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# Amit Acharya's blog

## Variational principles for nonlinear pde systems via duality

Thu, 2021-08-19 16:35 - Amit AcharyaAmit 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.

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## An action for nonlinear dislocation dynamics

Thu, 2021-08-19 16:31 - Amit AcharyaAmit 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.

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## Analysis of a model of field crack mechanics for brittle materials

Sat, 2021-02-13 12:14 - Amit AcharyaLéo 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.

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## Rotations with constant Curl are constant

Tue, 2020-06-30 17:46 - Amit AcharyaJanusz 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.

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## A possible link between brittle and ductile failure by viewing fracture as a topological defect

Mon, 2020-05-25 09:54 - Amit AcharyaAmit 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.

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## Field Dislocation Mechanics and Phase Field Crystal models

Sun, 2020-05-17 14:43 - Amit AcharyaAmit 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.

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## A unification of finite deformation J2 Von-Mises plasticity and quantitative dislocation mechanics

Thu, 2020-03-26 09:22 - Amit Acharya- Read more about A unification of finite deformation J2 Von-Mises plasticity and quantitative dislocation mechanics
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## Computing with non-orientable defects: nematics, smectics and natural patterns

Mon, 2020-02-24 11:10 - Amit Acharya

(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.

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## Some preliminary observations on a defect Navier-Stokes system

Sun, 2019-09-22 12:08 - Amit AcharyaAmit Acharya Roger Fosdick

(To appear in Comptes Rendus - Me'canique)

Some implications of the simplest accounting of defects of compatibility in the velocity field on the structure of the classical Navier-Stokes equations are explored, leading to connections between classical elasticity, the elastic theory of defects, plasticity theory, and classical fluid mechanics.

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## Finite Element Computation of Finite Deformation Dislocation Theory

Mon, 2019-08-26 08:14 - Amit Acharyain Computer Methods in Applied Mechanics and Engineering

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## Continuum mechanics of moving defects in growing bodies

Wed, 2019-06-19 10:16 - Amit AcharyaAmit Acharya Shankar Venkataramani

(In Materials Theory)

Growth processes in many living organisms create thin, soft materials with an intrinsically hyperbolic

geometry. These objects support novel types of mesoscopic defects - discontinuity lines

for the second derivative and branch points - terminating defects for these line discontinuities.

These higher-order defects move "easily", and thus confer a great degree of

flexibility to thin hyperbolic elastic sheets. We develop a general, higher-order, continuum mechanical framework

from which we can derive the dynamics of higher order defects in a thermodynamically consistent

manner. We illustrate our framework by obtaining the explicit equations for the dynamics

of branch points in an elastic body.

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## On the Structure of Linear Dislocation Field Theory

Wed, 2018-11-07 14:00 - Amit AcharyaAmit Acharya Robin J. Knops Jeyabal Sivaloganathan

(In JMPS, 130 (2019), 216-244)

Uniqueness of solutions in the linear theory of non-singular dislocations, studied as a special case of plasticity theory, is examined. The status of the classical, singular Volterra dislocation problem as a limit of plasticity problems is illustrated by a specific example that clarifies the use of the plasticity formulation in the study of classical dislocation theory. Stationary, quasi-static, and dynamical problems for continuous dislocation distributions are investigated subject not only to standard boundary and initial conditions, but also to prescribed dislocation density. In particular, the dislocation density field can represent a single dislocation line.

It is only in the static and quasi-static traction boundary value problems that such data are sufficient for the unique determination of stress. In other quasi-static boundary value problems and problems involving moving dislocations, the plastic and elastic distortion tensors, total displacement, and stress are in general non-unique for specified dislocation density. The conclusions are confirmed by the example of a single screw dislocation.

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## Plasticity implies the Volterra formulation: an example

Tue, 2018-09-25 11:13 - Amit AcharyaA demonstration through an example is given of how the Volterra dislocation formulation in linear elasticity can be viewed as a (formal) limit of a problem in plasticity theory. Interestingly, from this point of view the Volterra dislocation formulation with discontinuous displacement, and non-square-integrable energy appears as a large-length scale limit of a smoother microscopic problem. This is in contrast to other formulations using SBV functions as well as the theory of Structured deformations where the microscopic problem is viewed as discontinuous and the smoother plasticity formulation appears as a homogenized large length-scale limit.

