Micromechanicsk of contact and interphase layers

• Autorzy: Stupkiewicz St.

Warianty tytułu

PL

Mikromechanika warstw kontaktowych i międzyfazowych

• Języki publikacji: EN

Abstrakty

EN

This thesis presents several aspects of micromechanics of interfaces, interface layers, and materials with propagating phase transformation fronts. Two application areas are addressed, namely contact of rough bodies and martensitic microstructures in shape memory alloys. The objective is to develop micromechanical modelling tools suitable for the analysis of this class of problems and also to provide several specific applications motivated by scientific and technological interest. Chapter 1 has an introductory character and outlines the micromechanical point of view adopted in this thesis, as well as the scope and the objectives of the work. Chapter 2 presents selected basic concepts, definitions, and relationships which are frequently referred to throughout the thesis. The interior-exterior decomposition and the compatibility conditions are introduced. Furthermore, elements of homogenization with the spccification for simple laminates, including explicit micro-macro transition relations, are provided as a basis for thc micromechanical analysis of evolving martensitic microstructures. The introduction to homogenization serves also as a reference for the micromechanical analysis of boundary layers carried out in Chapters 4 and 5. The first part of the thesis, Chapters 3, 4, 5, and 6, is concerned with the micromechanics of contact interactions of rough bodies. Chapter 3 is mostly devoted to modelling of evolution of real contact area in metal forming processes. An introductory discussion of thin homogeneous laycrs is also provided and constitutive relations in a mixed form are introduced for elastic, elasto-plastic, and rigid-plastic material models. This formalism is next applied to derive a phenomenological model of real contact area evolution which accounts for the effect of macroscopic plastic deformations on asperity flattening. The phenomena and effects discussed in Section 3.3 constitute one of thc motivations of the subsequent micromechanical analysis of contact boundary layers, which is presented in Chapters 4, 5, and 6. Boundary layers induced by micro-inhomogeneous boundary conditions are studied in Chapter 4. The notion of the macro- and micro-scale is introduced and the method of asymptotic expansions is applied in order to derive the equations of the corresponding macroscopic and microscopic boundary value problems. While contact of rough bodies is the main interest of this part of the thesis, two simpler, but closely related, cases of prescribedv micro-inhomogeneous tractions and displacements are considered in detail, in addition to the case of frictional contact of a rough body with a rigid and smooth obstacle. In Chapter 5, a micromechanical framework is developed for the analysis of the boundary layers discussed in Chapter 4. A special averaging operation is defined, and several properties of the corresponding averages of the boundary layer fields are derived. As an illustration, the framework is applied to analyse the boundary layer induced in an elastic body by a sinusoidal fluctuation of surface traction. The finite element analysis of contact boundary layers, carried out in Chapter 6, concludes the first part of the thesis. Implementation issues are discussed, and two representative asperity interaction problems of asperity ploughing and asperity flattening in elasto-plastic solids are analyzed. In the latter case, a real three-dimensional topography of a sand-blased surface is considered, and experimental verification of the developed finite element model is performed. In the numerical examples, attention is paid to the interaction of the homogeneous macroscopic deformation with the deformation inhomogeneities within the boundary layer, and the related effects of the macroscopic in-plane strain on the macroscopic contact response are studied. Chapter 7, 8, and 9, constituting the second part of the thesis, are concerned with moddeling of martensitic microstructures in shape memory alloys (SMA). Chapter 7 is a brief introduction to the topic. Basic concepts and phenomena are introduced, and the crystallographic theory of martensite is outlined for both the internally twinned and internally faulted martensites. In Chapter 8, micromechanical modelling of evolving laminated microstructures in SMA single crystals is carried out. The martensitic transformation under stress is assumed to proceed by the nucleation and growth of parallel martensitic plates. The corresponding micromechanical model is developed by combining a micro-macro transition scheme with a rateindependent phase transformation criterion based on the local thermodynamic driving force on the phase transormation front. Macroscopic constitutive rate-equations are derived for the case of an evolving rank-one laminate. Finally, the macroscopic pseudoelastic response of single crystals of Cu-based shape memory alloys is studied along with the corresponding evolution of the microstructure, including the effects related to detwinning. A simple model of the stress-induced martensitic transformation in macroscopically adiabatic conditions is also discussed. In the modelling and in the applications, full account is taken for distinct elastic anisotropy of the phases which leads to the redistribution of internal stresses and to the related softening effect during progressive transformation. In Chapter 9, an approach is developed for prediction of the microstructure of stress-induced martensitic plates at the initial instant of transformation. Microstructural parameters and the transformation stress are obtained as a solution of the minimization problem for load multiplier, and the predicted microstructures are, in general, different from those following from the classical crystallographic theory of martensite. The approach is then applied for CuZnAl single crystals undergoing stress-induced cubicto-monoclinic transformation, and the effects of the stacking fault energy, loading direction, and temperature on the predicted microstructures are studied.

