Deformation of a plane modelled by John's material with a rigid elliptical inclusion loaded by force and moment
DOI:
https://doi.org/10.21638/11701/spbu10.2023.303Abstract
An exact analytical solution is obtained for a non-linear problem of elasticity theory for a plane with a rigid elliptical inclusion. A concentrated force and a moment are applied at the center of inclusion. The elastic properties of the plane are modeled by John’s harmonic material. For this material methods of the theory of functions of a complex variable are using for solving nonlinear plane problems. Nominal stresses and displacements are expressed in terms of two analytical functions of a complex variable, which are determined from the boundary conditions on the contour of inclusion. The problems of the action of a concentrated force and moment on an elliptical core in a plane are considered separately. A comparison with a similar linear problem is made. The influence of the applied force and moment on the magnitude of stresses is studied depending on various parameters of the problem. Calculations of nominal stresses on the contour joining the plane with inclusion are performed.
Keywords:
non-linear plane problem, rigid elliptical inclusion, John's harmonic material, concentrated force and moment
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References
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References
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Malkov V. M. Vvedenie v nelineinuiu uprugost' [Introduction to non-linear elasticity]. St. Petersburg, Saint Petersburg State University Press, 2010, 276 p. (In Russian)
Mal'kov V. M., Mal'kova Yu. V. Modeling non-linear deformation of a plate with an elliptic inclusion by John's harmonic material. Vestnik of Saint Petersburg University. Mathematics, 2017, vol. 50, iss. 1, pp. 74–81. https://doi.org/10.3103/S1063454117010095
Varley E., Cumberbatch E. Finite deformation of elastic materials surrounding cylindrical holes. Journal of Elasticity, 1980, vol. 10, no. 4, pp. 341–405.
Ru C.,Q. On complex-variable formulation for finite plane elastostatics of harmonic materials. Acta Mechanica, 2002, vol. 156, no. 3–4, pp. 219–234.
Ru C.,Q., Schiavone P., Sudak L.,J., Mioduchowski A. Uniformity of stresses inside an elliptic inclusion in finite plane elastostatics. International Journal of Non-linear Mechanics, 2005, vol. 38, iss. 2–3, pp. 281–287.
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Mal'kov V. M., Mal'kova Yu. V. Deformatsiia plastiny s uprugim ellipticheskim vkliucheniem [Deformation of a plate with elliptic elastic inclusion]. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2015, vol. 2 (60), iss. 4, pp. 617–632. (In Russian)
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