Mechanics of Deformable Solids 
DOI: http://www.dx.doi.org/10.24866/2227-6858/2020-3-1

Syomka E.

ELEONORA SYOMKA, Lecturer, Military Training and Scientific Center of the Air Force,
ORCID: https://orcid.org/0000-0002-0194-6979, e-mail: semka_elya@mail.ru,
Air Force Academy named after Professor N.E. Zhukovsky and Yu. A. Gagarina
Voronezh, Russia

Elastoplastic state of a hollow sphere

Abstract: The problem of a hollow sphere with different pressure and temperature levels on the external and the internal boundaries has been considered for the model of an ideal compressible isotropic elastic body. An independent problem is solved within the theory of small strains. It is assumed that total deformations are equal to the sum of elastic and plastic deformations, plastic deformations and stresses are related by the relations of the associated flow law. Elastic deformations are determined by their correlation under the Duhamel-Neumann law. The plasticity condition which is independent of the first invariant of the stress tensor is considered. In determination of stress and strain in the plastic range a quasi-static approach is considered, that is, the law of changing the external parameters of the action (pressure and temperature at the boundaries of the sphere) to the values assumed in the calculations is not indicated.
In this article, the boundaries of the pressure drop and temperature for which the sphere will be in an elastic state are determined. It has been established that depending on the values of the external parameters of the impact, the plastic zone can occur on the inner or outer boundary of the sphere, or on the inner and outer boundaries simultaneously, or between the boundaries of the sphere. As an example, the graphs of the distributions of stresses, deformations, displacements when the plastic zone occupies a certain part located between the inner and outer boundary of the sphere are given. The stress hodograph graphs which are integral elements of verification of the solution of the problem are given.
Keywords: elastoplastic body, theory of plastic flow, temperature stresses, hollow sphere, permissible values of external parameters, thermoelastic elasticity, elastic-plastic boundary, equivalent stress.

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