This work presents a geometrically exact Kirchhoff-like electroelastic rod theory wherein the contribution of free space energy is also factored in. In addition to the usual mechanical variables such as the rod's centerline and cross-section orientation, three electric potential parameters are also introduced to account for the variation in electric potential within the rod's cross-section as well as along the rod length. The free space energy is included through an electric flux-like variable acting on the lateral surface of the rod. The governing equations of this rod are derived through balance of forces and moments (arising from both mechanical and electric sources) for mechanical variables, weighted integration of the three-dimensional electric displacement equation in the rod's cross-section for electric potential parameters and through the asymptotic expansion of a boundary integral equation for the electric flux variable. We also obtain the expressions of internal contact moment and conjugates of electric potential parameters in terms of the rod's aforementioned mechanical and electric variables. A specific choice of the three-dimensional electroelastic stored energy density is further taken catering to piezoelectic materials and dielectric elastomers and the rod's constitutive equations are rederived which exhibit interesting interplay between mechanical and electric variables. The presented work can help improve the design of soft electroelastic rod-like actuators and sensors.
The article will soon appear in IJSS which can also be accessed here: https://www.researchgate.net/publication/365476326_An_electroelastic_Kir...
Improved formulas of extensional and bending stiffnesses of isotropic rectangular nanorods are derived. These formulas reduce to the existing widely used formulas for a special choice of material parameters, i.e., when the surface Poisson's ratio and the bulk Poisson's ratio match thus highlighting the limitation of the existing formulas. To see the derivation, please visit this link: https://www.researchgate.net/publication/361464178_Improved_formulas_of_...
The motion of filament-like structures in fluid media has been a topic of interest since long. In this regard, a well known slender body theory exists wherein the fluid flow is assumed to be Stokesian while the filament is modeled as a Kirchhoff rod which can bend and twist but remains inextensible and unshearable. In this work, we relax the inextensibility and unshearability constraints on filaments, i.e., the filament is modeled as a special Cosserat rod. Starting with the boundary integral formulation of Stokes flow involving the filament's surface velocity and fluid traction that acts on the filament surface, the method of matched asymptotic expansion is used to first obtain a leading order representation of the boundary integral kernels in the filament's aspect ratio. We then substitute Fourier series expansion (in filament's circumferential coordinate) of both the filament's surface velocity and fluid traction in the aforementioned leading order representation and further linearize it in the rod's shear strains to reduce the two-dimensional boundary integral over the filament surface into a line integral over the filament's centerline. Upon further collecting the coefficients of sine and cosine terms, the zeroth order Fourier mode yields a line integral equation relating the rod's centerline velocity with the distributed fluid force that acts on the filament. The presence of line integral makes the relation non-local in nature. On the other hand, the first order Fourier mode yields a simpler local relation between the rod's angular velocity and the distributed fluid couple. The line integral equation is shown to reduce to the classical slender body theory when shear strains and axial strain are set to zero. The non-dimensional governing equations of the special Cosserat rod are also derived accounting for the distributed fluid force and distributed fluid couple in them which are solved to obtain the filament motion. The presented theory is demonstrated with an example problem of the tumbling of filaments in background shear flow. We show that for relatively shorter filaments where the effect of shear and axial stretch is more dominant, the obtained results deviate from the ones based on the classical slender body theory.
The article will soon appear in Mathematics and Mechanics of Solids which can also be accessed at the following link: https://www.researchgate.net/publication/358446135_A_slender_body_theory...
We present a theory for finite and spatial elastic deformation of rods under the influence of arbitrary magnetic field and boundary condition. The rod is modeled as a Kirchhoff rod and is assumed to have uniformly distributed array of uniaxial spheroidal paramagnetic inclusions embedded in it all pointing in the same direction in the undeformed state. The governing equations of the magnetoelastic rod are derived which are further non-dimensionalized and linearized to investigate buckling in such rods. Analytical expressions for the onset of buckling from the rod's trivial state are obtained in terms of loading parameters (applied magnetic field, axial load, torque) as well as geometric (inclusion orientation in the undeformed state) and material (ratio of bending and torsional stiffnesses) parameters for different combinations of boundary conditions. The buckled shape of the rod at the onset of buckling is also examined. The presented work can be useful in simulation and design of magnetic soft continuum robots.
The article will soon appear in IJSS and the same can also be accessed here: https://www.researchgate.net/publication/352713036_A_magnetoelastic_theo...
This work addresses the determination of yield surfaces for geometrically exact elastoplastic rods. Use is made of a formulation where the rod is subjected to an uniform strain field along its arc length, thereby reducing the elastoplastic problem of the full rod to just its cross-section. By integrating the plastic work and the stresses over the rod's cross-section, one then obtains discrete points of the yield surface in terms of stress resultants. Eventually, Lamé curves in their most general form are fitted to the discrete points by an appropriate optimisation method. The resulting continuous yield surfaces are examined for their scalability with respect to cross-section dimensions and also compared with existing analytical forms of yield surfaces.
