Research Areas:
Large body of work is related
to applying techniques in the forefront of theoretical
physics to problems in physics of materials and
condensed matter. In particular, statistical mechanics,
stochastic processes and dynamical systems have
been used to model physical phenomenon cutting
across disciplines. The emphasis has shifted from
equilibrium systems to nonequilibrium systems,
with particular emphasis on driven far-from equilibrium
systems. Areas of materials science include physics
of defects, both point defects and extended defects,
plasticity, phase transformation, quasicrystals,
and stick-slip dynamics. Contributions to condensed
matter physics include critical point phenomenon,
stochastic processes, physical properties of quasiperiodic
systems, chaos, fractals and dynamical systems.
Some of these techniques find applications in
problems as varied as 'identification of cancer
cells' to 'low level computer vision'.
Clustering of point defects:
Initial work on materials physics was concerned
with the problem of clustering of point defects
(vacancies) to extended defects such as vacancy
loops, stacking fault tetrahedra and voids under
quenched conditions. These problems constituted
the first stochastic approach to this area.
Dislocation dynamics and Collective behavior of dislocations:
Early work was focused on applying methods of
stochastic processes to dislocation dynamics.
Starting from a statistical description of dislocations
(a kind of extended Fokker-Planck equation which
is nonlinear and nonlocal), several features of
creep in simple materials were recovered. While the
mathematical structure is complicated, exact expressions
for the several cumulants have been calculated. This
equation also leads to quantitative expression
for mobile dislocation density. The
above approach forms the basis for deriving a model
that explains most features of the Portevin-Le
Chatelier (PLC) effect, an effect discovered more
than a century ago that lacked a coherent model
till early 80's. The Ananthakrishna (AK) model explains
the rich spatio-temporal dynamics of the PLC effect.
The work extending over twenty years of concerted
efforts explains a large number of generic features
of the PLC effect. The basic idea is that the
whole phenomenon is a consequence of nonlinear
interaction of different populations of dislocations
representing essentially, collective modes of
dislocations. The model exhibits several experimentally
observed features such as the existence of the
PLC effect in a window of strain rates and temperature,
a negative strain rate sensitivity normally assumed
in many theories, and different types of bands
observed in experiments. All these features are
a direct consequence of Hopf bifurcation followed
by a reverse Hopf bifurcation. The model also
predicts chaotic behavior which has been since
verified by analyzing the experimental signals.
The number of degrees of freedom estimated from
this analysis is the same as that used in the
model, thus confirming the dynamical basis of
the model. Further analysis of stress signal shows a crossover
in the nature of the dynamics from chaotic to
power law type of dynamics. This is only example
of such a crossover in the physics or metallugical
literature other than the well known example of
hydrodynamics. Again the AK model explains this
crossover. The model predicts
type A, B and C bands found with decreasing strain
rates. The dynamical approach provides a basis
for understanding the negative strain rate behaviour
as also the critical nature of type A serrations
reflected in power law statistics of stress drops.
This is the only model in the literature which
has the ability to explain the entire range of
dynamical and generic metallurgical features of
the phenomenon. A similar dynamical approach to
radiation induced effects on mechanical properties
is under active investigation.
Again, the AK model
forms the basis for explaining a long standing problem
of acoustic emission that accompanies plastic deformation.
The main reason for lack of any theoretical basis for
explaining acoustic emission during plastic deformation
is the necessity for describing simultaneously widely
separated time scales and length scale corresponding to
speed of sound and the imposed deformation rate. Since
the nature of acoustic emission spectrum is characteristic
of the PLC band type, the Ananthakrishna model for
the PLC effect provides a basis for understanding the
nature of acoustic emission spectrum observed during type
C to type A dislocation bands.
