Emergence of coherent localized structures in shear deformations of temperature dependent fluids

TL;DR

This paper investigates the nonlinear formation of localized shear structures in temperature-dependent fluids, combining linear stability analysis with the construction of similarity solutions to explain shear localization phenomena.

Contribution

It introduces a nonlinear model capturing shear localization in temperature-dependent fluids and constructs similarity solutions representing coherent localized structures.

Findings

Thermal softening influences shear localization

Localized states emerge as similarity solutions

The model explains formation of shear bands in materials

Abstract

Shear localization occurs in various instances of material instability in solid mechanics and is typically associated with Hadamard-instability for an underlying model. While Hadamard instability indicates the catastrophic growth of oscillations around a mean state, it does not by itself explain the formation of coherent structures typically observed in localization. The latter is a nonlinear effect and its analysis is the main objective of this article. We consider a model that captures the main mechanisms observed in high strain-rate deformation of metals, and describes shear motions of temperature dependent non-Newtonian fluids. For a special dependence of the viscosity on the temperature, we carry out a linearized stability analysis around a base state of uniform shearing solutions, and quantitatively assess the effects of the various mechanisms affecting the problem: thermal softening, momentum diffusion and thermal diffusion. Then, we turn to the nonlinear model, and construct localized states - in the form of similarity solutions - that emerge as coherent structures in the localization process. This justifies a scenario for localization that is proposed on the basis of asymptotic analysis in \cite{KT}.