Objectives:

• A curve framing system for a general open/closed space curve via hybrid frame is put forth, and then, based on this frame, a novel hybrid frame-based curvilinear cylindrical coordinate (CCC) system is introduced to map complex geometries of any arbitrarily curved pipe/rod. Similar Bishop- and Frenet-CCC system can be obtained as a special case of the hybrid-CCC system.

• A general higher-order theory for curved tube-like structures is proposed using hybrid-CCC system to analyze general wall deformation of the tubes in contrast to the classical Cosserat rod theories. The constitutive relation of hyperelastic material is considered.

• Furthermore, the weak-form finite element model for the proposed higher-order tube theory is developed and illustrated through various interesting numerical examples.

Overview and motivation:

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 center-line motion is of primary focus, here we consider cases were the lateral boundaries also undergo significant deformation. To tackle these problems, a generalized curvilinear cylindrical coordinate 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.

The commonly used Frenet frame is continuous only for C3 continuous curve. This geometric limitation restricts the use of Cosserat rod theories as well. To overcome this limitation, we adopted a very general method of framing a space curve, namely, the hybrid frame, in its primitive form, from the computer graphic applications, and advance it further for the use in the applied mechanics. The hybrid frame can also continuously frame any general closed curve which Bishop frame fails to do. The hybrid frame can represent the space curves with continuous tangent continuously, which could be out of the preview of the commonly used Frenet frame. Based on this general curve-framing, we have proposed a novel and more complete curvilinear cylindrical coordinate system to analyze circular or nearly circular curved pipes.

The natural covariant frame for the proposed hybrid curvilinear cylindrical coordinate system is not orthonormal, which could make the formulation of the rod theory very inefficient, and hence, its finite element model computationally very expensive. To tackle this problem, we have derived the kinematics of the rod- or pipe-like structures by using a Cartan's (orthonormal) moving frame using tools of exterior calculus. The use of the orthonormal moving frame makes the derivation of the governing equations, and it's finite element model very efficient as compared to the commonly used natural covariant coordinate system.

Further, again based on a very general approximation of deformation of the cross-section of the tube, we have developed the finite element model for the hyperelastic constitutive relation for compressible and incompressible material in which the incompressibility constraint has been applied by two different methods, namely the Lagrange multiplier method and the penalty method. Moreover, the use of Fourier bases in the displacement approximation makes the Gauss integration process very expensive for cross-sectional integration in FEM as large number of Gauss points would be required for accurate numerical integration. To overcome this problem, we have derived a Gauss like integration scheme for Fourier bases, which further reduces the computational time of the analysis.

A finite element computer code is developed to do such analyses utilizing nonlinear solving techniques such as arc-length method and Newton's method. we have demonstrated the application of the theory via various numerical examples of thick and thin walled tubes and compared the results with an exact solution (whenever available) or solution from the commercially available code for validation using shell or 3-D theories. The method is proved to very efficient as compared with 2-D shell finite elements or 3-D finite elements, especially in the case of radially symmetry problems.

Cylindrical tube with varying material properties under internal pressure:

Closed tube under internal pressure:

1. Arbind, A., J. N. Reddy, and A. R. Srinivasa. "A nonlinear 1-D finite element analysis of rods/tubes made of incompressible neo-Hookean materials using higher-order theory." International Journal of Solids and Structures 166 (2019): 1-21. 

2. Arbind, A., A. R. Srinivasa, and J. N. Reddy. "A higher-order theory for open and closed curved rods and tubes using a novel curvilinear cylindrical coordinate system." Journal of Applied Mechanics 85, no. 9 (2018): 091006. 

3. Arbind, A., and J. N. Reddy. "A one-dimensional model of 3-D structure for large deformation: a general higher-order rod theory." Acta Mechanica 229, no. 4 (2018): 1803-1831. 

4. Arbind, A., and J. N. Reddy. "Correction to: A one-dimensional model of 3-D structure for large deformation: a general higher-order rod theory." Acta Mechanica 229, no. 10 (2018): 4313-4317.