PL

Mikromechanika materiałów niejednorodnych pozwala przewidywać ich właściwości makroskopowe na podstawie znanych właściwości, mikrostruktury oraz mechanizmów deformacji w skali mikro. Jest więc atrakcyjnym i efektywnym narzędziem nowoczesnej mechaniki materiałów. Niniejsza rozprawa habilitacyjna jest poświęcona mikromechanicznemu modelowaniu warstw i powierzchni. W mechanice ośrodków ciągłych makroskopową powierzchnię o zerowej grubości można zazwyczaj traktować w skali mikro jako warstwę o grubości niezerowej, charakteryzującą się pewną mikrostrukturą. Celem analizy mikromechanicznej jest wtedy określenie makroskopowych właściwości takiej powierzchni w zależności od jej mikrostruktury i zjawisk zachodzących w skali mikro. W pierwszej części niniejszej rozprawy, w rozdziałach 3-6, powyższe podejście mikromechaniczne wykorzystano do analizy warstw kontaktowych. Mikrostrukturę warstwy kontaktowej tworzą w tym przypadku chropowatość oddziałujących powierzchni i związane z nią niejednorodności deformacji w warstwie wierzchniej . Mikromechanika powierzchni obejmuje również prowadzoną w różnych skalach analizę materiałów, które zawierają powierzchnie (warstwy) międzyfazowe i w których te powierzchnie zasadniczo wpływają na makroskopowe właściwości tych materiałów. Z taką sytuacją mamy do czynienia, na przykład, w materiale podlegającym przemianie fazowej, w której trakcie następuje propagacja frontów przemiany fazowej i związana z nią ewolucja mikrostruktury materiału. Analizie mikromechanicznej i modelowaniu ewolucji warstwowych struktur martenzytycznych, naprężeniowo indukowanych w kryształach stopów z pamięcią kształtu, poświęcona jest druga część niniejszej rozprawy, rozdziały 7-9. Unikalne zachowanie i właściwości tych materiałów, podlegających martenzytycznej przemianie fazowej, wynikają ze zjawisk zachodzących w skali mikro na frontach przemiany fazowej. Podstawowym celem niniejszej pracy jest opracowanie metod mikromechanicznej analizy warstw i powierzchni. Podejście mikromechaniczne jest niezwykle atrakcyjne, gdyż pozwala przewidywać właściwości makroskopowe przy wykorzystaniu znanych i lepiej określonych praw i właściwości w skali mikro. Celem pracy jest również rozwiązanie, z wykorzystaniem opracowanych narzędzi, konkretnych zagadnień z zakresu stosowanej mechaniki materiałów. Zastosowania opisane w pracy dotyczą dwóch obszarów tematycznych (mikromechanika warstw kontaktowych oraz ewolucja mikrostruktur martenzytycznych w stopach z pamięcią kształtu). Choć zjawiska leżące u ich podstaw są zdecydowanie różne, w obu przypadkach zasadniczym elementem, którego nie można pominąć przy próbach modelowania, są powierzchnie i warstwy, a także zjawiska zachodzące w tych warstwach. Wspólne ujęcie obu obszarów zainteresowań w niniejszej rozprawie pozwoliło na poszerzenie zakresu analizowanych konfiguracji (warstwy jednorodne i niejednorodne, warstwy o grubości infinitezymalnej lub skończonej, układy o znanej lub nieznanej mikrostrukturze). Cechą wspólną wszystkich analizowanych przypadków jest również centralna rola warunków zgodności (Rozdział 2.4) w opisie mechaniki warstw i powierzchni. Szczegółowe wnioski płynące z niniejszej pracy podano na końcu każdego rozdziału. Otrzymane wyniki w pełni potwierdzają znane zalety podejścia mikromechanicznego. Zjawiska w skali mikro, które poddaje się analizie w celu opisania zjawisk i wyznaczenia efektywnych właściwości własności skali makro, są zazwyczaj lepiej poznane i łatwiejsze w opisie. Opis mikromechaniczny wymaga też wprowadzania mniejszej liczby parametrów materiałowych, dodatkowo mających jasną interpretację fizyczną. W pracy wskazano również na ograniczenia podejścia mikromechanicznego. Dokładność opisu zależy od dokładności, z jaką jesteśmy w stanie scharakteryzować mikrostrukturę i zachowanie w skali mikro. Ponadto, modelowanie mikromechaniczne w.ymaga często znaczących nakładów obliczeniowych, co wskazuje na potrzebę równoległego rozwijania modeli fenomenologicznych, które w możliwie dużym stopniu powinny korzystać z przesłanek płynących z mikromechaniki. Układ pracy jest następujący. Rozdział 1 stanowi wstęp zawierający motywację, zakres oraz cel badań. Rozdział 2 zawiera te podstawowe (i zazwyczaj dobrze znane) elementy współczesnej mikromechaniki, które są wykorzystane w kolejnych częściach pracy: rozkład symetrycznego tensora na składowe wewnętrzną i zewnętrzną względem wyróżnionej powierzchni; podstawowe elementy teorii homogenizacji; warunki zgodności na powierzchni nieciągłości; równania przejścia mikro-makro dla prostego laminatu dwufazowego. W rozdziałach 3-6 oraz 8-9 zamieszczono oryginalne wyniki badań własnych, częściowo opublikowane w pracach (124, 128-133). Rozdział 7 jest krótkim wprowadzeniem do mikrostruktur martenzytycznych. Rozdział 10 zawiera podsumowanie, wnioski oraz perspektywy dalszych badań.

