Mikromechaniczne modelowanie metali i stopów o wysokiej wytrzymałości właściwej

Kowalczyk-Gajewska, K.

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Mikromechaniczne modelowanie metali i stopów o wysokiej wytrzymałości właściwej

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Kowalczyk-Gajewska K.

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Micromechanical modelling of metals and alloys of high specific strength

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Abstrakty

The thesis reports the research effort aimed at the micromechanical description of various phenomena characteristic for elastic-viscoplastic deformations of polycrystalline materials. The attention has been focused on metals and alloys of high specific strength, in which the inelastic deformation at the local level is constrained by an insufficient number of easy deformation modes and the presence of lamellar substructure. The monograph consists of seven chapters. The first chapter has an introductory character. It outlines the motivation and scope of the thesis and indicates fields of applicability of the results obtained. In Chapter 2 a model of a single crystal deforming by slip and twinning is proposed, together with a new reorientation scheme formulated in order to account for appearance of twin-related orientations in polycrystalline aggregates. In the second part of the chapter an implementation of this model within known scale transition schemes is discussed. The validation of the proposed framework, when modelling the overall response and texture evolution for polycrystalline materials of high specific strength, is presented. In Chapter 3 a theoretical analysis of bounds and self-consistent estimates of overall properties of polycrystals of low symmetry, particularly those characterized by the constrained deformation at the local level, is performed. In the study two invariant decompositions of Hooke's tensors are employed. In Chapter 4 predictions of different extensions of the self-consistent method applicable to non-linear viscoplastic crystals of strong anisotropy are analysed. In Chapter 5 a micromechanical three-scale model of polycrystals of lamellar substructure is discussed, together with its extension to the large strain framework. An influence of the confinement effects induced by lamellar substructure on the overall response of polycrystals is evaluated. In Chapter 6 a new method of sequential linearisation of elastic-viscous response is presented. The procedure developed is applied to extend the self-consistent averaging scheme to elastic-viscoplastic heterogeneous materials. The last chapter recapitulates the most important conclusions and includes an outlook for future research employing the developed modelling tools.

Niniejsza rozprawa poświęcona jest mikromechanicznemu modelowaniu metali i stopów o wysokiej wytrzymałości właściwej. Metale i stopy charakteryzujace sie wysokim stosunkiem wytrzymałości i sztywności do gęstości materiału (np. stopy magnezu i tytanu, zwiazki miedzymetaliczne Ti-Al), wykazują zwykle niską ciągliwość i formowalność. Te niepożądane cechy ograniczają ich potencjalnie liczne zastosowania w przemyśle (Appel andWagner [6], Agnew et al. [2], Proust et al. [169], Mróz [141], Lasalmonie [110]). Wskazana kombinacja własności jest wynikiem mikrostruktury. W większości opisywanych materiałów mamy do czynienia z niską symetrią sieci (np. symetria heksagonalna A3 w przypadku stopów Mg i Ti lub symetria tetragonalna dla y-TiAl), a w konsekwencji z niewystarczajacą liczbą łatwych systemów poślizgu uruchamiajacych się podczas deformacji niesprężystych. W wielu przypadkach nie jest spełniony warunek Taylora, to znaczy liczba niezależnych łatwych systemów poślizgu, które są dostępne dla materiału, jest mniejsza od pięciu. Brak ten może być częściowo zrekompensowany przez inny mechanizm deformacji plastycznej - bliźniakowanie, Christian and Mahajan [36]. Analogiczny zespół zjawisk zachodzących na poziomie mikro jest obserwowany w przypadku stopów cyrkonu lub zwiazków miedzymetalicznych Fe-Al. Materiały te nie wykazują wysokiej wytrzymałości własciwej ale charakteryzują się wyjątkową odpornością na korozje. Dla wielu z wyżej wspomnianych metali i zwiazków miedzymetalicznych częste jest również występowanie substruktur lamelarnych powstałych na skutek obróbki termicznej, bądź tworzących się podczas bliźniakowania mechanicznego. Pojawienie się tego typu mikrostruktury wpływa na aktywność poszczególnych modów deformacji promując te sposród nich, które mają korzystną orientację względem geometrii laminatu (Lebensohn et al. [113], Proust et al. [169]). Oba fakty, to jest niespełnienie warunku Taylora przez zbiór możliwych łatwych systemów poślizgu oraz występowanie kierunkowych efektów typu Halla-Petcha związanych z istnieniem substruktury lamelarnej, wskazują na istnienie wiezów nałożonych na deformację niesprężysta na poziomie mikro pojedynczego ziarna. Na skutek powyższych cech omawianych materiałów, w procesach formowania wyrobów, np. walcowania, tworzą sią silne tekstury krystalograficzne, co na poziomie makroskopowym manifestuje się znaczącą anizotropią właściwości. Celem rozprawy jest opracowanie różnych metod mikromechanicznej analizy sprężysto-(lepko)plastycznych polikryształów metali i stopów o wysokiej sztywności i wytrzymałości właściwej. Oryginalne aspekty prezentowanych rezultatów to: - opis konstytutywny pojedynczego ziarna uwzgledniający sprzężenia zachodzące pomiędzy mechanizmami poślizgu i bliźniakowania, - opracowanie nowego schematu reorientacji ziarna na skutek bliźniakowania, słuzącego modelowaniu ewolucji tekstury krystalograficznej, - adaptacja zaproponowanego modelu pojedynczego ziarna deformującego się przez poślizg i bliżniakowanie w ramach różnych schematów przejścia mikro-makro i weryfikacja jego przewidywań pod względem makroskopowej odpowiedzi materiału oraz ewolucji tekstury krystalograficznej, - wykorzystanie niezmienników tensora IV-ego rzędu wynikających z rozkładów inwariantnych takiego tensora do znalezienia nowych zależności opisujących standardowe oszacowania właściwości makroskopowych (oszacowania Voigta i Reussa, Hashina-Shtrikmana, estymator wewnętrznie- zgodny), - sformułowanie warunków i dowód istnienia wewnętrznie-zgodnego oszacowania właściwości makroskopowych dla szerokiej klasy polikryształów z więzami nałożonymi na deformacje na poziomie lokalnym, w przypadku liniowych praw konstytutywnych, - numeryczne studium wpływu więzów nałożonych na deformację wynikających z niespełnienia warunku Taylora, jednokierunkowości mechanizmu bliźniakowania oraz występowania substruktur lamelarnych, na makroskopowa odpowiedz polikryształów metali opisanych nieliniowymi prawami konstytutywnymi, - sformułowanie trójskalowego modelu polikryształu o substrukturze lamelarnej w zakresie duzych deformacji niesprezystych, - opracowanie nowej metody przejscia mikro-makro dla sprezysto-lepkoplastycznych materiałów niejednorodnych bazujacej na zaproponowanej sekwencyjnej linearyzacji nieliniowej odpowiedzi materiału. Rozprawa składa sie z siedmiu rozdziałów. Rozdział pierwszy wskazuje motywacje do podjecia dyskutowanego zagadnienia, omawia cel i zakres podjętych badań oraz ich znaczenie, zawiera również uwagi dotyczące notacji zastosowanej w monografii. Ostatni rozdział podsumowuje najważniejsze rezultaty i płynące z nich wnioski oraz wskazuje kierunki możliwych dalszych badań. Praca uzupełniona jest trzema aneksami omawiającymi istotne narzędzia wykorzystane w modelowaniu: rozkład spektralny i harmoniczny tensora czwartego rzędu typu Hooke'a (Rychlewski [176, 178]) rozwiazanie Eshelby'ego oraz podstawy modelu wewnętrznie-zgodnego (self-consistent) ciał niejednorodnych (Eshelby [49], Hill [67], Li and Wang [120]), a także zagadnienia związane z implementacją numeryczną proponowanego podejścia. Monografię zamyka spis cytowanej literatury.

