Numerical aspects of mathematical modeling of crystal imperfection

Povstenko, Y.

Artykuł - szczegóły

Tytuł artykułu

Numerical aspects of mathematical modeling of crystal imperfection

Autorzy

Povstenko Y.

Wybrane pełne teksty z tego czasopisma

http://oficynawydawnicza.ewsie.edu.pl

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Warianty tytułu

Numeryczne aspekty matematycznego modelowania niedoskonałości kryształu

Języki publikacji

Abstrakty

In this paper, a survey of studies concerning mathematical modeling of crystal imperfections in solids is presented. The emphasis is placed on describing imperfections in nonlocal elastic continuum. Nonlocal theory reduces to the classical theory of elasticity in the long wave-length limit and to the atomic lattice theory in the short wave-length limit.

W artykule przedstawiono przegląd badań dotyczących modelowania matematycznego niedoskonałości sieci krystalicznej w ciele stałym. Nacisk kładzie się na opis niedoskonałości w nielokalnie odkształcalnym continuum. Teoria nielokalna w granicy długofalowej sprowadza się do klasycznej teorii sprężystości, a w granicy krótkofalowej – do teorii sieci atomowej.

Słowa kluczowe

imperfections of crystal lattice imperfections modeling nonlocal theory of elasticity

niedoskonałości sieci krystalicznej modelowanie niedoskonałości nielokalna teoria sprężystości

Wydawca

Oficyna Wydawnicza Europejskiej Uczelni

Czasopismo

Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie

Rocznik

2013

Tom

Nr 1(5)

Strony

1--13

Opis fizyczny

Bibliogr. 100 poz.

Twórcy

autor

Povstenko Y.

Europejska Wyższa Szkoła Informatyczno-Ekonomiczna w Warszawie, ul. Białostocka 22, 03-741 Warszawa

