Chapter 5
Analyzing the Problem of a Spherical
Cavity Expansion in a Medium
with Mohr-Coulomb-Tresca’s Plasticity
Condition
Vasiliy L. Kotov, Elena Yu. Linnik, and Tatiana A. Sabaeva
Abstract An analytical solution of the one-dimensional problem of a spherical
cavity expanding at a constant velocity from a point in a half-space occupied by a
plastic medium has been obtained. Impact compressibility of the medium is described
using linear Hugoniot’s adiabat. Plastic deformation obeys the Mohr-Coulomb yield
criterion with constraints on the value of maximum tangential stresses according
to Tresca’s criterion. In the assumption of rigid-plastic deformation (the elastic
precursor being neglected), incompressibility behind the shockwave front and the
equality of the propagation velocities of the fronts of the plastic wave and the
plane shockwave defined by linear Hugoniot’s adiabat, a boundary-value problem
for a system of two first-order ordinary differential equations for the dimensionless
velocity and stress depending on the self-similar variable is formulated. A closedform solution of this problem has been obtained in the form of a stationary running
wave—a plastic shockwave propagating in an unperturbed half-space. The solution
is a generalization of the earlier obtained analytical solution for a medium with the
Mohr-Coulomb plasticity condition. A formula for determining a critical pressure (a
minimal pressure required for the nucleation of a cavity, accounting for internal friction in the framework of Mohr-Coulomb yield criterion), which is a generalization
of the earlier solution for an ideal plastic medium with Tresca’s criterion, has been
derived. The resulting critical pressure was compared with a numerical solution in a
full formulation at cavity propagation velocities close to zero in a wide range of the
parameters of the Mohr-Coulomb yield criterion. The approximation inaccuracy of
the introduced formula does not exceed 6% for the internal friction coefficient varying
over the entire permissible range and the initial value of yield strength changing by
three orders of magnitude. The effect of constraining the limiting value of maximal
tangential stresses on the distribution of dimensionless stresses behind the shockwave
front has been examined. Formulas for determining the range of cavity expansion
velocities, within which a simple solution for a medium with Tresca’s plasticity
condition is applicable, have been derived. The obtained solution can be used for
V. L. Kotov (B) · E. Yu. Linnik · T. A. Sabaeva
Research Institute for Mechanics, National Research Lobachevsky State University of Niznhy
Novgorod, Gagarin ave. 23, 603950 Nizhny Novgorod, Russia
e-mail: vkotov@inbox.r
© Springer Nature Switzerland AG 2021
F. dell’Isola and L. Igumnov (eds.), Dynamics, Strength of Materials and Durability
in Multiscale Mechanics, Advanced Structured Materials 137,
https://doi.org/10.1007/978-3-030-53755-5_5
67
Analyzing the Problem of a Spherical
Cavity Expansion in a Medium
with Mohr-Coulomb-Tresca’s Plasticity
Condition
Vasiliy L. Kotov, Elena Yu. Linnik, and Tatiana A. Sabaeva
Abstract An analytical solution of the one-dimensional problem of a spherical
cavity expanding at a constant velocity from a point in a half-space occupied by a
plastic medium has been obtained. Impact compressibility of the medium is described
using linear Hugoniot’s adiabat. Plastic deformation obeys the Mohr-Coulomb yield
criterion with constraints on the value of maximum tangential stresses according
to Tresca’s criterion. In the assumption of rigid-plastic deformation (the elastic
precursor being neglected), incompressibility behind the shockwave front and the
equality of the propagation velocities of the fronts of the plastic wave and the
plane shockwave defined by linear Hugoniot’s adiabat, a boundary-value problem
for a system of two first-order ordinary differential equations for the dimensionless
velocity and stress depending on the self-similar variable is formulated. A closedform solution of this problem has been obtained in the form of a stationary running
wave—a plastic shockwave propagating in an unperturbed half-space. The solution
is a generalization of the earlier obtained analytical solution for a medium with the
Mohr-Coulomb plasticity condition. A formula for determining a critical pressure (a
minimal pressure required for the nucleation of a cavity, accounting for internal friction in the framework of Mohr-Coulomb yield criterion), which is a generalization
of the earlier solution for an ideal plastic medium with Tresca’s criterion, has been
derived. The resulting critical pressure was compared with a numerical solution in a
full formulation at cavity propagation velocities close to zero in a wide range of the
parameters of the Mohr-Coulomb yield criterion. The approximation inaccuracy of
the introduced formula does not exceed 6% for the internal friction coefficient varying
over the entire permissible range and the initial value of yield strength changing by
three orders of magnitude. The effect of constraining the limiting value of maximal
tangential stresses on the distribution of dimensionless stresses behind the shockwave
front has been examined. Formulas for determining the range of cavity expansion
velocities, within which a simple solution for a medium with Tresca’s plasticity
condition is applicable, have been derived. The obtained solution can be used for
V. L. Kotov (B) · E. Yu. Linnik · T. A. Sabaeva
Research Institute for Mechanics, National Research Lobachevsky State University of Niznhy
Novgorod, Gagarin ave. 23, 603950 Nizhny Novgorod, Russia
e-mail: vkotov@inbox.r
© Springer Nature Switzerland AG 2021
F. dell’Isola and L. Igumnov (eds.), Dynamics, Strength of Materials and Durability
in Multiscale Mechanics, Advanced Structured Materials 137,
https://doi.org/10.1007/978-3-030-53755-5_5
67