Wpływ mikorstruktury na własności kompozytów sprężystych, piezoelektrycznych i termosprężystych

• Autorzy: Gambin B.
• Języki publikacji: PL

Abstrakty

PL

Celem pracy jest przedstawienie wyników badań własnych autorki dotyczących wyznaczania efektywnych własności materiałów kompozytowych: sprężystych, piezoelektrycznych i termosprężystych. Pojęcie materiału kompozytowego (złożonego) zdefiniowano zgodnie z matematyczną teorią homogenizacji. Dzięki tej teorii można określić w sposób jednoznaczny związek pomiędzy mikrostrukturą i makroskopowymi (efektywnymi, zastępczymi) własnościami materiału. Badania, oparte o różne metody teorii homogenizacji i przedstawione w rozprawie, dotyczą następujących klas materiałów. Kompozyty sprężyste. Wyprowadzone zostaną nielokalne związki konstytutywne dla liniowego ośrodka sprężystego, w których nielokalność jawnie zależy od rozmiarów mikrostruktury. Pokazany będzie wpływ mikrostruktury na zmianę profilu fali przejścia przez warstwę z materiału typu FGM oraz opisane będzie zachowanie się makroskopowe ośrodka o periodycznie-stochastycznym rozkładzie szczelin. Kompozyty piezoelektryczne. Dzięki wprowadzeniu nowych pojęć i udowodnieniu kilku ważnych twierdzeń, możliwe będzie wyznaczenie własności fizycznie nieliniowych kompozytów piezoelektrycznych, dokonanie analizy zagadnień projektowania optymalnego piezoelektrycznych kompozytów gradientowych, podanie nowych charakterystyk sprzężenia elektromechanicznego, a także sfrormułowanie pewnych nowych ograniczeń na liniowe własności piezoelektryczne i obliczenie efektu warstwy brzegowej. Zostaną także wyprowadzone formuły typu Murata na tzw. laminację wielokrotną. Kompozyty termosprężyste. Zastosowanie metod homogenizacji stochastycznej oraz wprowadzenie nowego funkcjonału, który opisuje fizycznie nieliniowe oddziaływanie pól sprężystych z dodatkowym polem skalarnym, umożliwi znalezienie formuł opisujących efektywne stałe termosprężyste. Otrzymane wyniki teoretyczne będą wyjaśnione i zilustrowane przykładami numerycznymi. Merytorycznie rozprawa jest jednorodna tematycznie wszystkie rozdziały związane są z modelowaniem wpływu mikrostruktury na zachowanie się kompozytów. Z punktu widzenia metod użytych do wykonania postawionego celu należy wydzielić Rozdział 6, w którym użyto metod homogenizacji stochastycznej. Z powodu specyfiki języka używanego w metodzie stochastycznej homogenizacji, rozdział ten zawiera dużo nowych definicji i pojęć niezbędnych do zrozumienia przedstawionych wyników. Ze względu na szeroki zakres analizowanych zagadnień, literatura stanowiąca podstawę przeprowadzonych badań będzie omówiona w odpowiednich rozdziałach. Treść rozprawy jest w dużej części podsumowaniem oryginalnych wyników badań autorki prowadzonych w Instytucie Podstawowych Problemów Techniki PAN w latach 1989-2006. Część wyników opublikowano w pracach, których współautorami byli koledzy z Pracowni Metod Wariacyjnych i Biomechaniki.

Słowa kluczowe

PL

inżynieria materiałowa; wytrzymałość materiałów; mikrostruktura; kompozyty; kompozyty sprężyste; kompozyty piezoelektryczne; kompozyty termosprężyste

EN

mechanics of materials; microstructure; composites

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Prace Instytutu Podstawowych Problemów Techniki PAN
• Rocznik: 2006
• Tom: nr 12
• Strony: 1--183
• Opis fizyczny: Bibliogr. 178 poz., il.

Twórcy

autor

Gambin B.

