Abstract
In this chapter basic features of isotropic versus anisotropic initial yield criteria are discussed. Two ways to account for anisotropy are presented: the explicit and implicit formulations. The explicit description of anisotropy is rigorously based on well-established theory of common invariants (Sayir, Goldenblat–Kopnov, von Mises, Hill). The implicit approach involves linear transformation tensor of the Cauchy stress that accounts for anisotropy to enhance the known isotropic criteria to be able to capture anisotropy, hydrostatic pressure insensitivity, and asymmetry of the yield surface (Barlat, Plunckett, Cazacu, Khan). The advantages and differences of both formulations are critically presented. Possible convexity loss of the classical Hill’48 yield surface in the case of strong orthotropy is examined and highlighted in contrast to unconditionally stable von Mises–Hu–Marin’s criterion. Various transitions from the orthotropic yield criteria to the transversely isotropic ones are carefully distinguished in the light of irreducibility or reducibility to the isotropic Huber–von Mises criterion in the transverse isotropy plane and appropriate symmetry class of tetragonal symmetry (classical Hill’s formulation) or hexagonal symmetry (hexagonal Hill’s or von Mises–Hu–Marin’s). The new hybrid formulation applicable for some engineering materials based on additional bulge test is also proposed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Altenbach, H., Bolchoun, A., Kolupaev, V.A.: Phenomenological yield and failure criteria. In: Altenbach, H., Öchsner, A. (eds.) Plasticity of Pressure-Sensitive Materials. Springer, Heidelberg (2014)
Barlat, F., Lian, J.: Plastic behavior and stretchability of sheet metals. Int. J. Plast. 5(1), 51 (1989)
Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E.: Plane stress function for aluminium alloy sheets—part I: theory. Int. J. Plast. 19, 1297–1319 (2003)
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)
Boehler, J.P., Sawczuk, A.: Equilibre limite des sols anisotropes. J. Mécanique 9, 5–33 (1970)
Cazacu, O., Barlat, F.: A criterion for description of anisotropy and yield differential effects in pressure-insensitive materials. Int. J. Plast. 20, 2027–2045 (2004)
Cazacu, O., Planckett, B., Barlat, F.: Orthotropic yield criterion for hexagonal close packed metals. Int. J. Plast. 22, 1171–1194 (2006)
Chen, W.F., Han, D.J.: Plasticity for Structural Engineers. Springer, Berlin (1995)
Chu, E.: Generalization of Hill’s 1979 anisotropic yield criteria. In: Proceedings of the NUMISHEETS’89, pp. 199–208 (1989)
Davies, E.A.: The Bailey flow rule and associated yield surface. Trans. ASME E28(2), 310 (1961)
Drucker, D.C.: Relation of experiments to mathematical theories of plasticity. J. Appl. Mech. 16, 349–357 (1949)
Dunand, M., Maertens, A.P., Luo, M., Mohr, D.: Experiments and modeling of anisotropic aluminum extrusions under multi-axial loading—part I: plasticity. Int. J. Plast. 36, 34–49 (2012)
Dvorak, G.J., Bahei-El-Din, Y.A., Macheret, Y., Liu, C.H.: An experimental study of elastic-plastic behavior of a fibrous boron-aluminum composite. Int. J. Mech. Phys. Solids 36, 655–687 (1988)
Ganczarski, A., Lenczowski, J.: On the convexity of the Goldenblatt-Kopnov yield condition. Arch. Mech. 49(3), 461–475 (1997)
Ganczarski, A., Skrzypek, J.: Modeling of limit surfaces for transversely isotropic composite SCS-6/Ti-15-3. Acta Mechanica et Automatica 5(3), 24–30 (2011) (in Polish)
