Modeling of non-isothermic viscoelastic-plastic behavior of flexible reinforced plates
DOI:
https://doi.org/10.7242/1999-6691/2020.13.3.28Keywords:
flexible plates, crisscrossed reinforcement, non-isothermal deformation, viscoelastic plasticity, dynamic loading, Maxwell-Boltzmann body model, Ambartsumyan theory, explicit numerical schemeAbstract
Based on a step-by-step procedure over time, a numerical-analytical model of the nonisothermal viscoelastic-plastic behavior of crisscrossed-reinforced material is constructed. The components of the composition are isotropic; their viscoelastic deformation is described by the Maxwell - Boltzmann model of the body, and instantaneous inelastic behavior is described by the relations of the theory of elastoplastic deformation with isotropic hardening. In this case, the temperature dependence of the loading function and material constants is taken into account. The conditions of the viscoelastic deformation, the beginning of unloading, neutral and active viscoelastic-plastic loading of the thermosensitive materials of the composition are formulated. Relationship between the mechanical and thermophysical characteristics of the composite material is taken into account. The structural relationships of the thermophysical component of the problem are presented. The developed mathematical model is focused on the use of explicit schemes for the numerical integration of viscoelastic-plastic and thermophysical problems. The initial-boundary problem of nonisothermal viscoelastic-plastic bending deformation of fibrous plates is formulated. The poor resistance of such thin-walled constructions to transverse shear is described in the framework of Ambardzumyan's theory. The geometric nonlinearity of the problem is taken into account in the Karman approximation. In the transverse direction, the temperature is approximated by a second-order polynomial. To reduce the three-dimensional heat conduction problem to two-dimensional relations, the method of additional boundary conditions is used. The dynamic bending of a flat-crisscrossed-reinforced fiberglass plate under the influence of an air blast wave is investigated. It is shown that in the absence of external thermal heating in the process of oscillations, this structure heats by only 2-3ºС. In such cases, the calculation can reasonably be carried out without taking into account the temperature effect. In the presence of intense thermal loading of the structure, the heat sensitivity of the materials of the composition must be taken into account. It has been demonstrated that in the presence of a temperature field inhomogeneous over the thickness of the plate, the shape and magnitude of the residual deflection significantly depend on which face surface an external dynamic load is applied to.
Downloads
References
Mouritz A.P., Gellert E., Burchill P., Challis K. Review of advanced composite structures for naval ships and submarines // Compos. Struct. 2001. Vol. 53. P. 21-42. http://dx.doi.org/10.1016/S0263-8223(00)00175-6">http://dx.doi.org/10.1016/S0263-8223(00)00175-6
Bannister M. Challenger for composites into the next millennium – a reinforcement perspective // Compos. Appl. Sci. Manuf. 2001. Vol. 32. P. 901-910. https://doi.org/10.1016/S1359-835X(01)00008-2">https://doi.org/10.1016/S1359-835X(01)00008-2
Soutis C. Fibre reinforced composites in aircraft construction // Progr. Aero. Sci. 2005. Vol. 41. P. 143-151. https://doi.org/10.1016/j.paerosci.2005.02.004">https://doi.org/10.1016/j.paerosci.2005.02.004
Qatu M.S, Sullivan R.W., Wang W. Recent research advances on the dynamic analysis of composite shells: 2000–2009 // Compos. Struct. 2010. Vol. 93. P. 14-31. https://doi.org/10.1016/j.compstruct.2010.05.014">https://doi.org/10.1016/j.compstruct.2010.05.014
Gill S.K., Gupta M., Satsangi P. Prediction of cutting forces in machining of unidirectional glass-fiber-reinforced plastic composites // Front. Mech. Eng. 2013. Vol. 8. P. 187-200. https://doi.org/10.1007/s11465-013-0262-x">https://doi.org/10.1007/s11465-013-0262-x
Vasiliev V.V., Morozov E. Advanced mechanics of composite materials and structural elements. Elsever, 2013. 832 p.
Соломонов Ю.С., Георгиевский В.П., Недбай А.Я., Андрюшин В.А. Прикладные задачи механики композитных цилиндрических оболочек. М.: Физматлит, 2014. 408 с.
Gibson R.F. Principles of composite material mechanics. Taylor & Francis Group, LLC, 2016. 700 p. https://doi.org/10.1201/b19626">https://doi.org/10.1201/b19626
Димитриенко Ю.И. Механика композитных конструкций при высоких температурах. М.: Физматлит, 2018. 448 с.