Słowa kluczowe

EN

micromechanics; continuum mechanics

PL

mikromechanika; mikromechanika powierzchni; mechanika ośrodków ciągłych; mikromechanika materiałów niejednorodnych

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Prace Instytutu Podstawowych Problemów Techniki PAN
• Rocznik: 2005
• Tom: nr 2
• Strony: 3--244
• Opis fizyczny: Bibliogr. 161poz.

Twórcy

autor

Stupkiewicz St.

Bibliografia

• [1] R. Abeyaratne and J.K. Knowles. On the driving traction acting on a surface of strain discontinuity in a continuum. J. Mech. Phys. Solids, 38:345–360, 1990.
• [2] R. Abeyaratne and J.K. Knowles. On the kinetics of an austeniteto-martensite phase transformation induced by impact in a Cu–Al–Ni shape-memory alloy. Acta Mater., 45(4):1671–1683, 1997.
• [3] J. Aboudi. Mechanics of Composite Materials. Elsevier, Amsterdam, 1991.
• [4] M. Andrade, M. Chandrasekaran, and L. Delaey. The basal plane stacking faults in 18R martensite of copper base alloys. Acta Metall., 32(10):1809–1816, 1984.
• [5] B. Avitzur and Y. Nakamura. Analytical determination of friction resistance as a function of normal load and geometry of surface irregularities. Wear, 107:367–383, 1986.
• [6] A. Azarkhin and O. Richmond. A model of ploughing by a pyramidal indenter – upper bound method for stress-free surfaces. Wear, 157:409–418, 1992.
• [7] J.M. Ball, C. Chu, and R.D. James. Hysteresis during stress-induced variant rearrangement. J. Physique IV, 5(C8):245–251, 1995.
• [8] J.M. Ball and R.D. James. Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal., 100:13–50, 1987.
• [9] A.A. Bandeira, P. Wriggers, and P. de Mattos Pimenta. Numerical derivation of contact mechanics interface laws using a finite element approach for large 3D deformation. Int. J. Num. Meth. Engng., 59:173–195, 2003.
• [10] N. Bay. Friction stress and normal stress in bulk metal forming processes. J. Mech. Working Technol., 14:203–224, 1987.
• [11] K. Bhattacharya. Wedge-like microstructure in martensites. Acta Metall. Mater., 39:2431–2444, 1991.
• [12] K. Bhattacharya. Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Continuum Mech. Thermodyn., 5:205–242, 1993.
• [13] K. Bhattacharya. Microstructure of martensite: why it forms and how it gives rise to the shape-memory effect. Oxford University Press, Oxford, 2003.
• [14] K. Bhattacharya and G. Dolzmann. Relaxed constitutive relations for phase transforming materials. J. Mech. Phys. Solids, 48:1493–1517, 2000.
• [15] K. Bhattacharya and R.D. James. A theory of thin films of martensitic materials with application to microactuators. J. Mech. Phys. Solids, 47:531–576, 1999.
• [16] F.P. Bowden and D. Tabor. The Friction and Lubrication of Solids. Clarendon Press, Oxford, 1953.
• [17] J.S. Bowles and J.K. MacKenzie. The crystallography of martensitic transformations I and II. Acta Metall., 2:129–137 and 138–147, 1954.
• [18] J.L. Bucaille, E. Felder, and G. Hochstetter. Mechanical analysis of the scratch test on elastic and perfectly plastic materials with the three-dimensional finite element modeling. Wear, 249:422–432, 2001.
• [19] R. Buczkowski and M. Kleiber. A stochastic model of rough surfaces for finite element contact analysis. Comp. Meth. Appl. Mech. Engng., 169:43–59, 1999.
• [20] S. Chakravorty and C.M. Wayman. Electron microscopy of internally faulted Cu–Zn–Al martensite. Acta Metall., 25:989–1000, 1977.
• [21] J.M. Challen and P.L.B. Oxley. An explanation of the different regimes of friction and wear using asperity deformation models. Wear, 53:229–243, 1979.
• [22] C. Chu. Hysteresis and microstructures: a study of biaxial loading on compound twins of copper-aluminum-nickel single crystals. PhD thesis, University of Minnesota, 1993.
• [23] M.G. Cooper, B.B. Mikic, and M.M. Yovanovich. Thermal contact conductance. Int. J. Heat Mass Transfer, 12:279–300, 1969.
• [24] S.C. Cowin and M.M. Mehrabadi. Anisotropic symmetries of linear elasticity. Appl. Mech. Rev., 48(5):247–285, 1995.
• [25] E.A. de Souza Neto, K. Hashimoto, D. Peri´c, and D.R.J. Owen. A phenomenological model for frictional contact accounting for wear effects. Phil. Trans. R. Soc. Lond. A, 354:819–843, 1996.
