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Research Papers

A Higher-Order Theory for Open and Closed Curved Rods and Tubes Using a Novel Curvilinear Cylindrical Coordinate System

[+] Author and Article Information
A. Arbind, A. R. Srinivasa, J. N. Reddy

Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843

Manuscript received May 15, 2018; final manuscript received May 20, 2018; published online June 14, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(9), 091006 (Jun 14, 2018) (11 pages) Paper No: JAM-18-1289; doi: 10.1115/1.4040335 History: Received May 15, 2018; Revised May 20, 2018

In this study, the governing equation of motion for a general arbitrary higher-order theory of rods and tubes is presented for a general material response. The impetus for the study, in contrast to the classical Cosserat rod theories, comes from the need to study bulging and other deformation of tubes (such as arterial walls). While Cosserat rods are useful for rods whose centerline motion is of primary focus, here we consider cases where the lateral boundaries also undergo significant deformation. To tackle these problems, a generalized curvilinear cylindrical coordinate (CCC) system is introduced in the reference configuration of the rod. Furthermore, we show that this results in a new generalized frame that contains the well-known orthonormal moving frames of Frenet and Bishop (a hybrid frame) as special cases. Such a coordinate system can continuously map the geometry of any general curved three-dimensional (3D) structure with a reference curve (including general closed curves) having continuous tangent, and hence, the present formulation can be used for analyzing any general rod or pipe-like 3D structures with variable cross section (e.g., artery or vein). A key feature of the approach presented herein is that we utilize a non-coordinate “Cartan moving frame” or orthonormal basis vectors, to obtain the kinematic quantities, like displacement gradient, using the tools of exterior calculus. This dramatically simplifies the calculations. By the way of this paper, we also seek to highlight the elegance of the exterior calculus as a means for obtaining the various kinematic relations in terms of orthonormal bases and to advocate for its wider use in the applied mechanics community. Finally, the displacement field of the cross section of the structure is approximated by general basis functions in the polar coordinates in the normal plane which enables this rod theory to analyze the response to any general loading condition applied to the curved structure. The governing equation is obtained using the virtual work principle for a general material response, and presented in terms of generalized displacement variables and generalized moments over the cross section of the 3D structure. This results in a system of ordinary differential equations for quantities that are integrated across the cross section (as is to be expected for any rod theory).