Тарнопольский Ю.М., Жигун И.Г., Поляков В.А. Пространственно-армированные композиционные материалы: Справочник. М.: Машиностроение, 1987. 224 с.
Kazanci Z. Dynamic response of composite sandwich plates subjected to time-dependent pressure pulses // Int. J. Non Lin. Mech. 2011. Vol. 46. P. 807-817. https://doi.org/10.1016/j.ijnonlinmec.2011.03.011">https://doi.org/10.1016/j.ijnonlinmec.2011.03.011
Morinière F.D., Alderliesten R.C., Benedictus R. Modelling of impact damage and dynamics in fibre-metal laminates – A review // Int. J. Impact Eng. 2014. Vol. 67. P. 27-38. https://doi.org/10.1016/j.ijimpeng.2014.01.004">https://doi.org/10.1016/j.ijimpeng.2014.01.004
Vena P., Gastaldi D., Contro R. Determination of the effective elastic-plastic response of metal-ceramic composites // Int. J. Plast. 2008. Vol. 24. P. 483-508. https://doi.org/10.1016/j.ijplas.2007.07.001">https://doi.org/10.1016/j.ijplas.2007.07.001
Brassart L., Stainier L., Doghri I., Delannay L. Homogenization of elasto-(visco) plastic composites based on an incremental variational principle // Int. J. Plast. 2012. Vol. 36. P. 86-112. https://doi.org/10.1016/j.ijplas.2012.03.010">https://doi.org/10.1016/j.ijplas.2012.03.010
Ахундов В.М. Инкрементальная каркасная теория сред волокнистого строения при больших упругих и пластических деформациях // Механика композитных материалов. 2015. Т. 34, № 5. С. 419-432. (English version https://doi.org/10.1007/s11029-015-9509-4">https://doi.org/10.1007/s11029-015-9509-4)
Янковский А.П. Применение явного по времени метода центральных разностей для численного моделирования динамического поведения упругопластических гибких армированных пластин // Вычисл. мех. сплош. сред. 2016. Т. 9, № 3. С. 279-297. https://doi.org/10.7242/1999-6691/2016.9.3.24">https://doi.org/10.7242/1999-6691/2016.9.3.24
Янковский А.П. Моделирование вязкоупругопластического деформирования гибких армированных пластин с учетом слабого сопротивления поперечному сдвигу // Вычисл. мех. сплош. сред. 2019. Т. 12, № 1. С. 80-97. https://doi.org/10.7242/1999-6691/2019.12.1.8">https://doi.org/10.7242/1999-6691/2019.12.1.8
Безухов Н.И., Бажанов В.Л., Гольденблат И.И., Николаенко Н.А., Синюков А.М. Расчеты на прочность, устойчивость и колебания в условиях высоких температур / Под ред. И.И. Гольденблата. М.: Машиностроение, 1965. 567 с.
Белл Дж. Экспериментальные основы механики деформируемых твердых тел. Часть II. Конечные деформации. М.: Мир, 1984. 432 с.
Композиционные материалы. Справочник / Под ред. Д.М. Карпиноса. Киев: Наукова думка, 1985. 592 с.
Бондарь В.С. Неупругость. Варианты теории. М.: Физматлит, 2004. 144 с.
Нагди П.М., Мерч С.А. О механическом поведении вязко-упруго-пластических тел // Прикл. механика: Тр. Америк. об-ва инж.-механ. Сер. Е. 1963. T. 30, № 3. C. 3-12. (English version https://doi.org/10.1115/1.3636556">https://doi.org/10.1115/1.3636556)
Паймушин В.Н., Фирсов В.А., Гюнал И., Егоров А.Г., Каюмов Р.А. Теоретико-экспериментальный метод определения параметров демпфирования на основе исследования затухающих изгибных колебаний тест-образцов. 3. Идентификация характеристик внутреннего демпфирования // Механика композитных материалов. 2014. Т. 50, № 5. С. 883-902. (English version https://doi.org/10.1007/s11029-014-9451-x">https://doi.org/10.1007/s11029-014-9451-x)
Reissner E. The effect of transverse shear deformation on the bending of elastic plates // J. Appl. Mech. 1945. Vol. 12. P. A69-A77.