Słowa kluczowe

micromechanics alloy

stopy metali mikromechanika

Wydawca

Instytut Podstawowych Problemów Techniki PAN

Czasopismo

Prace Instytutu Podstawowych Problemów Techniki PAN

Rocznik

2011

Tom

nr 1

Strony

1--299

Opis fizyczny

Bibliogr. 229 poz.

Twórcy

autor

Kowalczyk-Gajewska K.

Bibliografia

[1] S. R. Agnew and O. Duygulu. Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B. Int. J. Plasticity, 21:1161–1193, 2005.
[2] S.R. Agnew, M.H. Yoo, and C.N. Tom´e. Application of texture simulation to understanding mechanical behavior of Mg and solid solution alloys containing Li or Y. Acta Mater., 49:4277–4289, 2001.
[3] I. Agote, J. Coleto, M. Guti´errez, A. Sargsyan, M. Garc´ia de Cortazar, M.A. Lagos, I.P. Borovinskaya, A.E. Sytschev, V.L. Kvanin, N.T. Balikhina, S.G. Vadchenko, K. Lucas, A. Wisbey, and L. Pambaguian. Microstructure and mechanical properties of gamma TiAl based alloys produced by combustion synthesis + compaction route. Intermetallics, 16: 1310–1316, 2008.
[4] L. Anand. Single-crystal elasto-viscoplasticity: application to texture evolution in polycrystalline metals at large strain. Comput. Methods Appl. Mech. Engrg., 193:5359–5383, 2004.
[5] L. Anand and M. Kothari. A computational procedure for rate independent crystal plasticity. J. Mech. Phys. Solids, 44(4):525–558, 1996.
[6] F. Appel and R. Wagner. Microstructure and deformation of two-phase γ-titanium aluminides. Mater. Sci. Engng. R., 22:187–268, 1998.
[7] R. J. Asaro. Crystal plasticity. J. Applied Mechanics, 50:921–934, 1983.
[8] R. J. Asaro and A. Needleman. Textured development and strain hardening in rate dependent polycrystals. Acta Metall., 33(6):923–953, 1985.
[9] S. Asgari, E. El-Danaf, S. R. Kalidindi, R. D. Doherty, and C. Necker. Strain hardening regimes and microstructural evolution during large strain compression of low stacking fault energy fcc alloys that form deformation twins. Metal. Mater Trans. A, 28:1781–1795, 1997.
[10] A. Bartels, H. Kestler, and H. Clemens. Deformation behavior of diﬀerently processed γ-titanium aluminides. Mater. Sci. Engng. A, 329-331: 153–162, 2002.
[11] Z.S. Basinski, M.S. Szczerba, M. Niewczas, J.D. Embury, and S.J. Basinski. The transformation of slip dislocations during twinning of copperaluminum alloy crystals. Rev. Metall., 94:1037–1043, 1997.
[12] J. G. Berryman. 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.
[13] J. G. Berryman. Bounds and estimates for elastic constants of random polycrystals of laminates. Int. J. Solids Structures, 42:3730–3743, 2005.
[14] M. Berveiller and A. Zaoui. An extension of the self-consistent scheme to the plastically ﬂowing polycrystals. J. Mech. Phys. Solids, 26:325, 1979.
[15] I.J. Beyerlein and C. N. Tom`e. Modeling transients in the mechanical response of copper due to strain path changes. Int. J. Plasticity, 23: 640–664, 2007.
[16] J. F. Bishop and R. Hill. A theory of the plastic distortion of a polycrystalline aggregate under combined stresses. Philos. Mag., 42(Ser. VII): 414–427, 1951.
[17] J. F. Bishop and R. Hill. A theoretical derivation of the plastic properties of a polycrystalline face-centred metal. Philos. Mag., 42(Ser. VII):1298–1307, 1951.
[18] T. B¨ohlke and A. Bertram. Isotropic orientation distributions of cubic crystals. J. Mech. Phys. Solids, 49:2459–2470, 2001.
[19] M. Bornert, R. Masson, P. Ponte Casta˜neda, and A. Zaoui. Secondorder estimates for the eﬀective behaviour of viscoplastic polycrystalline materials. J. Mech. Phys. Solids, 49:2737–2764, 2001.
[20] R. Brenner, R.A. Lebensohn, and O. Castelnau. Elastic anisotropy and yield surface estimates of polycrystals. Int. J. Solids Structures, 46:3018–3026, 2009.
[21] R. A. Brockman. Analysis of elastic-plastic deformation in TiAl polycrystals. Int. J. Plasticity, 19:1749–1772, 2003.
[22] A. Broohm, P. Zattarin, and P. Lipinski. Prediction of mechanical behaviour of inhomogeneous and anisotropic materials using an incremental scheme. Arch. Mech., 6:949–967, 2000.
[23] B. Budiansky. The elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids, 13:223–227, 1965.
[24] B. Budiansky and T. T. Wu. Theoretical prediction of plastic strains of polycrystals. Proc. 4 th U.S. Nat. Congr. Appl. Mech., pages 1175–1185, 1962. ASME, New York.
[25] H. J. Bunge. Texture Analysis in Material Science. Mathematical Methods. London: Butterworths, 1982.
[26] W. T. Burzyński. Studium nad hipotezami wytężenia. Lwów, 1928. (także Dzieła Wybrane, t. I, PWN, Warszawa 1982).
[27] E. P. Busso and G. Cailletaud. On the selection of active slip systems in crystal plasticity. Int. J. Plasticity, 21:2212–2231, 2005.
[28] G. Cailletaud. A micromechanical approach to inelastic behaviour of metals. Int. J. Plasticity, 8:55–73, 1992.
[29] J.L. Chaboche, P. Kanoute, and A. Roos. On the capabilities of mean-ﬁeld approaches for the description of plasticity in metal-matrix composites. Int. J. Plasticity, 21:1409–1434, 2005.
[30] P. Chadwick, M. Vianello, and S. C. Cowin. A new proof that the number of linear elastic symmetries is eight. J. Mech. Phys. Solids, 49:2471–2492, 2001.
[31] M. Cherkaoui, H. Sabar, and M. Berveiller. Micromechanical approach of the coated inclusion problem and applications to composite materials. J. Eng. Mater. Technol., 116:274–278, 1994.
