A gradient thermodynamic theory of self-organization

In this paper we present a gradient theory of internal variables in a thermodynamic context of the Gibbs free energy density phi. A fundamental point of the theory is that phi is a function of three different classes of internal variables:(a) tensorial dissipative and local;(b) vectorial dissipative and non-local and (c) vectorial inviscid and non-local. These classes obey different types of evolution equations. The ones pertaining to the non-local variables are partial differential equations of the diffusion-reaction type. We associate the inviscid non-local variables with energy release mechanisms and show that they lead to patterned deformation, otherwise known as self-organization. We conclude by giving a solution to the problem of a flat plate in nominal axial tension and derive various types of deformation patterns that result from small but unavoidable experimental deviations from this loading.