Amit Acharya's blog

Equations not arising from a variational principle, as well as ones that arise from functionals that are not bounded above or below, can be endowed with one with favorable properties. The concept, and some examples are listed below:

Gabriel Dante Lima-Chavez,        Amit Acharya,          Manas V. Upadhyay

A geometrically nonlinear theory for field dislocation thermomechanics based entirely on measurable state variables is proposed. Instead of starting from an ordering-dependent multiplicative decomposition of the total deformation gradient tensor, the additive decomposition of the velocity gradient into elastic, plastic and thermal distortion rates is obtained as a natural consequence of the conservation of the Burgers vector. Based on this equation, the theory consistently captures the contribution of transient heterogeneous temperature fields on the evolution of the (polar) dislocation density. The governing equations of the model are obtained from the conservation of Burgers vector, mass, linear and angular momenta, and the First Law. The Second Law is used to deduce the thermodynamical driving forces for dislocation velocity. An evolution equation for temperature is obtained from the First Law and the Helmholtz free energy density, which is taken as a function of the following measurable quantities: elastic distortion, temperature and the dislocation density (the theory allows prescribing additional measurable quantities as internal state variables if needed). Furthermore, the theory allows one to compute the Taylor-Quinney factor, which is material and strain rate dependent. Accounting for the polar dislocation density as a state variable in the Helmholtz free energy of the system allows for temperature solutions in the form of dispersive waves with finite propagation speed, despite using Fourier’s law of heat conduction as the constitutive assumption for the heat flux vector.

Fluid Mechanics:

Variational dual solutions for incompressible fluids

Amit Acharya, Bianca Stroffolini, Arghir Zarnescu

Nonlinear Elasticity:

A hidden convexity of nonlinear elasticity

Siddharth Singh, Janusz Ginster, Amit Acharya

This paper is dedicated to Professor Nasr Ghoniem on the occasion of his retirement.

(To appear in Journal of Materials Science: Materials Theory)

Abstract

A continuum mechanical model of coupled dislocation based plasticity and fracture at finite deformation is proposed. Motivating questions and target applications of the model are sketched.

The mid-surface scaling invariance of bending strain measures proposed in [1] is discussed in light of the work of [3].

preprint