• [26] J. Dutkiewicz, H. Kato, S. Miura, U. Messerschmidt, and M. Bartsch. Structure changes during pseudoelastic deformation of CuAlMn single crystals. Acta Mater., 44(11):4597–4609, 1996.
• [27] 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.
• [28] ENLUB: Development of new environmentally acceptable lubricants, tribological tests and models for European sheet forming industry. FP5-GROWTH project, http://www.enlub.com.
• [29] J.D. Eshelby. Energy relations and the energy momentum tensor in continuum mechanics. In M.F. Kanninen et al., editors, Inelastic Behaviour of Solids, pages 77–114. McGrow-Hill, New York, 1970.
• [30] B. Feeny, A. Guran, N. Hinrichs, and K. Popp. A historical review on dry friction and stick-slip phenomena. Appl. Mech. Rev., 51:321–341, 1998.
• [31] F. Feyel and J.-L. Chaboche. FE2 multiscale approach for modelling the elastoviscoplastic behaviouir of long fibre SiC/Ti composite materials. Comp. Meth. Appl. Mech. Engng., 183:309–330, 2000.
• [32] A. Ga lka, J.J. Telega, and R. Wojnar. Thermodiffusion in heterogeneous elastic solids and homogenization. Arch. Mech., 46(3):267–314, 1994.
• [33] B.P. Gearing, H.S. Moon, and L. Anand. A plasticity model for interface friction: application to sheet metal forming. Int. J. Plast., 17(2):237–271, 2001.
• [34] A.E. Giannakopoulos. The influence of initial elastic surface stresses on instrumented sharp indentation. Trans. ASME J. Appl. Mech., 70:638–643, 2003.
• [35] J.A. Greenwood and J.B.P. Williamson. Contact of nominally flat surfaces. Proc. R. Soc. Lond. A, 295:300–319, 1966.
• [36] G. Guenin, M. Morin, P.F. Gobin, W. Dejonghe, and L. Delaey. Elastic Cu–Zn–Al near the martensitic transformation temperature. Scripta Metall., 11:1071–1075, 1977.
• [37] M.E. Gurtin. Two-phase deformations of elastic solids. Arch. Ration. Mech. Anal., 84:1–29, 1983.
• [38] Z. Handzel-Powier˙za, T. Klimczak, and A. Polijaniuk. On the experimental verification of the Greenwood-Williamson model for the contact of rough surfaces. Wear, 154:115–124, 1992.
• [39] K.F. Hane. Bulk and thin film microstructures in untwinned martensites. J. Mech. Phys. Solids, 47:1917–1939, 1999.
• [40] K.F. Hane and T.W. Shield. Symmetry and microstructure in martensites. Phil. Mag. A, 78(6):1215–1252, 1998.
• [41] K.F. Hane and T.W. Shield. Microstructure in the cubic to monoclinic transition in titanium-nickel shape memory alloy. Acta Mater., 47(9):2603–2617, 1999.
• [42] K.F. Hane and T.W. Shield. Microstructure in a cubic to orthorhombic transition. J. Elasticity, 59:267–318, 2000.
• [43] K.F. Hane and T.W. Shield. Microstructure in the cubic to trigonal transition. Mater. Sci. Eng., A291:147–159, 2000.
• [44] R. Hill. The Mathematical Theory of Plasticity. Oxford University Press, Oxford, U.K., 1950.
• [45] R. Hill. Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids, 11:357–372, 1963.
• [46] R. Hill. On constitutive macro-variables for heterogeneous solids at finite strain. Proc. R. Soc. Lond. A, 326:131–147, 1972.
• [47] R. Hill. Interfacial operators in the mechanics of composite media. J. Mech. Phys. Solids, 31(4):347–357, 1983.
• [48] H. Horikawa, S. Ichinose, K. Morii, S. Miyazaki, and K. Otsuka. Orientation dependence of β1 – β’1 stress-induced martensitic transformation in a Cu–Al–Ni alloy. Metall. Trans. A, 19A:915–923, 1988.
• [49] Y. Huo and I. Müller. Nonequilibrium thermodynamics of pseudoelasticity. Continuum Mech. Thermodyn., 5(3):163–204, 1993.
• [50] H. Ike. Plastic deformation of surface asperities associated with bulk deformation of metal workpiece in contact with rigid tool. In M. Raous, M. Jean, and J.J. Moreau, editors, Contact Mechanics, pages 275–286. Plenum Press, New York, 1995.
• [51] R.D. James. Finite deformation by mechanical twinning. Arch. Ration. Mech. Anal., 77:143–176, 1981.
• [52] Q. Jiang and H. Xu. Microobservation of stress induced martensitic transformation in CuAlNi single crystals. Acta Metall. Mater., 40(4):607–613, 1992.
• [53] K.L. Johnson. The correlation of indentation experiments. J. Mech. Phys. Solids, 18:115–126, 1970.
• [54] K.L. Johnson. Contact Mechanics. Cambridge University Press, 1985.
• [55] K.L. Johnson. The application of shakedown principles in rolling and sliding contact. Eur. J. Mech. A/Solids, 11:155–172, 1992.
