Thick-walled spherical shell problem

Introduction. Cylindrical and spherical shells are extensively used in engineering. They face internal and/or external pressure and heat. Stresses and strains distribution in elastoplastic shells has been studied by many scientists. Numerous works involve the use of the von Mises yield conditions, maximum shear stress, maximum reduced stress. These condi- tions do not include the dependence on the first invariant of the stress tensor and the sign of the third invariant of the stress deviator. In some cases, it is possible to obtain numerical-analytical solutions for stresses, displacements and de- formations for bodies with spherical and cylindrical symmetry under axisymmetric thermal and force action.

Materials and Methods. The problem on the state of a thick-walled elastoplastic shell is solved within the framework of the theory of small deformations. A plasticity condition is proposed, which takes into account the dependence of the stress tensor on three independent invariants, and also considers the sign of the third invariant of the stress deviator and translational hardening of the material. A disconnected thermoelastoplastic problem is being solved. To estimate the stresses in the region of the elastic state of a spherical shell, an equivalent stress is introduced, which is similar to the selected plasticity function. The construction of the stress vector hodograph is used as a method for verification of the stress state.

Results. The problem has an analytical solution for linear plasticity functions. A solution is obtained when the strength- ening of the material is taken into account. Analytical and graphical relationships between the parameters of external action for the elastic or elastoplastic states of the sphere are determined. For a combined load, variants are possible when the plastic region is generated at the inner and outer boundaries of the sphere or between these boundaries.

Discussion and Conclusions. The calculation results have shown that taking into account the plastic compressibility and the dependence of the plastic limit on temperature can have a significant impact on the stress and strain state of a hollow sphere. In this case, taking into account the first invariant of the stress tensor under the plasticity condition leads to the fact that not only the pressure drop between the outer and inner boundaries of the spherical shell, but the pressure values at these boundaries, can vary within a limited range. In this formulation of the problem, when there is only thermal action, the hollow sphere does not completely pass into the plastic state. The research results provide predicting the behavior of an object (a hollow sphere) that experiences centrally symmetric distributed power and thermal external influences.

1. Chakrabarty J. Theory of Plasticity. Oxford: Elsevier Butterworth-Heinemann; 2006. 882 p.

2. Parkus G. Neustanovivshiesya temperaturnye napryazheniya [Unsteady temperature stresses]. Moscow: Fizmatlit; 1963. 252 p. (In Russ.)

3. Gamer U. On the elastic-plastic deformation of a sphere subjected to a spherically symmetrical temperature field. Journal of Thermal Stresses. 1988;11(3):159–173.

4. Dats EP, Mokrin SN, Murashkin EV. Raschet nakoplennoi ostatochnoi deformatsii v protsesse <nagreva- okhlazhdeniya> uprugoplasticheskogo shara [The calculation of the accumulated residual strain in the heating-cooling process of elasto-plastic ball]. Bulletin of the Yakovlev Chuvash State Pedagogical University. Series: Mechanics of Limit State. 2012;4:123–132. (In Russ.)

5. Murashkin EV, Dats EP. Termouprugoplasticheskoe deformirovanie mnogosloinogo shara [Thermoelasto- plastic deformation of multilayer ball]. Izvestiya RAN. Mekhanika Tverdogo Tela = Mechanics of Solids. 2017;5:30–(In Russ.)

6. Dats EP, Murashkin EV, Velmurugan R. Vychislenie neobratimykh deformatsii v polom uprugoplastich- eskom share v usloviyakh nestatsionarnogo temperaturnogo vozdeistviya [On computing irreversible strains of the hol- low ball under unsteady thermal action]. Bulletin of the Yakovlev Chuvash State Pedagogical University. Series: Me- chanics of Limit State. 2015;3:168–175. (In Russ.)

7. Kovalev AV, Khvostov IG. Ob opredelenii napryazhenii i peremeshchenii v uprugom prostranstve, oslab- lennom sfericheskoi polost'yu, s uchetom temperatury [On the determination of stresses and displacements in an elastic space weakened by a spherical cavity taking into account temperature]. Bulletin of the Yakovlev Chuvash State Peda- gogical University. Series: Mechanics of Limit State. 2014;2:29–35. (In Russ.)

8. Burenin AA, Murashkin EV, Dats EP. Residual stresses in AM fabricated ball during a heating pro- cess. AIP Conference Proceedings. 2018;1959(1):070008. URL: https://doi.org/10.1063/1.5034683

9. Syomka E. Uprugoplasticheskoe sostoyanie pologo shara [Elastoplastic state of a hollow sphere]. The Far Eastern Federal University: School of Engineering Bulletin. 2020;3:3–12. (In Russ.)

10. Burenin AA, Tkacheva AV. Kusochno-lineinye plasticheskie potentsialy kak sredstvo raschetov ploskikh neustanovivshikhsya temperaturnykh napryazhenii [Piecewise linear plastic potentials as a means of calculating plane unsteady temperature stresses]. Izvestiya RAN. Mekhanika Tverdogo Tela = Mechanics of Solids. 2020;6:40–49. (In Russ.)

11. Burenin AA, Tkacheva AV, Scherbatyuk GA. Ispol'zovaniyu kusochno-lineinykh plasticheskikh potentsi- alov v nestatsionarnoi teorii temperaturnykh napryazhenii [The use of piecewise linear plastic potentials in the nonsta- tionary theory of temperature stresses]. Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences. 2018;22(1):23–39. (In Russ.)

12. Dats EP, Murashkin EV, Tkacheva AV, et al. Temperaturnye napryazheniya v uprugoplasticheskoi trube v zavisimosti ot vybora usloviya plastichnosti [Temperature stresses in an elastic-plastic pipe depending on the selection of the plasticity condition]. Izvestiya RAN. Mekhanika Tverdogo Tela = Mechanics of Solids. 2018;1:32–43. (In Russ.)

13. Dats EP, Murashkin EV. Temperaturnye napryazheniya v usloviyakh toroidal'noi simmetrii [Thermal stresses under toroidal symmetry]. Bulletin of the Yakovlev Chuvash State Pedagogical University. Series: Mechanics of Limit State. 2019;2:57–70. (In Russ.)

14. Aleksandrova NN, Artemov MA, Baranovskii ES, et al. On stress/strain state in a rotating disk. AMCSM_2018 IOP Conf. Series: Journal of Physics: Conf. Series. 2019;1203:012001. URL: http://doi.org/10.1088/1742-6596/1203/1/012001

15. Semka EV, Artemov MA, Babkina YN, et al. Mathematical modeling of rotating disk states. In: Proc. Conf. 2019 Applied Mathematics, Computational Science and Mechanics: Current Problems, Voronezh, Russian Feder- ation. 2020;1479:012122.

16. Ishlinskii AYu, Ivlev DD. Matematicheskaya teoriya plastichnosti [Mathematical theory of plasticity]. Moscow: Fizmatlit; 2001. 704 p. (In Russ.)

17. Khan Kh. Teoriya uprugosti. Osnovy lineinoi teorii i ee primenenie [Theory of elasticity. Fundamentals of linear theory and its application]. Moscow: Mir; 1988. 343 p. (In Russ.)