Boudinage and folding as an energy instability in ductile deformation

We present a theory for the onset of localization in layered rate- and temperature-sensitive rocks, in which energy-related mechanical bifurcations lead to localized dissipation patterns in the transient deformation regime. The implementation of the coupled thermomechanical 2-D finite element model comprises an elastic and rate-dependent von Mises plastic rheology. The underlying system of equations is solved in a three-layer pure shear box, for constant velocity and isothermal boundary conditions. To examine the transition from stable to localized creep, we study how material instabilities are related to energy bifurcations, which arise independently of the sign of the stress conditions imposed on opposite boundaries, whether in compression or extension. The onset of localization is controlled by a critical amount of dissipation, termed Gruntfest number, when dissipative work by temperature-sensitive creep translated into heat overcomes the diffusive capacity of the layer. Through an additional mathematical bifurcation analysis using constant stress boundary conditions, we verify that boudinage and folding develop at the same critical Gruntfest number. Since the critical material parameters and boundary conditions for both structures to develop are found to coincide, the initiation of localized deformation in strong layered media within a weaker matrix can be captured by a unified theory for localization in ductile materials. In this energy framework, neither intrinsic nor extrinsic material weaknesses are required, because the nucleation process of strain localization arises out of steady state conditions. This finding allows us to describe boudinage and folding structures as the same energy attractor of ductile deformation.