Abstract
The problem of the mechanics of cracks in an orthotropic plate is considered. A general solution form is proposed as a generalized Papkovich–Neuber representation for the plane problem of the theory of elasticity of an orthotropic body. This representation allows us to write the general solution in displacements through two vector potentials satisfying the generalized harmonic equations. The dependences between the vector potentials of Papkovich–Neuber and the stress function are presented. It is shown that there is a complex-valued solution form through four analytic functions of complex variables associated with coefficients that are the roots of the characteristic equation corresponding to the generalized biharmonic equation. It is proved the statement that the operator of the problem is written only through conjugate analytic functions of complex variables, and the general solution is written through arbitrary linear combinations of four functions of complex variables. A general form of solutions with a given singularities is presented, including representations for both cracks in Mode I and cracks in Mode II. The conditions are analyzed that make it possible to obtain generalized non-singular solutions for cracks of Mode I and II. Finally, we establish the conditions for the regularization of singular solutions through the solutions of the generalized Helmholtz equations that correspond to a particular version of the gradient theory of elasticity.
Similar content being viewed by others
REFERENCES
C. T. Sun, Fracture Mechanics (Elsevier, Amsterdam, 2012).
K. Friedrich, Application of Fracture Mechanics to Composite Materials (Elsevier Sci., Amsterdam, 1989).
S. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body (Holden-Day, San Francisco, 1977).
N. I. Muskhelishvili and J. R. M. Radok, Some Basic Problems of the Mathematical Theory of Elasticity (Cambridge Univ. Press, Cambridge, 1953).
I. Tsukrov and M. Kachanov, ‘‘Anisotropic material with arbitrarily oriented cracks and elliptical holes: Effective elastic moduli,’’ Int. J. Fract. 92, L9–L14 (1998).
G. C. Sih, Mechanics of Fracture Initiation and Propagation (Kluwer Academic, Dordrecht, 1991).
M. Fakoor and N. M. Khansari, ‘‘Mixed mode I/II fracture criterion for orthotropic materials based on damage zone properties,’’ Eng. Fract. Mech. 153, 407–420 (2016).
V. V. Vasiliev and E. V. Morozov, Advanced Mechanics of Composite Materials and Structural Elements, 3rd ed. (Elsevier, Amsterdam, 2013).
M. Fakoor, ‘‘Augmented strain energy release rate (ASER): A novel approach for investigation of mixed-mode I/II fracture of composite materials,’’ Eng. Fract. Mech. 179, 177–189 (2017).
M. H. Sadd, Elasticity, Theory, Applications and Numerics (Elsevier, Amsterdam, 2014).
G. P. Cherepanov, Mechanics of Brittle Fracture (McGraw-Hill, New York, 1979).
T. L. Anderson, Fracture Mechanics: Fundamentals and Applications (CRC, Boca Raton, FL, 2017).
D. P. Miannay, Fracture Mechanics (Springer, Berlin, 2012).
E. C. Aifantis, ‘‘On the role of gradient in the localization of deformation and fracture,’’ Int. J. Eng. Sci. 30, 1279–1299 (1992).
A. Carpinteri and M. Paggi, ‘‘Asymptotic analysis in linear elasticity: From the pioneering studies by Wieghardt and Irwin until today,’’ Eng. Fract. Mech. 76, 1771–1784 (2009).
G. C. Sih and X. S. Tang, ‘‘Scaling of volume energy density function reflecting damage by singularities at macro-, meso- and microscopic level,’’ Theor. Appl. Fract. Mech. 43, 211–231 (2005).
S. A. Lurie, D. B. Volkov-Bogorodsky, and V. V. Vasiliev, ‘‘A new approach to non-singular plane cracks theory in gradient elasticity,’’ Math. Comput. Appl. 24 (4), 93 (2019). https://doi.org/10.3390/mca2404009
X.-L. Gao and S. K. Park, ‘‘Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem,’’ Int. J. Solids Struct. 44, 7486–7499 (2007).
V. V. Vasiliev and S. A. Lurie, ‘‘Nonlocal solutions to singular problems of mathematical physics and mechanics,’’ Mech. Solids 53, 135–144 (2018).
Yu. N. Rabotnov, Mechanics of a Deformable Solid Body (Nauka, Moscow, 1989) [in Russian].
A. I. Lurie, ‘‘On the theory of the system of linear differential equations with the constant coefficients,’’ Tr. Leningr. Prom. Inst. 6, 31–36 (1937).
P. F. Papkovich, ‘‘Solution générale des équations différentielles fondamentales de l’élasticité, exprimeé par trois fonctiones harmoniques,’’ C. R. Acad. Sci. (Paris) 195, 513–515 (1932).
H. Neuber, ‘‘Ein neuer ansatz zur lösung raümlicher probleme der elastizitätstheorie,’’ Z. Angew. Math. Mech. 14, 203–212 (1934).
H. Bateman and A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.
Funding
This work was supported by the Russian Science Foundation under grant 20-41-04404 issued to the Institute of Applied Mechanics of Russian Academy of Sciences.
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by A. V. Lapin)
Rights and permissions
About this article
Cite this article
Lurie, S.A., Volkov-Bogorodskiy, D.B. On Regularization of Singular Solutions of Orthotropic Elasticity Theory. Lobachevskii J Math 41, 2023–2033 (2020). https://doi.org/10.1134/S199508022010011X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S199508022010011X