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## A design principle for actuation of nematic glass sheets

Mon, 2018-09-10 11:12 - Amit Acharya(in Journal of Elasticity)

A continuum mechanical framework is developed for determining a) the class of stress-free deformed shapes and corresponding director distributions on the undeformed configuration of a nematic glass membrane that has a prescribed spontaneous stretch field and b) the class of undeformed configurations and corresponding director distributions on it resulting in a stress-free given deformed shape of a nematic glass sheet with a prescribed spontaneous stretch field. The proposed solution rests on an understanding of how the Lagrangian dyad of a deformation of a membrane maps into the Euleriandyad in three dimensional ambient space. Interesting connections between these practical questions of design and the mathematical theory of isometric embeddings of manifolds, deformations between two prescribed Riemannian manifolds, and the slip-line theory of plasticity are pointed out.

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## Stress of a spatially uniform dislocation density field

Mon, 2018-09-10 11:07 - Amit Acharya(To appear in Journal of Elasticity).

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## On Weingarten-Volterra defects

Sat, 2017-10-14 16:00 - Amit AcharyaAmit Acharya

(in Journal of Elasticity)

The kinematic theory of Weingarten-Volterra line defects is revisited, both at small and finite deformations. Existing results are clarified and corrected as needed, and new results are obtained. The primary focus is to understand the relationship between the disclination strength and Burgers vector of deformations containing a Weingarten-Volterra defect corresponding to different cut-surfaces.

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## Professor Walter Noll

Thu, 2017-06-08 11:01 - Amit AcharyaBelow is a message from the Secretary of the Society for Natural Philosophy. A formative influence of our field passes on.

Dear Members of the Society for Natural Philosophy.

We are sorry to be the bearers of sad news, but on Tuesday June 6th, 2017, Walter Noll passed away at his home in Pittsburgh surrounded by his wife Marilyn and two children, Victoria and Peter. He was a founding member and a past Chairman of the Society. He made major contributions to the fields of continuum mechanics and thermodynamics. There will be a memorial service at Carnegie Mellon University, which has yet to be scheduled.

Secretary,

Wladimir Neves.

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## Fracture and singularities of the mass-density gradient field

Sun, 2017-01-22 14:44 - Amit AcharyaTo appear in Journal of Elasticity

A continuum mechanical theory of fracture without singular fields is proposed. The primary

contribution is the rationalization of the structure of a `law of motion' for crack-tips, essentially

as a kinematical consequence and involving topological characteristics. Questions of compatibility

arising from the kinematics of the model are explored. The thermodynamic driving force

for crack-tip motion in solids of arbitrary constitution is a natural consequence of the model.

The governing equations represent a new class of pattern-forming equations.

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## Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions

Sun, 2016-05-22 20:23 - Amit AcharyaAmit Acharya, Gui-Qiang Chen, Siran Li, Marshall Slemrod, and Dehua Wang

(To appear in Archive for Rational Mechanics and Analysis)

We are concerned with underlying connections between fluids,

elasticity, isometric embedding of Riemannian manifolds, and the existence of

wrinkled solutions of the associated nonlinear partial differential equations. In

this paper, we develop such connections for the case of two spatial dimensions,

and demonstrate that the continuum mechanical equations can be mapped into

a corresponding geometric framework and the inherent direct application of

the theory of isometric embeddings and the Gauss-Codazzi equations through

examples for the Euler equations for fluids and the Euler-Lagrange equations

for elastic solids. These results show that the geometric theory provides an

avenue for addressing the admissibility criteria for nonlinear conservation laws

in continuum mechanics.