Bibliografia

[1] Barenblatt G. I., The mathematical theory of equilibrium cracks in brittle fracture, “Adv. Appl.Mech.” 7, 1962, p. 55-129.
[2] Beex L. A. A., Peerlings R. H. J., Geers M. G. D., A quasicontinuum methodology for multiscale analysis of discrete microstructural models, “Int. J. Numerical Meth. Engng.” 87, 2011, p. 701-718.
[3] Braun O.M., Kivshar Yu. S., Nonlinear dynamics of the Frenkel–Kontorova model, “Phys. Rep.” 306, 1998, p. 1-108.
[4] Braun O.M., Kivshar Yu. S., The Frenkel–Kontorova Model: Concepts, Methods and Applications, Springer-Verlag, Berlin 2004.
[5] Bullough R., Tewary V. K., Lattice theory of dislocations, [in:] Dislocations in Solids, vol. 2, ed. F. R. N. Nabarro, North Holland, Amsterdam 1979, p. 1-66.
[6] Catlow C. R. A. (ed.), Computer Modelling in Inorganic Crystallography, Academic Press, London 1997.
[7] Cherepanov G. P. (ed.), Fracture: A Topical Encyclopedia of Current Knowledge, Krieger, Melbourne 1998.
[8] Cuevas J., Sánchez-Rey B., Eilbeck J. C., Russels F.M., Interaction of moving discrete breathers with interstitial defects, “Discrete Continuous Dynamical Systems” Ser. S, 4, 2011, p. 1057-1067.
[9] Damask A. C., Diens G. J., Point Defects in Metals, Gordon and Breach Science Publishers, New York 1963.
[10] Di Paola M., Failla G., Pirrotta A., Sofi A., Zingales M., The mechanically based nonlocal elasticity: an overview of main results and future challenges, “Phil. Trans. Roy. Soc. A” 371, 2013, 20120433.
[11] Di Paola M., Zingales M., Long-range cohesive interactions of non-local continuum faced by fractional calculus, “Int. J. Solids Struct.” 45, 2008, p. 5642-5659.
[12] Dienes G. J., Paskin A., Computer modeling of cracks, [in:] Atomistics of Fracture, eds. R.M. Latanision, J. R. Pickens, Plenum Press, New York 1983, p. 671-704.
[13] Doyama M., Cotteril R.M. J., Atomic configurations of disclinations by computer simulation, “Phil.Mag.” 50, 1984, p. L7-L10.
[14] Duesbery M. S., Modeling of the dislocation core, [in:] Dislocations in Solids, vol. 8, Basic Problems and Applications, ed. F. R. N. Nabarro, North-Holland, Amsterdam1989, p. 67-173.
[15] Dugdale D. S., Yielding of steel sheets containing slits, “J. Mech. Phys. Solids” 8, 1960, p. 100-104.
[16] Edelen D. G. B., Nonlocal field theories, [in:] Continuum Physics, vol. 4, Polar and Nonlocal Field Theories, ed. A. C. Eringen, Academic Press, New York 1976, p. 75-204.
[17] Elliot T. B. (ed.), New Research on Semiconductors, NOVA Science Publishers, New York
[18] Eringen A. C., Edge dislocation in nonlocal elasticity, “Int. J. Engng. Sci.” 15, 1977, p. 177-183.
[19] Eringen A. C., Nonlocal Continuum Field Theories, Springer, New York 2002.
[20] Eringen A. C., Nonlocal polar field theories, [in:] Continuum Physics, vol. 4, Polar and Nonlocal Field Theories, ed. A. C. Eringen, Academic Press, New York 1976, p. 205-267.
[21] Eringen A. C., Screw dislocation in nonlocal elasticity, “J. Phys. D: Appl. Phys.” 10, 1977, p. 671-678.
[22] Eringen A. C., Ari N., Nonlocal stress field at Griffith crack, “Cryst. Latt. Def. Amorph. Mat.” 10, 1983, p. 33-38.
[23] Eringen A. C., Speciale C. G., Kim B. S., Crack-tip problem in nonlocal elasticity, “J. Mech. Phys. Solids” 25, 1977, p. 339-355.
[24] Fischmeister H. F., Exner H. E., Poech M.-H., Kohlhoff S., Gumbsch P., Schmauder S., Sigl L. S., Spiegler R., Modelling fracture processes in metals and composite materials, “Z. Metallk.” 80, 1989, p. 839-846.
[25] Gao H., Klein P., Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bonds, “J. Mech. Phys. Solids” 46, 1998, p. 187-218.
[26] Harding J. H., Computer simulation of defects in ionic solids, “Rep. Progr. Phys.” 53, 1990, p. 1403-1466.
