Theoretical Solution for Cavity Expansion in Crushable Soil
Publication: International Journal of Geomechanics
Volume 21, Issue 7
Abstract
This paper proposed a theoretical solution for cavity expansion in crushable soils. The constitutive relations of the crushable soils were described by the breakage mechanics model that explains the grain crushing induced grain size redistribution. The governing partial differential equations (PDEs) for the cavity expansion issue were formulated through the equations of equilibrium, constitutive relations, continuity conditions, and drainage conditions. The similarity transformation method was utilized to transform the PDEs to first-order linear ordinary differential equations, for which the numerical solutions were then obtained through the Runge–Kutta method. The effective stress, breakage, and specific volume around cylindrical and spherical cavities were given. The limit expansion pressure was particularly discussed through parametric analyses. The results showed that the normalized limit expansion pressure increases as the normalized critical comminution pressure increases when and tends to a constant value when . The increase of the normalized bulk modulus and critical state friction coefficient M led to the increase of limit expansion pressure, whereas the decrease of the ratio between bulk modulus and shear modulus δ, grading index , and coupling angle ω resulted in the increase of limit expansion pressure. Moreover, the limit expansion pressure was not sensitive to the initial specific volume υ0. The proposed solution could be used to interpret the issue of the pile end-bearing capacity in crushable soils.
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Acknowledgments
The work was supported by the National Natural Science Foundation of China (Grant/Award Nos. 51978105 and 52027812), Chongqing Youth Top Talent Plan (Grant/Award No. CQYC201905069), and Chongqing Technology Innovation and Application Development Special General Project (cstc2019jscx-msxmX0107).
Notation
The following symbols are used in this paper:
- Ap, Aq
- intermediate variable defined in Eqs. (39) and (40);
- a0, a
- initial and created cavity radius;
- B
- breakage;
- BB, Bq
- intermediate variable defined in Eqs. (43) and (44);
- Bp
- breakage at the EP boundary;
- Cp, Cq, CB
- intermediate variable defined in Eqs. (45)–(47);
- C1
- integration constant;
- Dr
- relative density of the soil;
- dλ
- nonnegative multiplier;
- EB
- breakage energy;
- Ec
- critical breakage energy;
- emax, emin
- maximum and minimum void ratio of the soil;
- e0
- initial void ratio of the soil;
- F(x, B)
- current cumulative GSD by mass;
- Fu(x)
- final cumulative GSD by mass;
- F0(x)
- initial cumulative GSD by mass;
- f
- yield function;
- G
- shear modulus;
- K
- bulk modulus;
- k
- k 1 for cylindrical cavity and k 2 for spherical cavity;
- k1, k2, k3
- three intermediate variables defined in Eq. (22);
- M
- friction coefficient at the critical state;
- NLP
- normalized limit expansion pressure;
- n0
- initial porosity of the soil;
- p′
- mean effective stress;
- critical comminution pressure;
- mean effective stress at the EP boundary;
- initial mean effective stress;
- q
- deviatoric stress;
- qp
- deviatoric stress at the EP boundary;
- r
- radial position of soil particle;
- rp
- radius of the plastic zone;
- t
- time;
- uw
- pore pressure of the soil;
- uw0
- initial pore pressure of the soil;
- uwp
- pore pressure of soil at the EP boundary;
- ur
- radial displacement of the soil;
- v
- radial expansion velocity of the soil;
- vp
- radial velocity at the EP boundary;
- x
- grain size;
- δ
- ratio between bulk modulus and shear modulus;
- ɛp, ɛq
- total volumetric and shear strains;
- elastic and plastic volumetric strains;
- elastic and plastic shear strains;
- radial and circumferential strain;
- η
- similarity variable;
- μ
- Poisson’s ratio of soil;
- effective radial stress at the cavity wall;
- effective radial and circumferential stress;
- effective radial and circumferential stress at the EP boundary;
- υ
- specific ratio of the soil;
- υp
- specific ratio of the soil at the EP boundary;
- υ0
- initial specific ratio of the soil;
- φ
- internal frictional angle;
- ω
- coupling angle; and
- ϑ
- grading index.
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History
Received: Sep 14, 2020
Accepted: Feb 10, 2021
Published online: Apr 21, 2021
Published in print: Jul 1, 2021
Discussion open until: Sep 21, 2021
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