Compatibility conditions for discrete planar structures

Abstract

Compatibility conditions are investigated for planar network structures consisting of nodes and connecting bars. These conditions restrict the elongations of bars and are analogous to the compatibility conditions of deformation in continuum mechanics. Two problems are considered: the discrete problem for structures with prescribed lengths and its linearization, the discrete problem of prescribed elongations. These problems approximate two continuum problems, the nonlinear continuum problem of given Cauchy Green tensor and its linearization, the continuum problem of prescribed strain. The requirement that the deformations remain planar imposes solvability conditions of all four problems. For triangulated structures, compatibility conditions for the nonlinear problem are expressed as a polynomial equation in the lengths of edges of the star domain surrounding each interior node. In the linearized discrete problem, the compatibility conditions become linear relations for the elongations of edges on the same domains. The continuum limits of the compatibility conditions for both discrete problems are proved to be the compatibility conditions for the continuum problems.The compatibility equations may be summed along a closed curve to give conditions supported on a strip along the curve. Similarly, for continuous materials, the compatibility equation for the prescribed strain problem may be integrated along a closed curve to provide an integral condition, analogous to how the prescribed Green tensor problem may be integrated to give the Gauss-Bonnet integral formula.Compatibility conditions are investigated for general trusses such as plates connected by girders or triangulated domains with holes or missing edges. Compatibility conditions on general trusses may be non-local. There may be rigid trusses without compatibility conditions in contrast to continuous materials. The number of compatibility conditions is the number of bars that may be removed from a structure and still keep it rigid. This number measures the resilience of the structure. The compatibility equations around a hole involve the edges in the neighborhood surrounding the hole. An asymptotic density of compatibility conditions for periodically damaged triangular structures is found to be sensitive to the number and location of the removed edges.