Ganczarski, A., Skrzypek, J.: Mechanics of Novel Materials (in Polish). Wydawnictwo Politechniki Krakowskiej, Kraków (2013)
Ganczarski, A., Skrzypek, J.: Constraints on the applicability range of Hill’s criterion: strong orthotropy or transverse isotropy. Acta Mech. 225, 2568–2582 (2014)
Goldenblat, I.I., Kopnov, V.A.: Obobshchennaya teoriya plasticheskogo techeniya anizotropnyh sred, pp. 307–319. Sbornik Stroitelnaya Mehanika, Stroĭizdat, Moskva (1966)
Guest, J.J.: On the strength of ductile materials under combined stress. Philos. Mag. 50, 69–132 (1900)
Haigh, B.F.: The strain-energy function and the elastic limit. Eng. Lond. 109, 158–160 (1920)
Hencky, H.: Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufen Nach-Spannungen. ZAMM 4, 323–334 (1924)
Herakovich, C.T., Aboudi, J.: Thermal effects in composites. In: Hetnarski, R.B. (ed.) Thermal Stresses V, pp. 1–142. Lastran Corporation Publishing Division, Rochester (1999)
Hershey, A.V.: The plasticity of an isotropic aggregate of anisotropic face—centred cubic crystals. J. Appl. Mech. 21(3), 241–249 (1954)
Hill, R.: A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A193, 281–297 (1948)
Hill, R.: The Mathematical Theory of Plasticity. Oxford University Press, Oxford (1950)
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, pp. 259–298. Syracuse University Press, Syracuse (1964)
Hosford, W.F.: Texture Strengthening. Met. Eng. Q. 6, 13–19 (1966)
Hosford, W.F.: A generalized isotropic yield criterion. Trans. ASME E39(2), 607–609 (1972)
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, 34–40, 49–50, 61–62, 80–81, Lwów, Pisma, Vol. II, PWN, Warszawa 1956, 3–20 (1904)
Ishlinskiĭ, A.Yu.: Gipoteza prochnosti formoizmeneniya, p. 46. University, Mekh, Uchebnye Zapiski Mosk (1940)
Jackson, L.R., Smith, K.F., Lankford, W.T.: Plastic flow in anisotropic sheet steel. Am. Inst. Min. Metall. Eng. 2440, 1–15 (1948)
Khan, A.S., Kazmi, R., Farrokh, B.: Multiaxial and non-proportional loading responses, anisotropy and modeling of Ti-6Al-4V titanium alloy over wide ranges of strain rates and temperatures. Int. J. Plast. 23, 931–950 (2007)
Khan, A.S., Liu, H.: Strain rate and temperature dependent fracture criteria for isotropic and anisotropic metals. Int. J. Plast. 37, 1–15 (2012)
Khan, A.S., Yu, S., Liu, H.: Deformation enhanced anisotropic responses of Ti-6Al-4V alloy, part II: a stress rate and temperature dependent anisotropic yield criterion. Int. J. Plast. 38, 14–26 (2012)
Kowalsky, U.K., Ahrens, H., Dinkler, D.: Distorted yield surfaces—modeling by higher order anisotropic hardening tensors. Comput. Math. Sci. 16, 81–88 (1999)
Korkolis, Y.P., Kyriakides, S.: Inflation and burst of aluminum tubes. part II: an advanced yield function including deformation-induced anisotropy. Int. J. Plast. 24, 1625–1637 (2008)
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
Lode, W.: Der Einfluss der mittleren Hauptspannung auf der Fliessen der Metalle, Forschungs arbeiten auf dem Gebiete des Ingenieurusesen, 303 (1928)
Luo, M., Dunand, M., Moth, D.: Experiments and modeling of anisotropic aluminum extrusions under multi-axial loading—part II: ductile fracture. Int. J. Plast. 32–33, 36–58 (2012)
Malinin, N.N., Rżysko, J.: Mechanika Materiałów. PWN, Warszawa (1981)
von Mises, R.: Mechanik der festen Körper im plastisch deformablen Zustand, Götingen Nachrichten. Math. Phys. 4(1), 582–592 (1913)