[32] G. Y. Chin, W. F. Hosford, and D. R. Mendorf. Accommodation of constrained deformation in f.c.c. metals by slip and twinning. Proc. R. Soc. A, 309:433–456, 1969.
[33] R. M. Christensen. Viscoelastic properties of heterogeneous media. J. Mech. Phys. Solids, 17:23–41, 1969.
[34] R. M. Christensen. Mechanics of Composite Materials. Dover Publications, 2005.
[35] R.M. Christensen and K. H. Lo. On the eﬀective moduli of composite materials: eﬀect of ﬁber length and geometry at dilute concentrations. Z. Angew. Math. Phys., 24:581, 1979.
[36] J. W. Christian and S. Mahajan. Deformation twinning. Progress in Materials Science, 39:1–157, 1995.
[37] S. C. Cowin and M. M. Mehrabadi. Anisotropic symmetries of linear elasticity. Appl. Mech. Rev., 48(5):247–285, 1995.
[38] S.C. Cowin, G. Yang, and M.M. Mehrabadi. Bounds on the eﬀective anisotropic elastic constants. J. Elasticity, 57:1–24, 1999.
[39] G. de Botton and P. Ponte Casta˜neda. Variational estimates for the creep behavior of polycrystals. Proc. R. Soc. Lond. A, 448:121–142, 1995.
[40] L. Delannay, R. E. Loge, Y. Chastel, and P. Van Houtte. Prediction of intergranular strains in cubic metals using a multisite elastic-plastic model. Acta Mater., 50:5127–5138, 2002.
[41] I. Doghri, L. Adam, and N. Bilger. Mean-ﬁeld homogenization of elastoviscoplastic composites based on a general incrementally aﬃne linearization method. Int. J. Plasticity, 26:219–238, 2010.
[42] B. J. Duggan, M. Hatherly, W. B. Hutchinson, and P. T. Wakeﬁeld. Deformation structures and textures in cold-rolled 70:30 brass. Metal. Sci., 12:343–351, 1978.
[43] M. K. Duszek-Perzyna and P. Perzyna. Analysis of anisotropy and plastic spin eﬀects on localization phenomena. Archive of Applied Mechanics, 68: 352–374, 1998.
[44] G.J. Dvorak and Y. Benveniste. On transformation strains and uniform ﬁelds in multiphase elastic media. Proc. R. Soc. Lond. A, 437:291—-310, 1992.
[45] E. El-Danaf, S. R. Kalidindi, R. D. Doherty, and C. Necker. Deformation texture transition in brass: critical role of micro-scale shear bands. Acta Mater., 48:2665–2673, 2000.
[46] E. El-Danaf, S. R. Kalidindi, and R. D. Doherty. Inﬂuence of deformation path on the strain hardening behavior and microstructure evolution in low sfe fcc metals. Int. J. Plasticity, 17:1245–1265, 2001.
[47] A. El Omri, A. Fennan, F. Sidoroﬀ, and A. Hihi. Elastic-plastic homogenization for layered composites. Eur. J. Mech. A/Solids, 19:585–601, 2000.
[48] A. T. English and G. Y. Chin. On the variation of wire texture with stacking fault energy in f.c.c. metals and alloys. Acta Metall., 13:1013, 1965.
[49] J.D. Eshelby. The determination of the elastic ﬁeld of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. A, 241:376–396, 1957.
[50] F. D. Fischer, T. Schaden, F. Appel, and H. Clemens. Mechanical twins, their development and growth. Eur. J. Mech. A/Solids, 22:709–726, 2003.
[51] S. Forte and M. Vianello. Symmetry classes for elasticity tensors. J. Elasticity, 43:81–108, 1996.
[52] S. Forte and M. Vianello. Functional bases for transversely isotropic and transversely hemitropic invariants of elasticity tensor. Q. Jl. Mech. Appl. Math., 51:543–552, 1998.BIBLIOGRAPHY 273
[53] W. Gambin. Plasticity of crystals with interacting slip systems. Engng. Trans., 39(3–4):303–324, 1991.
[54] W. Gambin. Reﬁned analysis of elastic–plastic crystals. Int. J. Solids Structures, 29(16):2013–2021, 1992.
[55] W. Gambin and K. Kowalczyk. Plastyczność metali (Plasticity of Metals). Oﬁcyna Wydawnicza PW, 2003.
[56] P. Gilormini, M.V. Nebozhyn, and P. Ponte Casta˜neda. Accurate estimates for the creep behaviour of hexagonal polycrystals. Acta Mater., 49: 329–337, 2001.
[57] Z. Hashin. Theory of mechanical behaviour of heterogeneous media. Appl. Mech. Rev., 17:1–9, 1964.
[58] Z. Hashin. The inelastic inclusion problem. Int. J. Engng Sci., 7:11–36, 1969.
[59] Z. Hashin and S. Shtrikman. On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids, 10:335–342, 1962.
[60] Z. Hashin and S. Shtrikman. A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids, 10:343–352, 1962.
[61] K. S. Havner. Finite Plastic Deformation of Crystalline Solids. Cambridge University Press, 1992.
[62] A.V. Hershey. The elasticity of an isotropic aggregate of anisotropic cubic crystals. J. Appl. Mech., 21:236–240, 1954.
[63] E. Herve and A. Zaoui. Modelling the eﬀective behavior of non-linear matrix-inclusion composites. Eur. J. Mech. A/Solids, 9:505–516, 1990.
[64] W. Heye and G. Wassermann. The formation of the rolling textures in fcc metals by slip and twinning. Scripta Metall., 2:205–207, 1968.
[65] R. Hill. Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids, 11:357–372, 1963.
[66] R. Hill. Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids, 13:89–101, 1965.
[67] R. Hill. A self-consistent mechanics of composite materials. J. Mech. Phys. Solids, 13:213–222, 1965.
[68] R. Hill and J. R. Rice. Constitutive analysis of elastic–plastic crystals at arbitrary strain. J. Mech. Phys. Solids, 20:401–413, 1972.
[69] J. W. Hutchinson. Bounds and self-consistent estimates for creep of polycrystalline materials. Proc. R. Soc. Lond. A, 348:101–127, 1976. 7–860, 2001.
[70] J.W. Hutchinson. Elastic-plastic behavior of polycrystalline metals and composites. Proc. Roy. Soc. Lond. A, 319:247–272, 1970.
[71] J.W. Hutchinson. Creep and plasticity of hexagonal polycrystals as related to single crystal slip. Metal. Trans. A, 8A:1465–1469, 1977.