The article will soon appear in Computational Mechanics which can also be accessed at the following link: https://www.researchgate.net/publication/346484659_Geometrically_exact_e...
A finite element formulation is presented for a direct approach to model elastoplastic deformation in slender bodies using the special Cosserat rod theory. The direct theory has additional plastic strain and hardening variables, which are functions of just the rod's arc-length, to account for plastic deformation of the rod. Furthermore, the theory assumes the existence of an effective yield function in terms of stress resultants, i.e., force and moment in the cross-section and cross-section averaged hardening parameters. Accordingly, one does not have to resort to the three-dimensional theory of elastoplasticity during any step of the finite element formulation. A return map algorithm is presented in order to update the plastic variables, stress resultants and also to obtain the consistent elastoplastic moduli of the rod. The presented FE formulation is used to study snap-through buckling in a semi-circular arch subjected to an in-plane transverse load at its mid-section. The effect of various elastoplastic parameters as well as pre-twisting of the arch on its load-displacement curve are presented.
The article will soon appear in IJNME and the same can be accessed at the following link: https://www.researchgate.net/publication/344344096_A_finite_element_form...
We present a singularity free formulation and its efficient numerical implementation for the spatial deformation of Kirchhoff rods having uniformly distributed electrostatic charge. Due to the presence of continuously distributed charge, the governing equations of the Kirchhoff rod become a system of integro-differential equations which is singular at every arc-length. We show that this singularity is of removable type which, ones removed, makes the system well defined everywhere. No cutoff length or mollifier is used to remove this singularity. An efficient finite difference scheme is presented for the numerical solution of this singularity free system of equations. We show that the presented numerical scheme turns out to be computationally efficient compared to an alternate approach in which the uniformly distributed charge is modeled by placing equivalent lumped charge at discrete locations along the rod. The scheme is demonstrated through an example problem of supercoiling in a charged elastic ring when twist is inserted in it.
The article will soon appear in Computer Methods in Applied Mechanics and Engineering and the same can also be accessed at the following link: https://www.researchgate.net/publication/341232833_A_singularity_free_approach_for_Kirchhoff_rods_having_uniformly_distributed_electrostatic_charge
A Cosserat rod based continuum approach is presented to obtain phonon dispersion curves of flexural, torsional, longitudinal, shearing and radial breathing modes in chiral nanorods and nanotubes. Upon substituting the continuum wave form in the linearized dynamic equations of stretched and twisted Cosserat rods, we obtain analytical expression of a coefficient matrix (in terms of the rod's stiffnesses, induced axial force and twisting moment) whose eigenvalues and eigenvectors give us frequencies and mode shapes, respectively, for each of the above phonon modes. We show that, unlike the case of achiral tubes, these phonon modes are intricately coupled in chiral tubes due to extension-torsion-inflation and bending-shear couplings inherent in them. This coupling renders the conventional approach of obtaining stiffnesses from the long wavelength limit slope of dispersion curves redundant. However, upon substituting the frequencies and mode shapes (obtained independently from phonon dispersion molecular data) in the eigenvalue-eigenvector equation of the above mentioned coefficient matrix, we are able to obtain all the stiffnesses (bending, twisting, stretching, shearing and all coupling stiffnesses corresponding to extension-torsion, extension-inflation, torsion-inflation and bending-shear couplings) of chiral nanotubes. Finally, we show unusual effects of the single-walled carbon nanotube's chirality as well as stretching and twisting of the nanotube on its phonon dispersion curves obtained from the molecular approach. These unusual effects are accurately reproduced in our continuum formulation.
The article will soon appear in Mathematics and Mechanics of Solids and it can also be accessed at the following link: https://www.researchgate.net/publication/333237562_Phonons_in_chiral_nan...
We present an efficient numerical scheme based on asymptotic numerical method for continuation of spatial equilibria of special Cosserat rods. Using quaternions to represent rotation, the equations of static equilibria of special Cosserat rods are posed as a system of thirteen first order ordinary differential equations having cubic nonlinearity. The derivatives in these equations are further discretized to yield a system of cubic polynomial equations. As asymptotic-numerical methods are typically applied to polynomial systems having quadratic nonlinearity, a modified version of this method is presented in order to apply it directly to our cubic nonlinear system. We then use our method for continuation of equilibria of the follower load problem and demonstrate our method to be highly efficient when compared to conventional solvers based on the finite element method. Finally, we demonstrate how our method can be used for computing the buckling load as well as for continuation of postbuckled equilibria of hemitropic rods.
The article will soon appear in Computer Methods in Applied Mechanics and Engineering. It can also be accessed at the following link: https://www.researchgate.net/publication/322750460_An_asymptotic_numeric...
