Mathematical Foundations of Limit Criterion for Anisotropic Materials

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Abstract

In the paper a new proposition of limit state criteria for anisotropic solids exhibiting different strengths at tension and compression is presented. The proposition is based on the concept of energetically orthogonal decompositions of stress state introduced by Rychlewski. The concept of stress state dependent parameters describing the influence of certain stress modes on the total measure of material effort was firstly presented by Burzynski. The both concepts are reviewed in the paper. General formulation of a new limit criterion as well as its specification for certain elastic symmetries is given. It is compared with some of the other known limit criteria for anisotropic solids. General methodology of acquiring necessary data for the criterion specification is presented. The ideas of energetic and limit state orthogonality are discussed - their application in representation of the quadratic forms of energy and limit state criterion as a sum of square terms is shown

[1] T. Fras, Z. Nowa k, P. Perzyn a, R.B. Pechersk i, Identification of the model describing viscoplastic behaviour of high strength metals, Inverse Problems in Science and Engineering 19, 1, 17-30 (2011).

[2] R.B. Pecherski, Relation of microscopic observations to constitutive modelling of advanced deformations and fracture initiation of viscoplastic materials, Archives of Mechanics 35, 257-277 (1983).

[3] J.J. Gilman, Electronic Basis of the Strength of Materials, Cambridge University Press, Cambride, UK, 2003.

[4] K. Nalepka, R.B. Pechersk i, Modelling of the interatomic interactions in the Copper crystal applied in the structure (111)Cu-(0001) Al2O3, Archives of Metallurgy and Materials 54, 511-522 (2009).

[5] K.T. Nalepka, R.B. Pechersk i, The Strength of the interfacial bond in the ceramic matrix composites Al2O3-Ni, Mechanics and Control 29, 132-137 (2010).

[6] K. Nalepka, Symmetry-based approach to parametrization of embedded-atom-method interatomicpotentials, Computational Materials Science 56, 100-107 (2012).

[7] K. Nalepka, Efficient approach to metal/metal oxide interfaces within variable charge model, The European Physical Journal B 85, 45 (2012).

[8] J.C. Maxwell, Origins of Clerk Maxwell’s electric ideas as described in familiar letters to William Thomson, Cambridge University Press, 1937.

[9] E. Beltrami, Opere matem. 4, Mailand, 180-189 (1920).

[10] M.T. Huber, Własciwa praca odkształcenia jako miara wytezenia materiału. Czasopismo Techniczne, 15, Lwów, 1904; see also: Specific work of strain asameasure of material effort, Archives of Mechanics 56, 3, 173-190 (2004).

[11] W. Thomson, (Lord Kelvin) Elements ofamathematical theory of elasticity, Philosophical Transactions of the Royal Society 166, 481-498 (1856).

[12] J. Rychlewski, On Hooke’s law [in Russian]. Prikladnaya Matematikai Mekhanika, 48, 420-435 (1984); see also: On Hooke’s law. Journal of Applied Mathematics and Mechanics 48, 303-314 (1984).

[13] J. Rychlewski, Elastic energy decomposition and limit criteria [in Russian]. Uspekhi mekhaniki, 7, 51-80 (1984); see also: Elastic energy decomposition and limit criteria. Engineering Transactions 59, 1, 31-63 (2011).

[14] W. Burzynski, Studium nad hipotezami wytezenia, Akademia Nauk Technicznych, Lwów, 1928; see also: Selected passages from Włodzimierz Burzynski’s doctoral dissertation Study on material effort hypotheses. Engineering Transactions 57, 3-4, 185-215 (2009).

[15] R. Mises, Mechanik der plastischen Formanderung von Kristallen. Zeitschrift f¨ur Angewandte Mathematik und Mechanik 8, 161-185 (1928).

[16] R. Mises, Mechanik der festen K¨orper im plastisch deformablen Zustand. Nachrichten von der Gesellschaft der Wissenschaften zu G¨ottingen, Mathematisch-Physikalische Klasse 1, 582-592 (1913).

[17] R. Hill, Atheory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society of London 193, 281-297 (1948).

[18] O. Hoffman, The brittle strength of orthotropic materials. Journal of Composite Materials 1, 200-206 (1967).