[72] T. Iwakuma and S. Nemat-Nasser. Finite elastic-plastic deformation of polycrystalline metals. Proc. R. Soc. Lond. A, 394:87–119, 1984.
[73] L. Jiang and J. J. Jonas. Eﬀect of twinning on the ﬂow behavior during strain path reversals in two Mg (+Al, Zn, Mn) alloys. Scripta Mater., 58: 803–806, 2008.
[74] L. Jiang, J.J. Jonas, R.K. Mishra, A.A. Luo, A.K. Sachdev, and S. Godet. Twinning and texture development in two Mg alloys subjected to loading along three diﬀerent strain paths. Acta Mater., 55:3899–3910, 2007.
[75] S. R. Kalidindi. Incorporation of twinning in crystal plasticity models. J. Mech. Phys. Solids, 46:267–290, 1998.
[76] S. R. Kalidindi. Modeling anisotropic strain hardening and deformation textures in low stacking fault energy fcc metals. Int. J. Plasticity, 17: 83[1] S. R. Agnew and O. Duygulu. Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B. Int. J. Plasticity, 21:1161–1193, 2005.
[77] S. R. Kalidindi and L. Anand. Macroscopic shape change and evolution of crystallographic texture in pre-textured fcc metals. J. Mech. Phys. Solids, 42:459–490, 1994.
[78] S. R. Kalidindi, C. A. Bronkhorst, and L. Anand. Crystallographic texture evolution in bulk deformation processing of fcc metals. J. Mech. Phys. Solids, 40:537–569, 1992.
[79] I. Karaman, H. Sehitoglu, A. J. Beaudoin, Y. I. Chumlyakov, H. J. Maier, and C. N. Tom´e. Modeling the deformation behavior of Hadﬁeld steel single and polycrystals due to twinning and slip. Acta Mater., 48:2031– 2047, 2000.
[80] I. Karaman, H. Sehitoglu, Y. I. Chumlyakov, H. J. Maier, and I. V. Kireeva. The eﬀect of twinning and slip on the Bauschinger eﬀect of Hadﬁeld steel single crystals. Metal. Mater. Trans. A, 32A:695–706, 2001.
[81] G. C. Kaschner, C. N. Tom´e, I. J. Beyerlein, S. C. Vogel, D. W. Brown, and R. J. McCabe. Role of twinning in the hardening response of zirconium during temperature reloads. Acta Mater., 54:2887–2896, 2006.
[82] G. C. Kaschner, C. N. Tom´e, R. J. McCabe, A. Misra, S. C. Vogel, and D. W. Brown. Exploring the dislocation/twin interactions in zirconium. Mater. Sci. Eng. A, 463:122–127, 2007.
[83] R. Kiryk. Micromechanical model of viscoplastic polycrystalline materials (in Polish). PhD Thesis 38/1992, IFTR Reports, 1992.
[84] R. Kiryk and H. Petryk. A self-consistent model of rate dependent plasticity of polycrystals. Arch. Mech., 50:247–263, 1998.
[85] K. Kishida, H. Inui, and M. Yamaguchi. Deformation of PST crystals of a TiAl/Ti3Al two-phase alloy at 1000oC. Intermetallics, 7:1131–1139, 1999.
[86] G. Kneer. Die elastischen Konstanten quasiisotroper Vielkristallaggregate. Physica Status Solidi (b), 3:K331–K335, 1963.
[87] G. Kneer. Uber die berechnung der Elastizitatsmoduln vielkristalliner Aggregate mit textur. Phys. Stat. Sol., 9, 1965.
[88] U. F. Kocks, C. N. Tom´e, and H.-R. Wenk. Texture and Anisotropy. Cambridge University Press, II edition, 2000.
[89] U.F. Kocks and H. Chandra. Slip geometry in partially constrained deformation. Acta Metall., 30:695, 1982.
[90] U.F. Kocks and H. Mecking. Physics and phenomenology of strain hardening: the FCC case. Progress in Materials Science, 48:171–273, 2003.
[91] J. Koike. Enhanced deformation mechanisms by anisotropic plasticity in polycrystalline Mg alloys at room temperature. Metal. Mater. Trans. A, 36A:1689–1696, 2005.
[92] Y. Koizumi, T. Nakano, and Y. Umakoshi. Plastic deformation and fracture behaviour of Ti3Al single crystals deformed at high temperature under cyclic loading. Acta Mater., 47:2019–2029, 1999.
[93] R. Kouddane, A. Molinari, and G.R. Canova. Proceedings of the International Seminar MECAMAT’91, Fontenblau/France/7-8 August 1991, chapter Self-consistent modelling of heterogeneous viscoelastic and elastoviscoplastic materials, pages 129–141. A.A.Balkema, 1993.
[94] K. Kowalczyk. Evolution of plastic anisotropy of the polycrystalline materials in large deformation processes. Engng. Trans., 49(4):537–571, 2001.
[95] K. Kowalczyk and W. Gambin. Model of plastic anisotropy evolution with texture-dependent yield surface. Int. J. Plasticity, 20:19–54, 2004.
[96] K. Kowalczyk and J. Ostrowska-Maciejewska. Energy-based limit conditions for transversally isotropic solids. Arch. Mech., 54(5-6):497–523, 2002.
[97] K. Kowalczyk-Gajewska. Micromechanical modelling of polycrystalline materials at large plastic deformations. (Modelowanie materiałów polikrystalicznych w zakresie dużych deformacji plastycznych). CD Proceedings of I Congress of Polish Mechanics, 2007. In Polish.
[98] K. Kowalczyk-Gajewska. Multiscale modelling of intermetallics of lamellar microstructure at ﬁnite strains. IUTAM Symposium Multi-Scale Plasticity of Crystalline Materials, Nov. 5-9, 2007. Book of Abstracts, pages 79–80, 2007.
[99] K. Kowalczyk-Gajewska. Bounds and self-consistent estimates of overall properties for random polycrystals described by linear constitutive laws. Arch. Mech., 61(6):475–503, 2009.
[100] K. Kowalczyk-Gajewska. Modelling of texture evolution in metals accounting for lattice reorientation due to twinning. Eur. J. Mech. Solids/A, 29:28–41, 2010.
[101] K. Kowalczyk-Gajewska. Comparison of three-scale and two-scale modelling of polycrystalline materials with lamellar substructure. CD Proceedings of IV European Conference on Computational Mechanics, ECCM 2010, Paris, May 16-21, 2010.