Bibliografia

• [1] Abddaimi Y., Michaille G., Licht C, Stochastic homogenization for an integral functional of a quasiconvex function with linear growth, Asymptotic Anal., 1997, 15, 183-202.
• [2] Adler J.P., The Geometry of Random Fields, Wiley, Chichester, 1981.
• [3] Adler P.M., Thovert J.-F., Real porous media: Local geometry and macroscopic properties, Appl. Mech. Reviews, 1998, 51, 537-585.
• [4] Adler P.M., Thovert J.-F., Fracture and Fracture Networks, Kluwer, Dordrecht, 1999.
• [5] Akcoglu M.A., Krengel U., Ergodic theorems for superadditive processes, J. Reine Angew. Math., 1981, 323, 53-67.
• [6] Allaire G., Schape Optimization by the Homogenization Method, Springer, New York, 2002.
• [7] Allaire G., Homogenization and two-scale convergence, SIAM J. Math. Anal, 1992, 23, 1482-1518.
• [8] Allaire G., Briane M., Multiscale convergence and reiterated homogenization, Proc. R. Soc. Edinburgh, 1996, 126A, 297-342.
• [9] Andrews K.T., Wright S., Stochastic homogenization of elliptic boundary-value problem with Lp-data, Asymptotic AnaL, 1998, 17, 165-184.
• [10] Aoubiza B., Crolet J.M., Meunier A., On the mechanical characterization of compact bonę structure using the homogenization theory, J. Biomechanics, 1996, 29, 1539-1547.
• [11] Attouch H., Variational Convergence for Functions and Operators, Pitman, Boston, 1984.
• [12] Avellaneda M., Olson T., Effective medium theories and efFective electro-mechanical coupling factors for piezoelectric composites, J. Int. Mat. Struct., 4, 1993.
• [13] Aze D., Epi-convergence et dualite. Applications a'la convergence des variables primales et duales pour des suites d'optimization convexe, Publications AVA-MAC, Universite de Perpignan, Report No. 84-12, 1984.
• [14] Bachvalov N.S., Panasenko G.P. Osrednienie processov v periodyczeskich sredach, Izd. Nauka, Moskwa 1984 (w j. rosyjskim)
• [15] Bahr U., Gambin B., Scaterring of an elastic wave from a heterogeneous materiał, Arch. Mech., 1977, 29, 6, 769-783.
• [16] Bendsoe M.P., Optimization of Structural Topology, Shape, and Materiał, Springer-Verlag, Berlin, 1995.
• [17] Bendsoe M.P., Kikuchi N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods AppL Mech. Eng., 1988, 71, 197-224.
• [18] Bendsoe M.P., Sigmund O., Topology Optimization, Theory, Methods and Applications, Springer-Verlag, Berlin, 2003.
• [19] Bendsoe M.P., Guedes J.M., Haber R.B., Pedersen P., Taylor J.E., An analytical model to predict optimal materiał properties in the context of optimal structural design, Trans. ASME J. AppL Mech., 1994, 61, 4, 930-937.
• [20] Bensoussan A., Lions J., Papanicolaou G., Asymptotic Analysis for Periodic Structure, North Holland, Amsterdam, 1978.
• [21] Bensoussan A., Lions J., Papanicolaou G., Boundary Layer Analysis in Homogenization of Diffusion Eąuation with Dirichlet Conditions, K. Ito (red.), J. Willey and Sons, New-York, 1978.
• [22] Benveniste Y., The determination of the elastic and electric fields in a piezoelec-tric inhomogeneity, J. AppL Phys., 1992, 72, 1086-1095.
• [23] Benveniste Y., Universal relations in piezoelectric composites with eigenstress and polarization fields, Part I: Binary media - local fields and effective behavior, J. AppL Mech., 1993, 60, 265-269; Part II: Multiphase media - effective behavior, ibid., 270-275.
• [24] Berggren S.A., Lukkassen D., Meidell A., Simula L., On stiffness properties of sąuare honeycombs and other unidirectional composites, Composites, Part B, 2001,32,503-511.
• [25] Berggren S.A., Lukkassen D.,Meidell A., Simula L., Some methods for calculating stifFness properties of periodic structures, Applications of Mathematics, 2003, 48, 2, 97-110.
• [26] Bhattacharya K., A novel approach to the application of ferroelectric thin films to micro-actuation, XXI International Congress of Theoretical and Applied Me-chanics, Warszawa, 2004.
