539.3

orthotropic materials, equilibrium equations, displacement, stresses, shear modules.

The model of an orthotropic deformable body based on the representation of stresses in terms of displacements is considered. The method of integration of three equations of the elastic equilibrium is used, based on the elimination of separate displacements. Problems related to the elimination of unnecessary functions from the representation of the general solution of the equations of the theory of elasticity are considered. Criteria are found that determine such a class of orthotropic materials that their stress-strain state can be expressed in terms of two functions. One function satisfies the equation of the second order in partial derivatives, and the other of the fourth order. It is established that the equation of the fourth order, in the general case, is not decomposed into two operator factors. Criteria were found for the expansion of a fourth-order equation into the product of two second-order equations. An equation has been written that must be satisfied by the elastic constants of an orthotropic material. The expression of deformations and stresses by introduced harmonic functions was written down.

1. Lekhnitskii S. G. Theory of elasticity of an anisotropic body. Moscow: Mir Publishers, 1981, 430 p.
2. Rand O, Rovenski V. Analytical Methods in Anisotropic Elasticity, Springer Science & Business Media, 2007, 452 p. ISBN 978-0-8176-4420-8.
3. Ting T. C. T. Anisotropic elasticity theory and applications, New York, Oxford, Oxford university press, 1996.
4. Revenko V. P. Presentation of a general 3D solution of equations of elasticity theory for a wide class of orthotropic materials // Scientific journal of the Ternopil national technical university. No. 3 (95). 2019. P. 49–54. URL: https://doi.org/10.33108/visnyk_tntu2019.03.04.
5. Elliot H. A. Axial symmetric stress distributions in allotropic hexagonal crystals. The problem of the plane and related problems, Math. Proc. Cambridge Phil. Soc. 45. No. 4. 1949. Р. 621–630.
6. Hu H. C. On the the three-dimensional problems of elasticity of a transversely isotropic body, Data Sci. Sinica, Vol. 2, 1953, pp.145–151.
7. Baida É. N., General solution of the equilibrium equations of anisotropic and isotropic bodies, Izv. Vyssh. Uchebn. Zaved., Stroit. Arkhitekt, No. 6, 1968, pp. 17–27.
8. Timoshenko S. P. and Goodier J. N. Theory of Elasticity. New York: McGraw-Hill. 1970. 586 pp.
9. Revenko V. P. Solving the three-dimensional equations of the linear theory of elasticity // Int. Appl. Mech. Vol. 45, 2009, pp. 730-741. URL: https://doi.org/10.1007/s10778-009-0225-4.
10. Revenko V. P., Bakulin V. N. Solving equations of 3D elasticity for orthotropic bodies. Journal of Physics: Conference Series: Materials Science and Engineering 927 012052 IOP Publishing, 2020, pp. 7. doi:10.1088/1757-899X/927/1/012052.

1. Lekhnitskii S. G. Theory of elasticity of an anisotropic body. Moscow: Mir Publishers, 1981, 430 p.
2. Rand O, Rovenski V. Analytical Methods in Anisotropic Elasticity, Springer Science & Business Media, 2007, 452 p. ISBN 978-0-8176-4420-8.
3. Ting T. C. T. Anisotropic elasticity theory and applications, New York, Oxford, Oxford university press, 1996.
4. Revenko V. P. Presentation of a general 3D solution of equations of elasticity theory for a wide class of orthotropic materials // Scientific journal of the Ternopil national technical university. No. 3 (95). 2019. P. 49–54. URL: https://doi.org/10.33108/visnyk_tntu2019.03.04.
5. Elliot H. A. Axial symmetric stress distributions in allotropic hexagonal crystals. The problem of the plane and related problems, Math. Proc. Cambridge Phil. Soc. 45. No. 4. 1949. Р. 621–630.
6. Hu H. C. On the the three-dimensional problems of elasticity of a transversely isotropic body, Data Sci. Sinica, Vol. 2, 1953, pp.145–151.
7. Baida É. N., General solution of the equilibrium equations of anisotropic and isotropic bodies, Izv. Vyssh. Uchebn. Zaved., Stroit. Arkhitekt, No. 6, 1968, pp. 17–27.
8. Timoshenko S. P. and Goodier J. N. Theory of Elasticity. New York: McGraw-Hill. 1970. 586 pp.
9. Revenko V. P. Solving the three-dimensional equations of the linear theory of elasticity // Int. Appl. Mech. Vol. 45, 2009, pp. 730-741. URL: https://doi.org/10.1007/s10778-009-0225-4.
10. Revenko V. P., Bakulin V. N. Solving equations of 3D elasticity for orthotropic bodies. Journal of Physics: Conference Series: Materials Science and Engineering 927 012052 IOP Publishing, 2020, pp. 7. doi:10.1088/1757-899X/927/1/012052.