[102] K. Kowalczyk-Gajewska and J. Ostrowska-Maciejewska. Mechanics of the 21st Century, Proceedings of the 21st International Congress of Theoretical and Applied Mechanics Warsaw, Poland, 15-21 August 2004, chapter On the invariants of the elasticity tensor for orthotropic materials. Springer (e-book), 2004.
[103] K. Kowalczyk-Gajewska and J. Ostrowska-Maciejewska. The inﬂuence of internal restrictions on the elastic properties of anisotropic materials. Arch. Mech, 56:205–232, 2004.
[104] K. Kowalczyk-Gajewska and J. Ostrowska-Maciejewska. Review on spectral decomposition of Hooke’s tensor for all symmetry groups of linear elastic material. Engng. Trans., 57:145–183, 2009.
[105] K. Kowalczyk-Gajewska and A. Roos. Intermetallics with lamellar microstructure: ﬁnite strain modelling. Modelling of Microstructure Media II. Abstracts. Łódź, 19.09.2008, pages 17–19, 2008.
[106] K. Kowalczyk-Gajewska, Z. Mróz, and R. B. Pęcherski. Micromechanical modelling of polycrystalline materials under non-proportional deformation paths. Arch. Metall. Mater., 52:181–192, 2007.
[107] E. Kroner. Berechnung der elastischen Konstanten des Vielkristalls den Konstanten des Einkristalls. Zeitschrift fur Physik A, 151:504–518, 1958.
[108] E. Kr¨oner. Zur plastischen Verformung des Vielkristalls. Acta Met., 9: 155–161, 1961.
[109] N. Lahellec and P. Suquet. On the eﬀective behavior of nonlinear inelastic composites: I. Incremental variational principles. J. Mech. Phys. Solids, 55:1932–1963, 2007.
[110] A. Lasalmonie. Intermetallics: Why is it so diﬃcult to introduce them in gas turbine engines? Intermetallics, 14:1123–1129, 2006.
[111] N. Laws and R. McLaughlin. Self-consistent estimates for viscoelastic creep compliance of composite materials. Proc. R. Soc. Lond. A, 359: 251–273, 1978.
[112] R. A. Lebensohn and C. N. Tom´e. A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconium alloys. Acta Metall. Mater., 41: 2611–2624, 1993.
[113] R. A. Lebensohn, H. Uhlenhut, C. Hartig, and H. Mecking. Plastic ﬂow of γ-TiAl-based polysynthetically twinned crystals: micromechanical modelling and experimental validation. Acta Mater., 46:229–236, 1998.
[114] R.A. Lebensohn, Y. Liu, and P. Ponte Casta˜neda. On the accuracy of the self-consistent approximation for polycrystals: comparison with full-ﬁeld numerical simulations. Acta Mater., 52:5347–5361, 2004.
[115] E. H. Lee. Elastic–plastic deformation at ﬁnite strains. J. Appl. Mech., 36(1), 1969.
[116] T. Leﬀers and D. Juul Jensen. The relation between texture and microstructure in rolled fcc materials. Text. Microstruct., 14–18:933–952, 1991.
[117] T. Leﬀers and R.K. Ray. The brass-type texture and its deviation from the copper-type texture. Progress in Materials Science, 54:351–396, 2009.
[118] T. Leﬀers and P. Van Houtte. Calculated and experimental orientation distributions of twin lamellae in rolled brass. Acta metall., 37:1191–1198, 1989.
[119] V.M. Levin. On the coeﬃcients of thermal expansion of heterogeneous media. Mekh. Tverd. Tela (In Russian), 88, 1968.
[120] S. Li and G. Wang. Introduction to Micromechanics and Nanomechanics. World Scientiﬁc, 2008.
[121] P. Lipinski, J. Krier, and M. Berveiller. Elastoplasticite des metaux en grandes deformations: Comportement global et evolution de la structure interne. Rev Phys. Appl., 25:361, 1990.
[122] X.Y. Lou, M. Li, R.K. Boger, S.R. Agnew, and R.H. Wagoner. Hardening evolution of AZ31B Mg sheet. Int. J. Plasticity, 23:44–86, 2007.
[123] S. Mahajan and G.Y. Chin. Twin-slip, twin-twin and slip-twin interactions in Co-8 wt.% Fe alloy single crystals. Acta Metall., 21:173–179, 1973.
[124] E. B. Marin and P. R. Dawson. Elastoplastic ﬁnite element analyses of metal deformations using polycrystal constitutive model. Comput. Methods Appl. Mech. Engrg., 165:23–41, 1998.
[125] W. T. Marketz, F. D. Fischer, F. Kauﬀmann, G. Dehm, T. Bidlingmaier, A. Wanner, and H. Clemens. On the role of twinning during room temperature deformation of γ-TiAl based alloys. Mater. Sci. Eng. A, 329-331: 177–183, 2002.
[126] R. Masson and A. Zaoui. Self-consistent estimates of the rate-dependent elasto-plastic behaviour of polycrystalline materials. J. Mech. Phys. Solids, 47:1543–1568, 1999.
[127] R. Masson, M. Bornert, P. Suquet, and A. Zaoui. An aﬃne formulation for the prediction of the eﬀective properties of non-linear composites and polycrystals. J. Mech. Phys. Solids, 48:1203–1227, 2000.
[128] K. K. Mathur and P. R. Dawson. On modelling the development of the crystallograﬁc texture in a bulk forming processes. Int. J. Plasticity, 5: 67–94, 1989.
[129] R. J. McCabe, G. Proust, E.K. Cerreta, and A. Misra. Quantitative analysis of deformation twinning in zirconium. Int. J. Plasticity, 25:454– 472, 2009.
[130] R. McLaughlin. A study of the diﬀerential scheme for composite materials. Int. J. Engrg. Sci., 15:237–244, 1977.
[131] H. Mecking, U. F. Kocks, and Ch. Hartig. Taylor factors in materials with many deformation modes. Scripta Materialia, 35:465–471, 1996.
[132] M. Mehrabadi, S.C. Cowin, and J. Jaric. Six-dimensional orthogonal tensor representation of the rotation about an axis in three dimensions. Int. J. Solids Structures, 32:439–449, 1995.
[133] S. Mercier and A. Molinari. Homogenization of elastic-viscoplastic heterogeneous materials: Self-consistent and Mori-Tanaka schemes. Int. J. Plasticity, 25:1024–1048, 2009.