A general framework is presented to model coupled thermo-elasto-plastic deformations in the theory of special Cosserat rods. The use of the one-dimensional form of the energy balance in conjunction with the one-dimensional entropy balance allows us to obtain an additional equation for the evolution of a temperature-like one-dimensional field variable together with constitutive relations for this theory. Reduction to the case of thermoelasticity leads us to the well known nonlinear theory of thermoelasticity for special Cosserat rods. Later on, additive decomposition is used to separate the thermoelastic part of the strain measures of the rod from their plastic counterparts. We then present the most general quadratic form of the Helmholtz energy per unit rod's undeformed length for both hemitropic and transversely isotropic rods. We also propose a prototype yield criterion in terms of forces, moments and hardening stress resultants as well as the associative flow rules for the evolution of plastic strain measures and hardening variables.
The article will soon appear in Mathematics and Mechanics of solids and can also be accessed at the following link: https://www.researchgate.net/publication/322113320_A_thermo-elasto-plast...
We present a continuum formulation to obtain simple expressions demonstrating the effects of surface residual stress and surface elastic constants on extensional and torsional stiffnesses of isotropic circular nanorods. Unlike the case of rectangular nanorods, we show that the stiffnesses of circular nanorods also depend on surface residual stress components. This is attributed to non-zero surface curvature inherent in circular nanorods. We further analyze their asymptotic limits in the limit of the nanorod's radius approaching both zero and infinity which correspond to surface dominated and bulk dominated regimes respectively. Finally, we use the recently proposed Helical Cauchy-Born rule and perform molecular statics calculations to obtain axial force, twisting moment and stiffnesses of the tungsten nanorod. The results from molecular statics calculations match most accurately with our formulae when compared to existing formulae by others. The article will appear soon in Mathematics and Mechanics of Solids and can also be accessed at the following link: https://www.researchgate.net/publication/322015816_Effect_of_surface_ela...
We demonstrate intersting extension-torsion-inflation coupling in intrinsically twisted chiral tubes. In particular, we show that by tuning its intrinsic twist and the material constants, such tubes can show negative poison effect on being stretched. Similarly, such tubes can overwind initially when stretched - the same unusual behavior has been reported earlier when a DNA molecule is stretched. The article got published recently in the Journal of Elasticity and is available at the following link: http://link.springer.com/article/10.1007%2Fs10659-017-9623-8
Job title: Research Associate/Postdoc
Minimum qualification: PhD in Solid Mechanics/Mathematics
Research area: Thermoelastic Modeling of nano and Contunuum Rods – A Molecular Approach
Salary: Rs 36000 per month + 30% HRA
Walk in interview: 3rd of Nov 2016 in Department of Applied Mechanics, IIT Delhi
Contact person: Prof. Ajeet Kumar, ajeetk@am.iitd.ac.in
See the attachment for more details.
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 IITD-IRD-143-2016.pdf | 18.12 KB |
A PhD/Postdoc position is available for Indian nationals in my group to work in the broad area of "Molecular origin of elastic/plastic deformations in nanorods". This is a collaborative project with Germany and may require a semester or two stay in Germany. The PhD applicant should have good background in mathematics and solid mechanics. The postdoc candidate should preferably also be familiar with molecular modeling of materials. Interested candidates are encouraged to email me at ajeetk@am.iitd.ac.in (or ajeet3856@gmail.com) alongwith with their CV.
Dr Ajeet Kumar
Dept. of Applied Mechanics
IIT Delhi
You may be interested in reading our following article: http://link.springer.com/article/10.1007/s10659-016-9586-1. We show the importance of incorporating material nonlinearity for accurate simulation of nanorods and nanotubes. The linear material laws are shown to give completely erroneous results. The nonlinear material laws for nanorods were obtained using the recently proposed "Helical Cauchy-Born rule". We also discuss how surface stress affects buckling in such nanostructures.
Dear Friends,
You may like to read one of my articles which recently got accepted in the Journal of Elasticity. It is titled "A helical Cauchy-Born rule for special Cosserat rod modeling of nano and continuum rods". The Cauchy-Born rule was primarily introduced for constitutive modeling of bulk crystals as three-dimensional elastic bodies. Arroyo & Belytschko later proposed exponential CB rule for shell-like nanostructures. Here, we have proposed a novel "helical Cauchy-Born rule" to derive nonlinear constitutive laws of rod-like nanostructures from atomic level computations.
The article is now available at the folliwng link: http://link.springer.com/article/10.1007%2Fs10659-015-9562-1
Ajeet Kumar
IIT Delhi
We recently published a paper in International Journal of Solids and Structures titled "A rod model for three dimensional deformations of single walled carbon nanotubes".(paper attached)
http://www.sciencedirect.com/science/article/pii/S0020768311002149
There are several research papers dealing with continuum modeling of a nanotube using shell theory. Longer nanotubes, however, appear more as a one dimensional rod but there is almost no work towards modeling of a nanotube using rod theory. The objective of this paper is to model a nanotube using rod theory. It also highlights challenges and associated future research plans. Not to mention, a one dimensional rod model is advantageous both from theoretical and computational viewpoint.
Comments and feedback most welcome!