**Abstract**

**Problem Statement (Relevance):** The current yield criteria enable to predict the behavior of anisotropic materials under plastic deformation conditions. However, they do not appear to allow for the physical basis of anisotropy, i.e. the crystalline structure of the material and the texture formed under severe plastic strain. Therefore, it would not be possible to solve the inverse problem using the above criteria. In other words, we cannot use the above criteria to determine what crystalline structure, which has to be obtained in semi-finished products, would be most adequate to the requirements of metal forming processes. **Objectives: **The objective is to derive a plasticity criterion and constitutive assumptions of the plasticity theory for orthotropic material allowing for the lattice constants and the parameters of the predominant crystallographic structure typical of the stress state. **Methods Applied: **The plasticity criterion was derived based on the specific distortion strain energy and using some tensor calculus and invariant theory elements. The constitutive assumptions of the anisotropic medium plasticity theory were proved to be correct through experiment. The experiment was based on the comparison between the hardening curve obtained as a result of the tensile testing of longitudinal and transverse samples and the calculated curve. **Originality: **Basic equations of the orthotropic medium plasticity theory were developed, which explicitly account for the crystallographic nature of anisotropy. **Findings: **It was established that in order to determine the relationships between stress and strain in different directions, one needs to build a hardening curve in just one direction. And then, based on the texture orientation and the crystal lattice parameters, one can obtain the hardening curves for the other directions. In this case the calculation error does not appear to be more than 2-3%. **Practical Relevance:** The proposed variant of the plasticity theory allows to ensure improved processability and performance of the products by designing the texture components.

**Keywords**

Anisotropy, yield criterion, crystal lattice, texture, crystallographic orientation, hardening curve, tension test.

1. Truszkowski W. (2001), The Plastic Anisotropy in Single Crystals and Polycrystalline Metals, Springer, Netherlands.

2. Hutchinson W.B., Oscarsson A. and Karlsson A. (1989) Control of microstructure and earing behaviour in aluminium alloy AA 3004 hot bands, Mater. Sci. Tech., 5, pp. 1118-1127.

3. Banabic D, Bunge H.J., Pohlandt K. and Tekkaya A.E. (2000), Formability Of Metallic Materials: Plastic Anisotropy, Formability Testing, Forming Limits, Springer, Berlin, Germany.

4. Engler O. and Hirsch J., (2002) Texture control by thermomechanical processing of AA6xxx Al-Mg-Si sheet alloys for automotive applications - a review, Materials Science and Engineering A, 336, pp. 249-262.

5. Hirsch J., Al-Samman T. (2013) Superior light metals by texture engineering: Optimized aluminum and magnesium alloys for automotive applications, ActaMaterialia, 61, pp. 818–843.

6. Banabic D. (2010), Sheet Metal Forming Processes. Constitutive Modelling and Numerical Simulation, Springer, Berlin, Germany.

7. Mises R. 1928. Mechanik der plastischen Formanderung von Kristallen. ZAMM. 8, 161-185.

8. Hill R. 1948. A theory of the yield and plastic flow of anisotropic metals. Proc. Roy. Soc. London. Ser A. 193, pp. 281-297.

9. Woodthorpe, J., Pearce, R. 1970. The anomalous Behavior of Aluminum Sheet under Balance Biaxial Tension. Int. J. Mech. Sci. 12, pp. 341-347.

10. Hill R. (1979) Theoretical plasticity of textured aggregates, Math. Proc. Cambridge Philos. Soc., pp. 179-191.

11. Barlat F., Lian J. 1989. Plastic Behavior and Stretchability of Sheet Metals. Part 1: Yield Function for Orthotropic Sheets under Plane Stress Conditions. Int. J. Plasticity. 5, pp. 51-66.

12. Aryshenskii Yu.М., Kaluzhskii I.I. and Uvarov V.V. Some issues of the orthotropic medium plasticity theory. Izvestiya vuzov. AviatsionnayaTekhnika [Proceedings of Russian universities: Aviation Engineering], 1969, no. 2, pp. 15-18. (In Russ.)

13. Aryshenskii Yu.М., Grechnikov F.V. and Аryshenskii V.Yu. Identifying sheet anisotropy requirements depending on further forming. Kuznechno-shtampovochnoe roizvodstvo [Die forging], 1990, no. 3, pp. 16-19. (In Russ.)

14. Barlat F., Lege D.J., Brem J.C. 1991. A six-component yield function for anisotropic materials. Int. J. Plasticity. 7, pp. 693-712.

15. Karafillis A.P., Boyce M.C. 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids. 41, pp. 1859-1886.

16. Barlat F., Brem J.C., Yoon J.W., Chung K., Dick R.E., Lege D.J., Pourboghrat F., Choi S.-H., Chu E. 2003. Plane stress yield function for aluminum alloy sheet. Part 1: Theory. Int. J. Plasticity. 19, pp. 1297-1319.

17. Barlat F., Aretz H., Yoon J.W., Karabrin M.E., Brem J.C. and Dick R.E. (2005), “Linear transformation based anisotropic yield functions”, Int. J. Plasticity, 21, pp. 1009-1039.

18. Bron F., Besson J. 2003. A yield function for anisotropic materials. Application to aluminum alloys. Int. J. Plasticity. 20, pp. 937-963.

19. Cazacu O., Barlat F. 2001. Generalization of Drucker's yield criterion to orthotropy. Mathematics and Mechanics of Solids. 6, pp. 613-630.

20. Cazacu O., Barlat F. 2003. Application of representation theory to describe yeilding of anisotropic aluminum alloys. Int. J. of Engng. Sci. 41, pp. 1367-1385.

21. Soare S. Banabic D. 2008. About mechanical data required to describe the anisotropy of thin sheets to correctly predict the earing of deep-drawn cups. Int. J. Plasticity. 4, pp. 34-37.

22. Hosford W.F. Mechanical Behavior of Materials. New-York, Cambridge University Press. 2005.

23. Adamesku R.A., Geld P.V., Mityushov E.A. The anisotropy of the physical properties of metals. Moscow: Metallurgiya, 1985, 136 p. (In Russ.)

24. Grechnikov F.V. Deformation of anisotropic materials. M.: Mashinostroenie, 1998, 446 p. (In Russ.)

25. Hill R. The Mathematical Theory of Plasticity. New-York, Oxford University Press. 1950.

26. Aryshenskii Yu.M. The theory of anisotropic sheet metal stamping. Publishing House of Saratov University, 1973, 112 p. (In Russ.)