Pure torsion problem in tensor notation

[1] EN 206-1:2013, Concrete – Part 1: Specification, performance, production and conformity. [2] Holicky, M., Vorlicek. M., Fractile estimation and sampling inspection in building, Acta Polytechnica, CVUT Praha, 1992.Vol. 32. Issue 1, 87-96. [3] Sual I. Gass, and Arjang A. Assad, An annotated timeline of operation research. An informal history. Springer Science and Business Media, 2005. [4] Harrison. T.A., Crompton, S., Eastwood. C., Richardson. G., Sym, R., Guidance on the application of the EN 206-1 conformity rules, Quarry Products Association, 2001. [5] Czarnecki, L. et al., Concrete according to PN EN 206-1The comment Collective work, Polish Cement and PKN, Cracow, Poland, 2004. [6] Lynch, B., Kelly, J., McGrath, M., Murphy, B., Newell, J., The new Concrete Standards, An Introduction to EN 206-1, The Irish Concrete Society, 2004. [7] Montgomery, D.C., Introduction to Statistical Quality Control, 5th Ed., Wiley, 2005. [8] Taerwe, L., Evaluation of compound compliance criteria for concrete strength, Materials and Structures, 1998, 21, pp. 13-20. [9] Catarino, J.M.R., Statistical criteria for acceptance of materials performance of concrete standards ENV 206;1993 and prEN206:1997. Proceedings of 12th ERMCO Congress, Vol. 1, Lisbon, 1998. [10] Brunarski, L., The mathematical basis for shaping the strength of materials conformity criteria, Scientific Work of the Institute of Building Technology, Publishing of the Institute of Building Technology, Warsaw 2009 (in Polish). [11] Szczygielska, E., Tur, V., The study of the conformity criterion for compressive strength of concrete based on order statistics, Civil Engineering and Architecture, 2013, Vol. 12, Issue 3, 223-230. [12] Taerwe, L., Caspeele, R., Conformity control of concrete: some basic aspects, Proceedings of 4-th International Probabilistic Symposium, Ghent, Belgium, 2006, 57-70. [13] Skrzypczak, I., Analysis of criteria of quality assessment for concrete and their influence on risk of the producer and the recipient, The Publisher of the Rzeszow University of Technology,, Rzeszow, Poland, 2013 (in Polish). [14] Evans, J., Lindsay, W., The Management and Control of Quality, West Publishing Company, 1989. [15] Banovac, E., Pavlovic, D., Vistica N., Mathematical aspects of acceptance sampling procedure. International Journal of Mathematical Models and Methods in Applied Sciences, 2012, Issue 5, Vol. 6, 625-633. [16] Dumicic, K., Bahovec, V., Kurnoga Zivadinovic, N., Studying an OC Curve of an Acceptance Sampling Plan: A Statistical Quality Control Tool, Proceedings of the 7th WSEAS International Conference on Mathematics & Computers in Business & Economics, Cavtat, Croatia, 2006, 1-6. Budownictwo i Architektura 18(1) (2019) 57-69 DOI: 10.24358/Bud-Arch_19_181_06


Introduction
The tensor notation used here is in accordance with the notation applied in the publications [1][2]. The description is known as index notation, although there are some variants to it. Prof. Jan Rychlewski consequently used the dyadic notation during his lectures on continuum mechanics at the Institute of Fundamental Technical Problems in Warsaw at the end of 1970s. In the monograph [3], vectors and tensors are written as boldface letters without indices. In this case, an additional notation with regard to components is necessary.
There are numerous monographs where tensors are used, albeit without the application of a uniform and consistent method. Let us recall here Sokolikoff's repeatedly reprinted monograph [4], for instance.
The Einstein summation convention (1916) concerns the geometrical/physical objects with upper and lower indices where there are various pairs of identical indices and where one is upper and the other one -lower. This implies summation over the paired indices over the whole range of their values.
In Fig. 1.a-b, the same vector u  is drawn as an oriented segment in two different dimensional coordinate systems. Although the base vectors are singular, they are different. In Fig. 1.a and Fig.1.b, the coordinate lines are the same, however, the projections of the u  vector differ significantly. In the case of Fig. 1.a, the parallel projection takes place, and in the case of Fig. 1  In Fig 1., the transformation of skew coordinates into a Cartesian system is also suggested. Such transformation reveals that in the case of the Cartesian coordinate system there is no distinction between the contravariant and covariant bases and, as a consequence, the summation convention plays only the role of an "alternator".
The skew coordinate systems were investigated by Kaliski, [5].
The metric tensor includes the information on the coordinate system and is obtained by the scalar product of base vectors n m mn Additionally, n m n m g   . (iii) Where mn g is the metric tensor of an arbitrary coordinate system with the base vectors m g . n m  is a metric tensor for a Cartesian coordinate system, also known as Kronecker's delta. Here, the m, n indices can run over 1, 2, 3, for instance. The metric tensor allows to increase or decrease an index which leads to associated tensors in the following way mk nk The u  vector coefficients can be found as a scalar product of the u and base vectors, according to the following formula: In the linear theory of elasticity there are the following groups of problems: -deformation, strain tensor n m  , -constitutive relations, stress tensor j i  , -equilibrium equation, -boundary conditions or initial-boundary conditions, -compatibility relation. This group set will be used below. The pure torsion problem of a bar belongs to the classical approaches in mechanics. It can be found in [4], [6][7]. One of the latest publications on the theory of elasticity is a monograph [8]. In the field of the continuum mechanics, the book [10] can be worth to mention.

