Mechanics of Solids (about journal) Mechanics of Solids
A Journal of Russian Academy of Sciences
 Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936

Russian Russian English English About Journal | Issues | Guidelines | Editorial Board | Contact Us
 


IPMech RASWeb hosting is provided
by the Ishlinsky Institute for
Problems in Mechanics
of the Russian
Academy of Sciences
IssuesArchive of Issues2011-3pp.400-415

Archive of Issues

Total articles in the database: 9179
In Russian (. . ): 6485
In English (Mech. Solids): 2694

<< Previous article | Volume 46, Issue 3 / 2011 | Next article >>
D.V. Semenova and K.B. Ustinov, "Asymptotic Computation of the Inhomogeneity Energy in a Body in an External Stress Field," Mech. Solids. 46 (3), 400-415 (2011)
Year 2011 Volume 46 Number 3 Pages 400-415
DOI 10.3103/S0025654411030071
Title Asymptotic Computation of the Inhomogeneity Energy in a Body in an External Stress Field
Author(s) D.V. Semenova (Moscow Institute of Physics and Technology (State University), Institutskii per.9, Dolgoprudny, Moscow Region, 141700 Russia, d--semenova@mail.ru)
K.B. Ustinov (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str.1, Moscow, 119526 Russia, ustinov@ipmnet.ru)
Abstract We consider a problem on an ellipsoidal inhomogeneity in an infinitely extended homogeneous isotropic elastic medium. The inhomogeneity differs from the ambient body in the elastic moduli (Poisson's ratio ν and shear modulus μ) and in that it has intrinsic strains. We use the equivalent inclusion method to write out expressions for the Helmholtz and Gibbs free energy of the inhomogeneity as quadratic forms in the intrinsic strains and strains at infinity. The general expressions for the coefficients of these quadratic forms are written out as three rank four tensors characterizing the contribution to the energy by the plastic strain (εp2), by the strain at infinity (ε02), and (only for the Gibbs energy) by the cross term ε0 εp.

The solutions for some important particular cases are written out by asymptotically expanding the general solution in small parameters. The parameters determining the type of the asymptotic behavior are the inclusion semiaxis ratio and the inclusion-to-matrix elastic modulus ratio. We consider various physical situations arising when studying needle- and disk-shaped inclusions and characterized by various order of passage to the limit. We analyze the applicability domains of the asymptotics by reduction to a single small parameter. For the case in which the parameters considerably differ from unity simultaneously, we single out seven domains corresponding to various relations between these parameters. We construct the diagrams of relative error isolines for the resulting tensors corresponding to the energy contribution from various strain tensor components.
Keywords elasticity, inhomogeneity, inclusion, energy, concentration, Eshelby tensor, small parameter
References
1.  J. D. Eshelby, "The Determination of the Elastic Fields of an Ellipsoidal Inclusion, and Related Problems," Proc. Roy. Soc. London. Ser. A. 241 (1226), 376-396 (1957).
2.  T. Mura, Micromechanics of Defects in Solids (Martinus Nijhoff Publ., Hague, 1982).
3.  S. K. Kanaun and V. M. Levin, Self-Consistent Methods for Composites, Vol. 1, Static Problems (Springer, New York, 2008).
4.  Yu. F. Kovalenko and R. L. Salganik, "Fractured Inhomogeneities and Their Influence on Effective Mechanical Characteristics," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 5, 76-86 (1977) [Mech. Solids (Engl. Transl.)].
5.  R. L. Salganik, "Transport Processes in Bodies with a Large Number of Cracks," Inzh. Fiz. Zh. 27 (6), 1069-1075 (1974) [J. Engng Phys. Thermophys. (Engl. Transl.) 27 (6), 1534-1538 (1974)].
6.  W. B. Russel and A. Acrwos, "On the Effective Moduli of Composite Materials: Slender Rigid Inclusions at Dilute Concentrations," ZAMP 23 (3), 434-463 (1972).
7.  W. B. Russel, "On the Effective Moduli of Composite Materials: Effect of Fiber Length and Geometry at Dilute Concentrations," ZAMP 24 (4), 581-600 (1973).
8.  K. B. Ustinov, "On the determination of effective elastic characteristics of two-phase media. The case of isolated inhomogeneities having the shape of an ellipsoid of revolution," Usp. Mekh. 2 (2), 126-168 (2003).
9.  I. Sevostianov and M. Kachanov, "Compliance Tensors of Ellipsoidal Inclusions," Int. J. Fract. 96, L3-L7 (1999).
10.  K. B. Ustinov and R. V. Goldstein, "On Application of Classical Eshelby Approach to Calculating Effective Elastic Moduli of Dispersed Composites," Int. J. Fract. 147 (1-4), 55-66 (2007).
11.  K .B. Ustinov, "Computation of the Energy of an Inhomogeneity: Asymptotics and Their Scope," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 103-113 (2010) [Mech. Solids. (Engl. Transl.) 45 (2), 238-246 (2010)]
12.  L. J. Walpole, "On the Overall Elastic Moduli of Composite Materials," J. Mech. Phys. Solids 17 (4), 235-251 (1969).
13.  J. R. Bristow, "Microcracks and the Static and Dynamics Elastic Constants of Annealed and Heavily Coldworked Metals," British J. Appl. Phys. 11, 81-85 (1960).
14.  T. T. Wu, "The Effort of Inclusion Shape on the Elastic Moduli of a Two-Phase Material," Int. J. Solids Struct. 2, 1-8 (1966).
Received 20 June 2009
Link to Fulltext http://www.springerlink.com/content/kx3656t218212403/
<< Previous article | Volume 46, Issue 3 / 2011 | Next article >>
Orphus SystemIf you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter

101 Vernadsky Avenue, Bldg 1, Room 246, 119526 Moscow, Russia (+7 495) 434-3538 mechsol@ipmnet.ru https://mtt.ipmnet.ru
Founders: Russian Academy of Sciences, Branch of Power Industry, Machine Building, Mechanics and Control Processes of RAS, Ishlinsky Institute for Problems in Mechanics RAS
© Mechanics of Solids
webmaster
Rambler's Top100