  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
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V.I. Gorbachev, "Effective Elastic Coefficients of an Inhomogeneous Solids," Mech. Solids. 53 (4), 454463 (2018) 
Year 
2018 
Volume 
53 
Number 
4 
Pages 
454463 
DOI 
10.3103/S0025654418040118 
Title 
Effective Elastic Coefficients of an Inhomogeneous Solids 
Author(s) 
V.I. Gorbachev (Lomonosov Moscow State University, Leninskie Gory, Moscow, 119991, Russia, vigorby@mail.ru) 
Abstract 
The first special boundary value problem (SBVP) of the theory of elasticity for an inhomogeneous body is considered. The effective elasticity coefficients are found from the solution of the SBVP. They form a fourthrank tensor, namely the tensor of effective elasticity moduli that makes it possible to express the volume average stresses via the mean deformations. It is shown that the solution of the first SBVP and hence the effective coefficients of elasticity are expressed in terms of the integrals of the Green tensor. The integrals of the Green tensor with respect to one of the variables are called the structural functions. The auxiliary equations, the solutions of which are determined by the functional dependence of the elastic characteristics on the coordinates, have been obtained for these structural functions. It is shown that in the case when the elastic moduli are periodic functions of one, two, or three coordinates, then the structural functions, far from the body boundary, are also periodic functions of the same coordinates. The structural functions are transformed to be equal to zero on the all body border approaching the boundary. In other words, in an inhomogeneous body with a periodic structure, it is possible to distinguish the boundary layer, which separates the regions of periodic values of structural functions from nonperiodic ones. The thickness of this layer is of the order of the characteristic size of the periodicity cell. The effective tensors are found due to the structural functions. It is proved that the tensor of effective elastic moduli satisfies all the conditions of symmetry and positive definiteness. The case of an infinite plate with nonuniform thickness is considered in detail. 
Keywords 
mechanics of solids, theory of elasticity, inhomogeneous body, mechanics of composites, effective properties 
References 
1.  V.V. Vasiliev,
Mechanics of Structures Made of Composite Materials
(Mashinostroyeniye, Moscow, 1988) [in Russian]. 
2.  N. Pagano,
"The Role of Effective Moduli in the Study of the Elastic Properties of Layered Composites,"
in Mechanics of Composite Materials and Structures (Mir, Moscow, 1978), Vol. 2, pp. 1337 [in Russian]. 
3.  Z. Hashin and B.W. Rosen,
"The Elastic Moduli of FiberReinforced Materials,"
Trans ASME. J. Appl. Mech.
31 (2), 223232 (1964). 
4.  B.E. Pobedrya,
Mechanics of Composite Materials
(Izdat. MGU, Moscow, 1984) [in Russian]. 
5.  B.E. Pobedrya,
Lectures on Tensor Analysis
(Izdvo MGU, Moscow, 1979) [in Russian]. 
6.  V.I. Gorbachev,
"Green Tensor Method for Solving Boundary Value Problems of the Theory of Elasticity for Inhomogeneous Media,"
in Computational Mechanics of Solid
(NCBI AN USSR, Moscow, 1991), pp. 6176 [in Russian]. 
7.  V.I. Gorbachev,
"Averaging of Linear Problems in the Mechanics of Composites with Nonperiodic Inhomogeneities,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 1, 3137 (2001)
[Mech. Sol. (Engl. Transl.) 36 (1), 2429 (2001)]. 
8.  V.I. Gorbachev,
"Homogenization processes in the inhomogeneous bodies,"
in Elasticity and Nonelasticity. Materials of the Intern. Scientific Symposium on the Problems of Mech. Def. Solid Body Dedicated Tothe 90th Anniversary of A.A. Ilyushin (Izdvo MGU, Moscow, 2001), pp. 294296 [in Russian]. 
9.  V.I. Gorbachev,
"Integral Formulae in the Coupled Problem of the Thermoelasticity of an Inhomogeneous Body. Application in the Mechanics of Composite Materials,"
Prikl. Mat. Mekh.
78 (2), 192208 (2014)
[J. Appl. Math. Mech. (Engl. Transl.)
78 (2), 192208 (2014)]. 
10.  V. Novatsky,
Theory of Elasticity
(Mir, Moscow, 1970) [in Russian]. 
11.  V. Ketch and P. Teodorescu,
Introduction to the Theory of Generalized Functions with Applications in Engineering
(Mir, Moscow, 1978) [in Russian] (Wiley, New York, 1978). 
12.  V.I. Gorbachev,
"Engineering Resistance Theory of Heterogeneous Rods made of Composite Materials,"
Vest. Mos. Gos. Tekh. Univ. Ser. Est. Nauk.
Herald of the Bauman Moscow State Technical University Series Natural Sciences
No. 6, 5672 (2016). 
13.  V.I. Gorbachev,
"Integral Formulas of Solutions of the Basic Linear Differential Equations of Mathematical Physics with Variable Factors,"
Chebysh. Sborn.
18 (3), 138160 (2017). 
14.  V.I. Gorbachev,
"Averaging Equations of Mathematical Physics with Coefficients Dependent on Coordinates and Time,"
Nanosc. Techn. Int. J.
8 (4), 345353 (2017). 
15.  V.I. Gorbachev and L.V. Olekhova,
"Effective Properties of a Nonuniform Beam under Torsion,"
Vestnik Moskov. Univ. Ser. I. Mat. Mekh.
62 (5), 4148 (2007)
[Moscow Univ. Mech. Bull. (Engl. Transl.) 62 (5), 123130 (2007)]. 
16.  E.I. Grigolyuk and L.A. Filshtinskii,
Perforated Plates and Shells
(Nauka, Moscow, 1970) [in Russian]. 
17.  N.S. Bakhvalov and G.P. Panasenko,
Homogenisation: Averaging Processes in Periodic Media
(Nauka, Moscow, 1984) [in Russian]. 
18.  G.M. Fihtengol'tz,
Course of Differential and Integral Calculus, Vol. 3 (Moscow, Nauka, 1966) [in Russian]. 
19.  B.E. Pobedrya and V.I. Gorbachev,
"On Static Problems for Elastic Composites,"
Vestnik Moskov. Univ. Ser. I. Mat. Mekh.
No. 5, 101111 (1977). 
20.  L.P. Khoroshun,
"Dependencies Between Stresses and Strains in Layered Media,"
Prikl. Mech.
2 (2), 1419 (1966). 
21.  I.M. Lifshitz and L.N. Rosenzweig,
"On the Theory of Elastic Properties of Polycrystals,"
J. Exp. Theor. Phys.
16 (2), 967980 (1946). 

Received 
11 February 2018 
Link to Fulltext 
https://link.springer.com/article/10.3103/S0025654418040118 
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