Shear banding as a dissipative structure from a thermodynamic viewpoint

Abstract

Granular materials are now known to be an illustration of complex materials as they display emergent macroscopic properties when loaded. An initially homogenous response can bifurcate into a heterogeneous one with the appearance of a rich variety of structured kinematical patterns. The shear banding that ensues illustrates a symmetry-breaking transition with multiple choices of macroscopic behaviours, a common feature of dynamical complex systems. Even though the phenomenon has been studied for decades, this regime transition remains mostly mysterious in geomaterials, with no convincing arguments that could link it to the underlying microscopic mechanisms. The paper investigates this issue by invoking the fundamental minimum entropy production theorem established by Prigogine in the past century to seek any connection with the second-order work theory in the mechanics of failure. A general equation linking the derivatives of the entropy of a mechanical system to the second-order work is thus inferred, which leads to a thermodynamic interpretation of bifurcations in the failure behaviour of granular materials under a given loading. This is verified through discrete element simulations that highlight the fundamental role played by the elastic energy stored within a granular material before a bifurcation occurs, which also corresponds to a minimization of the entropy production. The analysis suggests a new interpretation of the intriguing shear banding phenomenon as a bifurcation with the emergence of ordered dissipative structures germane to nonequilibrium thermodynamics of open systems.