Engineering Transactions, 70, 3, pp. 239–286, 2022
10.24423/EngTrans.2210.20220809

Parametrization of Cauchy Stress Tensor Treated as Autonomous Object Using Isotropy Angle and Skewness Angle

Andrzej Grzegorz ZIÓŁKOWSKI
Institute of Fundamental Technological Research Polish Academy of Sciences
Poland

Intrinsic features (eigenproperties) of the Cauchy stress tensor are discussed. Novelty notions of isotropy and skewness mode angles are introduced for the improved parametric description of spherical (isotropic) and deviatoric (anisotropic) components of stress tensor. The skewness angle is defined with pure shear employed as a comparison reference mode upon observing that pure shear states can be interpreted as elementary (atomic) blocks of any macroscopic deviatoric stress state. An original statistical-physical interpretation of the stress tensor orthogonal invariants is provided. A micromechanical explanation for observed decrease of the stress tensor anisotropy factor values, measured in terms of the tensor orbit diameter, with stress deviator diverging from pure shear mode, is proposed. Explicit reasons explaining why biaxial experimental layouts (simple shear and/or planar shear) are insufficient for the comprehensive characterization of materials properties submitted to complex stress states loadings are presented. New explicit formulas for the triaxiality factor valid for biaxial stress states are delivered.
Keywords: Cauchy stress; oriented geometrical object; isotropy angle; skewness angle; isomorphic cylindrical coordinates; pure shear, comparison reference state; anisotropy factor; biaxial tests; simple shear; planar shear; triaxiality factor.
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References

Bai Y., Wierzbicki T., A new model of metal plasticity and fracture with pressure and Lode dependence, International Journal of Plasticity, 24(6): 1071–1096, 2008, doi: 10.1016/j.ijplas.2007.09.004.

Blinowski A., Rychlewski J., Pure shears in the mechanics of materials, Mathematics and Mechanics of Solids, 3(4): 471–503, 1998, doi: 10.1177/108128659800300406.

Burzyński W., Theoretical foundations of the hypotheses of material effort, Engineering Transactions, 56(3): 269–305, 2008 (English translation of original paper in Polish, Teoretyczne podstawy hipotez wytezenia, Czasopismo Techniczne, 47: 1–41, 1929), https://et.ippt.pan.pl/index.php/et/article/view/199/143, accessed 31.01.2022.

Cauchy A., Research on the balance and internal movement of solid or fluid bodies, elastic or non-elastic [in French: Recherches sur l’equilibre et le mouvement interieur des corps solides ou fluides, elastiques ou non elastiques], Bulletin de la Societe Philomathique, pp. 9–13, 1823. Also published in Oeuvres compl`etes: Series 2 (Cambridge Library Collection – Mathematics, pp. 300–304), Cambridge: Cambridge University Press, 2009, doi: 10.1017/CBO9780511702518.038.

Davies E.A., Connelly F.M., Stress distribution and plastic deformation in rotating cylinders of strain-hardening material, Journal of Applied Mechanics, 26(1): 25–30, 1959, doi: 10.1115/1.4011918.

Eugster S.R., Glocker C., On the notion of stress in classical continuum mechanics, Mathematics and Mechanics of Complex Systems, 5(3–4): 299–338, 2017, doi: 10.2140/memocs.2017.5.299.

Gołąb S. (Author), Lepa E. (Translator), Tensor Calculus, Elsevier, Amsterdam, 1974.

Haigh B.P., The strain-energy function and the elastic limit, Engineering, 109: 158–160, 1920.

Itskov M., Tensor Algebra and Tensor Analysis for Engineers: with Applications to Continuum Mechanics, 5th ed., Springer, 2019.

Leckie F.A., Dal Bello D.J., Strength and Stiffness of Engineering Systems, Springer, 2009.

Lode W., Experiments on the influence of the mean principal stresses on the flow of the metals iron, copper and nickel [in German: Versuche uber den Einfluss der mittleren

Hauptspannungen auf das Fließen der Metalle Eisen, Kupfer und Nickel], Zeitschrift furPhysik, 36(11–12): 913–939, 1926, doi: 10.1007/BF01400222.

Malvern L., Introduction to the Mechanics of a Continuous Medium, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1969.

