Exact solutions to two contact problems with unknown contact domains on the elliptic punch penetration in the transversely isotropic elastic half-space are compared. In the first one – a “traditional problem” – the half-space boundary is parallel to the isotropy planes. There can be an axial symmetry case for a circular punch. In the second – a “nontraditional problem” – the half-space boundary is perpendicular to the isotropy planes. Here, the axial symmetry is impossible in principle: even for a circular punch, the contact domain will be elliptical. The contact domains and the forces required for the penetration of the punch at a given value are compared for both problems. The transversely isotropic body model is applicable for describing lots of materials which elastic parameters have been measured in recent decades: composites, ceramics, carbon fiber, graphite-epoxy, glass-epoxy, wood, aged concretes, some metals (titanium, cobalt, zinc), some semiconductors and rocks. They are widely used in the engineering and industry
elasticity theory, contact problems, transversely isotropic half-space, exact solution
Введение. Пионером в области исследования трансверсально изотропных тел считается Эллиот [1, 2]. В контактных задачах теории упругости для трансверсально изотропного полупространства традиционно рассматривался случай, когда область контакта параллельна плоскостям изотропии [3, 4]. Случай, когда область контакта или трещины перпендикулярна плоскостям изотропии, назван «нетрадиционным» [5, 6]. В случае «нетрадиционной» ориентации плоскостей изотропии рассматривались задачи для полосовой, клиновидной, эллиптической и заранее неизвестной областей контакта [7–10].
1. Elliot, H.A. Three-dimensional stress distributions in hexagonal crystals. Proceedings of Cambridge Philosophical Society, 1948, vol. 44, pp. 522–533.
2. Elliot, H.A. Axial symmetric stress distributions in aeolotropic hexagonal crystals. The problem of plane and related problems. Proceedings of Cambridge Philosophical Society,1949, vol. 45, pp. 621–630.
3. Grilitskiy, D.V., Shelestovskiy, B.G. Osesimmetrichnaya kontaktnaya zadacha termouprugosti dlya transversal´no izotropnogo poluprostranstva. [The axisymmetric contact problem of thermoelasticity for a transversely isotropic half-space.] Prikladnaya mekhanika, 1973, vol. 6, iss. 8, pp. 3-8 (in Russian).
4. Ding, Haojiang; Chen, Weiqiu; Zhang, L. Elasticity of transversely isotropic materials. Dordrecht : Springer, 2006, 435 p.
5. Fabrikant, V. I. Non-traditional contact problem for transversely isotropic half-space. Quarterly Journal of Mechanics and Applied Mathematics, 2011, vol. 64, no. 2, pp. 151–170.
6. Fabrikant, V. I. Non-traditional crack problem for transversely-isotropic body. European Journal of Mechanics - A / Solids, 2011, vol. 30, pp. 902–912.
7. Davtyan, D.B., Pozharskiy, D.A. Deystvie polosovogo shtampa na transversal´no izotropnoe poluprostranstvo. [The action of a strip punch on a transversely isotropic half-space.] Journal of Applied Mathematics and Mechanics, 2012, vol. 76, iss. 5, pp. 783–794 (in Russian).
8. Pozharskiy, D.A., Davtyan, D.B., Artamonova, E.A. Klinovidnyy shtamp na transversal´no izotropnom poluprostranstve. [Wedge-shaped punch on a transversely isotropic half-space.] Izvestiya vuzov. Severo-Kavkazskiy region. Estestvennye nauki. 2013, no. 1, pp. 31–33 (in Russian).
9. Pozharskiy, D.A., Davtyan, D.B. Trekhmernaya kontaktnaya zadacha dlya transversal´no izotropnogo tela. [Three-dimensional contact problem for a transversely isotropic solid.] Vestnik of DSTU, 2013, no. 7/8, pp. 22–26 (in Russian).
10. Davtyan, D.B., Pozharskiy, D.A. Deystvie ellipticheskogo shtampa na transversal´no izotropnoe poluprostranstvo. [Action of an elliptic punch on a transversally isotropic half-space.] Mechanics of Solids, 2014, no. 5, pp. 117–126 (in Russian).
