Abstract
Commonly used orthotropic Hill’s criterion of plastic flow initiation (Hill in Proc R Soc Lond A 193:281–297, 1948) suffers from some constraints and inconsistencies, which are of two different origins. Firstly, in case of high orthotropy degree, the quadratic form corresponding to Hill’s criterion may change type from convex and closed elliptic to concave and open hyperbolic in the deviatoric stress space (Ottosen and Ristinmaa in the mechanics of constitutive modeling, Elsevier, Amsterdam, 2005). Secondly, application of classical Hill’s criterion to transversely isotropic materials shows a discrepancy between Hill’s limit curves in the transverse isotropy plane and the Huber-von Mises prediction for isotropic materials (Huber 1904; von Mises 1913). The basic result of the present paper is to propose the new transversely isotropic von Mises–Hu–Marin’s-type criterion of hexagonal symmetry that is free from both constraints. The new enhanced Hu–Marin’s-type limit surface represents an elliptic cylinder, the axis of which is proportional to stress/strength, in contrast to Hill’s-type limit surface possessing the hydrostatic axis. Hence, this condition does not exhibit the deviatoricity property, which is a price for coincidence with the Huber–von Mises condition in the transverse isotropy plane, but with cylindricity ensured for an arbitrarily high orthotropy degree. The hybrid-type transversely isotropic Hu–Marin’s criterion of mixed symmetry based on additional biaxial bulge test, capable of fitting experimental findings for some complex composites, is also proposed. Application of this criterion has been verified for a unidirectional SiC/Ti composite examined by Herakovich (Thermal stresses V, Lastran Corp. Publ. Division, pp 1–142, 1999).
Article PDF
Similar content being viewed by others
References
Berryman J.G.: Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries. J. Mech. Phys. Solids 53, 2141–2173 (2005)
Betten J.: Applications of tensor functions to the formulation of yield criteria for anisotropic materials. Int. J. Plast. 4, 29–46 (1988)
Drucker, D.C.: A more fundamental approach to plastic stress-strain relations. In: Proceedings of 1st US National Congress on Applied Mechanics, Chicago, pp. 487–491 (1951)
Ganczarski A., Lenczowski J.: On the convexity of the Goldenblat-Kopnov yield condition. Arch. Mech. 49, 461–475 (1997)
Ganczarski A., Skrzypek J.: Modeling of limit surfaces for transversely isotropic composite SCS-6/Ti-15-3 (in Polish). Acta Mech. et Autom. 5, 24–30 (2011)
Goldenblat, I.I., Kopnov, V.A.: A generalized theory of plastic flow of anisotropic metals (in Russian). Stroit. Mekh. pp. 307–319 (1966)
Herakovich, C.T., Aboudi, J.: Thermal effects in composites. In: Hetnarski, R.B. (ed.) Thermal Stresses V, Lastran Corp. Publ. Division, pp. 1–142 (1999)
Hill R.: A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297 (1948)
Hill R.: The Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950)
Hill R.: Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 13, 89–101 (1965)
Hosford, W.F., Backhofen, W.A.: Strength and plasticity of textured metals. In: Backhofen, W.A., Burke, J., Coffin, L., Reed, N., Weisse, V. (eds.) Fundamentals of Deformation Processing, Syracuse Univ. Press, pp. 259–298 (1964)
Hu Z.W., Marin J.: Anisotropic loading functions for combined stresses in the plastic range. J. Appl. Mech. 22, 1 (1956)
Huber, M.T.: Właściwa praca odkształcenia jako miara wytȩżenia materiału. Czas. Techn. 22, Lwów; Pisma, vol.II, PWN, Warszawa 1956 (1904)