Acoustic Emission:
One common theme connecting several topics in materials
science is the generation of acoustic emission during
widely different deformation processes such as, fracture,
plastic deformation, peeling of an adhesive tape,
martensitic transformation etc. However there is a lack of
clarity in attempting to model what constitutes the dissipation
in the form of acoustic energy. Our approach has been to use
Rayleigh dissipation function that depends on local strain
rate. Using this functional form acoustic emission during
the varied process mentioned above has been explained.
Dynamical systems:
Work in this area is largely a result of the need to use
dynamical systems approach to understand the PLC effect.
Time series analysis of the stress-time charts from the
PLC experiments has been carried out for a large number
of samples, strain rates and initial conditions. As
experimental time series are usually short and noisy, a
new method was developed that is appropriate for this
situation. This has has been applied to several stress - time
series. Several analytical studies on the AK model have been
carried out. For example the model equations have been reduced
to Ginzburg-Landau form with a view to extract order parameter
equations and also to map theoretical parameters with the
experimental ones. One major spin off arising from detailed theoretical
studies of the model is its connection to an unsolved problem in global
bifurcation theory, namely incomplete approach to homoclinicity.
It turns out that the AK model helps to resolve this long standing
problem. The model has also thrown up several possible solutions
to other areas, like the connection to turbulence, characterization
of critical state in terms of slow manifold approach. One feature
displayed by the model is the generic feature of many mixed mode
oscillatory systems that exhibit period doubling and their reversal,
and crossover to period adding sequences. Yet they have an unimodal
one dimensional map with a long tail. This crossover has been
explained by imposing a few generic constraints on the map period
doubling and their reversal, and crossover to period adding
sequences have been recovered. Some interesting non-typical
scaling relations for certain types of periodic states have been
derived. Recently, we have investigated the possibility of
projecting low dimensional chaos out of a spatio-temporal chaotic system.
Stochastic processes:
This is another area where substantial contribution has been made.
Indeed, stochastic approach has been used to understand clustering
of point defects as also dislocation dynamics. Langevin dynamics
has been used to understand the nature of enhanced fluctuation in
the neighborhood of an yield drop. Other topics studied specific
to stochastic processes include solution of nonlinear Langevin
equation, subdiffusive behavior, application of stochastic processes
to transport of neutrons through shield material, and Kramer's escape
rate for the case of subdivided barrier. In the last example, there
is an optimal barrier subdivision which maximizes the escape rate.
This problem has also been cast as a biased random walk to obtain
closed form solutions which permits a better understanding of the
effects of barrier subdivision. Recently, the Landauers `blow torch'
conjecture has been studied with emphasis on the calculation of escape
rate. Diffusion in confined systems such as zeolites is one area where
these techniques have been applied. In particular, separation of
mixtures has been shown to be much more effective by coupling the blow
torch effect with levitation effect. Recently some of these techniques
have been applied to model biological systems. One area that combines
stochastic processes, soft condensed matter and friction is the recent
work on sheared colloidal liquids where it has been shown that sheared
colloidal liquids show all features of stochastic resonance.
Quasicrystals:
Substantial amount of work has been carried out
on electronic structure of quasicrystals. Other
properties such as critical magnetic properties
and diffusion in quasicrystals have also been
analyzed.
Fractal and multifractals:
Application of concept of fractals and multifractals
to understand the influence of self similar structure
on physical properties has been a theme followed
over the years. To name a few examples, multifractal
characterization of wave functions, mechanical
properties of polymer blends and detection of
cancer cells, and characterization of plume structure
in hydrodynamic turbulence. Some work on fractal
growth of surface fracture has also been carried
out. Multifractal measures has been used to describe
the transition from the type B bands in the PLC
effect to type A propagating bands. Increased
levels of multifractality is reflected in this
region of strain rates.