• [56] H. Kato, J. Dutkiewicz, and S. Miura. Superelasticity and shape memory effect in Cu–23at.%Al–7at.%Mn alloy single crystals. Acta Metall. Mater., 42(4):1359–1365, 1994.
• [57] A.G. Khachaturyan. Some questions concerning the theory of phase transformations in solids. Fiz. Tverd. Tela, 8(9):2709–2717, 1966. English translation: Sov. Phys. Solid State, 8:2163–2168, 1967.
• [58] A.G. Khachaturyan. Theory of Structural Transformations in Solids. John Wiley and Sons, New York, 1983.
• [59] Y. Kimura and T.H.C. Childs. Surface asperity deformation under bulk plastic straining conditions. Int. J. Mech. Sci., 41:283–307, 1999.
• [60] A. Klarbring. Derivation and analysis of rate boundary-value problems of frictional contact. Eur. J. Mech. A/Solids, 9(1):53–85, 1990.
• [61] R.V. Kohn. The relaxation of a double well energy. Continuum Mech. Thermodyn., 3:193–236, 1991.
• [62] K. Komvopoulos, N. Saka, and N.P. Suh. The mechanism of friction in boundary lubrication. Trans. ASME J. Tribol., 107:452–462, 1985.
• [63] J. Korelc. Automatic generation of numerical codes with introduction to AceGen 4.0 symbolic code generator. Available at http://www.fgg.uni-lj.si/Symech/, 2000.
• [64] J. Korelc. Computational Templates. User manual. Available at http://www.fgg.uni-lj.si/Symech/, 2000.
• [65] J. Korelc. Multi-language and multi-environment generation of nonlinear finite element codes. Engineering with Computers, 18:312–327, 2002.
• [66] D.A. Korzekwa, P.R. Dawson, and W.R.D. Wilson. Surface asperity deformation during sheet forming. Int. J. Mech. Sci., 34(7):521–539, 1992.
• [67] V. Kouznetsova, W.A.M. Brekelmans, and F.P.T. Baaijens. An approach to micro-macro modelling of heterogeneous materials. Comp. Mech., 27:37–48, 2001.
• [68] I.V. Kragelsky, M.N. Dobychin, and V.S. Kombalov. Friction and Wear – Calculation Methods. Pergamon Press, Oxford, 1982.
• [69] P. Krasniuk and S. Stupkiewicz. Evolution of real contact area in the presence of bulk plastic deformation: the effect of strain hardening. Internal Report ENLUB/2002/1, IPPT PAN, 2002.
• [70] L. Krstulović-Opara, P. Wriggers, and J. Korelc. A C1-continuous formulation for 3D finite deformation frictional contact. Comp. Mech., 29(1):27–42, 2002.
• [71] S. Kucharski, T. Klimczak, A. Polijaniuk, and J. Kaczmarek. Finite element model for the contact of rough surfaces. Wear, 177:1–13, 1994.
• [72] M. Landa, V. Novák, P. Sedlák, and P. Šittner. Ultrasonic characterization of Cu–Al–Ni single crystals lattice stability in the vicinity of the phase transition. Ultrasonics, 42:519–526, 2004.
• [73] Y.-H. Lee and D. Kwon. Estimation of biaxial surface stress by instrumented indentation with sharp indenters. Acta Mater., 52:1555–1563, 2004.
• [74] C. Lexcellent, B.C. Goo, Q.P. Sun, and J. Bernardini. Characterization, thermomechanical behaviour and micromechanical-based constitutive model of shape-memory Cu–Zn–Al single crystals. Acta Mater., 44(9):3773–3780, 1996.
• [75] R. Luciano and J.R. Willis. Boundary-layer corrections for stress and strain fields in randomly heterogeneous materials. J. Mech. Phys. Solids, 51:1075–1088, 2003.
• [76] G. Maciejewski, S. Stupkiewicz, and H. Petryk. Elastic micro-strain energy at the austenite–twinned martensite interface. Arch. Mech., 57(4):277–297, 2005.
• [77] H. Morawiec. Stopy z pamięcią kształtu i ich zastosowanie. In W.K. Nowacki, editor, Podstawy termomechaniki materiałów z pamięcią kształtu, pages 7–54. IPPT PAN, Warszawa, 1996.
• [78] Z. Mróz and S. Stupkiewicz. Constitutive model of adhesive and ploughing friction in metal forming processes. Int. J. Mech. Sci., 40:281–303, 1998.
• [79] I. Müller and H. Xu. On the pseudo-elastic hysteresis. Acta Metall. Mater., 39(3):263–271, 1991.
• [80] V. Novák, P. Šittner, D. Vokoun, and N. Zárubová. On the anisotropy of martensitic transformations in Cu-based alloys. Mater. Sci. Eng., A273–275:280–285, 1999.
• [81] V. Novák, P. Šittner, and N. Zárubová. Anisotropy of transformation characteristics of Cu-base shape memory alloys. Mater. Sci. Eng., A234–236:414–417, 1997.
• [82] J.T. Oden and J.A.C. Martins. Models and computational methods for dynamic friction phenomena. Comp. Meth. Appl. Mech. Engng., 52:527–634, 1985.
• [83] L. Orgeas and D. Favier. Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression. Acta Mater., 46(15):5579–5591, 1998.
