5 Analyzing the Problem of a Spherical Cavity Expansion …
69
dynamics. The existing closed-form solutions were mainly obtained in the assumption of Tresca’s yield condition. This is explained by the fact that the problematic
part of the applications of the methodology based on analyzing the cavity expansion problem consists mainly of problems of impact deformation of metals and
alloys, where the model of elastic-ideally-plastic media with Tresca’s or Mises yield
conditions is widely used.
The present paper introduces a formula derived for determining a critical pressure
(a minimal pressure required for the nucleation of a cavity), accounting for internal
friction in the framework of the Mohr-Coulomb yield criterion, which is a generalization of the earlier solution for an elastic-ideally-plastic medium with Tresca’s
criterion. The obtained solution can be used for analyzing spherical cavity expansion
problems in porous materials and geo-materials.
Papers (Grigoryan 1960; Bazhenov and Kotov 2008; Bazhenov et al. 2009) present
experimental and theoretical data testifying to the limited nature of yield strengths of
soils under high pressures. As is demonstrated in (Bazhenov et al. 2009), the nonlinear
dependence of yield strength on pressure, with the account of the scatter of the data
and measuring inaccuracies, can be represented as a double-link broken line—a linear
relation under low pressures, as it is assumed in the Mohr-Coulomb plasticity condition, and a limited maximal value of yield strength under high pressures according
to Tresca’s condition. In this connection, effective analyses of problems of penetration of rigid bodies into soft soils, using the cavity expansion methodology, require
an analytical solution of the spherical cavity expansion problem in a medium with
Mohr-Coulomb-Tresca’s plasticity condition.
5.2 Formulation of an Initial Boundary-Value Problem
for a System of Partial Differential Equations
A mathematical model of an elastoplastic medium is described by a system of differential equations expressing laws of continuity and change of kinetic momentum,
which, in spherical Eulerian coordinates, will be written as:
ρ
∂υ
∂r
+ 2
υ
r
= −
∂ρ
∂t
+ υ
∂ρ
∂r
,
∂σ r
∂r
+ 2
(σ r − σ θ )
r
= −ρ
∂υ
∂t
+ υ
∂υ
∂r
,
(5.1)
where ρ is density in a deformed state, υ is velocity, σ r and σ θ are radial and
circumferential components of Cauchy stress tensor (which are assumed positive
in compression), r is radial coordinate.
Equation system (5.1) is closed by the relation between pressure and volumetric
strain, which is linear or close to linear for small strains
p = f (θ ) ≡ K θ + O
θ
2
,
(5.2)
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dynamics. The existing closed-form solutions were mainly obtained in the assumption of Tresca’s yield condition. This is explained by the fact that the problematic
part of the applications of the methodology based on analyzing the cavity expansion problem consists mainly of problems of impact deformation of metals and
alloys, where the model of elastic-ideally-plastic media with Tresca’s or Mises yield
conditions is widely used.
The present paper introduces a formula derived for determining a critical pressure
(a minimal pressure required for the nucleation of a cavity), accounting for internal
friction in the framework of the Mohr-Coulomb yield criterion, which is a generalization of the earlier solution for an elastic-ideally-plastic medium with Tresca’s
criterion. The obtained solution can be used for analyzing spherical cavity expansion
problems in porous materials and geo-materials.
Papers (Grigoryan 1960; Bazhenov and Kotov 2008; Bazhenov et al. 2009) present
experimental and theoretical data testifying to the limited nature of yield strengths of
soils under high pressures. As is demonstrated in (Bazhenov et al. 2009), the nonlinear
dependence of yield strength on pressure, with the account of the scatter of the data
and measuring inaccuracies, can be represented as a double-link broken line—a linear
relation under low pressures, as it is assumed in the Mohr-Coulomb plasticity condition, and a limited maximal value of yield strength under high pressures according
to Tresca’s condition. In this connection, effective analyses of problems of penetration of rigid bodies into soft soils, using the cavity expansion methodology, require
an analytical solution of the spherical cavity expansion problem in a medium with
Mohr-Coulomb-Tresca’s plasticity condition.
5.2 Formulation of an Initial Boundary-Value Problem
for a System of Partial Differential Equations
A mathematical model of an elastoplastic medium is described by a system of differential equations expressing laws of continuity and change of kinetic momentum,
which, in spherical Eulerian coordinates, will be written as:
ρ
∂υ
∂r
+ 2
υ
r
= −
∂ρ
∂t
+ υ
∂ρ
∂r
,
∂σ r
∂r
+ 2
(σ r − σ θ )
r
= −ρ
∂υ
∂t
+ υ
∂υ
∂r
,
(5.1)
where ρ is density in a deformed state, υ is velocity, σ r and σ θ are radial and
circumferential components of Cauchy stress tensor (which are assumed positive
in compression), r is radial coordinate.
Equation system (5.1) is closed by the relation between pressure and volumetric
strain, which is linear or close to linear for small strains
p = f (θ ) ≡ K θ + O
θ
2
,
(5.2)