Abstract
A boundary singular integral equation of the plane problem was constructed using an approach based on the representation of the unknown Lekhnitskii complex potentials in the form of Cauchy type integrals with unknown densities on the boundary of the region occupied by the body. The contours of the holes and cuts and the shape of the outer boundary are exactly or approximately represented in the form of a sequence of straight and curved (in the form of elliptical arcs) boundary elements. The unknown densities on the boundary elements are approximated by a linear combination of some regular functions or complex functions that have a known singularity. In the numerical solution of the integral equation by the collocation method or by the least-squares method and in the subsequent calculations of the stress–strain state, the integrals of all types along the boundary elements are calculated analytically, which significantly increases the accuracy of the results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. A. Fil’shtinskii, “Elastic Equilibrium of a Plane Anisotropic Medium Weakened by Arbitrary Curved Cracks: The Limiting Transition to an Isotropic Medium,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 5, 91–97 (1976).
N. I. Ioakimidis and P. S. Theocaris, “The Problem of the Simple Smooth Crack in an Infinite Anisotropic Elastic Medium,” Int. J. Solids Struct. 13 (4), 269–278 (1977).
M. P. Savruk, P. N. Osiv, and I. V. Prokopchuk, Numerical Analysis of Plane Problems of Crack Theory (Naukova Dumka, Kiev, 1989) [in Russian].
V. N. Maksimenko, “Application of the Method of Influence Functions in Problems of the Theory of Cracks for Anisotropic Plates,” Prikl. Mekh. Tekh. Fiz. 34 (3), 128–137 (1993) [J. Appl. Mech. Tech. Phys. 34 (3), 410–418 (1993)].
A. M. Lin’kov, Complex Method of Boundary Integral Equations of Elasticity Theory (Nauka, St. Petersburg, 1999) [in Russian].
V. N. Maksimenko and E. G. Podruzhin, “Bending of Finite Anisotropic Plates Containing Smooth Holes and through Curved Cuts,” Sib. Zh. Indust. Mat. 9 (4), 125–135 (2006).
T. V. Hromadka and C. Lai, The Complex Variable Boundary Element Method in Engineering Analysis (Springer-Verlag, New York, 1987).
S. G. Lekhnitskii, Anisotropic Plates (Gostehteoretizdat, Moscow, 1957; Gordon and Breach, New York, 1968).
G. N. Savin, Stress Distribution Near Holes (Naukova Dumka, Kiev, 1968) [in Russian].
D. I. Sherman, “On the Solution of the Plane Elastic Problem for an Anisotropic Medium,” Prikl. Mat. Mekh. 6 (6), 509–514 (1942).
A. D. Dement’ev, “Calculation of Stress Intensity Factors at the Tip of a through Crack from Strain Measurement Data,” Uch. Zap. TsAGI 18 (5), 83–88 (1987).
S. A. Kaloerov and E. S. Goryanskaya, “Two-Dimensional Stress–Strain State of a Multiconnected Anisotropic Body,” in The Mechanics of Composites, Vol. 7: Stress Concentration (A.S.K., Kiev, 1998), pp. 10–26 [in Russian].
N. I. Muskhelishvili, Singular Integral Equations (Nauka, Moscow, 1968; Dover, New York, 1992).
M. P. Savruk, Fracture Mechanics and Strength of Materials: Handbook, Vol. 2: Stress Intensity Factors in Cracked Bodies (Naukova Dumka, Kiev, 1988) [in Russian].
S. P. Timoshenko and J. Goodier, Theory of Elasticity, (McGraw-Hill, New York (1970).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Tyagnii.
Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 56, No. 4, pp. 202–214, July–August, 2015.
Rights and permissions
About this article
Cite this article
Tyagnii, A.V. Boundary element solution of the plane elasticity problem for an anisotropic body with free smooth boundaries. J Appl Mech Tech Phy 56, 715–725 (2015). https://doi.org/10.1134/S0021894415040197
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0021894415040197