• [27] Bielski W., Nonstationary Flows of Viscous Fluids Through Porous Elastic Media. Homogenization Method, Publ. Inst. Geophys. Pol. Acad. Sci., A-29 (388), Warszawa, 2005.
• [28] Bielski W., Bytner S., Homogenization of piezoelectric medium, Abstracts of the 27th Polish Solid Mechanics Conference, August 29 - September 3, 1988, Rytro.
• [29] Bisegna P., Luciano R., Variational bounds for the overall properties of piezoelectric composites, J. Mech. Phys. Solids, 1996, 44, 583-602.
• [30] Bisegna R, Luciano R., On methods for bounding the overall properties of pe-riodic piezoelectric fibrous composites, J. Mech. Phys. Solids, 1997, 45, 8, 1329-1356.
• [31] Bisegna P., Luciano R., Bounds on the off-diagonal coefficients of the constitutive tensor of a composite materiał, Mech. Res. Comm., 1996, 23, 239-245.
• [32] Bourgeat A., Mikelić A., Wright S., Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 1994, 456, 19-51.
• [33] Boutin C, Auriault J.L., Ralyeigh seat ter ing in elastic composite materials, Int. J. Engng Sci., 1993, 227, 1669-1683.
• [34] Braides A., Omogeneizzazione di integrali non coercivi, Ricerche di Mat., 1983, XXXII, 2, 348-368.
• [35] Buryachenko V.A., Rammerstofer F.G., Local effective thermoelastic properties of graded random structure matrbc composites, Arch. of Appl. Mech., 2001, 71, 249-272.
• [36] Buryachenko V.A., Rammerstofer F.G., On the thermo-elasto-statics of composites with coated randomly distributed inclusions, Int. J. Solids and Structures, 2000, 37, 3177-32.
• [37] Buryachenko V.A., Effective strength properties of elastic physically nonlinear composites, w: Marigo J.J, Rousselier G., (red.), Proceedings of the MECAMAT Conference on Micromechanics of Materials. Editions Eyrolles, Paris, 1993, 567 -578.
• [38] Buryachenko V.A., Multiparticle effective field and related methods in micromechanics of composite materials, Applied Mechanics Reviews, 2001, 54, 1-47.
• [39] Boccardo L., Murat F., Homogeneisation de problemes ąuasi-lineaire, w: Atti del Convegno Studia di Probierni - Limite delia Analisi Funzionale, Bressanone, 7-9 Settembre, Pitagora Editrice, Bologna, 1982, 13-51.
• [40] Boivin D., Depauw J., Spectral homogenization of reversible random walks on Zd in a random enviroment, Stochastic Processes and their Appl, 2003, 104, 29-56.
• [41] Bouchitte G., Convergence et relaxation de fouctionnalles convexes du calcul des variations a croissance lineaire, Annales Fac. Sci. Toulouse, 1987, 8, 7-36.
• [42] Bouchitte G., Suąuet R, Homogenization, plasticity and yield design, w: Composite Media and Homogenization Theory, G. Dal Maso, G.F. Dell' Antonio (red.), Birkhauser, Boston, 1991, 107-133.
• [43] Bourgeat A., Mikelić A., Wright S., Stochastic two-scale convergence in themean and applications, J. Reine Angew. Math., 1994, 456, 19-51.
• [44] Braides A., Omogeneizzazione di integrali non coercivi, Ricerche di Mat, 1983, 32, 348-68.
• [45] Braides A., Defranceschi A., Homogenization of Multiple Integrals, Clarendon Press, Oxford, 1998.
• [46] Bytner S., Gambin B., Homogenization of first strain-gradient body, Mech. Teor. Stos., 1988, 26, 3, 423-429.
• [47] Cioranescu D., Donato R, An Introduction to Homogenization, Oxford University Press, Oxford, 1999.
• [48] Chen TY, Piezoelectric properties of multiphase fibrous composites: some theore-tical results, J. Mech. Phys. Solids, 1993, 41, 1781-1794.
• [49] Christensen R.M., Mechanics of Composite Materials, Moscow, Mir, 1982 (w j. rosyjskim).
• [50] Cowin S.C., The mechanical properties of cancellous bonę, w: Bonę Mechanics, S.C. Cowin (red.), CRC Press, Inc. Boca Raton, Florida, 1989, 129-157.
• [51] Crolet J.M., Aoubiza B., Meunier A., Compact bonę: numerical simulation of mechanical characteristics, J. Biomechanics, 1993, 26, 677-687.
• [52] Dacorogna B., Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, 1989.
• [53] Dal Maso G., Modica L., Nonlinear stochastic homogenization, Ann. Mat. Pura ed Applicata IV, 1986, 144, 347-389.
