Amit Acharya's blog

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

Uditnarayan Kouskiya                    Amit Acharya

We demonstrate the feasibility of a scheme to obtain approximate weak solutions to (inviscid) Burgers equation in conservation and Hamilton-Jacobi form, treated as degenerate elliptic problems. We show different variants recover non-unique weak solutions as appropriate, and also specific constructive approaches to recover the corresponding entropy solutions.

Siddharth Singh          Janusz Ginster        Amit Acharya

A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler-Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding, primal problem is ill-posed and demonstrates an explosive ‘Hadamard instability;’ this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response.  

The fully nonlinear (geometric and material) system of Field Dislocation Mechanics is reviewed to establish an exact analogy with the equations of ideal magnetohydrodynamics (ideal MHD) under suitable physically simplifying circumstances. Weak solutions with various conservation properties have been established for ideal MHD recently by Faraco, Lindberg, and Szekelyhidi using the techniques of compensated compactness of Tartar and Murat and convex integration; by the established analogy, these results would seem to be transferable to the idealization of Field Dislocation Mechanics considered. A dual variational principle is designed and discussed for this system of PDE, with the technique transferable to the study of MHD as well.

Amit Acharya

A methodology for defining variational principles for a class of PDE models from continuum mechanics is demonstrated, and some of its features explored. The scheme is applied to quasi-static and dynamic models of rate-independent and rate-dependent, single crystal plasticity at finite deformation.

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 first 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 field. The
vectorial crack field degrades the elastic modulus in a physically justified 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 field 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 field, 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.

 In J. Mech. Phys. Solids

See Rajat Arora's blog entry for abstract and preprint

Chiqun Zhang         Amit Acharya        Alan C Newell          Shankar C Venkataramani

 (in Physica, D)

Defects, a ubiquitous feature of ordered media, have certain universal features, independent of the underlying physical system, reflecting their topological, as opposed to energetic properties. We exploit this universality, in conjunction with smoothing defects by "spreading them out," to develop a modeling framework and associated numerical methods that are applicable to computing energy driven behaviors of defects across the amorphous-soft-crystalline materials spectrum. Motivated by ideas for dealing with elastic-plastic solids with line defects, our methods can handle order parameters that have a head-tail symmetry, i.e. director fields, in systems with a continuous translation symmetry, as in nematic liquid crystals, and in systems where the translation symmetry is broken, as in smectics and convection patterns. We illustrate our methods with explicit computations.

Preprint