[27] Hirth J. P., Lothe J., Theory of Dislocations, McGraw-Hill, New York 1968.
[28] Hirth J. P., Moono Rhee, Zbib H., Modeling of deformation by a 3D simulation of multiple, curved dislocations, “J. Computer-AidedMater. Design” 3, 1996, p. 164-166.
[29] Hsieh R. K. T., Volume defects in nonlocal micropolar elasticity, “Int. J. Engng. Sci.” 20, 1982, p. 261-270.
[30] Hull D., Bacon D. J., Introduction to Dislocations, 4th Edition, Pergamon Press, Oxford 2001.
[31] Jiang L. Y., Cheng J., Non-local theory for cracks in laminated media, “Theor. Appl. Fracture Mech.” 34, 2000, p. 235-242.
[32] Johnson R. A., Brown E., Point defects in copper, “Phys. Rev.” 127, 1962, p. 446-454.
[33] Kanzaki H., Point defect in face-centred cubic lattice – I distortion around defects, “J. Phys. Chem. Solids” 2, 1957, p. 24-36.
[34] Kelly A., Groves G.W., Kidd P., Crystallography and Crystal Defects, Wiley, Chichester 2000.
[35] Kohlhoff S., Gumbsch P., Fischmeister H. F., Crack propagation in b.c.c. crystals studied with a combined finite-element and atomistic model, “Phil. Mag. A” 64, 1991, p. 851-878.
[36] Kolesnikova A. L., Romanov A. E., Circular dislocation-disclination loops and their application to boundary problem solution in the theory of defects, Ioffe Physico-Technical Institute, Leningrad, Preprint No. 1019, 1986. (In Russian).
[37] Kolesnikova A. L., Romanov A. E., Representations of elastic fields of circular dislocation and disclination loops in terms of spherical harmonics and their application to various problems of the theory of defects, “Int. J. Solids Struct.” 47, 2010, p. 58-70.
[38] Kontorova T. A, Frenkel J. I., Theory of plastic deformation and twinning, “J. Exp. Theor. Phys. (Zhurnal Eksperimentalnoj i Teoreticheskoj Fiziki)” 8, 1938, p. 89-95. (In Russian).
[39] Kotomin E. A., Eglitis R. I., Borstel G., Jacobs P.W.M., Modeling of point defects, polarons and excitons in ferroelectric perovskites, [in:] Computational Materials Science. Vol. 187 of NATO Science Series, Series III: Computer and Systems Sciences, eds. R. Catlow, E. Kotomin, IOS Press Ohmsha, Washington 2003, p. 291-307.
[40] Kovács I., Vörös G., Lattice defects in nonlocal elasticity, “Physica B” 96, 1979, p. 111-115.
[41] Kröner E., Elasticity theory with long-range cohesive forces, “Int. J. Solids Struct.” 3, 1967, p. 731-742.
[42] Kröner E., Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer, Berlin 1958. 11
[43] Kroupa F., Dislocation loops, [in:] Theory of Crystal Defects, ed. B. Gruber, Academia, Prague 1966, p. 275-316.
[44] Krumhansl J. A., Some considerations on the relation between solid state physics and generalized continuum mechanics, [in:] Mechanics of Generalized Continua, ed. E. Kröner, Springer, Berlin 1968, p. 298-331.
[45] Kunin I. A., Theory of Elastic Media with Microstructure. Nonlocal Theory of Elasticity, Nauka, Moscow 1973. (In Russian).
[46] Likhachev V. A., Khairov R. Yu., Introduction to the Theory of Disclinations, Leningrad University Press, Leningrad 1975. (In Russian).
[47] Marcinkowski M. J., Sree Harsha K. S., Properties of finite circular dislocation glide loops, “J. Appl. Phys.” 39, 1968, p. 1775-1783.
[48] Moriarty J. A, Vitek V, Bulatov V. V., Yip S., Atomistic simulations of dislocations and defects, “J. Computer-Aided Mater. Design” 9, 2002, p. 99-132.
[49] Mura T.,Micromechanics of Defects in Solids, Martinus Nijhoff Pub., Dordrecht 1987.
[50] Nabarro F. R. N., Dislocations in a simple cubic lattice, “Proc. Phys. Soc. London” 59, 1947, p. 56-272.
[51] Nabarro F. R. N., Theory of Crystal Dislocations, Clarendon Press, Oxford 1967.
[52] Ortiz M., Philips R., Nanomechanics of defects in solids, “Adv. Appl. Mech.” 59, 1999, p. 1217-1233.