Богданович А.Е. Нелинейные задачи динамики цилиндрических композитных оболочек. Рига: Зинатне, 1987. 295 с.
Абросимов Н.А., Баженов В.Г. Нелинейные задачи динамики композитных конструкций. Н. Новгород: Изд-во ННГУ, 2002. 400 с.
Баженов В.А., Кривенко О.П., Соловей Н.А. Нелинейное деформирование и устойчивость упругих оболочек неоднородной структуры. Модели, методы, алгоритмы, малоизученные и новые задачи. М.: Книжный дом «ЛИБРОКОМ», 2012. 336 с.
Амбарцумян С.А. Теория анизотропных пластин. Прочность, устойчивость и колебания. М.: Наука, 1987. 360 с.
Reddy J.N. Mechanics of laminated composite plates and shells: Theory and analysis. CRC Press, 2004. 831 p.
Андреев А. Упругость и термоупругость слоистых композитных оболочек. Математическая модель и некоторые аспекты численного анализа. Saarbrucken, Deutschland: Palmarium Academic Publishing, 2013. 93 c.
Белькаид К., Тати А., Бумараф Р. Простой конечный элемент с пятью степенями свободы в узле, основанный на теории сдвигового деформирования третьего порядка // Механика композитных материалов. 2016. Т. 52, № 2. С. 367-384. (English version https://doi.org/10.1007/s11029-016-9578-z">https://doi.org/10.1007/s11029-016-9578-z)
Whitney J.M., Sun C.T. A higher order theory for extensional motion of laminated composites // J. Sound Vib. 1973. Vol. 30. P. 85-97. https://doi.org/10.1016/S0022-460X(73)80052-5">https://doi.org/10.1016/S0022-460X(73)80052-5
Иванов Г.В., Волчков Ю.М., Богульский И.О., Анисимов С.А., Кургузов В.Д. Численное решение динамических задач упругопластического деформирования твердых тел. Новосибирск: Сиб. унив. изд-во, 2002. 352 с.
Houlston R., DesRochers C.G. Nonlinear structural response of ship panels subjected to air blast loading // Comput. Struct. 1987. Vol. 26. No. 1-2. P. 1-15. https://doi.org/10.1016/0045-7949(87)90232-X">https://doi.org/10.1016/0045-7949(87)90232-X
Zeinkiewicz O.C., Taylor R.L. The finite element method. Butterworth-Heinemann, 2000. Vol. 1. The basis. 707 p.
Фрейденталь А., Гейрингер Х. Математические теории неупругой сплошной среды. М.: Физматгиз, 1962. 432 с.
Деккер К., Вервер Я. Устойчивость методов Рунге–Кутты для жестких нелинейных дифференциальных уравнений. М.: Мир, 1988. 334 с.
Хажинский Г.М. Модели деформирования и разрушения металлов. М: Научный мир, 2011. 231 с.
Малмейстер А.К., Тамуж В.П., Тетерс Г.А. Сопротивление жестких полимерных материалов. Рига: Зинатне, 1972. 498 с.
Янковский А.П. Определение термоупругих характеристик пространственно армированных волокнистых сред при общей анизотропии материалов компонент композиции. 1. Структурная модель // Механика композитных материалов. 2010. Т. 46, № 5. С. 663-678. (English version https://doi.org/10.1007/s11029-010-9162-x">https://doi.org/10.1007/s11029-010-9162-x)
Янковский А.П. Моделирование процессов теплопроводности в пространственно-армированных композитах с произвольной ориентацией волокон // Прикладная физика. 2011. № 3. С. 32-38.
Грешнов В.М. Физико-математическая теория больших необратимых деформаций металлов. М.: Физматлит, 2018. 232 с.
Кудинов А.А. Тепломассообмен. М.: ИНФРА-М, 2012. 375 с.
Кудинов В.А., Кудинов И.В. Методы решения параболических и гиперболических уравнений теплопроводности / Под ред. Э.М. Карташова. М.: Книжный дом «ЛИБРОКОМ», 2012. 280 с.
Новицкий Л.А., Кожевников И.Г. Теплофизические свойства материалов при низких температурах. Справочник. М.: Машиностроение, 1975. 216 с.
Рихтмайер Р., Мортон К. Разностные методы решения краевых задач. М: Мир, 1972. 418 с.
Теплотехника / Под ред. В.Н. Луканина. 4-е изд., испр. М.: Высш. шк., 2003. 671 с.
