Structural Mechanics of Engineering Constructions and BuildingsStructural Mechanics of Engineering Constructions and Buildings1815-52352587-8700Peoples’ Friendship University of Russia (RUDN University)2496510.22363/1815-5235-2020-16-5-323-333Research ArticleMultitudes of Voigt - Reuss forks and Voigt - Christensen - Reuss tridentsErofeevVladimir T.<p>Doctor of Technical Sciences, Professor, Dean of the Faculty of Architecture and Civil Engineering, Head of the Department of Building Materials and Technologies, Academician of the Russian Academy of Architecture and Construction Sciences</p>tingaev.s1@gmail.comTyuryakhinAleksej S.<p>Doctor of Engineering, Associate Professor of the Department of Applied Mechanics of the Faculty of Architecture and Civil Engineering</p>tingaev.s1@gmail.comTyuryakhinaTatyana P.<p>graduate student of the Department of Building Materials and Technologies of the Faculty of Architecture and Civil Engineering.</p>tingaev.s1@gmail.comNational Research Ogarev Mordovia State University (National Research University)1512202016532333317112020Copyright © 2020, Erofeev V.T., Tyuryakhin A.S., Tyuryakhina T.P.2020<p>In the literature, there are many studies of the representative volume of a composite material, in particular, those calculated using the formulas of Christensen, Voigt and Reiss. The aim of this work is to study the features of evaluating the set of forks of effective modules. Methods. On the basis of solving the Lame problem (for a thick-walled sphere), a spherical model of a representative volume (cell) of a composite material with a granular (spherical) filler is compiled and the value of the effective modulus of elasticity of a two-phase composite is determined. The study of the obtained formula for the effective modulus, expressed in dimensionless quantities, for the cell material revealed its identity with the R.M. Christensens formula, expressed in dimensional values, for the bulk modulus of composites with a spherical filler. In this case, Christensens solution was previously obtained by a different method when he considered the polydisperse model of the composite. The dimensionless form of the function (effective module) of three dimensionless parameters made it possible in flat spaces (two coordinate planes) to construct graphical images of the function of the named modules according to Christensen, which are compared and combined in one figure with similar images of the functions of estimating the values of the modules (real composites) according to Voigt and Reiss. Graphical studies in relation to the spherical representative volume model show that in the flat space of the set of Voigt - Reuss forks, these forks are not narrowed, but they are partially filled by the flat space of the set of Christensen - Reiss forks. The graphs of the functions of the modules, at the same time, form, simultaneously with the sets of two-toothed forks, a set of Voigt - Christensen - Reiss trident forks (tridents), which, depending on the size of the intervals of the numbers of the studied parameters, have forks of different sizes. Results. Graphic illustrations of numerical examples have been obtained showing that for given values of the module of the matrix and filler and the volume fraction of the latter, it is possible to determine the effective volumetric module and shear module of two-phase composites, and to perform a comparison with the conclusions of the applied plan. The dimensionless form of the obtained expressions makes it possible to solve the inverse problems of the mechanics of polydisperse composites, for example, to determine the volume module of the composite components by the effective modulus obtained by mechanical testing of standard samples.</p>spherical model of a representative volume of materialLame’s problem for a thick-walled sphereeffective module of two-phase compositesVoigt - Reiss forkChristensen - Reiss forkсферическая модель представительного объема материалазадача Ламе для толстостенной сферыэффективные модули двухфазных композитоввилка Фойгта - Рейссавилка Кристенсена - Рейсса[Bobryshev A.N., Erofeev V.T., Kozomazov V.M. Fizika i sinergetika dispersno-neuporyadochennyh kondensirovannyh kompozitnyh sistem [Physics and synergetics of dispersively disordered condensed composite systems]. Saint Petersburg: Nauka Publ.; 2012. (In Russ.)][Gusev B.V., Kondrashenko V.I., Maslov B.P., Faysovich A.S. Formirovanie struktury kompozicionnyh materialov i ih svojstva [Formation of the structure of composite materials and their properties]. Moscow: Nauchnyj mir Publ.; 2006. (In Russ.)][Christensen R.M. Vvedenie v mekhaniku kompozitov [Introduction to the mechanics of composites]. Moscow: Mir Publ.; 1982. 