• [84] K. Otsuka, T. Ohba, M. Tokonami, and C.M. Wayman. New description of long period stacking order structures of martensites in β-phase alloys. Scripta Metall. Mater., 29:1359–1364, 1993.
• [85] K. Otsuka, H. Sakamoto, and K. Shimizu. Successive stress-induced martensitic transformations and associated transformation pseudoelasticity in Cu–Al–Ni alloys. Acta Metall., 27:585–601, 1979.
• [86] K. Otsuka and K. Shimizu. Morphology and crystallography of thermoelastic Cu–Al–Ni martensite analyzed by the phenomenological theory. Trans. Jap. Inst. Metals, 15:103–108, 1974.
• [87] K. Otsuka and C.M. Wayman, editors. Shape Memory Materials. Cambridge University Press, 1998.
• [88] K. Otsuka, C.M. Wayman, K. Nakai, H. Sakamoto, and K. Shimizu. Superelasticity effects and stress-induced martensitic transformations in Cu–Al–Ni alloys. Acta Metall., 24:207–226, 1976.
• [89] E. Patoor, A. Eberhardt, and M. Berveiller. Micromechanical modelling of superelasticity in shape memory alloys. J. Physique IV, C1:277–292, 1996.
• [90] V.J. Pauk and C. Woźniak. Plane contact problem for a half-space with boundary imperfections. Int. J. Sol. Struct., 36:3569–3579, 1999.
• [91] J. Peña, F.J. Gil, and J.M. Guilemany. Effect of microstructure on dry sliding wear behaviour in CuZnAl shape memory alloys. Acta Mater., 50:3115–3124, 2002.
• [92] P. Pedersen. Elasticity—Anisotropy—Laminates. Technical University of Denmark, 1997.
• [93] B.N.J. Persson. Sliding Friction. Springer Verlag, Berlin, 1998.
• [94] H. Petryk. Slip line field solutions for sliding contact. In Friction, Lubrication and Wear—Fifty Years On, volume II, pages 987–994, London, 1987. Proc. Instn. Mech. Engrs.
• [95] H. Petryk. Macroscopic rate-variables in solids undergoing phase transformation. J. Mech. Phys. Solids, 46:873–894, 1998.
• [96] H. Petryk and S. Stupkiewicz. Micromechanical modelling of stressinduced phase transition in shape memory alloys. Arch. Metall. Mater., 49(4):765–777, 2004.
• [97] H. Petryk, S. Stupkiewicz, and G. Maciejewski. Modelling of austenite/ martensite laminates with interfacial energy effect. In Proc. IUTAM Symp. on Size Effects on Material and Structural Behaviour at Micron- and Nano-scales. Hong Kong, 2004. (in print).
• [98] G. Pietrzak and A. Curnier. Large deformation frictional contact mechanics: continuum formulation and augmented Lagrangian treatment. Comp. Meth. Appl. Mech. Engng., 177(3–4):351–381, 1999.
• [99] M. Pitteri and G. Zanzotto. Generic and non-generic cubic-tomonoclinic transitions and their twins. Acta Mater., 46(1):225–237, 1998.
• [100] E. Pruchnicki. Hyperelastic homogenized law for reinforced elastomer at finite strain with edge effects. Acta Mech., 129:139–162, 1998.
• [101] L. Qian, X. Xiao, Q.P. Sun, and T. Yu. Anomalous relationship between hardness ans wear properties of a superelastic nickel-titanium alloy. Appl. Phys. Lett., 84:1076–1078, 2004.
• [102] E. Rabinowicz. Friction and Wear of Materials. John Wiley & Sons, Inc., New York, 1965.
• [103] B. Raniecki. Termomechanika pseudosprężystości materiałów z pamięcią kształtu. In W.K. Nowacki, editor, Podstawy termomechaniki materiałów z pamięcią kształtu, pages 55–140. IPPT PAN, Warszawa, 1996.
• [104] B. Raniecki and Ch. Lexcellent. RL-models of pseudoelasticity and their specifications for some shape memory alloys. Eur. J. Mech. A/Solids, 13:21–50, 1994.
• [105] B. Raniecki and Ch. Lexcellent. Thermodynamics of isotropic pseudoelasticity in shape memory alloys. Eur. J. Mech. A/Solids, 17:185–205, 1998.
• [106] B. Raniecki and K. Tanaka. On the thermodynamic driving force for coherent phase transformations. Int. J. Engng Sci., 32:1845–1858, 1994.
• [107] J.R. Rice. Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In A.S. Argon, editor, Constitutive Equations in Plasticity, pages 23–79. MIT Press, Cambridge, Mass., 1975.
• [108] C. Rodriguez and L.C. Brown. The thermal effect due to stressinduced martensite formation in β-CuAlNi single crystals. Metall. Trans. A, 11A:147–150, 1980.
• [109] P.L. Rodriguez, F.C. Lovey, G. Guenin, J.L. Pelegrina, M. Sade, and M. Morin. Elastic constants of the monoclinic 18R martensite of a Cu–Zn–Al alloy. Acta Metall. Mater., 41(11):3307–3310, 1993.