• [54] Dal Maso G., An Introduction to F-Convergence, Birkhauser, Boston, 1993.
• [55] Dal Maso G., Modica L., Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math., 1986, 368, 28-42.
• [56] Dunn M.L., Taya M., Micromechanics predictions of the effective electrostatic moduli of piezoelectric composites, Int. J. Solids Structures, 1993, 30, 161-175.
• [57] Dumontet H., Study of a boundary layer problem in elastic composite materials, Mathematical Modelling and Numerical Analysis,1986, 20, 2, 265-286.
• [58] Ekeland L, Temam R., Convex Analysis and Variational Problems, Amsterdam, 1976.
• [59] Fish J., Wen-Chen higher-order homogenization of initial-boundary value problems, J. Eng. Mech., 2001, 127, 1223-1230.
• [60] Francfort G.A., Homogenization and linear thermoelasticity, SI AM J. Math. Anai, 1983, 14, 696-708.
• [61] Gałka A., Gambin B., Telega J.J., Tokarzewski S., Macroscopic properties of compact bonę and its hierarchical structure, Acta Bioeng. Biomech., 2003, 5, Suplement 1, 155-160.
• [62] Gałka A., Gambin B., Telega J.J., Wielowarstwowe kompozyty piezoelektryczne: modelowanie i optymalizacja, II Sympozjon, Kompozyty. Konstrukcje warstwowe, Dolnośląskie Wydawnictwo Edukacyjne, Wrocław-Karpacz, 2002.
• [63] Gałka A., Gambin B., Telega J.J., Multi-layered piezocomposites: homogenization and optimization; Theoretical Foundations of Civil Engineering, Polish-Ukrainian Transactions; W. Szczęśniak (red.), Oficyna Wydawnicza Politechniki Warszawskiej, Dniepropietrowsk, 2003, 299-308.
• [64] Gałka A., Gambin B., Telega J.J., Variational bounds on the effective moduli of piezoelectric composites, Arch. Mech., 1998, 50, 4.
• [65] Gałka A., Telega J.J., Tokarzewski S., Nonlinear transport eąuation and macroscopic properties of microheterogeneous media, Arch. Mech., 1997, 49, 293-319.
• [66] Gałka A., Telega J.J., Wojnar R., Homogenization and thermopiezoelectricity, Mech. Res. Comm., 1992, 19, 315-324.
• [67] Gałka A., Telega J.J., Wojnar R., Some computational aspects of homogenization of thermopiezoelectric composites, Comp. Assisted Mech. Eng. Sci., 1996, 3, 2, 133-154.
• [68] Gałka A., Telega J.J., Wojnar R., ThermodhTusion in heterogeneous elastic solids and homogenization, Arch. Mech., 1994, 46, 3, 267-314.
• [69] Gambin B., Influence of boundary on mean wave in random medium, w: Continous Models of Discrete Sysstems, 1980, University of Waterloo Press, 551-564.
• [70] Gambin B., Long waves in random elastic composites, w: Continous Models of Discrete Sysstems, K.H. Anthony (red.), World Sc, 1992.
• [71] Gambin B., Opis własności ciał sprężystych losowo niejednorodnych, Politechnika Poznańska, Materiały dla Studiów Doktoranckich i Podyplomowych, 13, 1986.
• [72] Gambin B., Stochastic homogenization, Control and Cybern., 1994, 23, 4, 672-676.
• [73] Gambin B., Stochastic homogenization of the first gradient strain modelling of elasticity, J. Theor. Appl. Mech., 1997, 1, 35, 83-93.
• [74] Gambin B., Two-scale asymptotic expansion in the theory of random media, Abstracts of the 27th Polish Solid Mechanics Conference, August 29 - September 3, 1988, Rytro.
• [75] Gambin B., Gałka A., Boundary layer problem in piezoelectric composite, w: IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, D.F. Parker i A.H. England (red.), Kluwer Acad. Publish., 1995, 327-332.
• [76] Gambin B., Gałka A., Telega J. J., A graded materiał model based on homogenization; w: Structured Media, TRECOP'01, B.T. Maruszewski (red.), Wydawnictwo Politechniki Poznańskiej, 2002, 71-82.
• [77] Gambin B., Gałka A., Telega J. J., Tokarzewski S., Influence of anisotropy induced by microcracks on effective elastic properties, Engng. Trans., 2005, 53, 4, 363-374.
• [78] Gambin B., Króner E., Higher-order terms in the homogenized stress-starain relation of periodic elastic media, Phys. Stat. Sol., 1989, (b), 151, 513-519.