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References

Green, A. E. , Naghdi, P. M. , and Wenner, M. L. , 1974, “ On the Theory of Rods—I: Derivations From the Three-Dimensional Equations,” Proc. R. Soc. London. A. Math. Phys. Sci., 337(1611), pp. 451–483. [CrossRef]
Green, A. E. , Naghdi, P. M. , and Wenner, M. L. , 1974, “ On the Theory of Rods—II: Developments by Direct Approach,” Proc. R. Soc. London. A. Math. Phys. Sci., 337(1611), pp. 485–507. [CrossRef]
Antman, S. S. , 2005, Nonlinear Problems of Elasticity (Mathematical Sciences, Vol. 107), Springer, New York.
Simo, J. C. , 1985, “ A Finite Strain Beam Formulation. the Three-Dimensional Dynamic Problem—Part I,” Comput. Methods Appl. Mech. Eng., 49(1), pp. 55–70. [CrossRef]
Simo, J. C. , and Vu-Quoc, L. , 1986, “ A Three-Dimensional Finite-Strain Rod Model—Part II: Computational Aspects,” Comput. Methods Appl. Mech. Eng., 58(1), pp. 79–116. [CrossRef]
Simo, J. C. , and Vu-Quoc, L. , 1991, “ A Geometrically-Exact Rod Model Incorporating Shear and Torsion-Warping Deformation,” Int. J. Solids Struct., 27(3), pp. 371–393. [CrossRef]
Goyal, S. , 2006, “ A Dynamic Rod Model to Simulate Mechanics of Cables and DNA,” Ph.D. thesis, The University of Michigan, Ann Arbor, MI. https://deepblue.lib.umich.edu/handle/2027.42/126037
Arbind, A. , and Reddy, J. N. , 2016, “ Transient Analysis of Cosserat Rod With Inextensibility and Unshearability Constraints Using the Least-Squares Finite Element Model,” Int. J. Non-Linear Mech., 79, pp. 38–47. [CrossRef]
Kumar, A. , and Mukherjee, S. , 2011, “ A Geometrically Exact Rod Model Including in-Plane Cross-Sectional Deformation,” ASME J. Appl. Mech., 78(1), p. 011010. [CrossRef]
Arbind, A. , and Reddy, J. N. , 2017, “ A One-Dimensional Model of 3-d Structure for Large Deformation: A General Higher-Order Rod Theory,” Acta Mech., 229(4), pp. 1803–1831.
Arbind, A. , and Reddy, J. N. , 2018, “Errata: A One-Dimensional Model of 3D Structure for Large Deformation: A General Higher-Order Rod Theory,” Acta Mechanica (accepted).
Arbind, A. , and Reddy, J. N. , 2018, “ A General Higher-Order One-Dimensional Model for Large Deformation Analysis of Solid Bodies,” Comput. Methods Appl. Mech. Eng., 328, pp. 99–121. [CrossRef]
Bishop, R. L. , 1975, “ There is More Than One Way to Frame a Curve,” Am. Math. Mon., 82(3), pp. 246–251. [CrossRef]
Lee, I.-K. , 2000, “ Curve Reconstruction From Unorganized Points,” Comput. Aided Geometric Design, 17(2), pp. 161–177. [CrossRef]
Hanson, A. J. , and Hui, M. , 1995, “ Parallel Transport Approach to Curve Framing,” Indiana University, Indianapolis, IN, Technical Report No. TR425.
O'Neill, B. , 2014, Elementary Differential Geometry, Academic Press, New York.
Bauer, U. , and Polthier, K. , 2009, “ Generating Parametric Models of Tubes From Laser Scans,” Comput.-Aided Des., 41(10), pp. 719–729. [CrossRef]
Harley, F. , 1963, Differential Forms With Applications to the Physical Sciences by Harley Flanders, Vol. 11, Elsevier, New York.

Figures

Grahic Jump Location
Fig. 1

(a) layered tube and (b) functionally graded tube; tubes with radially varying material properties will require a polar coordinate system on the cross section. The approach presented in this paper is ideally suited to modeling such structures with radially varying properties.

Grahic Jump Location
Fig. 2

The approximate geometry of human aorta cannot be modeled with a Serret Frenet frame because of the lack of sufficient smoothness and regularity (i.e., the curvature, κ≠0, ∀ s) of the centerline. We have developed a generalized coordinate system that will allow for the use of interpolated curves for such geometries: (a) reference curve approximation from aorta scan (representative diagram), (b) approximate geometry of aorta, and (c) aortic aneurysm.

Grahic Jump Location
Fig. 3

Various structures and the Frenet and Bishop frames on their reference curves: (a) pipe with composite reference curve, (b) pipe with sinusoidal reference curve, (c) pipe with piece-wise reference curve, (d) Frenet frame for composite reference curve, (e) Frenet frame for sinusoidal curve, (f) Frenet frame for piece-wise space curve, (g) Bishop frame for composite reference curve, (h) Bishop frame for sinusoidal curve, and (i) Bishop frame for piece-wise space curve; Figs. (d)-(f) highlight the difficulties and discontinuities inherent in the Serret-Frenet frame. The Bishop frame overcomes these (see (g)-(i)) but it still faces a problem with general closed curves (see Fig. 4).

Grahic Jump Location
Fig. 4

(a) Closed tube, (b) Bishop frame, and (c) hybrid frame on its reference closed curve. Notice that the Bishop frame has a discontinuity at the point O of closure, however the hybrid frame does not.

Grahic Jump Location
Fig. 5

General curvilinear cylindrical coordinate system. A key point note is that other than for the Bishop-CCC system, the basis vectors êr, êθ, and ês are orthonormal, but the vector ês does not follow the s-coordinate line.

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