Modeling Mechanics @ Multiscale

Self-contact in closed and open Kirchhoff rods

Raushan Singh, Jaya Tiwari and Ajeet Kumar

We consider the problem of an impenetrable Kirchhoff rod in self-contact. The contact is assumed to be frictionless and of point-contact type. The self-contacted rod is modeled as a set of contact-free segments each having unknown length and connected in series with each other through frictionless hard-contact condition. The Landau–Lifshitz approach is used to express analytically the centerline of each contact-free segment in terms of unknown parameters which are obtained using the hard-contact condition at each contact point and boundary condition on the full rod. The presented formulation is applicable to both closed and open rods. For the open rod case, its two end cross-sections need not be parallel either. Numerical solution of following three example problems are presented both before and after self-contact: (i) twisting of a closed rod (ii) compression and twisting of a straight rod (iii) compression and twisting of a rod with opening angle. For the case of twisting of a closed rod in the pre-self-contact regime, we also derive a minimal set of three integral equations to obtain the ring’s buckled spatial configuration. We also point to a difficulty in numerical integration of the rod’s centerline torsion over the rod length whenever the torsion value becomes large locally: a finer discretization of the rod’s length is necessitated in such regions to obtain an accurate integral value. An analytical approximation of the torsion integral is derived in such regions to make the full numerical integration computationally efficient.

Journal article

Singh, R., Tiwari, J., & Kumar, A. (2021). Self-contact in closed and open Kirchhoff rods. International Journal of Non-Linear Mechanics, 137, 103786.

A singularity free approach for Kirchhoff rods having uniformly distributed electrostatic charge

Raushan Singh and Ajeet Kumar

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, once removed, makes the system well defined everywhere. No cut-off 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.

Journal article

Singh, R., & Kumar, A. (2020). A singularity free approach for Kirchhoff rods having uniformly distributed electrostatic charge. Computer Methods in Applied Mechanics and Engineering, 367, 113133.

---

An asymptotic numerical method for continuation of spatial equilibria of special Cosserat rods

Raushan Singh, D. Abhishek and Ajeet Kumar

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.

Journal article

Singh, R., Abhishek, D., & Kumar, A. (2018). An asymptotic numerical method for continuation of spatial equilibria of special Cosserat rods. Computer Methods in Applied Mechanics and Engineering, 334, 167-182.

---