• [79] Gambin B., Telega J.J., Effective properties of elastic solids with randomly di-stributed microcracks, Mech. Res. Comm., 2000, 27, 697-706.
• [80] Gambin B., Telega J.J., Nazarenko L., Stationary thermoelasticity and stochastic homogenization, Arch. Mech., 2002, 54, 319-345.
• [81] Glasser A.M., The use of transient FGM interlayers for joining advanced cera-mics. Composites, Part B, 1997, 28, 71-84.
• [82] Gibiansky L.V., Torąuato S., On the use of homogenization theory to design opti-mal piezocomposites for hydrophone applications, J. Mech. Phys. Solids, 1997, 45, 689-708.
• [83] Golden K., Papanicolaou G., Bounds for effective parameters of heterogeneous media by analytic continuation, Commun. Math. Phys., 1983, 90, 473-491.
• [84] Hetnarski R.B., Ignaczak J., Mathematical Theory of Elasticity, Taylor and Francis, New York, London, 2004.
• [85] Hetnarski R.B., Ignaczak J., Generalized thermoelasticity, J. Thermal Stresses, 1999, 22, 451-476.
• [86] Ichikawa K. (red), Functionally graded materials in the 2\th century, Kluwer Acad. Publ., Boston, Dodrecht, London, 2001.
• [87] Jemioło S., Telega J.J., Representation of tensor functions and applications in continuum mechanics, IFTR Reports, 3/1997.
• [88] Jikov V.V., Asymptotic problcms connected with the heat eąuation in perforated domains, Math. USSR - Sb., 1992, 71, 125-147.
• [89] Jikov V.V., Problems of function extension related to the theory of homogenization, Diff. Eqs., 1990, 26, 33-44.
• [90] Jikov V.V., Kozlov S.M., Oleinik O. A., Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
• [91] Jourdain B., Lelievre T., Le Bris C, Existence of solution for a micro-macro model of polymeric fluid: the FENE model, J. Func. Anal, 2004, 209, 162-193.
• [91] Khoroshun L.P., Methods of the theory of random functions in determining the macroscopic properties of microheterogeneous mec^a, Int. Appl. Mech., 1978, 14, 2, 3-17.
• [93] Khoroshun L.P., Nazarenko L.V., Thermoelasticity of orthotropic composites with ellipsoidal inclusions, Int. Appl. Mech., 1990, 26, 9, 3-10.
• [94] Koizumi M., FGM activities in Japan. Composites, Part B, 1997, 28, 1-4.
• [95] Komorowski T., Olla S., On the sector condition and homogenization of diffusion with a Gaussian drift, J. Func. Anal, 2003, 197, 179-211.
• [96] Kozlov S.M., Averaging of random operators, Math. USSR - Sb., 1980, 37, 167-180.
• [97] Kycia K., Nowicki A., Gail O., Hien T.D., Calculation of effective materiał tensors and the electromechanical coupling coefRcient of a type 1-3 composite transducer, Arch. Acoustics, 1996, 21, 371-386.
• [98] Lakes R., Foam structures with a negative Poisson's ratio, Science, 1987, 235, 1038-1040.
• [99] Larsen U.D., Sigmund O., Bouwstra S., Deisgn and fabrication of compliant micro-mechanisms and structures with negative Poisson's ratio, Journal of Mi-croelectronical Systems, 1997, 6, 99-106.
• [100] Laurent P.J., Approzimation et Optimisation, Hermann, Paris, 1972.
• [101] Lewiński T., Telega J.J., Plates, Laminates and Shells: Asymptotic Analysis and Homogenization, Singapore, World Scientific, 2000.
• [102] Licht C., Michaille G., Global-local subadditive ergodic theorems and ap-plication to homogenization in elasticity, Preprint Departement de Sciences Mathematiąues, Universte Mountpellier II, 4, 1997.
• [103] Lipton R., Homogenization of stresses fluctuations, failure criteria and design of functionally graded materials for strength and stiffness, J. Mech. Phys. Sol, 2001.
• [104] Liu G.R., Han X., Lam K.Y., Stress waves in functionally gradient materials and its use for materiał characterization, Composites: Part B, 1999, 30, 383-394.
• [105] Luciano R., Willis J.R., Boundary-layer corrections for stress and strain fields in randomly heterogeneous materials, J. Mech. Phys. Solids, 2003, 51, 1075-1088.