[19] S.W. Tsai, E.M. Wu, Ageneral theory of strength for anisotropic materials. Journal of Composite Materials 5, 58-80 (1971).

[20] R.M. Caddell, R.S. Raghav a, A.G. Atkins, Ayield criterion for anisotropic and pressure dependent solids such as orientedpolymers. Journal of Materials Science 8, 11, 1641-1646 (1973).

[21] V.S. Deshpande, N.A. Flec k, M.F. Ashb y, Effective properties of theoctet-truss lattice material. Journal of the Mechanics and Physics of Solids 49, 1747-1769 (2001).

[22] C. Li u, Y. Huang, M.G. Stou t, On the asymmetric yield surface of plastically orthotropic materials;aphenomenological study. Acta Materialia 45, 6, 2397-2406 (1997).

[23] R. Hill, Theoretical plasticity of textured aggregates. Mathematical Proceedings of the Cambridge Philosophical Society 85, 1, 179-191 (1979).

[24] W.F. Hosford, On yield loci of anisotropic cubic metals. Proceedings of the 7th North American Metalworking Conference, SME, Dearborn, MI., (1979).

[25] D. Banabic, Sheet metal forming processes, constitutive modelling and numerical simulation, Chapter 2, Springer Verlag, Berlin, Heidelberg, 2010.

[26] L.W. Hu, Modified Tresca’s yield condition and associated flow rules for anisotropic materials and applications. Journal of the Franklin Institute 265, 3, 187-204 (1958).

[27] A.P. Karafillis, M.C. Boyc e, Ageneral anisotropic yield criterion using bounds andatransformation weighting tensor. Journal of the Mechanics and Physics of Solids 41, 1859-1886 (1993).

[28] H.L. Schreyer, Q.H. Zu o, Anisotropic yield surfaces based on elastic projection operators. Journal of Applied Mechanics 62, 780-785 (1995).

[29] J.M. Luque- Raigón, R. Campoamor- Sturs- ber g, Aunified approach for plasticity yield criteria on the tangent space to the Cauchy tensor. Mathematics and Mechanics of Solids 17, 2, 83-103 (2012).

[30] P. Szeptynski, Some remarks on Burzynski’s failure criterion for anisotropic materials. Engineering Transactions 59, 2, 119-136 (2011).

[31] W. Olszak, W. Urbanowski, The plastic potential and the generalized distortion energy in the theory of non-homogeneous anisotropic elastic-plastic bodies. Archives of Mechanics 8, 671-694 (1956).

[32] I.I. Goldenblat, Theory of small elastic-plastic deformations of anisotropic media [in Russian], Izvestiya Akademii Nauk SSSR, Otd.Tehn. Nauk, 2, (1955).

[33] I.I. Goldenblat, Some problems of the mechanics of deformable media [in Russian], Gostekhizdat, Moscow, 1955.

[34] W. Olszak, J. Ostrowska- Maciejewsk a, The plastic potential in the theory of anisotropic elastic-plastic solids. Engineering Fracture Mechanics 21, 4, 625-632 (1985).

[35] J. Ostrowska- Maciejewsk a, J. Rychlewski, Plane elastic and limit states in anisotropic solids. Archives of Mechanics 40, 4, 379-386 (1988).

[36] W.A. Spitzig, R.J. Sober, O. Richmond, The Effect of hydrostatic pressure on the deformation behavior of maraging and HY-80 steels and its implications for plasticity theory. Metallurgical Transactions A 7A, 1704-1710 (1976).

[37] O. Richmond, W.A. Spitzi g, Pressure dependence and dilatancy of plastic flow. Theoretical and applied mechanics; Proceedings of the 15th International Congress, 377-386 (1980).

[38] C.D. Wilson, Acritical reexamination of classical metal plasticity. Journal of Applied Mechanics 69, 63-68 (2002).

[39] D.C. Drucke r, Plasticity theory, strength-differential (SD) phenomenon, and volume expansion in metals and plastics. Metallurgical Transactions 4, 667-673 (1973).

[40] O. Cazacu, F. Barlat, Generalization of Drucker’s yield criterion to orthotropy. Mathematics and Mechanics of Solids 6, 6, 613-630 (2001).