von Mises, R.: Mechanik der plastischen Formänderung von Kristallen. ZAMM 8(13), 161–185 (1928)
Nigam, H., Dvorak, G.J., Bahei-El-Din, Y.A.: An experimental investigation of elastic-plastic behavior of a fibrous Boron-Aluminum composite. I. Matrix-dominated mode. Int. J. Plast. 10, 23–48 (1933)
Nixon, M.E., Cazacu, O., Lebensohn, R.A.: Anisotropic response of high-purity \(\alpha \)-titanium: experimental characterization and constitutive modeling. Int. J. Plast. 26, 516–532 (2010)
Ottosen, N.S., Ristinmaa, M.: The Mechanics of Constitutive Modeling. Elsevier, Amsterdam (2005)
Raniecki, B., Mróz, Z.: Yield or martensitic phase transformation conditions and dissipative functions for isotropic, pressure-insensitive alloys exhibiting SD effect. Acta Mech. 195, 81–102 (2008)
Reuss, A.: Vereifachte Berechnung der plastischen Formänderungen in der Plastizitätstheorie. ZAMM 10(3), 266–274 (1933)
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. Mechanical Engineering Publications, London (1990)
Rymarz, Cz.: Continuum Mechanics (in Polish). PWN, Warszawa (1993)
Sayir, M.: Zur Fließbedingung der Plastizitätstheorie. Ingenierurarchiv 39, 414–432 (1970)
Schmidt, R.: Über den Zusammenhang von Spannungen und Formänderungen im Vestigungs-gebiet. Ing.-Arch. 3, 215–235 (1932)
Skrzypek, J., Ganczarski, A.: Anisotropic initial yield and failure criteria including temperature effect. In: Hetnarski, R.B. (ed.) Encyclopedia of Thermal Stresses, vol. A, pp. 146–159. Springer, Dordrecht (2014)
Sobotka, Z.: Theorie des plastischen Fliessens von anisotropen Körpern. Z. Angew. Math. Mechanik 49, 25–32 (1969)
Spencer, A.J.M.: Theory of invariants. In: Eringen, C. (ed.) Continuum Physics, pp. 239–353. Academic Press, New York (1971)
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(1), 3–38 (1993)
Tamma, K.K., Avila, A.F.: An integrated micro/macro modelling and computational methodology for high temperature composites. In: Hetnarski, R.B. (ed.) Thermal Stresses V, pp. 143–256. Lastran Corporation Publishing Division, Rochester (1999)
Tresca, H.: Mémoire sur l’écoulement des corps solids soumis á de fortes pressions. Comptes Rendus de l’Académie des Sciences 59, 754–758 (1864)
Tsai, S.T., Wu, E.M.: A general theory of strength for anisotropic materials. Int. J. Numer. Methods Eng. 38, 2083–2088 (1971)
Voyiadjis, G.Z., Thiagarajan, G.: An anisotropic yield surface model for directionally reinforced metal-matrix composites. Int. J. Plast. 11, 867–894 (1995)
Westergaard, H.M.: On the resistance of ductile materials to combined stresses in two and three directions perpendicular to one another. J. Frankl. Inst. 189, 627–640 (1920)
Yoshida, F., Hamasaki, H.M., Uemori, T.: A user-friendly 3D yield function to describe anisotropy of steel sheets. Int. J. Plast. 45, 119–139 (2013)
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
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Ganczarski, A.W., Skrzypek, J.J. (2015). Termination of Elastic Range of Pressure Insensitive Materials—Isotropic and Anisotropic Initial Yield Criteria. In: Skrzypek, J., Ganczarski, A. (eds) Mechanics of Anisotropic Materials. Engineering Materials. Springer, Cham. https://doi.org/10.1007/978-3-319-17160-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-17160-9_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-17159-3
Online ISBN: 978-3-319-17160-9
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)