This paper extends the discussion of the approximate method of integrating the equations of compressible fluid flow in the hodograph plane first presented by the author before the Sixth International Congress of Applied Mechanics, Paris, France, September, 1948. As an introduction to the discussion of the polygonal approximation method, fundamental fluid-flow equations are reviewed briefly. Determination of the flow function ψ by the “Method of Reflections” is described and an application of the method illustrated. How flow in the physical plane can be determined by superposition of solutions discussed is shown for the simpler incompressible case.

This paper is an abstract of a manuscript submitted by Mr. Bailey at the request of the Plasticity Committee of the Applied Mechanics Division through its chairman, Dr. Nadai. The paper is also an abridgment of Mr. Bailey’s paper read before the Institution of Mechanical Engineers, London, November 22, 1935. The purpose of this abstract is to indicate the wide range of topics treated fully in the original paper and to give a useful and adequate summary of this pioneer work by Mr. Bailey. In many cases, because of lack of space, the bases of the author’s conclusions are not discussed in this abstract nor are the mathematical derivations included. The reader interested in these details should consult the London paper. Footnotes have been added to this abstract mentioning references which are readily available and which will supplement and illuminate this abbreviated treatment. Comprehensive and somewhat different abstracts of the original London paper have also appeared in “Engineering,” November 29 and December 13, 1935, and in “The Engineer,” December 6, 1935.

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).

L. A. Pars (1964, A Treatise on Analytical Dynamics, Heinemann, London) has shown in his comprehensive treatise that acceleration dependent forces are not admissible in Newtonian mechanics. More recently, Zhechev (2007, On the Admissibility of Given Acceleration – Dependent Forces in Mechanics, Jr. of Applied Mechanics, ASME, 74 , Jan, pp. 107–111; 2007, Peculiarities of the use of Acceleration – Dependent Forces in Mechanical Problems, Proc. I. Mech. E, 221 Part K, pp. 497–503) has shown that the proof given by Pars is faulty and has concluded that acceleration dependent forces are admissible in Newtonian mechanics and in many cases such forces are useful in controlling mechanical systems. This brief technical note attempts to show that the matter is more complex and needs further discussion.