[134] S. Mercier, N. Jacques, and A. Molinari. Validation of an interaction law for the Eshelby inclusion problem in elasto-viscoplasticity. Int. J. Solids Structures, 42:1923–1941, 2005.
[135] A. Molinari. Large plastic deformation of crystalline aggregates, chapter Self-consistent estimates of plastic and viscoplastic polycrystalline materials, pages 173–245. Springer Wien New York, 1997.
[136] A. Molinari. Averaging models for heterogeneous viscoplastic and elastic viscoplastic materials. Transactions of the ASME, 124:62–70, 2002.
[137] A. Molinari and L. Toth. Tuning a self-consistent viscoplastic model by ﬁnite element results I: Modelling. Acta Metall. Mater., 42:2453–2458, 1994.
[138] A. Molinari, G.R. Canova, and S. Ahzi. Self-consistent approach of the large deformation polycrystal visco-plasticity. Acta Metall., 35:2983–2994, 1987.
[139] A. Molinari, S. Ahzi, and R. Kouddane. On the self-consistent modeling of elastic-plastic behavior of polycrystals. Mech. Mater., 26:43–62, 1997.
[140] T. Mori and K. Tanaka. Average stress in matrix and average elastic energy of materials with misﬁtting inclusions. Acta Metall., 21:571–574, 1973.
[141] Z. Mróz, editor. State of the Art Report on Knowledge-based Multicomponent Materials, volume 1, Intermetallics: Properties, Modelling and Applications. KMM, 2006.
[142] T. Nakano, B. Ogawa, Y. Koizumi, and Y. Umakoshi. Plastic deformation behaviour and dislocation structure in Ti3Al single crystals cyclically deforming by prism slip. Acta Mater., 46:4311–4324, 1998.
[143] M. V. Nebozhyn, P. Gilormini, and P. Ponte-Casta˜neda. Variational selfconsistent estimates for cubic viscoplastic polycrystals: the eﬀects of grain anisotropy and shape. J. Mech. Phys. Solids, 49:313–340, 2001.
[144] S. Nemat-Nasser. Averaging theorems in ﬁnite deformation plasticity. Mech. Mater., 31:493–523, 1999.
[145] S. Nemat-Nasser and M. Hori. Micromechanics: overall properties of heterogeneous materials. North-Holland Elsevier, 1999.
[146] S. Nemat-Nasser, L. Ni, and T. Okinaka. A constitutive model for fcc crystals with application to polycrystalline ofhc copper. Mech. Mater., 30:325–341, 1998.
[147] S. Nikolov, R.A. Lebensohn, and D. Raabe. Self-consistent modeling of large plastic deformation, texture and morphology evolution in semicrystalline polymers. J. Mech. Phys. Solids, 54:1350–1375, 2006.
[148] KMM NoE. DNRT 1-1.2-3(a) Complete results of experimental approach and characterization. Internal report, LMT Cachan, University of Metz, IPPT PAN and IMZ PAN, 2007.
[149] KMM NoE. DNRT 1-1.2-4(c) Texture measurements and microstructure determination of TiAl, FeAl materials processed by partners. Internal report, Universita Politecnica delle Marche (A. Manescu and A. Giuliani), 2007.
[150] KMM NoE. DNRT 1-1.3-6(c) Comparison of the predictions of the ONERA and IPPT models concerning the macroscopic mechanical response of TiAl (prepared by K. Kowalczyk-Gajewska and H. Petryk (IPPT PAN), Arjen Roos (ONERA), S. Mercier (University of Metz)). Internal report, 2008.
[151] R. W. Ogden. Non-linear Elastic Deformations. Dover Publication, Mineola, New York, 2 edition, 1997.
[152] H. Ogi, S. Kai, H. Ledbetter, R. Tarumi, M. Hirao, and K. Takashima. Titanium’s high-temperature elastic constants through the hcp–bcc phase transformation. Acta Mater., 52:2075–2080, 2004.
[153] J. Ostrowska-Maciejewska and J. Rychlewski. Generalized proper states for anisotropic elastic materials. Arch. Mech., 53(4-5):501–518, 2001.
[154] V. Paidar and M. Yamaguchi. Constrained deformation of a lamellar structure. Mater. Sci. Engng A, 462:460–464, 2007.
[155] A. Paquin, H. Sabar, and M. Berveiller. Integral formulation and selfconsistent modelling of elastoviscoplastic behavior of heterogeneous materials. Arch. Appl. Mech., 69:14–35, 1999.
[156] B. Peeters, M. Seefeldt, C. Teodosiu, S.R. Kalidindi, P. Van Houtte, and E. Aernoudt. Work-hardening/softening behaviour of b.c.c. polycrystals during changing strain paths: I. An integrated model based on substructure and texture evolution, and its prediction of the stress-strain behavior of an IF steel during two-stage strain paths. Acta Mater., 49:1607–1619, 2001.
[157] L. Peselnick and R. Meister. Variational method of determining eﬀective moduli of polycrystals: (A) hexagonal symmetry, (B) trigonal symmetry. J. Appl. Phys., 60:3120–3124, 1986.
[158] H. Petryk. General conditions for uniqueness in materials with multiple mechanisms of inelastic deformation. J. Phys. Mech. Solids, 48:367–396, 2000.
[159] H. Petryk and S. Stupkiewicz. A quantitative model of grain reﬁnement and strain hardening during severe plastic deformation. Mater. Sci. Engng. A, 444:214–219, 2007.
[160] D.C. Pham. Elastic moduli of perfectly random polycrystalline aggregate. Philos. Mag. A, 76:31–44, 1997.
[161] D.C. Pham. Asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals. Int. J. Solids Struct., 40: 4911–4924, 2003.
[162] O. Pierard and I. Doghri. An enhanced aﬃne formulation and the corresponding numerical algorithms for the mean-ﬁeld homogenization of elasto-viscoplastic composites. Int. J. Plasticity, 22:131–157, 2006.
[163] O. Pierard, J. LLorca, J. Segurado, and I. Doghri. Micromechanics of particle-reinforced elasto-viscoplastic composites: Finite element simulations versus aﬃne homogenization. Int. J. Plasticity, 23:1041–1060, 2007.