Maugin G., Continuum Mechanics Through the Eighteenth and Nineteenth Centuries, Historical Perspectives from John Bernoulli (1727) to Ernst Hellinger (1914), Springer, 2014.

Maugin G., The Thermomechanics of Plasticity and Fracture (Cambridge Texts in Applied Mathematics), Cambridge University Press, 1992.

Misses R.V., Mechanics of plastic deformation of crystals [in German: Mechanik der plastischen Formanderung von Kristallen], ZAMM – Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, 8(3): 161–185, 1928, doi: 10.1002/zamm.19280080302.

Murzewski J., A probabilistic theory of plastic and brittle behaviour of quasihomogeneous materials, Archives of Mechanics, 12(2): 203–227, 1960 (received December 1, 1958).

Müller I., A History of Thermodynamics: The Doctrine of Energy and Entropy, Springer, Berlin, 2007.

Novozhilov V.V., On the physical meaning of stress invariants used in the theory of plasticity, Applied Mathematics and Mechanics, 16(5): 617–619, 1952.

Novozhilov V.V., On relations between stresses and strains in non-linear elastic media, Applied Mathematics and Mechanics, 15(2): 183–194, 1951.

Ogden R.W., Non-linear Elastic Deformations, Dover Publications, Inc., Mineola, New York, 1997.

Ostrowska-Maciejewska J., Foundations and Applications of Tensor Calculus, IFTR Reports [in Polish: Podstawy i zastosowania rachunku tensorowego], Warsaw 2007, http://reports.ippt.pan.pl/IFTR Reports 1 2007.pdf.

Polyanin A.D., Manzhirov A.V., Handbook of Mathematics for Engineers and Scientists, Chapman and Hall/CRC, Taylor & Francis Group, Boca Raton, 2007.

Rychlewski J., On Hooke’s law, Journal of Applied Mathematics and Mechanics, 48(3): 303–314, 1984 (English translation from Russian original published in PMM, U.S.S.R.), doi: 10.1016/0021-8928(84)90137-0.

Rychlewski J., Elastic energy decomposition and limit criteria, Engineering Transactions, 59(1): 31–63, 2011 (English translation from Russian original published in Advances in Mechanics, 7: 51–80, 1984), http://et.ippt.pan.pl/index.php/et/article/view/159/100.

Rychlewski J., To estimate the anisotropy [in German: Zur Abschatzung der Anisotropie], ZAMM – Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, 65(6): 256–258, 1985, doi: 10.1002/zamm.19850650617.

Rychlewski J., On evaluation of anisotropy of properties described by symmetric secondorder tensors, Czechoslovak Journal of Physics B, 34(6): 499–506, 1984, doi: 10.1007/BF01595703.

Rychlewski J., Symmetry of Causes and Effects [in Polish: Symetria przyczyn i skutków], WN PWN, Warszawa, 1991.

Spencer A.J.M., Continuum Mechanics, Dover Publications Inc., Mineola, New York, 2004.

Tonolo A., Commemoration of Gregorio Ricci-Curbastro on the first centenary of his birth [in Italian: Commemorazione di Gregorio Ricci-Curbastro nel primo centenario della nascita, Rendiconti del Seminario Matematico della Universit`a di Padova], Reports of the Mathematical Seminar of the University of Padua, Vol. 23, pp. 1–24, 1954, http://www.numdam.org/item/RSMUP_1954__23__1_0.pdf, accessed 31.01.2022.

Truesdell C., Essays in the History of Mechanics, Springer-Verlag Inc., New York, 1968.

Voigt W., The Fundamental Physical Properties of the Crystals In Elementary Representation [in German: Die fundamentalen physikalischen Eigenschaften der Kristalle in elementarer Darstellung], Verlag Von Veit & Comp., Leipzig, 1898.

Westergaard H.M., On the resistance of ductile materials to combined stresses in two or three directions perpendicular to one another, Journal of the Franklin Institute, 189(5): 627–640, 1920, doi: 10.1016/S0016-0032(20)90373-3.

Ziółkowski A., Simple shear test in identification of constitutive behavior of materials submitted to large deformations – hyperelastic materials case, Engineering Transactions, 54(4): 251–269, 2006, doi: 10.24423/engtrans.423.2006.




DOI: 10.24423/EngTrans.2210.20220809