Справочник по композитным материалам / Под ред. Дж. Любина. М.: Машиностроение, 1988. Кн. 1. 448 с.
###
Mouritz A.P., Gellert E., Burchill P., Challis K. Review of advanced composite structures for naval ships and submarines. Compos. Struct., 2001, vol. 53, pp. 21-42. http://dx.doi.org/10.1016/S0263-8223(00)00175-6">http://dx.doi.org/10.1016/S0263-8223(00)00175-6
Bannister M. Challenger for composites into the next millennium – a reinforcement perspective. Compos. Appl. Sci. Manuf., 2001, vol. 32, pp. 901-910. https://doi.org/10.1016/S1359-835X(01)00008-2">https://doi.org/10.1016/S1359-835X(01)00008-2
Soutis C. Fibre reinforced composites in aircraft construction. Progr. Aero. Sci., 2005, vol. 41, pp. 143-151. https://doi.org/10.1016/j.paerosci.2005.02.004">https://doi.org/10.1016/j.paerosci.2005.02.004
Qatu M.S, Sullivan R.W., Wang W. Recent research advances on the dynamic analysis of composite shells: 2000–2009. Compos. Struct., 2010, vol. 93, pp. 14-31. https://doi.org/10.1016/j.compstruct.2010.05.014">https://doi.org/10.1016/j.compstruct.2010.05.014
Gill S.K., Gupta M., Satsangi P. Prediction of cutting forces in machining of unidirectional glass-fiber-reinforced plastic composites. Front. Mech. Eng., 2013, vol. 8, pp. 187-200. https://doi.org/10.1007/s11465-013-0262-x">https://doi.org/10.1007/s11465-013-0262-x
Vasiliev V.V., Morozov E. Advanced mechanics of composite materials and structural elements. Elsever, 2013. 832 p.
Solomonov Yu.S., Georgiyevskiy V.P., Nedbai A.Ya., Andryushin V.A. Prikladnyye zadachi mekhaniki kompozitnykh tsilindricheskikh obolochek [Applied problems of mechanics of composite cylindrical shells]. Moscow, Fizmatlit, 2014. 408 p.
Gibson R.F. Principles of composite material mechanics. Taylor & Francis Group, LLC, 2016. 700 p. https://doi.org/10.1201/b19626">https://doi.org/10.1201/b19626
Dimitriyenko Yu.I. Mekhanika kompozitnykh konstruktsiy pri vysokikh temperaturakh [Mechanics of composite structures at high temperatures]. Moscow, Fizmatlit, 2018. 448 p.
Tarnopol’skiy Yu.M., Zhigun I.G., Polyakov V.A. Prostranstvenno-armirovannyye kompozitsionnyye materialy: Spravochnik [Spatially reinforced composite materials: Reference Book]. Moscow: Mashinostroyeniye, 1987, 224 p.
Kazanci Z. Dynamic response of composite sandwich plates subjected to time-dependent pressure pulses. Int. J. Non Lin. Mech., 2011, vol. 46, pp. 807-817. https://doi.org/10.1016/j.ijnonlinmec.2011.03.011">https://doi.org/10.1016/j.ijnonlinmec.2011.03.011
Morinière F.D., Alderliesten R.C., Benedictus R. Modelling of impact damage and dynamics in fibre-metal laminates – A review. Int. J. Impact Eng., 2014, vol. 67, pp. 27-38. https://doi.org/10.1016/j.ijimpeng.2014.01.004">https://doi.org/10.1016/j.ijimpeng.2014.01.004
Vena P., Gastaldi D., Contro R. Determination of the effective elastic-plastic response of metal-ceramic composites. Int. J. Plast., 2008, vol. 24, pp. 483-508. https://doi.org/10.1016/j.ijplas.2007.07.001">https://doi.org/10.1016/j.ijplas.2007.07.001
Brassart L., Stainier L., Doghri I., Delannay L. Homogenization of elasto-(visco) plastic composites based on an incremental variational principle. Int. J. Plast., 2012, vol. 36, pp. 86-112. https://doi.org/10.1016/j.ijplas.2012.03.010">https://doi.org/10.1016/j.ijplas.2012.03.010
Akhundov V.M. Incremental carcass theory of fibrous media under larger elastic and plastic deformations. Mech. Compos. Mater., 2015, vol. 51, pp. 539-558. https://doi.org/10.1007/s11029-015-9509-4">https://doi.org/10.1007/s11029-015-9509-4
Yankovskii A.P. Applying the explicit time central difference method for numerical simulation of the dynamic behavior of elastoplastic flexible reinforced plates. J. Appl. Mech. Tech. Phy., 2017, vol. 58, pp. 1223-1241. https://doi.org/10.1134/S0021894417070112">https://doi.org/10.1134/S0021894417070112
Yankovskii A.P. Modelling the viscoelastic-plastic deformation of flexible reinforced plates with account of weak resistance to transverse shear. Vychisl. mekh. splosh. sred – Computational Continuum Mechanics, 2019, vol. 12, no. 1, pp. 80-97. https://doi.org/10.7242/1999-6691/2019.12.1.8">https://doi.org/10.7242/1999-6691/2019.12.1.8
Bezukhov N.I., Bazhanov V.L., Gol’denblat I.I., Nikolayenko N.A., Sinyukov A.M. Raschety na prochnost’, ustoychivost’ i kolebaniya v usloviyakh vysokikh temperatur [Calculations for strength, stability and fluctuations in high temperature conditions], ed. by I.I. Gol’denblatt. Moscow, Mashinostroyeniye, 1965. 567 p.