336 p. (In Russ.)][Vasiliev V.V., Protasov V.D., Bolotin V.V. Kompozitsionnye materialy [Composite material]: handbook. Moscow: Mashinostroenie Publ.; 1990. (In Russ.)][Pronina Y.G. Analytical solution for decelerated mechanochemical corrosion of pressurized elastic-perfectly plastic thick-walled spheres. Corrosion Science. 2015;90:161–167. DOI: 10.1016/j.corsci.2014.10.007.][Arya V.K. Analytical and finite element solutions of some problems using a viscoplastic model. Computers and Structures. 1989;33(4):957–967. DOI: 10.1016/0045-7949(89)90430-6.][Loghman A., Shokouhi N. Creep damage evaluation of thick-walled spheres using a long-term creep constitutive model. Journal of Mechanical Science and Technology. 2009;23(10). Article number: 2577. DOI: 10.1007/s12206-009-0631-x.][Loghman A., Ghorbanpour Arani A., Aleayoub S.M.A. Time-dependent creep stress redistribution analysis of thick-walled functionally graded spheres. Mechanics of Time-Dependent Materials. 2011;15(4):353–365. DOI: 10.1007/s11043-011-9147-8.][Cowper G.R. The elastoplastic thick-walled sphere subjected to a radial temperature gradient. Journal of Applied Mechanics. 1960;27(3):496–500. DOI: 10.1115/1.3644030.][Durban D., Baruch M. Analysis of an elasto-plastic thick walled sphere loaded by internal and external pressure. International Journal of Non-Linear Mechanics. 1977;12(1):9–22. DOI: 10.1115/1.3644030.][Parvizi A., Alikarami S., Asgari M. Exact solution for thermoelastoplastic behavior of thick-walled functionally graded sphere under combined pressure and temperature gradient loading. Journal of Thermal Stresses. 2016; 39(9):1152-1170. DOI: 10.1080/01495739.2016.1188614.][Chen Y.Z., Lin X.Y. An alternative numerical solution of thick-walled cylinders and spheres made of functionally graded materials. Computational Materials Science. 2010;48(3):640–647. DOI: 10.1016/j.commatsci.2010.02.033.][Vestyak V.A., Tarlakovskii D.V. Unsteady axisymmetric deformation of an elastic thick-walled sphere under the action of volume forces. Journal of Applied Mechanics and Technical Physics. 2015;56(6):984–994. DOI: 10.1134/S0021894415060085.][Sedova O.S., Pronina Y.G. Taking account of hydrostatic pressure in the modeling of corrosion of thick spherical shells. 2015 International Conference on Mechanics – Seventh Polyakhov’s Reading. 2015;7106771. DOI: 10.1109/POLYAKHOV.2015.7106771.][Loghman A., Moradi M. The analysis of time-dependent creep in FGPM thick walled sphere under electro-magneto-thermo-mechanical loadings. Mechanics of Time-Dependent Materials. 2013;17(3):315–329. DOI: 10.1007/s11043-012-9185-x.][Hashin Z. The elastic moduli of heterogeneous materials. J. Appl. Mech. 1962;29:143–150.][Voigt W. Lehrbuch der Kristallphysik. Berlin: Teubner; 1928.][Reuss A. Berechung der Fliessgrenze von Mischkristallen auf Grund der Plastizitatsbedingund. Z. Angew. Math. Und Mech. 1929;9(1):49–58.][Bezukhov N.I. Osnovy teorii uprugosti, plastichnosti i polzuchesti [Fundamentals of the theory of elasticity, plasticity and creep]. Мoscow: Vysshaya shkola Publ., 1968. (In Russ.)][Cherkasov V.D., Tyuryakhin A.S. Teoriya dvukhsvyaznykh modelei mikromekhaniki kompozitov [Theory of two-connected models of micromechanics of composites]: monograph. Saransk: Publishing House of the Mordovian University; 2009. (In Russ.)][Erofeev V.T., Tyuryakhin A.S., Erofeeva I.V. Relationships between carrier phase parameters and effective parameters in granular composite models. Construction mechanics and calculation of structures. 2018;(1):7–17. (In Russ.)][Erofeev V.T., Tyuryakhin A.S., Tyuryakhina T.P. Flat Space of Values of Volume Module of Grain Composite with Spherical Fill-Lem. International Journal of Civil Engineering and Technology. 2019;10(8):333–342.][Erofeev V.T., Tyuryakhin A.S., Tyuryakhina T.P System of ordered subsets of the volume modulus values of polydisperse composites with spherical inclusions. News of higher educational institutions. Construction. 2019; 6(726):5–7. DOI: 10.32683/0536-1052-2019-726-6-5-17. (In Russ.)][Erofeev V.T., Tyuryakhin A.S., Tyuryakhina T.P., Tingaev A.V. Effective modules of two-phase construction composites with grain filler. Structural Mechanics of Engineering Constructions and Buildings. 2019:15(6):407–414. http://dx.doi.org/10.22363/1815-5235-2019-15-6-407-414 (In Russ.)][Tarasyuk I.A., Kravchuk A.S. Calculation of Voigt – Reuss “Range” in the theory of elasticity of structural heterogeneous, average isotropic, composite bodies without application of variational principles. Apriori. Series: Natural and technical Sciences. 2014;3:1–18. Available from: https://elibrary.ru/item.asp?id=29744041 (accessed: 23.06.2020). (In Russ.)]