Coulomb's theory
We assume: I) the summation convention and tensor calculus in three-dimensional space will be implemented, II) the theory of small deformations and the principle of stiffness apply where i  stands for covariant derivative and in Cartesian coordinate system becomes to

III) material is isotropic and homogenous, Hooke's law is valid -
where 1 J is the first invariant of the strain tensor, IV) the pure torsion of straight rod of a constant circular section is considered, (Fig.2.), in Cartesian coordinate system we have are the principal axes in the sense of inertial moments, VI) the Coulomb's assumption is assumed -in case of pure torsion the cross-sections of the rod turn each other like infinitely rigid discs,

Fig.2. Cartesian coordinate system, internal force vectors and stress vectors
Pure torsion problem in tensor notation

VII) the Cauchy's relation is valid
the stress vector t  is related to stress tensor ij  by cutting the body (point) with plane which is oriented by outward normal vector n  at any chosen body point; in detail VIII) statics is analysed.

Deformation -displacements -geometric relations
Lets repeat again -in case of Cartesian coordinates, for tensors of valence one or two, there are no distinction between covariant, contravariant or shifted indices - where m =1, 2, 3, (three dimensions). In initial state, for an arbitrary point P of a rod cross-section, in polar description, its position is uniquely defined by position vector P   . After torsion, in the actual configuration, the point P rotates to the position P' and is depicted be vector ' P   . In analysed case, the difference between actual and initial configurations (P' and P) depicts the deformation, (Fig. 3.). Assumption VI) implies that it results the only two (of three) non zero displacement vector components

Sławomir Karaś
and a circular trajectory of movement Where: Additionally the polar coordinate system is introduced. By virtue of II) assumption, the rotational angle of a disc ) this implies as follows On the basis of (8) the Cartesian coordinates of P i P' can be written in the form (14.2) Using (11)

Strain tensor components
Having known displacement vector (7), (15) we can use (1) to determine the components of strain tensor in Cartesian coordinate system Pure torsion problem in tensor notation adequately we obtain 0 , 1 In (17) the above notation is introduced The strain tensor can be presented in the matrix form

Constitutive relation -stresses
By means of (2) which is valid for isotropic and homogenous material we can find the set of stress tensor components. Applying (19) we get Where G is the Kirchhoff modulus. The non zeros components are in only two cases 2 13 x G    and 1 23 x G   . (21) In matrix notation one arrives to

Equilibrium state
In the general case, the state of equilibrium expresses two conditions of resetting the resultant vectors of forces and moments acting on/in the body respectively. In the problem of pure torsion, the symmetry of stress tensor (21.1) leads to the first condition of the form The rod cross-section is characterized by normal vector n  which Cartesian components ( Fig. 1.) are reduced as follows Using (21) we can write the components of a stress vector (4) in the cross-section On the basis of assumption IV) and appropriately (19), (22) do (24) we obtain .
x const const -is torsion stiffness against rotation. Here o J is a centrifugal moment of inertia. Now, applying (15), (17), (21), (26) and (26.1), sequentially components of displacement vector as well as strain and stress tensors can be rewritten as follows Let us find extremes of shearing stresses in the rod cross-section. Now, the stress vector is written as a sum of normal and shearing stresses, see (21), (24) and Fig. 2, then we arrive at (30) The modulus of   can be calculated as . (31) Let us analyse vectors   and  or in other way (Fig. 4.) - . extr  occurs at the cross-section circuit, and its vector is tangential to the circuit.

Navier's error
Without success, the Coulomb's theory for cross-sections of any shape was used by Claude-Louis Navier. We will show that assumption VI) is valid only for bars with a circular cross-section.
Consider the equilibrium in the case of pure torsion. The edge surface is free of any load. On this cross-section border surface we select an arbitrary point B, Fig. 5.
The boundary surface is free of external loads, the stress vector is a zero one The conditions (36) correspond to three equations of equilibrium, which are explicit in form Equation (38) is equivalent to a condition Pure torsion problem in tensor notation geometric expression of solutions is a family of circles.
The problem of determining the function of deplanation , which is harmonic in open area of the cross-section, and on the edge of this area the function  fulfils the condition for a normal derivative (50) is a problem of the potential theory often referred to as the second boundary problem of Neumann.
The problem of beam bending with shearing in tensor notation was derived in the paper [10]. The review of the deformation measures -strain tensors -was carried out in [11].

Conclusions
Any, even a simple mechanical problem, can be consistently analysed by means of tensor notation. Nevertheless, the reader should be familiar with the fundamentals of the theory of elasticity in tensor calculus. This necessitates additional classes/lectures for students, or self-study in tensor calculus.
The problem of pure torsion was selected due to the clarity of the assumptions and a relatively small scope of the issue.
The problem discussed in the paper was presented during the Math-Bridge Camp, Muenster 2018. The discussion pointed out that for people not familiar with the tensor calculus, the problem is not clear. Majority of workshop attendees preferred the classic approach to the problem of torsion. Such conservatism is characteristic of academic teachers, while to students the tabula rasa principle applies, i.e. they learn tensor calculus without prejudice.
For over 15 years, in the 1970s and 1980s, the team of the Stereomechanics Department of Lublin University of Technology, headed by prof. Jerzy Grycz taught classes in the strength of materials as well as the theory of elasticity and plasticity in tensor calculus. The author was a member of this team, and the classic torsion problem posted was one of several issues developed by the author.