• [106] Łydżba D., Zastosowanie metody asymptotycznej homogenizacji w mechanice gruntów i skal, Oficyna Wydawnicza Politechniki Wrocławskiej, Wrocław, 2002.
• [107] Markworth A.J., Ramesh K.S., Parks W.P., Modelling studies applied to functionally graded materials, Journal of Materials Science, 1995, 30, 2183-2193.
• [108] Maugin G.A., Continuum Mechanics of Electromagnetic Solids, North-Holland, Amsterdam, 1988.
• [109] Maugin- G.A., Turbe N. Homogenization of piezoelectric composites via Bloch expansion, Int. J. Appl. Electrom,agnetics in Mat, 1991, 2, 135-140.
• [110] Messaoudi K., Michaille G., Stochastic homogenization of nonconvex integral functionals, Math. Modelling Numer. AnaL, 1995, 28, 329-356.
• [111] Milton G., The Theory of Composites, Cambridge University Press, 2002.
• [112] Milton G., Kohn R.V. J. Mech. Phys. Solids, 1988, 36, 6, 597-629.
• [113] Murat F., Compacite par compensation, Annali delia Scuola Norm. Sup. di Pisa, Classe di Scienze, Serie IV, 1978, 5, 489-508.
• [114] Murat F., Tartar L., H-convergence, w: Topics in mathematical modeling of com-posite materials, Birkhauser, 1997.
• [115] Nazarenko L.V., Thermoelastic properties of porous elastic materials, Int. Appl. Mech., 1997, 33, 2, 114-121.
• [116] Nelli Silva E.C., Nishivaki S., Kikuchi N., Design of piezocomposite materials and piezoelectric transducer using topology optimization. Part II, Arch. Comp. Meth. Eng., 1999, 6, 3, 191-222.
• [117] Nelli Silva E.C., Ono Fonseca J.S., Montero de Espinosa F., Crumm A.T., Brady G.A., Halloran J.W., Kikuchi N., Design of piezocomposite materials and piezoelectric transducer using topology optimization. Part I, Arch. Comp. Meth. Eng., 1999, 6, 2, 117-182.
• [118] Newnham R.E., www.personal.psu.edu/staff/r/e/renl
• [119] Nguetseng G., A generał coiwergence result for a functional related to the theory of homogenization, SI AM J. Math. Anal, 1989, 20, 608-623.
• [120] Nowacki W., Electromagnetic effects in deformable solids, PWN, Warszawa, 1980 (w j. polskim).
• [121] Oksendal B., Stochastic differential eąuations: An introduction with applications, Springer-Verlag, Berlin, 1992.
• [122] Oleinik O.A., Yosifian G.A., Shamaev A.S., Mathematical Problems of Strongly Nonhomogeneous Elastic Media, Izd. Moskovskogo Universiteta, Moskva, 1990 (w j. rosyjskim).
• [123] Ostoja-Starzewski M., On the admissibility of an isotropic, smooth elastic continuum, Arch. Mech., 2005, 57, 4, 345-355.
• [124] Pankov A., G-Convergence and Homogenization of Nonlinear Partial Differential Operators, Kluwer Academic Publishers, Dordrecht, 1997.
• [125] Paroni R., Homogenization of polycrystalline aggregates, Arch. Rat. Mech. Anal, 2000, 151, 311-337.
• [126] Papanicolaou G.C., Macroscopic properties of composites bubbly fluids, suspen-sions and related problems, In: Les Methodes de Uhomogeneisation: Theorie et Application en Physiąue, D. Bergman, J.L. Lions, G. Papanicolaou, L. Tar tar, E. Sanchez-Palencia (red.), Eyrolles, Paris 1982, 233-317.
• [127] Papanicolaou G.C., Varadhan S.R.S., Boundary value problems with rapidly oscillating random coefficients, In: Proc. Colloąuium on Random Fields: Rigo-rous Results in Statistical Mcchanics and Quantum Field Theory, Golloą. Mat. Soc. J. Bolyai, 1979, North-Holland, Amsterdam, 835-873.
• [128] Pardoux E., Piatnitski A.L., Homogenization of nonlinear random parabolic differential eąuation, Stochastic Processes and their AppL, 2003, 104, 1-17.
• [129] Pascali D., Sburlan S., Nonlinear Mappings of Monotone Type, SijthofF and No-ordhoff, Alphen aan den Rijn, The Netherlands, 1978.
• [130] Pellegrini Y.-P., Field distributions and effective-medium approximation for we-akly nonlinear media, Physical Review B, 2000, 61, 9365-9372.
• [131] Pellegrini Y.-P., Self-consistent effective-medium approximation for strongly nonlinear media, Physical Review B, 2001, 64, 1-9.
