Research Papers

A Geometrically Exact Rod Model Including In-Plane Cross-Sectional Deformation

[+] Author and Article Information
Ajeet Kumar

Department of Aerospace Engineering & Mechanics, University of Minnesota, Minneapolis, MN 55455

Subrata Mukherjee

Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853

J. Appl. Mech 78(1), 011010 (Oct 13, 2010) (10 pages) doi:10.1115/1.4001939 History: Received November 25, 2009; Revised May 18, 2010; Posted June 09, 2010; Published October 13, 2010; Online October 13, 2010

We present a novel approach for nonlinear, three dimensional deformation of a rod that allows in-plane cross-sectional deformation. The approach is based on the concept of multiplicative decomposition, i.e., the deformation of a rod’s cross section is performed in two steps: pure in-plane cross-sectional deformation followed by its rigid motion. This decomposition, in turn, allows straightforward extension of the special Cosserat theory of rods (having rigid cross section) to a new theory allowing in-plane cross-sectional deformation. We then derive a complete set of static equilibrium equations along with the boundary conditions necessary for analytical/numerical solution of the aforementioned deformation problem. A variational approach to solve the relevant boundary value problem is also presented. Later we use symmetry arguments to derive invariants of the objective strain measures for transversely isotropic rods, as well as for rods with inbuilt handedness (hemitropy) such as DNA and carbon nanotubes. The invariants derived put restrictions on the form of the strain energy density leading to a simplified form of quadratic strain energy density that exhibits some interesting physically relevant coupling between the different modes of deformation.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

A typical cross section of a rod undergoing in-plane cross-sectional deformation followed by rigid rotation: two possible decompositions

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Figure 2

A pictorial representation of the tractions generating the stress resultants conjugate to the in-plane cross-sectional strain measures

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Figure 3

Deformation of a cross section due to bending alone: (a) a rectangle becoming trapezoid (b) a circular cross section with its bending axis ek

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Figure 4

Surface deformation of an initially hollow circular cylinder: a=1+0.7 sin(0.5πs), b=1−0.7 sin(0.5πs), c=0




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