Stable disarrangement phases of elastic aggregates: a setting for the emergence of no-tension materials with non-linear response in compression

A recent field theory of elastic bodies undergoing non-smooth submacroscopic geometrical changes (disarrangements) provides a setting in which, for a given homogeneous macroscopic deformation F of the body, there are typically a number of different states G of smooth, submacroscopic deformation (disarrangement phases) available to the body. A tensorial consistency relation and the inequality G ≤ det F accommodates G determine the totality of disarrangement phases G corresponding to F, and it is natural to seek for a given F those disarrangement phases that minimize the Helmholtz free energy (stable disarrangement phases). We introduce these concepts in the particular context of continuous bodies comprised of many small elastic bodies (elastic aggregates) and in the context where disarrangements do not contribute to the Helmholtz free energy (purely dissipative disarrangements). In this setting, the Helmholtz free energy response (Formula presented.) of the pieces of the aggregate determines the totality of disarrangement phases corresponding to F, which necessarily includes the phase G = F (compact phase) in which every piece of the aggregate undergoes the given macroscopic deformation F. When the response function (Formula presented.) is isotropic and smooth, and when (Formula presented.) possesses standard semiconvexity and growth properties, the body also admits phases of the form (Formula presented.) (loose phases) with (Formula presented.) an arbitrary rotation, provided that (Formula presented.) satisfies the accommodation inequality (Formula presented.). Loose phases, when available, achieve the global minimum (Formula presented.) of the free energy and consequently are stable and stress-free. When (Formula presented.) has the specific form (Formula presented.), with (Formula presented.) given elastic constants, we determine all of the disarrangement phases corresponding to (Formula presented.). These include not only the compact and loose phases, but also disarrangement phases (Formula presented.) in which the stress (Formula presented.) is uniaxial or planar. Our main result (“stability implies no-tension”) is the assertion that every stable disarrangement phase for (Formula presented.) cannot support tensile tractions, and our treatment of elastic aggregates thus provides a natural setting for the emergence of no-tension materials whose response in compression is non-linear. Existing treatments of no-tension materials assume at the outset that the body cannot support tension and that the response in compression is linear.