ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2018, Vol. 59, No. 1, pp. 104–111.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
K.V. Berdnikov, V.V. Struzhanov.
EQUILIBRIUM STATE OF A SOFTENING ELASTOPLASTIC MEDIUM
WITH AN EXPANDING SPHERICAL CAVITY
K. V. Berdnikov and V. V. Struzhanov
Abstract: This paper describes the problem of a stress–strain state arising from expansion of a
spherical cavity under increasing internal pressure. The properties of a medium are described by a
single curve with a descending section (Hencky medium with softening) under the condition of non-
positivity of volume deformation. An iteration procedure for calculation of equilibrium parameters
is proposed. This procedure is based on the method of simple iterations. Numerical calculations
conﬁrming the developed technique are presented.
Keywords: Hencky medium, polar-symmetric deformation, softening, method of simple iterations,
It was shown in [1–6] that accounting for the supercritical stage of deformation (stage of strain softening of
the material) makes it possible to use deformation reserves of the material and, consequently, calculate the values
of ultimate bearing capacity of structural elements, which are the closest to real ones. However, the use of softening
characteristics in the calculation of strength complicates the formulation of corresponding constitutive relationships
for a complex stress state and in the study of mathematical problems, which is due to possible nonuniqueness and
instability of solutions to boundary-value problems with explicit account for softening [7, 8].
At the same time, there is a class of nondimensional problems, in which the stage of softening can be
described with hypotheses used for constructing some plasticity theories, such as deformation theory of plasticity
(Hencky model) under the condition that material is formed in constrained conditions when the volume strain is
nonpositive. In this case, the properties of the material and constitutive relationships are determined by a single
curve with a descending section . Another simpliﬁcation is possible in the case of polar-symmetric deformation.
In this case, it is possible to construct quite simply implementable iteration scheme for calculating stress–strain
states with a gradual increase in the load and system transition from one stable equilibrium state into another
arbitrarily close equilibrium state.
A spherical cavity of radius a is located in a space, whose material is elastoplastic and possesses the property
of strain softening. This cavity expands under the action of monotonically increasing inner pressure p. Because
of polar symmetry in the material surrounding the cavity, there are only radial displacements u(r)thatdampat
Institute of Engineering Science, Ural Branch, Russian Academy of Sciences, Ekaterinburg, 620049 Rus-
sia; firstname.lastname@example.org; email@example.com. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika,
Vol. 59, No. 1, pp. 120–128, January–February, 2018. Original article submitted July 15, 2016; revision submitted
January 9, 2017.
2018 by Pleiades Publishing, Ltd.