[164] P. Ponte Casta˜neda. Exact second-order estimates for the eﬀective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids, 44:827–862, 1996.
[165] P. Ponte Casta˜nenda. The eﬀective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids, 39:45–71, 1991.
[166] P. Ponte Casta˜nenda and P. Suquet. Nonlinear composites. Adv. Appl. Mech., 34:171–302, 1998.
[167] J. Pospiech. Eﬀects in the texture and microstructure in some metals of cubic and hexagonal symmetry caused by the change of the rolling direction. Acta Metal. Mater., 53:83–87, 2008.
[168] J. Pospiech, M. Ostaﬁn, and R. Schwarzer. The eﬀect of rolling geometry on the texture and microstructure in AZ31 and copper. Acta Metal. Mater., 51:37–42, 2006.
[169] G. Proust, C. N. Tom´e, and G. C. Kaschner. Modeling texture, twinning and hardening evolution during deformation of hexagonal materials. Acta Mater., 55:2137–2148, 2007.
[170] G. Proust, C.N. Tom´e, A. Jain, and S.R. Agnew. Modeling the eﬀect of twinning and detwinning during strain-path changes of magnesium alloy AZ31. Int. J. Plasticity, 25:861–880, 2009.
[171] Y.U. Qui and G.J. Weng. Elastic constants of a polycrystal with transversally isotropic grains, and the inﬂuence of precipitates. Mech. Mater., 12:1–15, 1991.
[172] B. Raniecki and Z. Mróz. Inelastic Solids and Structures (Antoni Sawczuk memorial volume), chapter On the strain-induced anisotropy and texture in rigid-plastic solids. Prineridge Press, Swansea, U.K., 1990.
[173] A. Roos, J.-L. Chaboche, L. Gelebart, and J. Crepin. Multiscale modeling of titanium aluminides. Int. J. Plasticity, 20:811—-830, 2004.
[174] Y. Rougier, C. Stolz, and A. Zaoui. Representation spectrale en visco`elasticit´e lin´eaire de mat´eriaux h´et´erog`enes. C. R. Acad. Sci. Paris, 316, Serie II:1517–1522, 1993.
[175] Y. Rougier, C. Stolz, and A. Zaoui. Self-consistent modelling of elasticviscoplastic polycrystals. C. R. Acad. Sci. Paris, 318, Serie II:145–151, 1994.
[176] J. Rychlewski. CEIIINOSSSTTUV. Mathematical structure of elastic bodies. Technical Report 217, Inst. Mech. Probl. USSR Acad. Sci., Moskva, 1983. In Russian.
[177] J. Rychlewski. Unconventional approach to linear elasticity. Arch. Mech., 47(2):149–171, 1995.
[178] J. Rychlewski. A qualitative approach to Hooke’s tensors. Part I. Arch. Mech., 52:737–759, 2000.
[179] J. Rychlewski. Elastic waves under unusual anisotropy. J. Mech. Phys. Solids, 49:2651–2666, 2001.
[180] J. Rychlewski. A qualitative approach to Hooke’s tensors. Part II. Arch. Mech., 53:45–63, 2001.
[181] H. Sabar, M. Berveiller, V. Favier, and S. Berbenni. A new class of micro-macro models for elastic-viscoplastic heterogeneous materials. Int. J. Solids Structures, 39:3257–3276, 2002.
[182] A. A. Salem, S. R. Kalidindi, and S. L. Semiatin. Strain hardening due to deformation twinning in α-titanium: Constitutive relations and crystal plasticity modeling. Acta Mater., 53:3495–3502, 2005.
[183] A.A. Salem, S.R. Kalidindi, R.D. Doherty, and S.L. Semiatin. Strain hardening due to deformation twinning in α-Titanium: Mechanisms. Metal. Mater. Trans. A, 37A:259–268, 2006.
[184] W. Schillinger, B. Lorenzen, and A. Bartels. Anisotropic mechanical behavior of textured γ-TiAl caused by the directionality of twinning. Mater. Sci. Engng. A, 329-331:644–648, 2002.BIBLIOGRAPHY 283
[185] J.C. Simo and T.J.R. Hughes. Computational Inelasticity. Springer, 1998.
[186] A. Staroselsky and L. Anand. Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning. J. Mech. Phys. Solids, 46: 671–696, 1998.
[187] A. Staroselsky and L. Anand. A constitutive model for hcp materials deforming by slip and twinning: application to magnesium alloy AZ31B. Int. J. Plasticity, 19:1843–1864, 2003.
[188] R. Staroszczyk. A multi-grain model for migration recrystallization in polar ice. Arch. Mech., 61:259–282, 2009.
[189] S. Stupkiewicz. Micromechanics of contact and interphase layers. Habilitation Thesis 2/2005, IFTR Reports, 2005.
[190] S. Stupkiewicz and H. Petryk. Modelling of laminated microstructures in stress-induced martensitic transformations. J. Phys. Mech. Solids, 50: 2303–2331, 2002.
[191] S. Stupkiewicz and H. Petryk. A bi-crystal aggregate model of pseudoelastic behaviour of shape-memory alloy polycrystals. Int. J. Mech. Sci., 52:219–228, 2010.
[192] P. Suquet. Homogenization Techniques for Composite Media, chapter Elements of homogenization for inelastic solid mechanics, pages 193–278. Springer-Verlag, Berlin, 1985.
[193] M. S. Szczerba, T. Bajor, and T. Tokarski. Is there a critical resolved shear stress for twinning in face-centered cubic crystals? Philos. Mag., 84:481–502, 2004.
[194] D.R.S. Talbot and J.R. Willis. Variational principles for inhomogeneous non-linear media. IMA J. Appl. Math., 35:39–54, 1985.
[195] K. Tanaka. Single-crystal elastic constants of γ-TiAl. Phil. Mag. Lett., 73:71–78, 1996.
[196] K. Tanaka and M. Koiwa. Single-crystal elastic constants of intermetallic compounds. Intermetallics, 4:29–39, 1996.
[197] K. Tanaka, K. Okamoto, H. Inui, Y. Minonishi, M. Yamaguchi, and M. Koiwa. Elastic constants and their temperature dependence for the intermetallic compound Ti3Al. Philos. Mag. A, 73:1475–1488, 1996.
[198] G.P. Tandon and G.J. Weng. A theory of particle-reinforced plasticity. ASME J. Appl. Mech., 55:126–135, 1988.