Jackson L.R., Smith K.F., Lankford W.T.: Plastic flow in anisotropic sheet steel. Am. Inst. Mining. Metall. Eng. 2440, 1–15 (1948)
Jastrzebski Z.D.: The Nature and Properties of Engineering Materials. Wiley, New York (1987)
Kowalewski, Z.L., Śliwowski, M.: Effect of cyclic loading on the yield surface evolution of 18G2A low-alloy steel. Int. J. Mech. Sci. 39, 1, 51–68 (1997)
Kowalsky U.K., Ahrens H., Dinkler D.: Distorted yield surfaces—modeling by higher order anisotropic hardening tensors. Comput. Mater. Sci. 16, 81–88 (1999)
Kuna-Ciskał H., Skrzypek J.: CDM based modelling of damage and fracture mechanisms in concrete under tension and compression. Eng. Fract. Mech. 71, 681–698 (2004)
Lankford, W.T., Low, J.R., Gensamer, M.: The plastic flow of aluminium alloy sheet under combined loads. Trans. AIME 171, 574; TP 2238, Met. Techn., Aug. 1947 (1947)
Lemaitre J., Chaboche J.-L.: Méchanique des Matériaux Solides. Dunod Publ., Paris (1985)
Love A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover Publ., New York (1944)
Malinin N.N, Rżysko J.: Mechanics of Materials (in Polish). PWN, Warszawa (1981)
von Mises, R.: Mechanik der festen Körper im plastisch-deformablen Zustand, Nachrichten der Gesellschaft der Wissenschaften zu Göttingen (1913)
von Mises R.: Mechanik der plastischen Formänderung von Kristallen. ZAMM 8, 161–185 (1928)
Mursa, K.S.: Examination of orthotropic metal sheets under uniaxial tension (in Russian). Izv. Vys. Ucheb. Zav., Mash. 6 (1972)
Nye J.F.: Physical Properties of Crystals. Their Representations by Tensor and Matrices. Clarendon Press, Oxford (1957)
Ottosen N.S., Ristinmaa M.: The Mechanics of Constitutive Modeling. Elsevier, Amsterdam (2005)
Rogers T.G.: Yield criteria, flow rules, and hardening in anisotropic plasticity. In: Boehler, J.P. (ed.) Yielding, Damage and Failure of Anisotropic Solids, pp. 53–79. Mech. Eng. Publ., London (1990)
Rymarz Cz.: Continuum Mechanics (in Polish). PWN, Warszawa (1993)
Sayir M.: Zur Fließbedingung der Plastiztätstheorie. Ingenievrarchiv 39, 414–432 (1970)
Skrzypek J., Ganczarski A. (2013) Anisotropic initial yield and failure criteria including temperature effect. In: Hetnarski, R. (ed.), Encyclopedia of Thermal Stresses, Springer Science+Business Media Dordrecht
Spencer A.J.M.: Theory of invariants. In: Eringen, C. (ed.) Continuum Physics, Academic Press, New York, pp. 239–353
Sun, C.T., Vaidya, R.S.: Prediction of composite properties from a representative volume element. Compos. Sci. Technol. 56, 171–179 (1996)
Szczepiński W.: On deformation-induced plastic anisotropy of sheet metals. Arch. Mech. 45, 3–38 (1993)
Tamma K.K., Avila A.F. (1999) An integrated micro/macro modelling and computational methodology for high temperature composites. In: Hetnarski R.B. (ed.) Thermal Stresses V, Lastran Corp. Publ. Division, Rochester, NY, pp. 143–256
Tsai S.T., Wu E.M.: A general theory of strength for anisotropic materials. Int. J. Numer. Methods. Eng. 38, 2083–2088 (1971)
Życzkowski M.: Anisotropic yield conditions. In: Lemaitre, J. (ed.) Handbook of Materials Behavior Models, pp. 155–165. Academic Press, San Diego (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Ganczarski, A.W., Skrzypek, J.J. Constraints on the applicability range of Hill’s criterion: strong orthotropy or transverse isotropy. Acta Mech 225, 2563–2582 (2014). https://doi.org/10.1007/s00707-014-1089-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-014-1089-1