newton-steps are solved using the singular value decomposition to get the one with the smallest coordinate deviation in underdeterminate situations. In contrast to rigidity theory the problem will not be linearised and the study of deformations with finite elongations is possible. For the discussion of deformation behaviour, in particular Poisson ratios, the analysis will be restricted to unique deformations. For the multidimensional cases, symmetry con- straints are imposed to select one path of the continuum of possible deformations. Transla- tional periodic systems can have further symmetries. All symmetries of one structure build a group. In the plane there are exact 17 different groups, called crystallographic plane groups. The group describe all symmetries of a system expressed by symmetry operations. They contain at least the translational operations, but can have more. The requirement that under deformation symmetries remain provides the possibility to reduce the dimen- sion of the solution space. The symmetries of the full group or of possible subgroups are retained. The implementation offers the generation of the system of bar equations and the symmetry relationships between symmetric equivalent joints in the desired group from the list of joint coordinates and the set of linked pairs of joints in the asymmetric unit. Finite deformations of the skeletal structures and their Poisson’s ratio as a function of the elongation are determined for a number of classes of known tessellations. The main result is that a large number of the Archimedean, 2-uniform and uniform tessellations by regular polygons and star polygons have auxetic mechanisms for large deformations when constraining symmetries. Two as yet unknown examples of auxetic mechanisms at infinitesimal deformations without the additional symmetry constraints could also be identified. The presented program to numerically calculate finite deformations of skeletal struc- tures provides the basis for a future statistical mechanics approach to the generally multi- dimensional deformation space. The approach can also be used to understand deformation mechanisms of disordered networks, and to search systematically for structural character- istics that correlate to the deformation behaviour.