• [132] Ponte Castańeda P., Telega J.J., Gambin B. (red.), Nonlinear Homogenization and Its Applications to Composites, Polycrystals and Smart Materials, Kluwer Academic Publishers, Dordrecht, Boston, London, 2004.
• [133] Rockafellar R.T., Convex Analysis, Princeton University Press, Princeton, 1970.
• [134] Reiter T., Dvorak G.J., Micromechanical models for graded composite materials: II. Thermomechanical loading, J. Mech. Phys. Solids, 1998, 46, 9, 1651-1673.
• [135] Reiter T., Dvorak G.J., Tvergaard V., Micromechanical models for graded composite materials, J. Mech. Phys. Solids, 1997, 45, 1281-1302.
• [136] Roberts A.P., Statistical reconstruction of three-dimensional porous media from two-dimensional images, Phys. Remew E, 1997, 56, 3203-3212.
• [137] Rockafellar R.T., Convex Analysis, Princeton University Press, Princeton, 1970.
• [138] Rogula D. (red.), Nonlocal Theory of Materiał Media, CISM Courses and Lectu-res, No. 268, Springer-Verlag, Wien/New York, 1982.
• [139] Ruan X., Danforth S.C., Safari A., Chou T.-W., A theoretical study of the coupling effects in piezoelectric ceramics, Int. J. Solids and Structures, 1999 36 465-487. ' '
• [140] Sab K., Determination de la resistance macroscopiąue d' une plaąue perforee aleatoirement, C. R. Acad. Sci. Paris, 1994, 319, 491-497.
• [141] Sab K., Homogenization of non-linear random media by a duality method. Application to plasticity, Asymptotic Analysis, 1994, 9, 311-336.
• [142] Sab K., On the homogenization and simulation of random materials, Eur. J Mech., A/Solids, 1992, 11, 585-607.
• [143] Sanchez-Hubert J., Sanchez-Palencia E., Introduction aux Methods Asymptotiąu-es et d L'homogeneisation, Masson, Paris, 1992.
• [144] Sanchez-Palencia E., Non-Homogeneos Media and Vibration Theory, Springer-Verlag, Berlin, 1980.
• [145] Sevostianov I., Levin V., Kachanov M., On the modeling and design of piezocomposites with prescribed properties, Arch. Appl. Mech., 2001, 71, 733-747.
• [146] Shulga S.B., Homogenization of nonlinear variational problems by means of two-scale convergence, Proc. Steklov Institute of Mathematics, 2002, 236, 357-364.
• [147] Sigmund O., Torąuato S., Aksay LA., On the design of 1-3 piezocomposites using topology optimization, J. Mater. Res., 1998, 13, 4, 1038-1048.
• [148] Sobczyk K. Fale stochastyczne, PWN, Warszawa, 1982.
• [149] Sobczyk K., Reconstruction of random materiał microstructures: patterns of ma-ximum entropy, it Probabilistic Engineering Mechanics , 2003, 18, 4, 278-287.
• [150] Sobczyk K. (red.) Random Materiał Microstructure, Modelling and Mechanical Behaviour, AMAS Lect. Notes 17, Inst. of Fundament. Tech. Res., Polish Aca-demy of Sciences, Warsaw, 2004.
• [151] Sobczyk K., Kirkner D.J. Sochastic modelling of microstructures, Birkhauser, Boston, 2001.
• [152] Soluch W. i inni, Wstęp do piezoelektroniki, Wydawnictwa Komunikacji i Łączności, Warszawa, 1980.
• [153] Spagnolo S., Convergence in energy for elliptic operators, w: Hubbard B. (red.), Proceedings of the Third Symposium on Numerical Solutions of Partial Differen-tial Eąuations, (College Park). Academic Press, New York, 1975, 469-498.
• [154] Talbot D.R.S., Willis J.R., Bounds and self-consistent estimates for the overall properties of nonlinear composites, JMJ4 J. Appl. Math., 1987, 39, 215-240.
• [155] Telega J.J., Homogenization of fissured elastic solids in the presence of unilateral conditions and friction, Computational Mech., 1990, 6, 109-127.
• [156] Telega J.J., Justification of a refined scaling of stifFnesses of Reissner plates with fine periodic structure, Math. Models and Math. in Appl. Sci., 1992, 2, 375-406.
• [157] Telega J.J., Piezoelectricity and homogenization. Application to biomechanics, w: Continuum Models and Discrete Systems, vol. 2, G.A. Maugiń (red.), Longman, Scientific and Technical, Harlow, Essex, 1991, 220-229.
• [158] Telega J.J., Bielski W., Flow in random porous media: effective models, Compu-ters and Geotech., 2003, 30, 271-288.
