Anchored boundary conditions for locally isostatic networks

Abstract

Finite pieces of locally isostatic networks have a large number of floppy modes because of missing constraints at the surface. Here we show that by imposing suitable boundary conditions at the surface the network can be rendered effectively isostatic. We refer to these as anchored boundary conditions. An important example is formed by a two-dimensional network of corner sharing triangles, which is the focus of this paper. Another way of rendering such networks isostatic is by adding an external wire along which all unpinned vertices can slide (sliding boundary conditions). This approach also allows for the incorporation of boundaries associated with internal holes and complex sample geometries, which are illustrated with examples. The recent synthesis of bilayers of vitreous silica has provided impetus for this work. Experimental results from the imaging of finite pieces at the atomic level need such boundary conditions, if the observed structure is to be computer refined so that the interior atoms have the perception of being in an infinite isostatic environment.

Rigidity of Linearly Constrained Frameworks

“…Motivated by numerous potential applications, notably in mechanical engineering, rigidity has also been considered for frameworks with various kinds of pinning constraints [4,8,13,14,16]. Most relevant to this paper is the work of Streinu and Theran [14] on slider-pinning, which we describe below.…”

Ring correlations in random networks

“…These new results on 2D glasses have opened up numerous opportunities to study the structure of glasses using actual atomic coordinates. Recent work on 2D glasses includes modeling of silica bilayers [13,14], ring distribution [15], medium-range order [16], suitable boundary conditions to recover missing constraints in the surface [17] and the refinement of experimental samples [18]. Rigidity theory has also uncovered a connection between 2D glasses and jammed disk packings [19,20].…”

Realizations of Isostatic Material Frameworks

“…To render Trihex isostatic, anchored boundary condition are used and the blue vertices are pinned (immobilized). [ 27 ] The pinned vertices are placed generically (not on an equilateral triangle) and all edges initially have almost, but not exactly, the same length. The other three vertices on the surface are removed because they do not change the rigidity of the network as each triangle is rigid.…”