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Basant Lal Sharma

Publications and source records attributed to Basant Lal Sharma.

At least 19 recordsLinked to original sources
arXiv2025

Instabilities in Colloidal Crystals on Fluid Membranes

The complex physics of self-assembly in colloidal crystals on deformable interfaces and surfaces poses interesting possibilities for the designability and synthesis of next-generation metamaterials. The goal of this article is to characterize instabilities arising in colloidal crystals assembled on fluid membranes. The colloidal particles are modeled as pair-wise interacting point particles, constrained to lie on a fluid membrane and yet free to reorganize, and the membrane's elastic energy is modeled via the Helfrich energy. We find that when a collection of particles is arranged on a planar membrane in some regular fashion -- such as periodic lattice -- then the regular configuration admits bifurcations to non-planar configurations. Using the Bloch-wave anstaz for the mode of instabilities, we present a parameteric analysis of the boundary between the stable and unstable regimes. We find that instabilities can occur through two distinct kinds of modes, when the parameters belong in certain physically interesting regimes, referred to as long-wavenumber modes ($L$ modes) and short-wavenumber modes ($S$ modes) in the article. We discuss some connections between these results and recent experiments, as well as the open problem of budding in biomembranes.

cond-mat.soft↗
arXiv2025

On discrete X-ray transform

We consider a discrete version of X-ray transform going back, in particular, to Strichartz (1982). We suggest non-overdetermined reconstruction for this discrete transform. Extensions to weighted (attenuated) analogues are given. Connections to the continuous case are presented.

math.FA↗
arXiv2025

Local bifurcation analysis of circular von-Kármán plate with Kirchhoff rod boundary

Symmetry based reduction is applied to the buckling of a circular von-Karman plate with Kirchhoff rod boundary, where a mismatch between the edge length and the perimeter of plate is treated as the bifurcation parameter. A nonlinear operator formulation describes the equilibrium of the elastic rod plate system. The critical points, as potential bifurcation points, are stated using the linearized operator. The symmetry of null space for each critical point is identified as a subgroup of the complete symmetry group of nonlinear problem, the equivariance associated with the nonlinear operator is used in this process. Sufficient evidence is provided for each critical point to be a bifurcation point for the symmetry reduced problem and post buckling analysis is carried out using Lyapunov Schmidt reduction. Bifurcation curves are obtained till quadratic order in bifurcation parameter away from each critical value. Theoretical results for bifurcation curves are validated against the numerical simulation based on a symmetry reduced finite element method for some illustrative examples of critical points. A numerical study is carried out for the dependence of the coefficient of quadratic term in the bifurcation parameter when structural parameters are varied in a neighborhood of four fixed sets of structural parameters. Numerical results based on a symmetry reduced finite element analysis confirm that the nonlinear solution agrees with the local theoretical behavior close to a critical point but deviates further away from it. Using these tools, two main conclusions are reached. First it is observed that the critical points of the linearized problem are indeed bifurcation points. Second, an alteration in the nature of bifurcation is observed during the parameter sweep study when the plate is in tension.

physics.class-ph↗
arXiv2025

Equilibrium of circular von-Kármán plate bonded with Kirchhoff rod

A circular von Karman plate is considered bonded at its boundary to a circular Kirchhoff rod via a hinge like junction. There is a mismatch of dimension between the rod and the plate boundary in their respective stress free configurations. The process of gluing of the inextensible circular rod to the edge of the extensible plate causes the system to develop internal stress in the natural planar configuration. For some critical values of this mismatch the rod plate system, as expected, shows buckling behaviour in the interior or at the boundary of plate. In this paper, the corresponding bifurcation problem is formulated using the mismatch parameter as a bifurcation parameter. The non linear equations of equilibrium are linearized near the homogeneously deformed state and necessary conditions of bifurcation are solved for the critical geometric mismatch that is when non trivial solutions of the linearized equations exist. These critical points are candidates to be classified as bifurcation points. Additionally, the semi analytic nature of the analysis is exploited to perform a parametric study, relative to structure parameters in the problem formulation, of the critical points in the problem and the characteristic features are summarised. It is found that the rod plays an important role in determining the buckling behaviour when the plate's dimension is smaller than the looped rod, as in that case the rod buckles and the plate can bend or stretch depending on the planar or non planar nature of buckled mode of the rod. When the plate is larger, the rod effectively stimulates a rigid Dirichlet boundary condition due to its inextensibility; the critical points in this case are found to coincide with the case when buckling in the sole plate as a system is considered.

physics.class-ph↗
arXiv2024

Inverse source problem for discrete Helmholtz equation

We consider multi-frequency inverse source problem for the discrete Helmholtz operator on the square lattice $\mathbb{Z}^d$, $d \ge 1$. We consider this problem for the cases with and without phase information. We prove uniqueness results and present examples of non-uniqueness for this problem for the case of compactly supported source function, and a Lipshitz stability estimate for the phased case is established. Relations with inverse scattering problem for the discrete Schrödinger operators in the Born approximation are also provided.

