Amit Acharya's blog

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.

 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

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.

in Computer Methods in Applied Mechanics and Engineering

Link to Rajat Arora's blog

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.

https://www.researchgate.net/publication/333877242_Continuum_mechanics_of_moving_defects_in_growing_bodies

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.

https://www.researchgate.net/publication/328792035_On_the_Structure_of_Linear_Dislocation_Field_Theory