[41] M.M. Mehrabad i, S.C. Cowi n, Eigentensors of linear anisotropic elastic materials, Quarterly Journal of Mechanics and Applied Mathematics 43, 1, 15-41 (1990).

[42] S. Sutcliffe, Spectral decomposition of the elasticity tensor, Transactions of the ASME 762, 59 (1992).

[43] J. Rychlewski, Unconventional approach to linear elasticity, Archives of Mechanics 47, 2, 149-171 (1995).

[44] I.M. Gelfand, Lectures on linear algebra [in Russian], Nauka, Moscow, 1966.

[45] K. Kowalczyk- Gajewska, J. Ostrowska- Ma - ciejewska, The influence of internal restrictions on the elastic properties of anisotropic materials, Archives of Mechanics 56, 3, 205-232 (2004).

[46] D.C. Drucker, W. Prager, Soil mechanics and plastic analysis for limit design. Quarterly of Applied Mathematics 10, 2, 157-165 (1952).

[47] T.C.T. Ting, Canalinear anisotropic elastic material have a uniform contraction underauniform pressure? Mathematics and Mechanics of Solids 6, 235-243 (2001).

[48] S. Federico, Volumetric-distortional decomposition of deformation and elasticity tensor. Mathematics and Mechanics of Solids 15, 672-690 (2010).

[49] K. Kowalczyk, J. Ostrowska- Maciejewsk a, R.B. Pechersk i, An energy-based yield criterion for solids of cubic elasticity and orthotropic limit state. Archives of Mechanics 55, 5-6, 431-448 (2003).

[50] J. Ostrowska-Maciejewsk a, R.B. Pechersk i, P. Szeptynski, Limit condition for anisotropic materials with asymmetric elastic range, Engineering Transactions 60, 2, 125-138 (2012).

[51] P. Szeptynski, R.B. Pechersk i, Proposition of a new yield criterion for orthotropic metal sheets accounting for asymmetry of elastic range [in Polish]. Rudyi Metale Niezelazne 57, 4, 243-250 (2012).

[52] K. Kowalczyk- Gajewska, J. Ostrowska- Ma - ciejewska, Review on spectral decomposition of Hooke’s tensor for all symmetry groups of linear elastic material. Engineering Transactions 57, 3-4, 145-183 (2009).

[53] A. Blinowski, J. Ostrowska- Maciejewsk a, On the elastic orthotropy, Archives of Mechanics 48, 1, 129-141 (1996).

[54] M. Nowak, J. Ostrowska- Maciejewsk a, R.B. Pechersk i, P. Szeptynski, Yield criterion accounting for the third invariant of stress tensor deviator. Part I. Derivation of the yield condition basing on the concepts of energy-based hypotheses of Rychlewski and Burzynski. Engineering Transactions 59, 4, 273-281 (2011).

[55] P. Szeptynski, Yield criterion accounting for the influence of the third invariant of stress tensor deviator. Part II: Analysis of convexity condition of the yield surface. Engineering Transactions 59, 4, 283-297 (2011).

[56] R.B. Pechersk i, P. Szeptynski, M. Nowa k, An extension of Burzynski hypothesis of material effort accounting for the third invariant of stress tensor, Archives of Metallurgy and Materials 56, 2, 503-508 (2011).

[57] S.K. Iyer, C.J. Lissende n, Multiaxial constitutive model accounting for the strength-differential in Inconel 718. International Journal of Plasticity 19, 2055-2081 (2003).

[58] D. Bigoni, A. Piccolroaz, Yield criteria for quasibrittle and frictional materials. International Journal of Solids and Structures 41, 2855-2878 (2004).

[59] C. Lexcellent et al., Experimental and numerical determinations of the initial surface of phase transformation under biaxial loadingin some polycrystalline shape-memory alloys. Journal of the Mechanics and Physics of Solids 50, 2717-2735 (2002).

[60] B. Raniecki, Z. Mróz, Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SDeffect. Acta Mechanica 195, 81-102 (2008).

[61] J. Podgórski, Limit state condition and the dissipation function for isotropic materials. Archives of Mechanics 36, 3, 323-342 (1984).

[62] J.P. Bardet, Lode dependences for isotropic pressure sensitive materials. Journal of Applied Mechanics 57, 498-506 (1990).

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