math.AP↗
arXiv2024

Surface waves in randomly perturbed discrete models

We study the propagation of surface waves across structured surfaces with random, localized inhomogeneities. A discrete analogue of Gurtin-Murdoch model is employed and surface elasticity, in contrast to bulk elasticity, is captured by distinct point masses and elastic constants for nearest-neighbour interactions parallel to the surface. Expressions for the surface wave reflectance and transmittance, as well as the radiative loss, are provided for every localized patch of point mass perturbation on the surface. As the main result in the article, we provide the statistics of surface wave reflectance and transmittance and the radiative loss for an ensemble of random mass perturbations, independent and identically distributed with mean zero, on the surface. In the weakly scattering regime, the mean radiative loss is found to be proportional to the size of the perturbed patch, to the variance of the mass perturbations, and to an effective parameter that depends on the continuous spectrum of the unperturbed system. In the strongly scattering regime, the mean radiative loss is found to depend on another effective parameter that depends on the continuous spectrum, but not on the variance of the mass perturbations. Numerical simulations are found in quantitative agreement with the theoretical predictions for several illustrative values of the surface structure parameters.

math-ph↗
arXiv2024

Effective dynamics in lattices with random mass perturbations

We consider a one-dimensional mono-atomic lattice with random perturbations of masses spread over a finite number of particles. Assuming Newtonian dynamics and linear nearest-neighbour interactions and allowing for a provision of pinning due to substrate interaction, we discuss a transient dynamics problem and a time-harmonic transmission problem. By a stochastic, multiscale analysis we provide asymptotic expressions for the displacement field that propagates through the random perturbations and for the time-harmonic transmission coefficients. These theoretical predictions are supported by illustrations of their agreements with numerical simulations.

math.PR↗
arXiv2023

Scattering of surface waves by inhomogeneities in crystalline structures

In current scientific and technological scenario, studies of transmittance of surface waves across structured interfaces have gained some wind amidst applications to metasurfaces, electronic edge-waves, crystal grain boundaries, etc. The results presented in the present article shed a light on the influence of material inhomogeneities on propagation of surface waves. Within the framework of classical mechanics, an analog of Gurtin-Murdoch model is employed where elastic properties on surface are assumed to be distinct from bulk. Restricting to scalar waves on prototype square lattice half-plane, particles on considered structured surface have piecewise-constant mass and surface force-constants across an interfacial point. Particles in bulk lattice interact with nearest neighbours in a way that involves unequal force-constants parallel to surface versus normal to it. A surface wave band exists for such lattice structure wherein the waveform decays exponentially inside half-plane. A formula for surface wave transmittance is given based on an exact solution on half-plane, and, thus, previous work of Sharma & Eremeyev 2019 Int. J. Eng. Sci. 143, 33-38 is extended. An explicit expression for fraction of energy influx leaked via bulk waves is a highlight. Included are graphical results for several illustrative values of surface structure parameters.

physics.class-ph↗
arXiv2023

Computational Modeling of Coupled Interactions of Fluid Membranes with Embedded Filaments

In this work, we present a computational formulation based on continuum mechanics to study the interaction of fluid membranes embedded with semiflexible filaments. This is motivated by systems in membrane biology, such as cytoskeletal networks and protein filaments aiding the cell fission process. We model the membrane as a fluid shell via the Helfrich-Canham energy and the filament as a one-dimensional Cosserat continuum. We assume the filament to be tethered to the surface of the membrane in a way that it is allowed to float on the surface freely. The novel filament-membrane coupling, which is anticipated to yield interesting physics, also gives rise to unique computational challenges, which we address in this work. We present validation results and apply the formulation to certain problems inspired by cellular biology.

math.NA↗
arXiv2023

Generalizing Parametrization Invariance in the Calculus of Variations

We revisit the notion of parametrization invariance while introducing certain weakened notions of invariance in the calculus of variations. In this work, we employ a straightforward approach in the classical setting and mostly restrict attention to functionals on one-dimensional domains. We establish a connection between parametrization invariant functionals and functionals embodying a weaker notion of invariance of their Lagrangian; we term this notion as T-Lagrangian analogous to the well-known idea of null Lagrangian. However, the Euler-Lagrange operator of a T-Lagrangian vanishes only along the tangential direction in the configuration space. On one-dimensional domain and for first- and second-order theories, we show that functional described by such a Lagrangian is necessarily a parametrization invariant functional modulo null Lagrangian. Keeping the motivation for partial differential equations, we also introduce and explore the notion of N-Lagrangian, with an invariance complementary to the case of T-Lagrangian, whose Euler-Lagrange operator vanishes along normal directions. We find that in a one-dimensional setting, every N-Lagrangian is simply a null Lagrangian.

math.CA↗
arXiv2020

Coupled electro-elastic deformation and instabilities of a toroidal membrane

We analyse here the problem of large deformation of dielectric elastomeric membranes under coupled electromechanical loading. Extremely large deformations (enclosed volume changes of 100 times and greater) of a toroidal membrane are studied by the use of a variational formulation that accounts for the total energy due to mechanical and electrical fields. A modified shooting method is adopted to solve the resulting system of coupled and highly nonlinear ordinary differential equations. We demonstrate the occurrence of limit point, wrinkling, and symmetry-breaking buckling instabilities in the solution of this problem. Onset of each of these "reversible" instabilities depends significantly on the ratio of the mechanical load to the electric load, thereby providing a control mechanism for state switching.

