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Research Papers

Computational Rod Model With User-Defined Nonlinear Constitutive Laws

[+] Author and Article Information
Soheil Fatehiboroujeni

Department of Mechanical Engineering,
University of California, Merced,
Merced, CA 95343
e-mail: sfatehiboroujeni@ucmerced.edu

Harish J. Palanthandalam-Madapusi

Mechanical Engineering,
Indian Institute of Technology,
Gandhinagar Palaj,
Gandhinagar 382355, India
e-mail: harish@iitgn.ac.in

Sachin Goyal

Department of Mechanical Engineering,
University of California, Merced,
Merced, CA 95343
e-mail: sgoyal2@ucmerced.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 28, 2017; final manuscript received July 25, 2018; published online August 22, 2018. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 13(10), 101006 (Aug 22, 2018) (8 pages) Paper No: CND-17-1525; doi: 10.1115/1.4041028 History: Received November 28, 2017; Revised July 25, 2018

Computational rod models have emerged as efficient tools to simulate the bending and twisting deformations of a variety of slender structures in engineering and biological applications. The dynamics of such deformations, however, strongly depends on the constitutive law in bending and torsion that, in general, may be nonlinear, and vary from material to material. Jacobian-based computational rod models require users to change the Jacobian if the functional form of the constitutive law is changed, and hence are not user-friendly. This paper presents a scheme that automatically modifies the Jacobian based on any user-defined constitutive law without requiring symbolic differentiation. The scheme is then used to simulate force-extension behavior of a coiled spring with a softening constitutive law.

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Figures

Grahic Jump Location
Fig. 1

The motion of each cross section of the rod at length s and time t is determined by tracking the transformations of the body-fixed frame âi(s,t) with respect to the inertial frame of reference êi

Grahic Jump Location
Fig. 2

The algorithm of the numerical scheme. The guessed solution for all spatial nodes is shown with Y*i. At each time-step, the linearized equation is integrated in space. The spatial integration is iterated and is used to update the guessed solution until it converges to the true solution Yi bounded by a small tolerance ε.

Grahic Jump Location
Fig. 3

Loading scenarios of compressing and stretching the helical rod

Grahic Jump Location
Fig. 4

Bending constitutive law that captures the relationship between the restoring moment q1 and curvature κ1. The rod chosen to be isotropic with a circular cross section. Therefore, the bending constitutive law for both planes of bending q1−κ1, and q2−κ2 are the same.

Grahic Jump Location
Fig. 5

Twisting constitutive law that captures the relationship between the restoring torque q3 and torsion κ3

Grahic Jump Location
Fig. 6

This diagram shows the relationship between the magnitude of the force along the axis of the helix faxial and the end-to-end distance ΔX

Grahic Jump Location
Fig. 7

The vibrations of the rod are calculated via three different descriptions of the constitutive law. The end-to-end distance of the rod ΔX is plotted versus time using the accurate description of the constitutive law, the least-square fitting, and the linear approximation of the constitutive law.

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