• [110] A.L. Roytburd. Martensitic transformation as a typical phase transformation in solids. In D. Seitz and D. Turnbull, editors, Solid State Physics, volume 33, pages 317–380. Academic Press, New York, 1978.
• [111] A.L. Roytburd. Modified Clausius-Clapeyron equation for phase transformation hysteresis in solids (in Russian). Fiz. Tverd. Tela, 25(1):33–40, 1983. English translation: Sov. Phys. Solid State, 25:17– 21, 1983.
• [112] A.L. Roytburd. Thermodynamics of polydomain heterostructures. I. Effect of macrostresses. J. Appl. Phys., 83(1):228–238, 1998.
• [113] A.L. Roytburd. Thermodynamics of polydomain heterostructures. II. Effect of microstresses. J. Appl. Phys., 83(1):239–245, 1998.
• [114] A.L. Roytburd and M.N. Pankova. The influence of external stresses on the orientation of the habit plane and substructure of stressinduced martensite plates in iron-based alloys. Fiz. Met. Metalloved., 59(4):769–779, 1985. English translation: Phys. Met. Metall., 59:131–140, 1985.
• [115] A.L. Roytburd and J. Slutsker. Thermodynamic hysteresis of phase transformation in solids. Physica B, 233:390–396, 1997.
• [116] A.L. Roytburd and J. Slutsker. Deformation of adaptive materials.Part I. Constrained deformation of polydomain crystals. J. Mech Phys. Solids, 47:2299-2329, 1999.
• [117] A.L. Roytburd and J. Slutsker. Deformatin of adaptive materials. Part III: Deformation of crystals with polytwin product phases. J. Mech. Phys. Solids, 49:1795–1822, 2001.
• [118] E. Sanchez-Palencia. Boundary layers and edge effects in composites. In E. Sanchez-Palencia and A. Zaoui, editors, Homogenization Techniques for Composite Media, volume 272 of Lecture Notes in Physics, pages 121–192. Springer, Berlin, 1987.
• [119] T.W. Shield. Orientation dependence of the pseudoelastic behavior of single crystals of Cu–Al–Ni in tension. J. Mech. Phys. Solids, 43:869–895, 1995.
• [120] J.C. Simo and T.J.R. Hughes. Computational Inelasticity. Springer-Verlag, New York, 1998.
• [121] V. Smyshlyaev and J.R. Willis. On the relaxation of a three-well energy. Proc. R. Soc. Lond. A, 455:779–814, 1999.
• [122] S. Stupkiewicz. Extension of the node-to-segment contact element for surface-expansion-dependent contact laws. Int. J. Num. Meth. Engng., 50:739–759, 2001.
• [123] S. Stupkiewicz. Augmented Lagrangian formulation and sensitivity analysis of contact problems. In E. Onate and D.R.J. Owen, editors, COMPLAS 2003, VII International Conference on Computational Plasticity, CIMNE, Barcelona, 2003. proceedings on CD.
• [124] S. Stupkiewicz. The effect of stacking fault energy on the formation of stress-induced internally faulted martensite plates. Eur. J. Mech. A/Solids, 23(1):107–126, 2004.
• [125] S. Stupkiewicz, J. Korelc, M. Dutko, and T. Rodič. Shape sensitivity analysis of large deformation frictional contact problems. Comp. Meth. Appl. Mech. Engng., 191(33):3555–3581, 2002.
• [126] S. Stupkiewicz and A. Marciniszyn. Modelling of asperity deformation in the thin-film hydrodynamic lubrication regime. In N. Bay, editor, Proc. 2nd Int. Conf. on Tribology in Manufacturing Processes ICTMP2004, pages 695–702, Nyborg, Denmark, June 2004.
• [127] S. Stupkiewicz and Z. Mr´oz. A model of third body abrasive friction and wear in hot metal forming. Wear, 231:124–138, 1999.
• [128] S. Stupkiewicz and Z. Mr´oz. Modelling of bulk deformation effects on real contact area and friction in metal forming processes. In Proc. EUROMECH 435 Friction and Wear in Metal Forming, pages 63–70, Valenciennes, France, June 2002.
• [129] S. Stupkiewicz and Z. Mróz. Phenomenological model of friction accounting for subsurface plastic deformation in metal forming. In J.A.C. Martins and M.D.P. Monteiro Marques, editors, Contact Mechanics, Solid Mechanics and its Applications, pages 179–186, Dordrecht, 2002. Kluwer Academic Publishers.
• [130] S. Stupkiewicz and Z. Mróz. Phenomenological model of real contact area evolution with account for bulk plastic deformation in metal forming. Int. J. Plast., 19(3):323–344, 2003.
• [131] S. Stupkiewicz and H. Petryk. Finite-strain micromechanical model of stress-induced martensitic transformations in shape memory alloys. Mater. Sci. Eng. (submitted).
