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Amit Acharya's blog

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On the Structure of Linear Dislocation Field Theory

Amit Acharya          Robin J. Knops         Jeyabal Sivaloganathan

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

 A 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

(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

It can be shown that the stress produced by a spatially uniform dislocation density field in a body comprising a linear elastic material under no loads vanishes. We prove that the same result does not hold in general in the geometrically nonlinear case. This problem of mechanics establishes the purely geometrical result that the curl of a sufficiently smooth two-dimensional rotation field cannot be a non-vanishing constant on a domain.

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On Weingarten-Volterra defects

Amit 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

Below 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

To 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

Amit 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

Amit 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

Amit 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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The metric-restricted inverse design problem

Amit 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

Amit 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.)

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Carlson - Mathematical Preliminaries and Continuum Mechanics 1991

I attach some class notes developed by the late Professor Donald Carlson from which many generations of students at the University of Illinois learnt Continuum Mechanics.

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Continuum mechanics of the interaction of phase boundaries and dislocations in solids

Amit Acharya         Claude Fressengeas

Springer Proceedings in Mathematics and Statistics on Differential Geometry and Continuum Mechanics, Vol. 137, pages 123-165. Ed: G. Q Chen, M. Grinfeld, R.J. Knops (Proceedings of  Workshop held at the Intl. Centre for Mathematical Sciences in Edinburgh, 2013.)

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Dislocation motion and instability

Yichao Zhu       Stephen J. Chapman       Amit Acharya

(to appear in Journal of the Mechanics and Physics of Solids)

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An observation on the experimental measurement of dislocation density

Amit Acharya and Robin J. Knops

(to appear in Journal of Elasticity)

The common practice of ignoring the elastic strain gradient in measurements of geometrically necessary dislocation (GND) density is critically examined. It is concluded that the practice may result in substantial errors. Our analysis points to the importance of spatial variations of the elastic strain field in relation to its magnitude in inferring estimates of dislocation density from measurements.

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A fundamental improvement to Ericksen-Leslie kinematics

Hossein Pourmatin     Amit Acharya       Kaushik Dayal

(to appear in Quarterly of Applied Mathematics)

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Jump condition for GND evolution

Acharya, A., Jump conditions for GND evolution as a constraint on slip transmission at grain boundaries, Philosophical Magazine, 87(8-9), 1349-1359, 2007 (see attachment)

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A mathematician's take on "what is light?"

Attached is an intriguing commentary on the scientific method through an example, written by my good friend, Luc Tartar. The specific example is that of trying to understand what 'light' might be, especially from a mathematician's point of view. The mathematician in this case is an extremely talented one, who also happens to actually understand a whole lot of physics and mechanics.

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