טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentNitecki Saar
SubjectThe Mechanical Behavior of 2-D Lattice of Bistable Elements
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Josef Givli


Abstract

This thesis aims at studying theoretically the mechanical behavior of 2D lattices comprised from bistable elements (springs). The model of a 2D lattice with bistable springs is of relevance to a wide range of physical phenomena, such as atomic models of shape memory materials, truss structures for engineering applications, mechanics of composites, mechanics of protein networks, and mechanical metamaterials. The methodology employed is a theoretical analysis that draws on ideas from studies on 1D bistable lattices, previously applied for investigating intriguing material behavior, and generalizing these models to account for the 2D arrangement of the bistable elements. Focus is put on revealing the wealth of possible equilibrium configurations, the evolution of patterns/fronts of Phase transition, and on stability of the equilibrium configurations. To this end, we study four different 2D lattices, all comprised from a repeating square block and differ in the diagonal springs that are present within the square block. In addition, we employ the trilinear approximation combined with the assumption of small displacements to enable analytical analysis and closed-form expressions. Further, we develop a unique approach, the “equivalent stiffness method”, that significantly generalizes the stability-analysis results of the 1D bistable chain. In short, this method enables representation of a 2D sub-lattice in the form of a 1D lattices. This approach may be simply extended to more complex lattices, such as 3D lattices or lattices that involve beams (with resistance to bending) instead of simple springs. Based on this approach we were able to show that, in terms of stability, the 2D bistable lattice fundamentally differs from the 1D chain. For example, an equilibrium configuration that involves two or more springs with negative stiffness is always unstable for the 1D chain. On the other hand, in the 2D lattice, configurations involving two or more springs with negative stiffness may be stable, even if the two springs lie on the same row or even if they share the same node. Finally, we perform a parametric study by means of extensive numerical simulations. The main purpose of the parametric study was to provide better understanding for the influence of the lattice parameters, such as size, stiffness of the springs, the lattice geometry, etc., on the micro-level as well as the macro-level behavior.