Microscopic mechanism of strain localisation in amorphous materials

A large class of amorphous or disordered materials ranging from "hard" bulk-metallic glasses to "soft" foams, exhibit strain localisation when subject to a sufficiently large deformation. This particular phenomenon often leads to material failure and thus obtaining a microscopic understanding of this process becomes very important. In recent work, we have studied the instability responsible for this process from a microscopic point of view using athermal, quasi-static simulations of binary Lennard-Jones glasses and continuum solid mechanics. The talk will start with an introduction to some of the basic ideas in amorphous elasticity & plasticity. The importance of non-affine motion, the notion of elementary plastic instabilities in the stress-strain response and the connection between non-affine response and eigenvalues & eigen-modes of the Hessian matrix will be discussed brief?y. Data obtained from numerical simulations show that the non-affine displacement field associated with a plastic instability undergoes a qualitative shift, changing from a quadrupolar field to a shear band as we strain the material. We will understand this transition using the theoretical formalism of Eshelby inclusion(s). An expression for the elastic energy of N inclusions dispersed and oriented randomly in an elastic medium subject to a global loading will be obtained. It will then be proven analytically that at sufficiently large values of strain, a state of minimal energy is when each of these N inclusions are equi-aligned and lie on a line oriented at 45° to the global compressive axes. It will be seen that this highly correlated arrangement of inclusions, is responsible for organising the non-affine flow into a shear band. A formula for yield-strain obtained from this calculation, will be presented. Extension of these ideas to account for finite temperature and finite strain-rates will also be discussed. In a second and shorter part of the talk, I will also present some of my Ph.D. work on laminar, hydraulic jumps and the connection to non-linear waves. Results from free-surface Navier-Stokes simulations of hydraulic jumps in both planar and circular geometries will be discussed and some interesting theoretical analysis will be presented in brief.