• [132] S. Stupkiewicz and H. Petryk. Modelling of laminated microstructures in stress-induced martensitic transformation. J. Mech. Phys. Solids, 50:2303–2331, 2002.
• [133] S. Stupkiewicz and H. Petryk. Micromechanical modelling of stressinduced martensitic transformation and detwinning in shape memory alloys. Journal de Physique IV, 115:141–149, 2004.
• [134] M. Suezawa and K. Sumino. Behaviour of elastic constants in Cu-Al-Ni alloy in the close vicinity of Ms-point. Scripta Metall., 10:789–792, 1976.
• [135] N.P. Suh and H.-C. Sin. The genesis of friction. Wear, 69:91–114, 1981.
• [136] Q.P. Sun, editor. Mechanics of Martensitic Phase Transformation in Solids. Kluwer Academic Publishers, 2002.
• [137] P.M. Suquet. Elements of homogenization for inelastic solid mechanics. In E. Sanchez-Palencia and A. Zaoui, editors, Homogenization Techniques for Composite Media, volume 272 of Lecture Notes in Physics, pages 193–278. Springer, Berlin, 1987.
• [138] M.P.F. Sutcliffe. Surface asperity deformation in metal forming processes. Int. J. Mech. Sci., 30(11):847–868, 1988.
• [139] W. Szczepiński. Wstęp do analizy procesów obróbki plastycznej. PWN, Warszawa, 1967.
• [140] P. Thamburaja and L. Anand. Polycrystalline shape-memory materials: effect of crystallographic texture. J. Mech. Phys. Solids, 49:709–737, 2001.
• [141] K. Varadi, Z. Neder, and K. Friedrich. Evaluation of the real contact areas, pressure distributions and contact temperatures during sliding contact between real metal surfaces. Wear, 200:55–62, 1996.
• [142] S. Vedantam and R. Abeyaratne. A Helmholz free-energy function for a Cu–Al–Ni shape memory alloy. Int. J. Non-Lin. Mech., 40:177–193, 2005.
• [143] P. Šittner. Private communication. 2004.
• [144] P. Šittner and V. Novák. Anisotropy of Cu-based shape memory alloys in tension/compression thermomechanical loads. Trans. ASME J. Eng. Mat. Tech., 121:48–55, 1999.
• [145] P. Šittner and V. Novák. Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals. Int. J. Plasticity, 16:1243–1268, 2000.
• [146] P. Šittner, V. Novák, and N. Zárubová. Martensitic transformations in [001] CuAlZnMn single crystals. Acta Mater., 46(4):1265–1281, 1998.
• [147] T. Wanheim, N. Bay, and A.S. Petersen. A theoretically determined model for friction in metal working processes. Wear, 28:251–258, 1974.
• [148] M.S. Wechsler, D.S. Lieberman, and T.A. Read. On the theory of the formation of martensite. Trans. AIME J. Metals, 197:1503–1515, 1953.
• [149] D.J. Whitehouse and J.F. Archard. The properties of random surfaces in contact. Proc. R. Soc. Lond. A, 316:97–121, 1970.
• [150] J.R. Willis. Variational and related methods for the overall properties of composites. In Advances in Applied Mechanics, volume 21, pages 1–78. Academic Press, New York, 1981.
• [151] W.R.D. Wilson. Friction models for metal forming in the boundary lubrication regime. Trans. ASME J. Eng. Mat. Technol., 113:60–68, 1991.
• [152] W.R.D. Wilson and W.M. Lee. Mechanics of surface roughening in metal forming processes. Trans. ASME J. Manuf. Sci. Engrg., 123(2):279–283, 2001.
• [153] W.R.D. Wilson and S. Sheu. Real area of contact and boundary friction in metal forming. Int. J. Mech. Sci., 30(7):475–489, 1988.
• [154] S. Wolfram. The Mathematica Book, 4th ed. Wolfram Media/Cambridge University Press, 1999.
• [155] Cz. Wozniak. Refined macrodynamics of periodic structures. Arch. Mech., 45:295–304, 1993.
• [156] P. Wriggers. Computational Contact Mechanics. Wiley, Chichester, 2002.
• [157] P. Wriggers, T. Vu Van, and E. Stein. Finite element formulation of large deformation impact-contact problems with friction. Comp. Struct., 37:319–331, 1990.
• [158] M. Yasunaga, Y. Funatsu, S. Kojima, K. Otsuka, and T. Suzuki. Measurement of elastic constants. Scripta Metall., 17:1091–1094, 1983.
• [159] X.Y. Zhang, Q.P. Sun, and S.W. Yu. A non-invariant plane model for the interface in CuAlNi single crystal shape memory alloys. J. Mech. Phys. Solids, 48:2163–2182, 2000.
• [160] O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method. Butterworth-Heinemann, Oxford, 5th edition, 2000.
• [161] F. Zimmermann and M. Humbert. Determination of the habit plane–
• 0 phase transformation induced by stress in Ti–5Al–2Sn–4Zr–4Mo–2Cr–1Fe. Acta Mater., 50:1735–1740, 2002.