[199] G. I. Taylor. Plastic strain in metals. Journal of Institute of Metals, 62: 307–324, 1938.284 BIBLIOGRAPHY
[200] C. N. Tom´e, R. A. Lebensohn, and U. F. Kocks. A model for texture development dominated by deformation twinning: application to zirconium alloy. Acta Metal. Mater., 39:2667–2680, 1991.
[201] C.N. Tom´e and R.A. Lebensohn. Manual for Code Visco-Plastic SelfConsistent (VPSC). Version 7b. Technical report, Los Alamos National Labolatory, May 2, 2007.
[202] C.N. Tom´e, R.A. Lebensohn, and C.T. Necker. Mechanical anisotropy and grain interaction in recrystallized aluminum. Metal. Mater. Trans. A, 33A:2635–2648, 2002.
[203] P. Van Houtte. Simulation of the rolling texture and shear texture of brass by the Taylor theory adapted for mechanical twinning. Acta Metal., 26: 591–604, 1978.
[204] P. Van Houtte. On the equivalence of the relaxed Taylor theory and the Bishop-Hill theory for partially constrained plastic deformation of crystal. Mater. Sci. Eng., 55:69–77, 1982.
[205] P. Van Houtte, L. Delannay, and I. Samajdar. Quantitative prediction of cold rolling textures in low carbon steel by means of the LAMEL model. Textures and Microtextures, 31:109–149, 1999.
[206] P. Van Houtte, L. Delannay, and S. R. Kalidindi. Comparison of two grain interaction models for polycrystal plasticity and deformation texture prediction. Int. J. Plasticity, 18:359–377, 2002.
[207] P. Van Houtte, S. Li, M. Seefeldt, and L. Delannay. Deformation texture prediction: from the Taylor model to the advanced Lamel model. Int. J. Plasticity, 21:589–624, 2005.
[208] L. J. Walpole. Advances in Applied Mechanics, volume 21, chapter Elastic Behavior of Composite Metarials: Theoretical Foundations, pages 169– 236. 1981.
[209] H. Wang, P.D.Wu, C.N.Tom´e, and Y. Huang. A ﬁnite strain elastic–viscoplastic self-consistent model for polycrystalline materials. J. Mech. Phys. Solids, 58:594–612, 2010.
[210] Y.D. Wang, A. Vadon, and J.J. Heizmann. Room temperature compression textures and deformation mechanisms of TiAl-46Al-2V alloy. Mater. Sci. Eng., A222:70–75, 1997.
[211] Y.N. Wang and J.C. Huang. The role of twinning and untwinning in yielding behavior in hot-extruded Mg–Al–Zn alloy. Acta Mater., 55:897– 905, 2007.
[212] H. Watanabe, T. Mukai, M. Sugioka, and K. Ishikawa. Elastic and damping properties from room temperature to 673 K in an AZ31 magnesium alloy. Scripta Mater., 51:291–295, 2004.
[213] J.P. Watt. Hashin-Shtrikman bounds on the eﬀective elastic moduli of polycrystals with trigonal (3, 3) and tetragonal (4 , 4¯, 4m) symmetry. J. Appl. Phys., 36:2879–2884, 1965.
[214] G.J. Weng. Self-consistent determination of time-dependent behavior of metals. ASME J. Appl. Mech., 48:41–46, 1981.
[215] M. Werwer and A. Cornec. Numerical simulation of plastic deformation and fracture in polysynthetically twinned (PST) crystals of TiAl. Comput. Mater. Sci., 19:97–107, 2000.
[216] M. Werwer and A. Cornec. The role of superdislocations for modeling plastic deformation of lamellar TiAl. Int. J. Plasticity, 22:1683–1698, 2006.
[217] M. Werwer, R. Kabir, A. Cornec, and K.-H. Schwalbe. Fracture in lamellar TiAl simulated with the cohesive model. Engng. Fracture Mech., 74: 2615–2638, 2007.
[218] K. Wierzbanowski. Computer simulation of rolling texture formation in hcp and orthorombic metals. Scripta Metall., 13:795, 1978.
[219] K. Wierzbanowski, A. Baczmański, and P. Lipinski. Niejednorodności odkształcenia w procesach przeróbki plastycznej. Seminarium poświęcone 70. rocznicy urodzin Profesora Zdzisława Jasieńskiego., chapter Modele sprężysto-plastyczne odkształcenia polikryształu i ich zastosowania., pages 173–190. IMIM PAN, 2005.
[220] J. R. Willis. Advances in Applied Mechanics, volume 21, chapter Variational and Related Methods for the Overall Properties of Composites, pages 2–79. 1981.
[221] J.R. Willis. Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids, 25:185–202, 1977.
[222] S. Wolfram. The Mathematica Book, 5th ed. Wolfram Media, 2003.
[223] X. Wu, S.R. Kalidindi, C. Necker, and A.A. Salem. Prediction of crystallographic texture evolution and anisotropic stress–strain curves during large plastic strains in high purity α-titanium using a Taylor-type crystal plasticity model. Acta Mater., 55:423–432, 2007.
[224] F. Xu, R.A. Holt, and M.R. Daymond. Modeling lattice strain evolution during uniaxial deformation of textured Zircaloy-2. Acta Mater., 56:3672– 3687, 2008.
[225] J. Xu and S-Q. Shi. Investigation of mechanical properties of ε-zirconium hydride using micro- and nano-indentation techniques. J. Nucl. Mater., 327:165–170, 2004.
[226] S. Yamanaka, K. Yoshioka, M. Uno, M. Katsura, H. Anada, T. Matsuda, and S. Kobayashi. Thermal and mechanical properties of zirconium hydride. J. Alloys Compunds, 293-295:23–29, 1999.
[227] M.H. Yoo and C.L. Fu. Physical constants, deformation twinning, and microcracking of titanium aluminides. Metal. Mater. Trans. A, 29A:49–63, 1998.
[228] K. Yoshida, R. Brenner, B. Bacroix, and S. Bouvier. Eﬀect of regularization of Schmid law on self-consistent estimates for rate-independent plasticity of polycrystals. Eur. J. Mech. A/Solids, 28:905–915, 2009.
[229] C. Zener. Elasticite et Anelasticite des Metaux. Dunod, Paris, 1955.

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