Calculation of deformations in nanocomposites using the block multipole method with the analytical-numerical account of the scale effects

D. B. Volkov-Bogorodskii   ; Yu. G. Evtushenko   ; V. I. Zubov   ; S. A. Lur'e

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 7, pp. 1302-1321

Résumé

Local scale effects for linear continuous media are investigated as applied to the composites reinforced by nanoparticles. A mathematical model of the interphase layer is proposed that describes the specific nature of deformations in the neighborhood of the interface between different phases in an inhomogeneous material. The characteristic length of the interphase layer is determined formally in terms of the parameters of the mathematical model. The local stress state in the neighborhood of the phase boundaries in the interphase layer is examined. This stress can cause a significant change of the integral macromechanical characteristics of the material as a whole if the interphase boundaries are long. Such a situation is observed in composite materials reinforced by microparticles and nanoparticles even when the volume concentration of the inclusions is small. A numerical simulation of the stress state is performed on the basis of the block analytical-numerical multipole method with regard for the local effects related to the special nature of the deformation of the interphase layer in the vicinity of the interface.

Bibliographie

[1] Thostenson E. T., Ren Z., Chou T. W., “Advances in the science and technology of carbon nanotubes and their composites: a review”, Composites Sci. and Technol., 6 (2001), 1899–1912 | DOI

[2] Cosserat E., Cosserat F., Theore des corps deformables, Hermann, Paris, 1909 | Zbl

[3] Mindlin R. D., “Micro- structure in linear elasticity”, Arch. Ration. Mech. and Analysis, 1 (1964), 51–78 | MR

[4] Toupin R. A., “Theories of elasticity with couple-stress”, Arch. Ration. Mech. and Analysis, 2 (1964), 85–112 | MR

[5] Aero E. L., Kuvshinskii E. B., “Osnovnye uravneniya teorii uprugosti sred s vraschatelnym vzaimodeistviem chastits”, Fiz. tverdogo tela, 2 (1960), 1399–1409 | MR

[6] Eshelby J. D., “The determination of the elastic field of an ellipsoidal inclusion and related problems”, Proc. Roy. Soc. London, 241:1226 (1957), 376–396 | DOI | MR | Zbl

[7] De Bum P., Kontinualnaya teoriya disloklinatsii, Mir, M., 1977

[8] Eshelbi D., Kontinualnaya teoriya dislokatsii, izd-vo Inostr. lit., M., 1963

[9] Kadic A., Edelen G. B., A Gauge theory of dislocation and disclination, New York, 1983

[10] Eringen A. K., Teoriya mikropolyarnoi uprugosti. Razrushenie, v. 2, Mir, M., 1975

[11] Belov P. A., Lure S. A., “Modeli deformirovaniya tverdykh tel i ikh analogi v teorii polya”, Izv. RAN. Mekhan. tverdogo tela, 1998, no. 3, 157–166 | MR

[12] Obraztsov I. F., Lure S. A., Yanovskii Yu. G., Belov P. A., “O nekotorykh klassakh modelei tonkikh struktur”, Izv. vuzov. Severo-Kavkazskii region, Estestvennye nauki (k 80-letiyu akad. I. I. Vorovicha), 2000, no. 3, 110–118, Rostov-na-Donu | Zbl

[13] Lure S. A., Belov P. A., Matematicheskie modeli mekhaniki sploshnoi sredy i fizicheskikh polei, VTs RAN, M., 2000

[14] Lure S. A., Belov P. A., Krivolutskaya I. I., “Ob odnoi modeli kogezionnykh vzaimodeistvii v sploshnykh sredakh”, Konstruktsii iz kompozitsionnykh materialov, no. 2, VIMI, M., 2000, 3–16 | MR

[15] Babeshko A. B., Lure S. A., Belov P. A., Yanovskii Yu. G., “Masshtabnye effekty v modelyakh sploshnykh sred”, Mekhan. kompozitnykh konstruktsii, 1:8 (2002), 71–82

