Abstract
The general solutions of some weakened systems of equations expressed in terms of stresses in the isotropic theory of elasticity are analyzed. These systems are not equivalent to the classical one and involve the equilibrium equations and only three of the six equations of compatibility (either diagonal or off-diagonal ones). In the framework of elasticity theory, an equivalence of the formulations of quasistatic boundary value problems based on such systems and expressed in terms of stresses is discussed.
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D. V. Georgievskii and B. E. Pobedrya, “The Number of Independent Compatibility Equations in the Mechanics of Deformable Solids,” Prikl. Mat. Mekh. 68(6), 1043–1048 (2004) [J. Appl. Math. Mech. 68 (6), 941–946 (2004)].
B. E. Pobedrya and D. V. Georgievskii, “Equivalence of Formulations for Problems in Elasticity Theory in Terms of Stresses,” Russ. J. Math. Phys. 13(2), 203–209 (2006).
B. E. Pobedrya, Numerical Methods in the Theory of Elasticity and Plasticity (Mosk. Gos. Univ., Moscow, 1995) [in Russian].
X. Markenscoff, “A Note of Strain Jump Conditions and Cesáro Integrals for Bonded and Slipping Inclusions,” J. Elasticity 45(1), 45–51 (1996).
V. Volterra, “Sur l’Équilibre des Corps Elastiques Multipliment Connexes,” Ann. l’École Norm. Sup. 24, 401–517 (1907).
E. Kroner, Kontinuumstheorie der Versetzungen und Eigenspannungen (Springer, Berlin, 1958).
E. Cesáro, “Sulle Formole del Volterra, Fondamentali Nella Teoria Delle Distorsioni Elastiche,” Rend. Acad. R. di Napoli. 12(1), 311–321 (1906).
P. G. Ciarlet, L. Gratie, and C. Mardare, “A Generalization of the Classical Cesáro-Volterra Path Integral Formula,” C. R. Acad. Sci. Paris. Ser. I. 347, 577–582 (2009).
M. M. Filonenko-Borodich, Theory of Elasticity (Fizmatgiz, Moscow, 1959) [in Russian].
B. E. Pobedrya, “A New Formulation of the Problem in Mechanics of a Deformable Solid Body under Stress,” Dokl. Akad. Nauk SSSR 253(2), 295–297 (1980) [Sov. Math. Dokl. 22 (2), 88–91 (1980)].
B. E. Pobedrya, “Static Problem in Stresses,” Vestn. Mosk. Univ. Ser. 1: Mat. Mekh., No. 3, o61–67 (2003) [Moscow Univ. Mech. Bull. 58 (3), 6–12 (2003)].
V. A. Kucher, X. Markenscoff, and M. V. Paukshto, “Some Properties of the Boundary Value Problem of Linear Elasticity in Terms of Stresses,” J. Elasticity 74(2), 135–145 (2004).
Sh. Li, A. Gupta, and X. Markenscoff, “Conservation Laws of Linear Elasticity in Stress Formulations,” Proc. Roy. Soc. London. Ser. A. Math. Phys. and Eng. Sci. 461(2053), 99–116 (2005).
N. M. Borodachev, “Solutions of Spatial Problem of Elasticity Theory in Stresses,” Prikl. Mekh. 42(8), 74–82 (2006) [Int. Appl. Mech. 42 (8), 849–878 (2006)].
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Original Russian Text © D. V. Georgievskii, 2012, published in Vestnik Moskovskogo Universiteta, Matematika, Mekhanika. 2012. Vol. 67, No. 6. pp 26–32.
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Georgievskii, D.V. General solutions of weakened equations in terms of stresses in the theory of elasticity. Moscow Univ. Mech. Bull. 68, 1–7 (2013). https://doi.org/10.3103/S0027133013010019
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DOI: https://doi.org/10.3103/S0027133013010019