Phase transformations:
Martensites are athermal transformations. They are unusual as
they exhibit features of second order transition such as the
precursor effect observed well above the transformation
temperature and power law statistics of acoustic emission (AE)
signals during thermal cycling. Another unusual feature is the
correlated, repetitive AE signals found during thermal cycling
in a restricted temperature domain. The latter has been shown
to be related to the growth and shrinkage of plates and hence
related directly to the shape memory effect. A model that
includes hydrodynamic nature of sound, long range interaction
of the transformed martensite phases, nucleation at defect sites
and an additional dissipation that mimics acoustic emission has
been designed. Morphological patterns resemble patterns observed
in experiments. The model reproduces well known features such as
jerky motion of the front and thermal hysteresis. The energy released
during cooling runs show a power law distribution of acoustic
emission signals as seen in experiments. The same model also explains
the correlated nearly repetitive acoustic emission signals under
thermal cycling in a small temperature interval as seen in
experiments. The associated martensite platelets grow and shrink
under thermal cycling and thus the model also explains the shape
memory effect for the first time from a microscopic angle. The
model also shows precursor effect. The model has also been extended
to understand several features of magneto-martensites. Finally,
dynamics of magneto-martensites has also been studied. Another
transformation that is generally believed to be athermal is the
beta to omega transformation. But this has never been established.
Recently, we have designed a phase field model that demonstrates
the beta to omega transformation occurs at nearly the speed of
sound. The model also predicts several other features such as
the existence of the omega phase in a certain domain of alloying
element etc.
Precursors, predictability
of failure in earthquake models:
Predicting time of failure is very important question both conceptually
as well as technologically, in particular, in the context of earthquake
predictability. As the magnitudes of earthquakes follow Gutenberg-Richters
law which is a power law, the system exhibits all length scales and time
scales. Hence predicting an individual earthquake event is considered
impossible. This statement clearly applies to model systems. This paradox
has been resolved by considering the well known Burridge-Knopoff model
for earthquakes by including a term that mimics acoustic emission. The
model predicts individual earthquake events within one percent of time
for the onset of the event. The increased precursor activity, which in
this case is acoustic emission, itself follows a power law behavior in
its approach to the failure point. This universal approach could be
applicable to other failure systems. The model also explains many experimental
features of AE signals observed in rock samples in laboratory that has not been explained so far.
Peeling of an Adhesive Tape:
Peeling is a kind of fracture that is important in the context of adhesion and falls into stick-slip systems like the PLC effect, i.e., measured force-velocity curve shows an unstable branch. The problem of peeling of an adhesive tape has been experimentally well studied for a long time but the equations of motion are beset with controversy as these do not produce stick-slip jumps. This has been resolved by showing that the equations of motion fall into the category of differential-algebraic set of equations and hence are singular requiring a special algorithm meant to solve differential algebraic equations. Once this method is used, the dynamical jumps follow naturally. Several experimental features are also explained. It turns out that the kinetic energy of stretched tape is missing and when it is included, the singular nature is lifted and natural stick-slip jumps follow. By including additional dissipative energy that mimics acoustic emission, the contact line dynamics is studied. Many features of acoustic emission automatically follow as also other experimental features such as the stuck-peeled configuration of the peel front. The model predicts chaotic dynamics in the mid range of pull velocities. The existence of chaotic dynamics has been verified though time series analysis of the acoustic emission signals.
Friction:
Dynamics of sliding friction has remained a difficult phenomenon to understand. A model had been developed that involves viscoelastic contribution, plastic deformation of asperities to explain the origin of stick-slip in dynamical condition. The model also provides a frame work for the velocity weakening friction law proposed in the literature. The approach is very general to the extent it can be extended to the nano-indentation, scratching tools etc.
Biophysics:
Biological systems exhibit rich dynamical and statistical features. Stochastic methods have been used to explain some unusual statistical features in gene expression. Langevin dynamical approach has been used to explain unusual statistics of motile particles. One of the areas of interest is nature of mechanical response of cells.
Thin film Growth:
As our laboratory has several experts in the area
of thin film growth and characterization, some
modeling efforts have been devoted to interpretation
of several experimental findings.
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