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

2018 ◽  
Vol 85 (9) ◽  
Author(s):  
A. Arbind ◽  
A. R. Srinivasa ◽  
J. N. Reddy
Keyword(s):  
Coordinate System ◽  
Cross Section ◽  
3D Structure ◽  
Order Theory ◽  
Rod Theory ◽  
Exterior Calculus ◽  
Cylindrical Coordinate ◽  
The Cross ◽  
General Material ◽  
Higher Order Theory

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).

Computational Investigation on the Dispersive Transport of Angiographic Contrast Injected Antegradely Into Flow in a Tube

2007 ◽  
Author(s):  
Qing Hao ◽  
Baruch B. Lieber
Keyword(s):  
Coordinate System ◽  
Cross Section ◽  
Single Molecule ◽  
Molecular Diffusion ◽  
Local Concentration ◽  
Dispersive Transport ◽  
Cross Sectional ◽  
Advection Dispersion ◽  
The Cross ◽  
The Difference

When a solute such as angiographic contrast is introduced into a solvent such as blood analog fluid flowing in a straight circular tube, it spreads under the combined action of molecule diffusion and the variation of velocity over the cross-section [8]. If two molecules are being carried in the flow, for example, one in the center and one near the wall, the rate of separation caused by the difference in bulk velocity will greatly exceed that caused by molecule motion. Given enough time, any single molecule would wander randomly throughout the cross section of the pipe because of molecular diffusion, and would sample at random all the advective velocities [4]. Therefore, Taylor [8] adopted the Lagrangian approach to the problem, casting the equations in a coordinate system that moves with the average velocity of the flow and replacing the molecular diffusion coefficient with a dispersion coefficient, and the local concentration with the cross sectional mean concentration. Recasting Taylor’s equation in an inertial coordinate system one obtained the so called advection-dispersion equation.

More on definable sets of p-adic numbers

1988 ◽  
Vol 53 (3) ◽  
pp. 912-920 ◽  
Author(s):  
Philip Scowcroft
Keyword(s):  
Cross Section ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Model Completeness ◽  
First Order ◽  
First Order Theory ◽  
Ordered Fields ◽  
Definable Sets ◽  
The Cross ◽  
Primitive Recursive

To eliminate quantifiers in the first-order theory of the p-adic field Qp, Ax and Kochen use a language containing a symbol for a cross-section map n → pn from the value group Z into Qp [1, pp. 48–49]. The primitive-recursive quantifier eliminations given by Cohen [2] and Weispfenning [10] also apply to a language mentioning the cross-section, but none of these authors seems entirely happy with his results. As Cohen says, “all the operations… introduced for our simple functions seem natural, with the possible exception of the map n → pn” [2, p. 146]. So all three authors show that various consequences of quantifier elimination—completeness, decidability, model-completeness—also hold for a theory of Qp not employing the cross-section [1, p. 453; 2, p. 146; 10, §4]. Macintyre directs a more specific complaint against the cross-section [5, p. 605]. Elementary formulae which use it can define infinite discrete subsets of Qp; yet infinite discrete subsets of R are not definable in the language of ordered fields, and so certain analogies between Qp and R suggested by previous model-theoretic work seem to break down.To avoid this problem, Macintyre gives up the cross-section and eliminates quantifiers in a theory of Qp written just in the usual language of fields supplemented by a predicate V for Qp's valuation ring and by predicates Pn for the sets of nth powers in Qp (for all n ≥ 2).

An Inverse Design Method of Buckling-Guided Assembly for Ribbon-Type 3D Structures

2019 ◽  
Vol 87 (3) ◽  
Author(s):  
Zheng Xu ◽  
Zhichao Fan ◽  
Yanyang Zi ◽  
Yihui Zhang ◽  
Yonggang Huang
Keyword(s):  
Cross Section ◽  
3D Structure ◽  
Experimental Studies ◽  
Functional Materials ◽  
Optimization Method ◽  
Inverse Design ◽  
Torsional Angle ◽  
3D Structures ◽  
Practical Applications ◽  
The Cross

Abstract Mechanically guided three-dimensional (3D) assembly based on the controlled buckling of pre-designed 2D thin-film precursors provides deterministic routes to complex 3D mesostructures in diverse functional materials, with access to a broad range of material types and length scales. Existing mechanics studies on this topic mainly focus on the forward problem that aims at predicting the configurations of assembled 3D structures, especially ribbon-shaped structures, given the configuration of initial 2D precursor and loading magnitude. The inverse design problem that maps the target 3D structure onto an unknown 2D precursor in the context of a prescribed loading method is essential for practical applications, but remains a challenge. This paper proposes a systematic optimization method to solve the inverse design of ribbon-type 3D geometries assembled through the buckling-guided approach. In addition to the torsional angle of the cross section, this method introduces the non-uniform width distribution of the initial ribbon structure and the loading mode as additional design variables, which can significantly enhance the optimization accuracy for reproducing the desired 3D centroid line of the target ribbon. Extension of this method allows the inverse design of entire 3D ribbon configurations with specific geometries, taking into account both the centroid line and the torsion for the cross section. Computational and experimental studies over a variety of elaborate examples, encompassing both the single-ribbon and ribbon-framework structures, demonstrate the effectiveness and applicability of the developed method.

