Amit Acharya's blog
Computing with non-orientable defects: nematics, smectics and natural patterns
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.
Some preliminary observations on a defect Navier-Stokes system
Amit 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.
Finite Element Computation of Finite Deformation Dislocation Theory
in Computer Methods in Applied Mechanics and Engineering
Continuum mechanics of moving defects in growing bodies
Amit 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.
On the Structure of Linear Dislocation Field Theory
Amit 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.
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.
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.
Stress of a spatially uniform dislocation density field
(To appear in Journal of Elasticity).
On Weingarten-Volterra defects
Amit Acharya
(in Journal of Elasticity)
The kinematic theory of Weingarten-Volterra line defects is revisited, both at small and fi nite deformations. Existing results are clari fied 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 di fferent cut-surfaces.