Amit Acharya's blog

A methodology for deriving dual variational principles for the classical Newtonian mechanics of mass points in the presence of applied forces, interaction forces, and constraints, all with a general dependence on particle velocities and positions, is presented. Methods for incorporating constraints are critically assessed. General theory, as well as explicitly worked out variational principles for a dissipative system (Lorenz) and a system with anholonomic constraints (Pars) are demonstrated. Conditions under which a (family of) dual Hamiltonian flows, as well as constants of motion, may be associated with a dissipative, and possibly constrained, primal system are specified.

link to paper

See Abhishek Arora's blog:

1. Mechanics of micropillar confined thin film plasticity

2. Interface-dominated plasticity and kink bands in metallic nanolaminates

See Siddharth Singh's blog:

Modeling of experimentally observed topological defects inside bulk polycrystals

See Uditnarayan Kouskiya's blog:

Hidden convexity in the heat, linear transport, and Euler's rigid body equations: A computational approach

Amit Acharya

A formal methodology for developing variational principles corresponding to a given nonlinear
pde system is discussed. The scheme is demonstrated in the context of the incompressible
Navier-Stokes equations, systems of first-order conservation laws, and systems of Hamilton-
Jacobi equations.

Amit Acharya

To appear in J. Mech. Phys Solids

Amit Acharya

An action functional is developed for nonlinear dislocation dynamics. This serves as a first step
towards the application of effective field theory in physics to evaluate its potential in obtaining
a macroscopic description of dislocation dynamics describing the plasticity of crystalline solids.
Connections arise between the continuum mechanics and material science of defects in solids,
effective field theory techniques in physics, and fracton tensor gauge theories.

The scheme that emerges from this work for generating a variational principle for a nonlinear
pde system is general, as is demonstrated by doing so for nonlinear elastostatics involving a stress
response function that is not necessarily hyperelastic.

Léo Morin       Amit Acharya

(in Computer Methods in Applied Mechanics and Engineering)

A computational model for arbitrary brittle crack propagation, in a fault-like layer within a 3-d
elastic domain, and its associated quasi-static and dynamic fields is developed and analyzed. It
uses an FFT-based solver for the balance of linear momentum and a Godunov-type projection-evolution
method for the crack evolution equation. As applications, we explore the questions of equilibria
and irreversibility for crack propagation with and without surface energy, existence of strength
and toughness criteria, crack propagation under quasi-static and dynamic conditions,
including Modes I, II and III, as well as multiaxial compressive loadings.

Janusz Ginster        Amit Acharya

to appear Archive for Rational Mechanics and Analysis

We address a problem that extends a fundamental classical result of
continuum mechanics from the time of its inception, as well as answers a fundamental
question in the recent, modern nonlinear elastic theory of dislocations.
Interestingly, the implication of our result in the latter case is qualitatively different
from its well-established analog in the linear elastic theory of dislocations.

Preprint

Amit Acharya

(to appear in Comptes Rendus Mécanique)

A continuum model of fracture that describes, in principle, the propagation and interaction of
arbitrary distributions of cracks and voids with evolving topology without a 'fracture criterion'
is developed. It involves a 'law of motion' for crack-tips, primarily as a kinematical consequence
coupled with thermodynamics. Fundamental kinematics endows the crack-tip with a topological
charge. This allows the association of a kinematical conservation law for the charge, resulting
in a fundamental evolution equation for the crack-tip field, and in turn the crack field. The
vectorial crack field degrades the elastic modulus in a physically justified anisotropic manner.
The mathematical structure of this conservation law allows an additive 'free' gradient of a scalar
field in the evolution of the crack field. We associate this naturally emerging scalar field with the
porosity that arises in the modeling of ductile failure. Thus, porosity-rate gradients affect the
evolution of the crack-field which, then, naturally degrades the elastic modulus, and it is through
this fundamental mechanism that spatial gradients in porosity growth affect the strain-energy
density and stress carrying capacity of the material - and, as a dimensional consequence related
to fundamental kinematics, introduces a length-scale in the model. A key result of this work is
that brittle fracture is energy-driven while ductile fracture is stress-driven; under overall shear
loadings where mean stress vanishes or is compressive, shear strain energy can still drive shear
fracture in ductile materials.

The paper can be found here

Amit Acharya           Jorge Vinals

(in Physical Review, B)

A new formulation of the Phase Field Crystal model is presented that is consistent with the necessary microscopic independence between the phase field, reflecting the broken symmetry of the phase, and both mass density and elastic distortion. Although these quantities are related in equilibrium through a macroscopic equation of state, they are independent variables in the free energy, and can be independently varied in evaluating the dissipation functional that leads to the model governing equations. The equations obtained describe dislocation motion in an elastically stressed solid, and serve as an extension of the equations of plasticity to the Phase Field Crystal setting. Both finite and small deformation theories are considered, and the corresponding kinetic equations for the fields derived.