Research on Synthesized Modular Modeling Based on General Mathematic Model of Arbitrary Rotary Helical Surface

2013 ◽  
Vol 579-580 ◽  
pp. 283-288
Author(s):  
Li Zhi Gu ◽  
Zong Zhi Zhou
Keyword(s):  
Mathematical Model ◽  
Coordinate System ◽  
Cross Section ◽  
Mathematic Model ◽  
Characteristic Line ◽  
Surface Model ◽  
Helical Surface ◽  
Modular Modeling ◽  
General Mathematical Model ◽  
The Cross

Based on the analysis of the helical surface movement forming principle, adopted synthesized modular modeling ideas, realized the construction of a complete helical surface mathematical model consisting of three core elements, element of the general mathematical model of rotator generatrix profile, element of the general mathematical model of characteristic line and element of the general mathematical model of transformation between the cross-section coordinate system and workpiece coordinate system. Generatrix of the linear-typed contour, the arc-typed contour, the higher order equation-typed contour, and the discrete point-typed contour was investigated systematically for the model. With the synthesized modular model, if the specific profile of a helical Component is given on the cross-section, it is of convenience and ease to obtain the concrete parametric model of the helical surface, simply by changing the parameter settings through the interface to acquire any types of helical surface model, thereby increasing the efficiency of modeling and providing technical support for the precise model of the helical surface with parameterization.

scholarly journals Influence of longitudinal force and internal pressure on the frequency of free vibrations of an underground oil pipeline

2020 ◽  
Vol 217 ◽  
pp. 01010
Author(s):  
Andrey Dmitriev ◽  
Vladimir Sokolov ◽  
Aleksey Bereznyov
Keyword(s):  
Cross Section ◽  
Large Diameter ◽  
Normal Component ◽  
Longitudinal Force ◽  
Free Vibrations ◽  
Oil Pipeline ◽  
Rod Theory ◽  
The Cross ◽  
Pipeline Wall ◽  
Inertia Forces

This paper is based on the equation obtained earlier by V.G. Sokolov to find the frequencies of natural vibrations of straight sections of large-diameter pipelines. In this work, to take into account the effect of hydrostatic pressure on the pipeline wall from oil flowing at different speeds, the solution obtained by M.A. Ilgamov and A.S. Volmyr is used. At the same time, the effect of a stationary fluid flow on the pipeline wall is taken into account in the equation written in forces for the last term of the normal component of inertia forces. The resulting modified equation allows determining the frequency characteristics of the pipeline both according to the rod theory (without taking into account the deformation of the cross section) and according to the theory of shells (taking into account the deformation of the cross section).

A New Approach to the Demonstration of Oxidase-Localization Using Tannic Acid Or Catechin

1973 ◽  
Vol 31 ◽  
pp. 698-699
Author(s):  
V. Mizuhira ◽  
Y. Futaesaku
Keyword(s):  
Cross Section ◽  
Tannic Acid ◽  
Elastic Fiber ◽  
The Other ◽  
New Approach ◽  
Tissue Block ◽  
Active Reaction ◽  
The Cross

Previously we reported that tannic acid is a very effective fixative for proteins including polypeptides. Especially, in the cross section of microtubules, thirteen submits in A-tubule and eleven in B-tubule could be observed very clearly. An elastic fiber could be demonstrated very clearly, as an electron opaque, homogeneous fiber. However, tannic acid did not penetrate into the deep portion of the tissue-block. So we tried Catechin. This shows almost the same chemical natures as that of proteins, as tannic acid. Moreover, we thought that catechin should have two active-reaction sites, one is phenol,and the other is catechole. Catechole site should react with osmium, to make Os- black. Phenol-site should react with peroxidase existing perhydroxide.

Energy Distribution in Electron Beams

1972 ◽  
Vol 30 ◽  
pp. 568-569
Author(s):  
Tamotsu Ohno
Keyword(s):  
Electron Beam ◽  
Energy Distribution ◽  
Cross Section ◽  
Integral Curve ◽  
Electron Beams ◽  
Electron Gun ◽  
Angular Divergence ◽  
Apparent Increase ◽  
Integral Width ◽  
The Cross

The energy distribution in an electron; beam from an electron gun provided with a biased Wehnelt cylinder was measured by a retarding potential analyser. All the measurements were carried out with a beam of small angular divergence (<3xl0-4 rad) to eliminate the apparent increase of energy width as pointed out by Ichinokawa.The cross section of the beam from a gun with a tungsten hairpin cathode varies as shown in Fig.1a with the bias voltage Vg. The central part of the beam was analysed. An example of the integral curve as well as the energy spectrum is shown in Fig.2. The integral width of the spectrum ΔEi varies with Vg as shown in Fig.1b The width ΔEi is smaller than the Maxwellian width near the cut-off. As |Vg| is decreased, ΔEi increases beyond the Maxwellian width, reaches a maximum and then decreases. Note that the cross section of the beam enlarges with decreasing |Vg|.

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