Multitudes of Voigt - Reuss forks and Voigt - Christensen - Reuss tridents

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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. Christensen’s formula, expressed in dimensional values, for the bulk modulus of composites with a spherical filler. In this case, Christensen’s 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.

About the authors

Vladimir T. Erofeev

National Research Ogarev Mordovia State University (National Research University)

Author for correspondence.

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

68 Bolshevistskaya St, Saransk, 430005, Russian Federation

Aleksej S. Tyuryakhin

National Research Ogarev Mordovia State University (National Research University)


Doctor of Engineering, Associate Professor of the Department of Applied Mechanics of the Faculty of Architecture and Civil Engineering

68 Bolshevistskaya St, Saransk, 430005, Russian Federation

Tatyana P. Tyuryakhina

National Research Ogarev Mordovia State University (National Research University)


graduate student of the Department of Building Materials and Technologies of the Faculty of Architecture and Civil Engineering.

68 Bolshevistskaya St, Saransk, 430005, Russian Federation


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Copyright (c) 2020 Erofeev V.T., Tyuryakhin A.S., Tyuryakhina T.P.

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