[16] Lurie S., Belov P., Volkov-Bogorodsky D., Tuchkova N., “Nanomechanical modeling of the nanostructures and dispersed composites”, Int. J. Comput. Mater. Sci., 28:3–4 (2003), 529–539 | DOI

[17] Lurie S., Belov P., Volkov-Bogorodsky D., “Multiscale modeling in the mechanics of materials: cohesion, interfacial interactions, inclusions and defects”, Analys. and Simulation of Multifield Problems, Lecture Notes in Applied and Computational Mechanics, 12, Springer, 2003, 101–110 | Zbl

[18] Lurie S., Belov P., Tuchkova N., “The application of the multiscale models for description of the dispersed composites”, Comput. Math. Sci. A, 36:2 (2004), 145–152 | MR

[19] Sedov L. I., “Matematicheskie metody postroeniya novykh modelei sploshnykh sred”, Uspekhi matem. nauk, 20:5(125) (1965), 121–180 | MR | Zbl

[20] Lantsosh K., Variatsionnye printsipy mekhaniki, Mir, M., 1965

[21] Miva M., “Influence of the diameters of particles on the modulus of elasticity of reinforced polymers”, Kobunshi Ronbunshu, 35:2 (1978), 125–129

[22] Grachev H. I., Evtushenko Yu. G., “Biblioteka programm dlya resheniya zadach optimalnogo upravleniya”, Zh. vychisl. matem. i matem. fiz., 19:2 (1979), 367–387 | MR | Zbl

[23] Novatskii V., Teoriya uprugosti, Nauka, M., 1975

[24] Mors F. M., Feshbakh G., Metody teoreticheskoi fiziki, v. 1, Izd-vo inostr. lit., M., 1958

[25] Vlasov V. I., “Ob odnom metode resheniya nekotorykh ploskikh smeshanykh zadach dlya uravneniya Laplasa”, Dokl. AN SSSR, 23:5 (1977), 1012–1015 | MR

[26] Vlasov V. I., Prudnikov A. P., “Asimptotika reshenii nekotorykh kraevykh zadach dlya uravneniya Laplasa pri deformirovanii oblasti”, Sovrem. probl. matem. Itogi nauki i tekhn., 20, VINITI AN SSSR, M., 1982, 3–36 | MR

[27] Vlasov V. I., Prudnikov A. P., “Metod resheniya kraevykh zadach dlya uravneniya Laplasa v oblastyakh s krivolineinoi granitsei”, Matematichki vestnik, 38:4 (1986), 617–623 | MR | Zbl

[28] Vlasov V. I., Volkov D. B., “Zadacha Dirikhle v kruge s uglovym vyrezom”, Soobscheniya po prikladnoi matematike, VTs AN SSSR, M., 1986 | MR

[29] Vlasov V. I., Volkov D. B., Prudnikov A. P., “Reshenie zadachi Dirikhle v oblasti s granitsei, soderzhaschei skruglennyi ugol”, Z. anal. Anwend., 6:1 (1987), 61–73 | MR | Zbl

[30] Vlasov V. I., Kraevye zadachi v oblastyakh s krivolineinoi granitsei, VTs AN SSSR, M., 1987 | MR

[31] Vlasov V. I., Kraevye zadachi v oblastyakh s krivolineinoi granitsei, Dis. $\dots$ dokt. fiz.-matem. nauk, VTs AN SSSR, M., 1990 | MR

[32] Vlasov V. I., Paltsev A. B., “Metod resheniya zadachi Dirikhle dlya oblastei s uzkoi schelyu”, Dokl. RAN, 330:2 (1993), 140–143 | Zbl

[33] Vlasov V. I., Rachkov A. B., “Nekotorye obobscheniya metoda multipolei”, Soobsch. po prikl. matem., VTs RAN, M., 1994