[53] Panasyuk V. V., Limit Equilibrium of Brittle Bodies with Cracks, Naukova Dumka, Kiev 1968. (In Russian).
[54] Peierls R. E., The size of a dislocation, “Proc. Phys. Soc. London” 52, 1940. p. 34-37.
[55] Podstrigach Ya. S., On one nonlocal theory of deformation of solids, “Appl. Mech. (Prikladnaya Mekhanika)” 3 (2), 1967, p. 71-76. (In Russian).
[56] Povstenko Y., Space-time-fractional heat conduction equation and the theory of thermoelasticity, [in:] 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey, 5-7 November 2008.
[57] Povstenko Y., Theories of thermal stresses based on space-time-fractional telegraph equations, “Comp. Math. Appl.” 64, 2012, p. 3321-3328.
[58] Povstenko Y., Theory of thermoelasticity based on the space-time-fractional heat conduction equation, “Phys. Scr. T” 136, 2009, 014017.
[59] Povstenko Y., Thermoelasticity which uses fractional heat conduction equation, “Math.Meth. Phys.-Mech. Fields” 51, 2008, p. 239-246. (See also “J. Math. Sci.” 62, 2009, p. 296-305).
[60] Povstenko Y. Z., Axisymmetric ring loading in nonlocal elastic space, “Int. J. Engng. Sci.” 39, 2001, p. 285-302.
[61] Povstenko Y. Z., Circular dislocation loops in non-local elasticity, “J. Phys. D: Appl. Phys.” 28, 1995, p. 105-111.
[62] Povstenko Y.Z., Circular rotational dislocation loop in nonlocal elastic medium, “Math. Meth. Phys.-Mech. Fields” 38, 1995, p. 95-98. (In Ukrainian). English translation: “J. Math. Sci.” 81, 1996, p. 3080-3083.
[63] Povstenko Y. Z., Fractional heat conduction equation and associated thermal stresses, “J. Thermal Stresses” 28, 2005, p. 83-102.
[64] Povstenko Y. Z., Imperfections in nonlocal elasticity, “J. Phys. (Paris)” 8, 1998, p. 309-316.
[65] Povstenko Y. Z., Modelling of crystal imperfections in non-local elastic continuum, [in:] Multiple Scale Analysis and Coupled Physical Systems: Saint-Venant Symposium, Press de l'école nat. des ponts et chaussées, Paris 1997, p. 535-542.
[66] Povstenko Y. Z., Nonlocal and gradient elasticity theories and their application to description of imperfections in solids, “Math. Meth. Phys-Mech. Fields” 46, 2003, p. 136-146. (In Ukrainian).
[67] Povstenko Y. Z., Point defect in a nonlocal elastic medium, “Math. Meth. Phys.-Mech. Fields” 41(3), 1998, p. 85-89. (In Ukrainian). English translation: “J. Math. Sci.” 104, 2001, p. 1501-1505.
[68] Povstenko Y. Z., Straight disclinations in nonlocal elasticity, “Int. J. Engng. Sci.” 33, 1995, p. 575-582.
[69] Povstenko Y. Z., Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation, “Int. J. Engng. Sci.” 43, 2005, p. 977-991.
[70] Povstenko Y. Z., Thermoelasticity based on fractional heat conduction equation, [in:] Proc. 6th Int. Congr. Thermal Stresses, May 26–29, 2005, Vienna, Austria, vol. 2, eds. F. Ziegler, R. Heuer, C. Adam, Vienna University of Technology, Vienna 2005, p. 501-504.
[71] Povstenko Y. Z., Two-dimensional axisymmentric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation, “Int. J. Solids Struct.” 44, 2007, p. 2324-2348.
[72] Povstenko Y. Z., Kubik I., Concentrated ring loading in a nonlocal elastic medium, “Int. J. Engng. Sci.” 43, 2005, p. 457-471.
[73] Povstenko Y. Z., Matkovskii O. A., Circular disclination loops in nonlocal elasticity, “Int. J. Solids Struct.” 37, 2000, p. 6419-6432.
[74] Raabe D., Computational Materials Science. The Simulation of Materials Microstructures and Properties, Wiley, Weinheim 1998.
[75] Ravi R. G., Imperfections and Impurities in Semiconductor Silicon, Wiley, New York 1981.
[76] Rhodes R. G., Imperfections and Active Centers in Semiconductors, Pergamon, London 1964.
[77] Rogula D., Introduction to nonlocal theory of material media, [in:] Nonlocal Theory of Material Media, ed. D. Rogula, Springer, Berlin 1982, p. 125-222.
