Education Journal, Vol. 2, Issue 2, Jun  2019, Pages 29-35; DOI: 10.31058/j.edu.2019.22002 10.31058/j.edu.2019.22002

A Straightforward Program for Formulating General Solid Mechanics Equations in Non Cartesian Geometries for Undergraduate Engineering Students

, Vol. 2, Issue 2, Jun  2019, Pages 29-35.

DOI: 10.31058/j.edu.2019.22002

Hector Ariel Di Rado 1 , Pablo Alejandro Beneyto 1*

1 Applied Mechanics Department, Faculty of Engineering, National Northeast University, Resistencia, Argentina

Received: 30 August 2019; Accepted: 20 September 2019; Published: 4 October 2019

Abstract

Whilst in physics science the use of convected coordinates, non Cartesian geometry or any other geometrical tool is widespread, in undergraduate engineering teaching of continuum mechanics usually a different path is underwent. Even if geometry generalization lie beneath the core of a more closely description of the actual distribution of strain, stress, and displacement, a concise and precise treatment of geometry evolution is frequently overruled in engineering undergraduate courses. The main scope of the present short communication is to provide an explanation to the aforementioned behavior and to put forward different situations in which the main drawbacks induced by the underlain mathematical tools involved in are outbalanced by an all encompassing approach in the development of the main field equation of solid mechanics. Hereinafter, main guidelines for introducing the most relevant equations of solid mechanics in curved space are presented as a natural and easy-to-comprehend extension of Cartesian coordinates. Specifically, a straightforward deduction of differential equilibrium equation in Plane Polar as well as Spherical Polar coordinates is carried out. Whether Euclidean or not, the whole problem boils down to design a simple program carrying out the necessary change of frame, rendering this program a teaching tool that worthwhile to be included in undergraduates courses.

Keywords

Engineering, Education, Geometry, Mathematics, Continuum Mechanics, Non-Euclidean, Convected Coordinates

1. Introduction

When an undergraduate engineering student face for the first time critical design constraints such as minimum weight, cost assessment, or mainly geometrical constrains, a more exact treatment of theory of structures encourages the migration from beam theory to elasticity theory [1]. Amongst all the abovementioned factors, the geometry issue is overwhelmingly the most relevant and, indeed, the convected coordinates lie beneath its core [2,3].

The implementation of convected coordinates in the formulation of main solid mechanics equations is a very old issue [10]. In fact, many of the most important text from circa the middle of past century include complete chapters with a detailed description of the topic [4,5,6]. Yet, the formalism wherein all mathematical artillery is addressed stands for a natural barrier for an undergraduate engineering student even if the whole text is devoted to them.

Whereas a concise and precise treatment of geometry in undergraduate programs is frequently underestimated [7] and possibly the underlain justification is the mathematical artillery involved in, throughout the present work a brief and at the same time accessible to non mathematicians description of the basic mathematical tools involved in convected formulation of main solid mechanic equations is enforced and specifically differential equilibrium equations for non Cartesian coordinates are written out. Furthermore, it will be put forward the advantage of presenting a general method for coordinate change of differential equations of motion, e.g. to polar coordinates, when it is checked against the framework using a different elemental volume according to the target coordinate system. Due to some argument may be furthered about the requirement that an engineer during his instruction must be able to manipulate the figure representation of the various efforts acting on elemental bodies. [8], the last part of the present paper will be devoted to contest this kind of argument.

2. Materials and Methods

2.1. Basic Mathematical Background

The following mathematical tools must be furnished to undergraduate engineering student if a solid mathematical background that could account for a thorough and in depth treatment of solid mechanics, is expected.

One main concept to deal with is a clear however broad definition of tensors in orthogonal and non orthogonal coordinates. Furthermore, when non orthogonal coordinates are involved, the notion of covariance pops up and covariant reference for vectors as well as covectors may be brought up. From author's standpoint, the notion of covector should be more desirable base on the fact that the consistency of the various scalar products, arisen in energy or virtual work equation, is preserved when the abovementioned non orthogonal transformations are intended. Besides, the student faces an important fact of reality:entities divide by distance are not equal to those involving products of distance. The former are covectors meanwhile the latter are vectors.

Along with vectors and covectors, the notion of metric tensor is unavoidable. Far from being a drawback, the metric tensor provides a simple arithmetization of the space geometric characteristics. In fact, by simple inspection, the metric tensor provides vital information about the space basis, whether vector or covector (i.e., if they are constants, unitary, etc.). It is of utmost importance for an engineering student recognizing that, e. g., the polar coordinates broadly used in solving axisymmetric problems, stand for a non constant system of reference wherein any derivative must involve vector or covector basis which in turn bring into consideration another important concept:covariant derivative and Christoffel symbols. This kind of derivative is frequently not included in basic calculus course though it is a natural concept not demanding mayor background (in contrast, for example, to Lie Derivative) and for this reason it inclusion is not whatsoever too strong a requirement.

