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Research Papers

# Constitutive-Law Modeling of Microfilaments From Their Discrete-Structure Simulations—A Method Based on an Inverse Approach Applied to a Static Rod Model

[+] Author and Article Information

Theoretical and Applied Mechanics,Cornell University,Ithaca, NY, 14853-1503ah353@cornell.edu

Sachin Goyal1

Mechanical and Aerospace Engineering,Cornell University,Ithaca, NY, 14853-1503sgoyal@umich.edu

Mechanical Engineering,Indian Institute of Technology, Gandhinagar, Gujarat, 382424, Indiaharish@iitgn.ac.in

Any change in the orientation of a cross-section from the undeformed configuration to the deformed configuration can be accomplished by a single Euler rotation about a unit vector $û$ by angle θ. In terms of the single Euler rotation parameters, $[R]=exp(θ[ũ])=[I]+[ũ] sinθ+[ũ]2(1-cosθ)$, where [I] is the identity matrix and $[ũ]$ is the skew-symmetric matrix corresponding to $û$.

Although this simplicity may not be readily obvious, all the simplicities that we impose in this section will become plausible with validation of the estimated constitutive law in Sec. 5 and analysis in Sec. 5

Note that with this choice of reference frame, the stress-free curvature vector $κ→$0 (s) = $0→$ since the discrete structure is straight in the undeformed (reference) state. This fact can be verified by satisfying the Serret-Frenet formula [15] in the reference state.

1

Corresponding author.

J. Appl. Mech 79(5), 051005 (Jun 22, 2012) (7 pages) doi:10.1115/1.4006449 History: Received November 30, 2010; Revised February 28, 2012; Posted March 26, 2012; Published June 22, 2012; Online June 22, 2012

## Abstract

Twisting and bending deformations are crucial to the biological functions of several microfilaments such as DNA molecules. Although continuum-rod models have emerged as efficient tools to describe the nonlinear dynamics of these deformations, a major roadblock in the continuum–mechanics-based description of microfilaments is the accurate modeling of the constitutive law, which follows from their atomistic-level structure and interactions. In this paper, we present a method for estimating the constitutive law using a static rod model and deformed configuration data generated from discrete-structure simulations. Furthermore, we illustrate the method on a filament with an artificial discrete-structure. We simulate its deformation in response to a prescribed loading using a multibody dynamics (MBD) solver. Using position data generated from the MBD solver, we first estimate the curvature of the filament, and subsequently use it to estimate the effective relationship between the restoring moment and curvature.

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## Figures

Figure 1

Flow chart illustrating the components of the estimation of the constitutive law. The red solid arrows represent the path of information flow for a forward model simulation, while the blue dashed arrows represent the path of information flow for inverse modeling to estimate the constitutive law. As seen from the chart, the inverse modeling framework uses data (or simulated data) and model information to estimate the constitutive law. The forward model is implemented in Hyperworks Motionview, while the inverse modeling is implemented in MATLAB.

Figure 2

Discrete structure of the filament. Two orthogonal views are shown, one along z-axis (top) and the other into the z-axis (bottom). Unit point masses are arranged in a cubic arrangement and five repeats of the cubic arrangement are shown. Each edge of the cubic cell has unit length and has linear springs of unit stiffness connecting the point masses in the undeformed (reference) configuration. In addition, the diagonal springs are placed on two opposite faces of the cube shown in the view along z-axis (top), but not on the other four faces of the cube.

Figure 3

Deformed shape of the discrete-structure cantilever in static equilibrium. The free end has a prescribed shear force, but has no tension and no bending moment. Since there are 30 cubic cells of 1 mm edge each, L = 30 mm. The cross-section-fixid reference frame âi(s) is also shown. The unit vector â3(s) points along the outward normal to the cross-section. The unit vector â1(s) is chosen in the plane of bending, while â2(s) is parallel to the axis of bending.

Figure 4

Estimated constitutive law

Figure 5

Deformed shape of the discrete-structure cantilever in static equilibrium (left). The free end has prescribed shear force, tension as well as bending moment. Centerline predicted from discrete-structure simulation in Hyperworks (dots) and continuum-rod simulation (solid curve) employing the estimated constitutive-law (right).

Figure 6

A schematic to derive the constitutive law ψ2 (κ2 , q2 ) = 0 from the restoring spring-forces developed in the discrete structure. The neutral surface is approximately half way between the top and bottom surfaces. The diagonal springs are not shown because they do not contribute to the restoring moment.

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