[78] Romanov A. E., Vladimirov V. I., Disclinations in crystalline solids, [in:] Dislocations in Solids, Vol. 9, ed. F. R. N. Nabarro, North-Holland, Amsterdam 1992, p. 191-402,
[79] Sah C. T., Bulk and interface imperfections in semiconductors, “Solid-State Electronics” 19, 1976, p. 975-990.
[80] Seif D., Ghoniem N.M., Multipolar elastic representation of defect field in metals, Proc. Multiscale Modeling Conf., Freiburg, Germany 2010.
[81] Shenoy V. B., Miller R., Tadmoe E. B., Rodney D., Phylips R., Ortiz M., An adaptive finite element approach to atomistic scale mechanics – the quasicontinuum method, “J. Mech. Phys. Solids” 47, 1999, p. 611-642.
[82] Shenoy V. B., Ortiz M., Philips R., The atomistic structure and energy of nascent dislocation loops, “Model. Simul. Mater. Sci. Engng.” 7, 1999, p. 603-619.
[83] Sinclair J. E., Gehlen P.C., Hoagland R. G., Hirth J. P., Flexible boundary conditions and nonlinear geometric effects in atomic dislocation modeling, “J. Appl. Phys.” 49, 1978, p. 3890-3897.
[84] Sneddon I. N., Lowengrub M., Crack Problems in the Classical Theory of Elasticity, Wiley, New York 1969.
[85] Tadmor E. B., Ortiz M., Philips R., Mixed atomistic and continuum models of deformation of solids, “Langmuir” 12, 1996, p. 4529-4534.
[86] Tadmor E. B., Philips R., Ortiz M., Quasicontinuum analysis of defects in solids, “Phil. Mag. A” 73, 1996, p. 1529-1563.
[87] Teodosiu C., Elastic Models of Crystal Defects, Springer, Berlin 1982.
[88] Tewary V. K., Lattice-statics model for edge dislocations in crystal, “Phil. Mag. A” 80, 2000, p. 1445-1452.
[89] Tewary V. K., Multiscale Green’s-function method for modeling point defects and extended defects in anisotropic solids: Application to a vacancy and free surface in copper, “Phys. Rev. B” 69, 2004, 094109.
[90] Tewary V. K., Multiscale modeling of point defects and free surfaces in semi-infinite solids, [in:] Modeling and Numerical Simulation of Material Behavior and Evolution, eds. E. Olevsky, V. Tikare, A. Zavaliangos, Materials Research Society, Warrendale 2002, p. 21.
[91] Tewordt L., Distortion of the lattice around an interstitial, a crowdion, and a vacancy in copper, “Phys. Rev.” 109, 1958, p. 61-68.
[92] Thomson R., Physics of Fracture, [in:] Atomistics of Fracture, eds. R.M. Latanision, J. R. Pickens, Plenum Press, New York 1983, p. 167-204.
[93] Thomson R., Zhou S. J., Carlsson A.E., Tewary V. K., Lattice imperfections studied by use of lattice Green’s functions, “Phys. Rev. B” 46, 1992, p. 613-622.
[94] Van Bueren. H. G., Defects in Solids, North-Holland Publishing Company, Amsterdam1960.
[95] Vitek V., Lejček L., Paidar V., Models of the cores of dislocations in metals and disclinations in liquid crystals, “Czech. J. Phys.” 45, 1995, p. 1003-1018.
[96] Volterra V., Sur l'equilibre des corps élastiques multiplement connexes, “Annal. Ecole Norm. Super.”, Ser. 3, 24, 1907, p. 401-517.
[97] Wang R., Chen G. L., Sun Z. Q., Application of nonlocal elasticity to the energetics for solute atoms in body-centered cubic transition metals with dislocation, “Met. Trans. A” 23, 1992, p. 3115-3120.
[98] Willardson R. K., Beer A. C., Weber E. R. (eds.), Semiconductors and Semimetals, Imperfections in III/V Materials, vol. 38, Academic Press, London 1993.
[99] Yavari A., Ortiz M., Bhattacharya K., A theory of unharmonic lattice statics for analysis of defective crystals, “J. Elast.” 86, 2007, p. 41-83.
[100] Zhou Z.-G., Han J.-C., Du S.-Y., Investigation of a Griffith crack subject to antiplane shear by using the non-local elasticity, “Int. J. Solids. Struct.” 36, 1999, p. 3891-3901

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