[34] Vlasov V. I., Volkov D. B., “Metod resheniya zadachi sopryazheniya garmonicheskikh funktsii”, Dokl. RAN, 344:2 (1995), 151–154 | MR | Zbl

[35] Vlasov V. I., Volkov D. B., “Metod multipolei dlya resheniya uravneniya Puassona v oblastyakh so skruglennym uglom”, Zh. vychisl. matem. i matem. fiz., 35:6 (1995), 867–892 | MR | Zbl

[36] Vlasov V. I., “Multipole method for solving some boundary value problems in complex-shaped domains”, Z. angew. Math. und Mech., 76:1 (1996), 279–282 | MR | Zbl

[37] Vlasov V. I., Volkov D. B., “Analytic-numerical method for solving the Poisson equation in domains with rounded comers”, Z. angew. Math. und Mech., 76:1 (1996), 573–574 | DOI | MR | Zbl

[38] Vlasov V. I., Paltsev A. B., “O primenenii metoda multipolei k raschetu elektricheskogo polya v lazere spetsialnoi konstruktsii”, Zh. vychisl. matem. i matem. fiz., 37:10 (1997), 1221–1236 | MR | Zbl

[39] Vlasov V. I., Volkov-Bogorodskii D. B., “Analitike-chislennyi metod dlya resheniya uravnenii Lyame v slozhnykh prostpanstvennykh oblastyakh”, Tezisy dokl., Izv. RAN. Ser. Mekhan. tverdogo tela, 1998, no. 5, 176 | MR

[40] Vlasov V. I., Volkov-Bogorodsky D. B., “Block multipole method for boundary value problems in complex-shaped domains”, Z. angew. Math. und Mech., 78:3 (1998), 1111–1112 | MR

[41] Vlasov V. I., “A meshless method for solving boundary value problems in 3D domains of complex shape”, Book of Abstracts, The Fourth Internat. Congress on Industr. and Appl. Math. Edinburgh (5–9 July 1999), 1999, 323

[42] Vlasov V. I., Skorokhodov S. L., “Metod multipolei dlya zadachi Dirikhle v dvusvyaznykh oblastyakh slozhnoi formy. I. Obschee opisanie metoda”, Zh. vychisl. matem. i matem. fiz., 40:11 (2000), 1633–1647 | MR | Zbl

[43] Volkov-Bogorodskii D. B., “Razrabotka blochnogo analitiko-chislennogo metoda resheniya zadach mekhaniki i akustiki”, Sb. tr. shkoly-seminara “Kompozitsionnye materialy”, IPRIM RAN, M., 2000, 44–56

[44] Volkov-Bogorodskii D. B., “O reshenii sistemy uravnenii Lyame metodom naimenshikh kvadratov na mnogoblochnoi strukture”, Dinamich. i tekhnol. probl. mekhan. konstruktsii i sploshnykh sred, Materialy VII Mezhdunar. simpoziuma (Yaropolets, 12–16 fevralya 2001 g.), MAI, M., 2001, 99–102

[45] Volkov-Bogorodskii D. B., “Primenenie blochnogo analitiko-chislennogo metoda multipolei k zadacham akustiki”, Vestn. MAI, 12:3 (2005), 51–60

[46] Volkov-Bogorodskii D. B., “Podkhod k zadacham o vzaimodeistvii akusticheskoi i uprugoi sredy s pomoschyu blochnogo metoda multipolei”, Dinamich. i tekhnol. probl. mekhan. konstruktsii i sploshnykh sred, Materialy VII Mezhdunar. simpoziuma (Yaropolets, 14–18 fevralya 2005 g.), MAI, M., 2005, 17–22

[47] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 1, Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1973; т. 2, Функции Бесселя, функции параболического цилиндра, ортогональные многочлены, 1974

[48] Bakhvalov N. S., Panasenko G. P., Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR | Zbl

[49] Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M., Chislennye metody, Nauka, M., 1987 | MR | Zbl