cond-mat.soft↗
arXiv2020

A dislocation-dipole in one dimensional lattice model

A family of equilibria corresponding to dislocation-dipole, with variable separation between the two dislocations of opposite sign, is constructed in a one dimensional lattice model. A suitable path connecting certain members of this family is found which exhibits the familiar Peierls relief. A landscape for the variation of energy has been presented to highlight certain sequential transition between these equilibria that allows an interpretation in terms of quasi-statically separating pair of dislocations of opposite sign from the viewpoint of closely related Frenkel-Kontorova model. Closed form expressions are provided for the case of a piecewise-quadratic potential wherein an analysis of the effect of an intermediate spinodal region is included.

cond-mat.stat-mech↗
arXiv2020

Variational principles of nonlinear magnetoelastostatics and their correspondences

We derive the equations of nonlinear magnetoelastostatics using several variational formulations involving the mechanical deformation and an independent field representing the magnetic component. An equivalence is also discussed, modulo certain boundary integrals or constant integrals, between these formulations using the Legendre transform and properties of Maxwell's equations. The second variation based bifurcation equations are stated for the incremental fields as well for all five variational principles. When the total potential energy is defined over the infinite space surrounding the body, we find that the inclusion of certain term in the energy principle, associated with the externally applied magnetic field, leads to slight changes in the Maxwell stress tensor and associated boundary conditions. On the other hand, when the energy contained in the magnetic field is restricted to finite volumes, we find that there is a correspondence between the discussed formulations and associated expressions of physical entities. In view of a diverse set of boundary data and nature of externally applied controls in the problems studied in the literature, along with a equally diverse list of variational principles employed in modeling, our analysis emphasizes care in the choice of variational principle and unknown fields so that consistency with other choices is also satisfied.

physics.class-ph↗
arXiv2020

Null Lagrangians in linear theories of micropolar type and few other generalizations of elasticity

In the context of linear theories of generalized elasticity including those for homogeneous micropolar media, quasicrystals, piezoelectric and piezomagnetic media, we explore the concept of null Lagrangians. For obtaining the family of null Lagrangians we employ the sufficient conditions of H. Rund. In some cases a non-zero null Lagrangian is found and the stored energy admits a split into a null Lagrangian and a remainder. However, the null Lagrangian vanishes whenever the relevant elasticity tensor obeys certain symmetry conditions which can be construed as an analogue of the Cauchy relations.

math-ph↗
arXiv2020

Null Lagrangians in Cosserat elasticity

In the framework of nonlinear theory of Cosserat elasticity, also called micropolar elasticity, we provide the complete characterization of null Lagrangians for three dimensional bodies as well as for shells. Using the Gibb's rotation vector for description of the microrotation, this task is possible by an application of a theorem stated by Olver and Sivaloganathan in `{the structure of null Lagrangians}' (Nonlinearity, {1}, 1988, pp. 389-398). A set of necessary and sufficient conditions is also provided for the elasticity tensors to correspond to a null Lagrangian in linearized micropolar theory.

math-ph↗
arXiv2020

Discrete scattering by two staggered semi-infinite defects: reduction of matrix Wiener-Hopf problem

As an extension of the discrete Sommerfeld problems on lattices, the scattering of a time harmonic wave is considered on an infinite square lattice when there exists a pair of semi-infinite cracks or rigid constraints. Due to the presence of stagger, also called offset, in the alignment of the defect edges the asymmetry in the problem leads to a matrix Wiener-Hopf kernel that cannot be reduced to scalar Wiener-Hopf in any known way. In the corresponding continuum model the same problem is a well known formidable one which possesses certain special structure with exponentially growing elements on the diagonal of kernel. From this viewpoint the present paper tackles a discrete analogue of the same by reformulating the Wiener-Hopf problem and reducing it to a finite set of linear algebraic equations; the coefficients of which can be found by an application of the scalar Wiener-Hopf factorization. The considered discrete paradigm involving lattice waves is relevant for modern applications of mechanics and physics at small length scales.

math-ph↗
arXiv2020

Discrete scattering by a pair of parallel defects

Scattering of a time harmonic anti-plane shear wave due to either a pair of crack tips or a pair of rigid constraint tips on square lattice is considered. The two problems correspond to the so called zero-offset case of scattering due to a pair of identical Sommerfeld screens. The peculiar structural symmetry allows the reduction of coupled equations to two scalar Wiener-Hopf equations and a total of four geometrically reduced problems on lattice half-plane. Exact solution of each problem for incidence from the bulk lattice, as well as from an associated lattice waveguide, is constructed. A suitable superposition of the four expressions is used to construct the solution of the main problem. The discrete paradigm involving the wave mode incident from the waveguide is relevant for modern applications where an investigation of mechanisms of electronic and thermal transport at nanoscale remains an interesting problem.

math-ph↗