Summarizing, the here suggested program should include the following minimum background:

Vectors, vector space.

Covectors, covector space.

Generalized tensor space

Metric tensor.

Covariant derivative and Christoffel symbols.

From Authors’ standpoint, taking for granted the previous list, a complete formulation of solid mechanics with focus on geometry concepts may be taken on board without mayor efforts on the undergraduate students’part.

3. Results and Discussion

Two examples will be given in order to prove the power of focusing in geometry concepts when it comes to deriving some of the main field equation of continuum mechanics [9].

3.1. Differential Equilibrium Equations in Plane Polar Coordinates

The derivation of differential equations in plane polar coordinates may be carried out mainly in three manners, (a) Evaluating the equilibrium of forces acting in an element according to Figure 1:

Figure 1. Polar differential object [11].

Where symbols r, θ and σ, stand for Polar coordinates and stress respectively.

(b) By substitution of the relationships between Cartesian coordinates and Polar coordinates in the differential equations in Cartesian coordinates, i.e.

         (1)

Where symbols x and y, stand for Cartesian coordinates respectively.

(c) By formulating the differential equation regarding covariant derivative of stress tensor

         (2)

Being, T, stress tensor and, the curvilinear reference vector. Taking the divergence of the stress tensor regarding the covector character of the gradient operator:

         (3)

In the former, stand for Christoffel Symbols of the second kind. These coordinates depend on the space metric tensor. Equation (3), although seemingly awkward, encompasses any space by simply reckoning the corresponding Christoffel symbols according to the metric. Being the case in point planar Polar coordinates, the symbols are

         (4)

Substituting equitation (4) on equation (3), gives:

         (5)

The former are the differential equilibrium equitation for Planar Polar coordinates with mass force ().

3.2. Differential Equilibrium Equations in Spherical Polar Coordinates

Carrying out equilibrium equations derivation in another geometrical space is a straightforward task whether method (c) of the previous Paragraph is considered.

For Spherical coordinates, the Christoffel symbols are:

         (6)

Again, substitution of equations (6) on equations (3), gives the pursuit system of equations:

         (7)

In the former, ϕ is the third spherical coordinate.

3.3. General Non Euclidean Coordinates

General coordinates equilibrium equations may be straightforward carried out redefining the group of equations (4) or (6), namely, simply calculating the Christoffel Symbols for the specific targeted geometrical space.

3.4. Implementation

The foregoing program for teaching solid mechanics is being implemented in Northeast National University- UNNE - Argentine for the last five years during the solid mechanic course imparted in the sixth semester of the civil engineering career. Lectures included the essential mathematic foundation as well as the aforecited basic guidelines for assessing strains from general geometry standpoint [9]. Some of the drawbacks arisen during the first year, i.e. the necessary minimum mathematical background, the selection of the specific exercises, etc., have been properly overcome. Furthermore, it is utterly suggested the inclusion of the mathematical tools mentioned in Paragraph [2] in solid mechanics contents rather than in calculus courses for its imminent use eases student comprehension.

Through final examination in the last two years students have been showed up very good outcomes that allow instructors bearing out the improvements either managing frame interchange or grasping the fundamentals of non standard geometries. Furthermore, this methodology paves the way specifically to further courses devoted to computational mechanics since the hallmark of programming is the capacity to standardizing processes, something certainly achieved here.

3.5. Discussion

A claim was pointed out in Paragraph 1, concerning the fact that the kind of representation like Figure 1 as starting point for the derivation of differential equations would be preferred to the derivation using curvilinear coordinates. And this argument was endured presumably based on the mathematical difficulties that the latter method brought about as well as certain degree of shade that the lack of element representation cast over the problem. Whereas this last comment may be attended by simply adding the solid element shape to geometric-based derivation as a complementary tool [9], the former should be rejected because any minimum mathematical background added to is certainly balanced with the prospect of encompassing general spaces as well as making viable its computational implementation.

4. Conclusions

An alternative program for teaching solid mechanics to engineering undergraduate students was presented as long as a minimum recommended content for solid mechanics lectures. This alternative deems the mathematical description of space geometry a cornerstone instrument in engineering education for it endows professors with a formidable, all encompassing and modern tool that furnishes a driving force capable of leading students through all conventional and non conventional solid shapes as well as leading the access to a better understanding of computational mechanics. The allegedly increase in mathematical costs should be ruled out when it is contrasted with students capabilities scaled up.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this article.

Copyright

© 2017 by the authors. Licensee International Technology and Science Press Limited. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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