Flügge S. (ed.) Encyclopedia of Physics. Vol. VIa/1. Mechanics of Solids I. Springer-Verlag, 1973. 813 p.
Karpinos D.M. (ed.) Kompozitsionnyye materialy. Spravochnik [Composite materials. Reference Book]. Kiev: Naukova dumka, 1985. 592 p.
Bondar’ V.S. Neuprugost'. Varianty teorii [Inelasticity. Variants of the theory]. Moscow, Fizmatlit, 2004. 144 p.
Naghdi P.M., Murch S.A. On the mechanical behavior of viscoelastic/plastic solids. J. Appl. Mech., 1963, vol. 30, no. 3, pp. 321-328. https://doi.org/10.1115/1.3636556">https://doi.org/10.1115/1.3636556
Paimushin V.N., Firsov V.A., Gyunal I., Egorov A.G., Kayumov R.A. Theoretical-experimental method for determining the parameters of damping based on the study of damped flexural vibrations of test specimens. 3. Identification of the characteristics of internal damping. Mech. Compos. Mater., 2014, vol. 50, pp. 633-646. https://doi.org/10.1007/s11029-014-9451-x">https://doi.org/10.1007/s11029-014-9451-x
Reissner E. The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech., 1945, vol. 12, pp. A69-A77.
Bogdanovich A.E. Nelineynyye zadachi dinamiki tsilindricheskikh kompozitnykh obolochek [Nonlinear problems of the dynamics of cylindrical composite shells]. Riga, Zinatne, 1987. 295 p.
Abrosimov N.A., Bazhenov V.G. Nelineynyye zadachi dinamiki kompozitnykh konstruktsiy [Nonlinear problems of dynamics composites designs]. Nizhniy Novgorod: Nizhniy Novgorod State University, 2002. 400 p.
Bazhenov V.A., Krivenko O.P., Solovey N.A. Nelineynoye deformirovaniye i ustoychivost’ uprugikh obolochek neodnorodnoy struktury. Modeli, metody, algoritmy, maloizuchennyye i novyye zadachi [Nonlinear deformation and stability of elastic shells of non-uniform structure. Models, methods, algorithms, the insufficiently studied and new problems]. M.: Knizhnyi dom “LIBROKOM”, 2012. 336 p.
Ambartsumyan S.A. Teoriya anizotropnykh plastin. Prochnost’, ustoychivost’ i kolebaniya [The theory of anisotropic plates. Strength, stability and fluctuations]. Moscow, Nauka, 1987. 360 p.
Reddy J.N. Mechanics of laminated composite plates and shells: Theory and analysis. CRC Press, 2004. 831 p.
Andreev A. Uprugost’ i termouprugost’ sloistykh kompozitnykh obolochek. Matematicheskaya model’ i nekotoryye aspekty chislennogo analiza [Elasticity and thermo-elasticity layered composite shells. Mathematical model and some aspects of the numerical analysis.]. Saarbrucken, Deutschland: Palmarium Academic Publishing, 2013. 93 p.
