78
V. L. Kotov et al.
Fig. 5.3 Critical pressure as a function of the initial value of yield strength: numerical solution
(marked with squares and rhombs, respectively) and solution using formulas (5.23), (5.24) and
(5.26) (solid lines and the dotted line)
5.6 An Analytical Solution of the Cavity Expansion
Problem in a Medium with a Linear Shock Adiabat
Experiments (Lagunov and Stepanov 1963; Dianov et al. 1977; Bragov and
Grushevskii 1993; Arlery et al. 2010; Bragov et al. 2005; Balandin et al. 2015)
show that dynamic compressibility of many media is characterized by Hugoniot’s
shock adiabat in the form of linear relation
U s = C 0 + su p ,
(5.28)
correlating plane shockwave velocity U s and velocity of the particles behind the wave
front u p . Here, C 0 is sonic velocity in a medium under zero pressure, s is constant.
Applying relation (5.28) and Rankin-Hugoniot’s conditions, a correlation between
stress σ r and volumetric strain θ is obtained, which will have the following form:
σ r = f 1 (θ ) ≡ ρ 0 C
2
0 θ (1 − sθ )
−2 (Lagunov and Stepanov 1963). In this form of the
relation, constant s characterizes compressive strength of the medium.
An analytical solution of the problem is constructed in the plastic yield region
limited by radii r = V t and r = ct in the following assumptions:
(a) the medium is assumed to be rigidly plastic (elastic deformation of the soft
soil medium is neglected), i.e., the plastic yield region adjoins the region of
unperturbed medium, where σ r = υ = 0, υ is velocity of the particles of the
medium;
(b) the solution is a plastic shockwave propagating through an unperturbed halfspace;
V. L. Kotov et al.
Fig. 5.3 Critical pressure as a function of the initial value of yield strength: numerical solution
(marked with squares and rhombs, respectively) and solution using formulas (5.23), (5.24) and
(5.26) (solid lines and the dotted line)
5.6 An Analytical Solution of the Cavity Expansion
Problem in a Medium with a Linear Shock Adiabat
Experiments (Lagunov and Stepanov 1963; Dianov et al. 1977; Bragov and
Grushevskii 1993; Arlery et al. 2010; Bragov et al. 2005; Balandin et al. 2015)
show that dynamic compressibility of many media is characterized by Hugoniot’s
shock adiabat in the form of linear relation
U s = C 0 + su p ,
(5.28)
correlating plane shockwave velocity U s and velocity of the particles behind the wave
front u p . Here, C 0 is sonic velocity in a medium under zero pressure, s is constant.
Applying relation (5.28) and Rankin-Hugoniot’s conditions, a correlation between
stress σ r and volumetric strain θ is obtained, which will have the following form:
σ r = f 1 (θ ) ≡ ρ 0 C
2
0 θ (1 − sθ )
−2 (Lagunov and Stepanov 1963). In this form of the
relation, constant s characterizes compressive strength of the medium.
An analytical solution of the problem is constructed in the plastic yield region
limited by radii r = V t and r = ct in the following assumptions:
(a) the medium is assumed to be rigidly plastic (elastic deformation of the soft
soil medium is neglected), i.e., the plastic yield region adjoins the region of
unperturbed medium, where σ r = υ = 0, υ is velocity of the particles of the
medium;
(b) the solution is a plastic shockwave propagating through an unperturbed halfspace;