Phase transitions and anisotropic responses of planar triangular nets under large deformation

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Abstract
Responses of triangular networks in large reversible deformation are studied analytically at zero temperature and by Monte Carlo simulation at nonzero temperature. Exact expressions for the elastic strain energy at zero temperature are derived for several models in which the network potential energy depends on either the length of the network element (i.e., central force interactions) and/or the area of each network triangle. For nets of Hookean spring elements having a nonzero force-free length, cubic terms arise in the strain energy through the sixfold symmetry of the network, and thereby break the symmetric response at small strain. Because of the symmetry of the two-body potential and the anisotropy of the network, pure compression of the Hookean spring net leads to a martensiticlike phase transition at all finite temperatures studied. Networks of elemental tethers or springs that have a zero force-free length balanced against a three-vertex potential energy that rises with decreasing triangle area (to emulate volume exclusion in polymer networks) do not undergo a phase transition, although inclusion of a maximum tether length (to model the polymer chains' contour limits) reveals a simple but distinct type of triangular net anisotropy.