Engineering Transactions, 70, 2, pp. 183-199, 2022

A Proposal of a Novel Geometrical Measure of Material Effort

Silesian University of Technology

A stress tensor may be presented as a surface delimited by a stress vector located at angles α1, α2, α3 in relation to axes x, y, z. Geometrically, it outlines a domain and is linked to the loading. In this study, the area of such a surface and the volume of the domain were determined, along with their cross-sections with reference to areas and circumferences. Different stresses were also compared. This article presents cases of uniaxial, biaxial, triaxial tension and pure shear for an isotropic solid body. The analysis of a stress tensor in this conceptual work does not involve any material features, yet yields interesting results, particularly in the case of pure shear and uniaxial tension.
Keywords: stress tensor; geometrical representation; material effort hypotheses
Full Text: PDF
Copyright © The Author(s). This is an open-access article distributed under the terms of the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0).


Derski W., Introduction to Continuum Mechanics [in Polish: Zarys mechaniki ośrodków ciągłych], PWN, Warszawa, 1975.

Chaves E.W.V, Notes on Continuum Mechanics, Springer Science+Business Media, 2013.

Jakowluk A., Breczko T., Continuum Mechanics [in Polish: Mechanika ośrodków ciągłych], Wydawnictwo Politechniki Białostockiej, Białystok, 1985.

Jastrzębski P., Mutermilch J., Orłowski W., Strength of Materials [in Polish: Wytrzymałość materiałów], Arkady, Warszawa, 1986.

Huber T., Specific work of strain as a measure of material effort (translated from the original paper in Polish), Archives of Mechanics, 56(3): 173–190, 2004.

Sendeckyj G.P., Empirical strength theories, [in:] Testing for Prediction of Material Performance in Structures and Components, R.S. Shane [Ed.], ASTM STP 515, American Society for Testing and Materials, pp. 171–179, 1972.

Burzyński W., Theoretical foundations of the hypotheses of material effort, Engineering Transactions, 56(3): 269–305, 2008 [Translated from the original paper in Polish: Teoretyczne podstawy hipotez wytężenia, Czasopismo Techniczne, 47: 1–41, 1929].

Hill R., A theory of the yielding and plastic flow of anisotropic metals, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 193(1033): 281–297, 1948, 10.1098/rspa.1948.0045.

Frąś T., Pęcherski R.B., Applications of Burzyński hypothesis of material effort for isotropic solids, Mechanics and Control, 29(2): 45–50, 2010.

Pęcherski R.B., Szeptyński P., Nowak M., An extension of Burzyński hypothesis of material effort accounting for the third invariant of stress tensor, Archives of Metallurgy and Materials, 56(2): 503–508, 2011, doi: 10.2478/v10172-011-0054-4.

Oller S., Car E., Lubliner J., Definition of a general implicit orthotropic yield crite- rion, Computer Methods in Applied Mechanics and Engineering, 192(7–8): 895–912, 2003, 10.1016/S0045-7825(02)00605-9.

Pęcherski R.B., Rusinek A., Fras T., Nowak M., Nowak Z., Energy-based yield con- dition for orthotropic materials exhibiting asymmetry of elastic range, Archives of Metal- lurgy and Materials, 65(2): 771–778, 2020, doi: 10.24425/amm.2020.132819.

Kostowski E., Heat Transfer [in Polish: Przepływ ciepła], Wydawnictwo Politechniki Śląskiej, Gliwice, 1991.

Madejski J., Theory of Heat Transfer [in Polish: Teoria wymiany ciepła], Wydawnictwo Uczelniane Politechniki Szczecińskiej, Szczecin, 1998.

Oberhettinger F., Magnus W., Implementation of Elliptic Functions in Physics and Technique [in Polish: Zastosowania funkcji eliptycznych w fizyce i technice], PWN, Warszawa, 1963.

DOI: 10.24423/EngTrans.1311.20220602