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## A microscopic continuum model for defect dynamics in metallic glasses

Fri, 2016-03-04 23:01 - Amit AcharyaAmit Acharya Michael Widom

To appear in *Journal of the Mechanics and Physics of Solids*

Motivated by results of the topological theory of glasses accounting for geometric frustration,

we develop the simplest possible continuum mechanical model of defect dynamics in metallic

glasses that accounts for topological, energetic, and kinetic ideas. A geometrical description

of ingredients of the structure of metallic glasses using the concept of local order based on

Frank-Kasper phases and the notion of disclinations as topological defects in these structures is

proposed. This novel kinematics is incorporated in a continuum mechanical framework capable

of describing the interactions of disclinations and also of dislocations (interpreted as pairs of

opposite disclinations). The model is aimed towards the development of a microscopic understanding

of the plasticity of such materials. We discuss the expected predictive capabilities of

the model vis-a-vis some observed physical behaviors of metallic glasses.

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## Microstructure in plasticity without nonconvexity

Tue, 2015-12-22 05:47 - Amit AcharyaAmit Das Amit Acharya Pierre Suquet

To appear in Special issue of Computational Mechanics on "Connecting Multiscale Mechanics to Complex Material Design"; Guest Editors: Wing Kam Liu, Jacob Fish, J. S Chen, Pedro Camanho; Issue dedicated to Ted Belytschko

A simplified one dimensional rate dependent model for the evolution of plastic distortion is obtained from a three dimensional mechanically rigorous model of mesoscale field dislocation mechanics. Computational solutions of variants of this minimal model are investigated to explore the ingredients necessary for the development of microstructure. In contrast to prevalent notions, it is shown that microstructure can be obtained even in the absence of non-monotone equations of state. In this model, incorporation of wave propagative dislocation transport is vital for the modeling of spatial patterning. One variant gives an impression of producing stochastic behavior, despite being a completely deterministic model. The computations focus primarily on demanding macroscopic limit situations, where a convergence study reveals that the model-variant including non-monotone equations of state cannot serve as effective equations in the macroscopic limit; the variant without non-monotone ingredients, in all likelihood, can.

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## Prof. Ray Ogden wins Rodney Hill Prize in Solid Mechanics

Sat, 2015-10-10 12:02 - Amit Acharya2016 Rodney Hill Prize in Solid Mechanics

Congratulations!

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## The metric-restricted inverse design problem

Thu, 2015-01-08 19:42 - Amit AcharyaAmit Acharya Marta Lewicka Mohammad Reza Pakzad

In Nonlinearity, 29, 1769-1797

We study a class of design problems in solid mechanics, leading to a variation on the

classical question of equi-dimensional embeddability of Riemannian manifolds. In this general new

context, we derive a necessary and sufficient existence condition, given through a system of total

differential equations, and discuss its integrability. In the classical context, the same approach

yields conditions of immersibility of a given metric in terms of the Riemann curvature tensor.

In the present situation, the equations do not close in a straightforward manner, and successive

differentiation of the compatibility conditions leads to a more sophisticated algebraic description

of integrability. We also recast the problem in a variational setting and analyze the infimum value

of the appropriate incompatibility energy, resembling "non-Euclidean elasticity". We then derive a

Γ-convergence result for the dimension reduction from 3d to 2d in the Kirchhoff energy scaling

regime. A practical implementation of the algebraic conditions of integrability is also discussed.

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## From dislocation motion to an additive velocity gradient decomposition, and some simple models of dislocation dynamics

Thu, 2014-05-01 12:27 - Amit AcharyaAmit Acharya Xiaohan Zhang

(Chinese Annals of Mathematics, 36(B), 2015, 645-658. Proceedings of the International Conference on Nonlinear and Multiscale Partial Differential Equations: Theory, Numerics and Applications held at Fudan University, Shanghai, September 16-20, 2013, in honor of Luc Tartar.)

## Scaling theory of continuum dislocation dynamics in three dimensions: Self-organized fractal pattern formation

Sat, 2014-02-08 14:07 - Amit AcharyaY. S. Chen, W. Choi, S. Papanikolaou, M. Bierbaum, J. P. Sethna

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