• [159] Telega J.J., Bielski W., Nonstationary flow through random porous media with elastic skeleton, w: Poromechanics II, J.-L. Auriault, C. Geindreau, P. Royer, J.-F. Bloch, C. Boutin, J. Lewandowska (red.), A.A. Balkema Publishers, Lisse, The Netherlands, 2002, 569-574.
• [160] Telega J.J., Bielski W., Stochastic homogenization and macroscopic modelling of composites and flow through porous media, Theoret. Appl. Mech., 2002, 28-29, 337-377.
• [161] Telega J.J., Gałka A., Gambin B., Effective properties of physically nonlinear piezoelectric composites, Arch. Mech., 1998, 50, 321-340.
• [162] Telega J.J., Gambin B., EfTective properties of an elastic body damaged by random distribution of microcracks, w: Continuum Models and Discrete Systems, K.Z. Markov (red.), 1996, World Scientific, Singapore, 300-307.
• [163] Telega J.J., Gambin B., Rola nowoczesnych metod matematycznych - wariacyjnych i asymptotycznych - w modelowaniu materiałów kompozytowych i konstrukcji, w: Nauki Techniczne u Progu XXI Wieku: wizja wybranych dyscyplin z perspektywy IPPT PAN, M. Kleiber (red.), Drukarnia Braci Grodzickich, Warszawa, 2002, 33-40.
• [164] Telega J.J., Gambin B., Stochastic homogenization of elastic perfectly plastic Hencky solids: influence of boundary conditions, Buli. Pol Acad. Sci Tech Sci., 2001, 49, 17-29.
• [165] Telega J.J., Lewiński T., Stiffness loss of cross-play laminates with interlaminar cracks, w: MEC AM AT 93, International Seminar on Micromechanics of Mateńals, Editions Eyrolls, Paris, 1993, 317-326.
• [166] Telega J.J., Tokarzewski S., Gałka A., Effective conductivity of non-linear two-phase media: homogenization and two-point Pade approximants, Acta Appl Math., 2000, 61, 295-315.
• [167] Telega J.J., Wojnar R., Electrokinetics in random deformable porous media, w: IUTAM Symposium on the Mechanics of Physicochemical and Electrochemical Interactions in porous Media, May 18-23, 2003, Rolduc, Kerkade, The Nether-lands, S.C. Cowin, J.M. Huyghe (red.), Kluwer, 2003.
• [168] Tiersten H.F., Eąuations for the extension and fłexure of relatively thin electro-elastic plates undergoing large electric fields, w: Mechanics of Electromagnetic Mateńals and Structures, J.S. Lee, GA. Maugin, Y. Shindo (red.), AMD-vol 161 ASME, New York, 1993, 21-34.
• [169] Torąuato S., Random Heterogeneous Mateńals: Micro structure and Macroscopic Properties, Springer, New York, 2002.
• [170] Torąuato S., Random heterogeneous media: microstructure and improved bounds on effective properties, Appl. Mech. Rev., 1991, 44, 37-76.
• [171] Turbe N, Maugin G.A, On the linear piezoelectricity composite materials, Math. Meth. in the Appl. Sci., 1991, 14, 403-412.
• [172] Szymczyk J, Woźniak Cz, A contribution to the modelling of periodically lami-nated elestic solids, Electronic Journal of Polish Agńcultural Uniuersities, Cwil Engineeńng, 9, 1, 2006. Dostępne http://www.ejpau.media.pl
• [173] Woźniak M, On the dynamie behavior of micro-damaged stratified media, Int. J. FracL, 1995, 73, 223-232.
• [174] Rychlewska J, Woźniak Cz, Modelling of functionally graded laminates revisi-ted, Electronic Journal of Polish Agńcultural Uniuersities, Civil Engineeńng, 9, 2, 2006. Dostępne http://www.ejpau.media.pl
• [175] Willis J.R., Variational estimates for the overall response of an inhomogeneous nonlinear dielectric, Homogenization and Effectwe Properties of Materials and Media, J.L. Ericksen, D. Kinderlehrer, R.V. Kohn, J.-L. Lions (red.), Springer-Verlag, New York 1986, 245-263.
• [176] Wright S., On the steady-state flow of an incompressible fluid through a randomly perforated porous medium, J. Diff. Eąuations, 1998, 146, 261-286.
• [177] Yoon H.S., Katz J.L., Measurements of elastic properties and micro-hardness, J. Biomech., 1976, 9, 459.
• [178] Zhikov V.V., On an extension of the method of two-scale convergence and its applications, Math. Sbornik, 2000, 191, 31-72.