Belkaid K., Tati A., Boumaraf R. A simple finite element with five degrees of freedom based on Reddy’s third-order shear deformation theory. Mech. Compos. Mater., 2016, vol. 52, no. 2, pp. 257-270. https://doi.org/10.1007/s11029-016-9578-z">https://doi.org/10.1007/s11029-016-9578-z
Whitney J.M., Sun C.T. A higher order theory for extensional motion of laminated composites. J. Sound Vib., 1973, vol. 30, pp. 85-97. https://doi.org/10.1016/S0022-460X(73)80052-5">https://doi.org/10.1016/S0022-460X(73)80052-5
Ivanov G.V., Volchkov Yu.M., Bogul’skiy I.O., Anisimov S.A., Kurguzov V.D. Chislennoye resheniye dinamicheskikh zadach uprugoplasticheskogo deformirovaniya tverdykh tel [Numerical solution of dynamic problems of elastoplastic deformation of solids]. Novosibirsk: Siberian university, 2002. 352 p.
Houlston R., DesRochers C.G. Nonlinear structural response of ship panels subjected to air blast loading. Comput. Struct., 1987, vol. 26, no. 1-2, pp. 1-15. https://doi.org/10.1016/0045-7949(87)90232-X">https://doi.org/10.1016/0045-7949(87)90232-X
Zeinkiewicz O.C., Taylor R.L. The finite element method. Butterworth-Heinemann, 2000. Vol. 1. The basis. 707 p.
Freudental A.M., Geiringer H. The mathematical theories of the inelastic continuum / Flugge S. (ed.) Handbuch der Physik. Band VI: Elastizitat und Plastizitat [Handbook of Physics. Vol. 6: Elasticity and Plasticity]. Springer, 1958. 642 p. Pp. 229‑433. https://doi.org/10.1007/978-3-642-45887-3_3">https://doi.org/10.1007/978-3-642-45887-3_3
Dekker K., Verwer J.G. Stability of Runge–Kutta methods for stiff nonlinear differential equation. Amsterdam-New York-Oxford: North-Holland, 1984. 307 p.
Khazhinskiy G.M. Modeli deformirovaniya i razrusheniya metallov [Model of deformation and fracture of metals]. Moscow, Nauchnyy mir, 2011. 231 p.
Malmeyster A.K., Tamuzh V.P., Teters G.A. Soprotivleniye zhestkikh polimernykh materialov [Resistance of rigid polymeric materials]. Riga, Zinatne, 1972, 498 p.
Yankovskii A.P. Determination of the thermoelastic characteristics of spatially reinforced fibrous media in the case of general anisotropy of their components. 1. Structural model. Mech. Compos. Mater., 2010, vol. 46, no. 5, pp. 451-460. https://doi.org/10.1007/s11029-010-9162-x">https://doi.org/10.1007/s11029-010-9162-x
Yankovskii A.P. Modelling of processes of thermal conductivity in spatially reinforced composites with any orientation of fibres]. Prikladnaya fizika – Applied Physics, 2011, no. 3, pp. 32-38.
Greshnov V.M. Greshnov V.M. Fiziko-matematicheskaya teoriya bol’shikh neobratimykh deformatsiy metallov [Physical and mathematical theory of large irreversible deformations of metals]. Moscow, Fizmatlit, 2018. 232 p.
Kudinov A.A. Teplomassoobmen [Heat and mass transfer]. Moscow, INFRA-M, 2012. 375 p.
Kudinov V.A., Kudinov I.V. Metody resheniya parabolicheskikh i giperbolicheskikh uravneniy teploprovodnosti [Methods for solving parabolic and hyperbolic heat equations], ed. by E.M. Kartashov. Moscow, Book House "LIBROCOM", 2012. 280 p.
Novitsky L.A., Kozhevnikov I.G. Teplofizicheskiye svoystva materialov pri nizkikh temperaturakh. Spravochnik [Thermophysical properties of materials at low temperatures. Handbook]. Moscow: Mashinostroyeniye, 1975. 216 p.
Richtmyer R.D., Morton K.W. Difference methods for initial-value problems. John Wiley & Sons, 1967. 405 p.
Lukanin V.N. (ed.) Teplotekhnika [Heat engineering]. 4th ed. Moscow, Vysshaya shkola, 2003. 671 p.
Lubin G. (ed.) Handbook of composites. New York: Van Nostrand Reinhold Company Inc., 1982. 786 p.
Downloads
Published
Issue
Section
License
Copyright (c) 2020 Computational Continuum Mechanics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
