## Abstract

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## Abbreviations

$$p_{\text{a}}$$, $$p_{\text{in}}$$, $$p_{\text{out}}$$ :

Axial stress, internal and external radial pressures

$$\varsigma$$ :

$$\varsigma = 1$$ for loading and $$\varsigma = - 1$$ for unloading

k :

$$k = 1$$ for a cylindrical cavity and $$k = 2$$ for a spherical cavity

r, θ, z :

Coordinates of the cylindrical coordinate system

r, θ, φ :

Coordinates of the spherical coordinate system

$$r_{0}$$ :

Initial value of the radial coordinate r

$$p^{\prime }$$, $$q$$ :

Mean effective stress and deviatoric stress

$$p^{\prime}_{\text{cs}}$$, $$q_{\text{cs}}$$ :

Mean effective stress and deviatoric stress at the critical-state

$$p$$ :

Mean total pressure

$$p_{0}$$, $$p^{\prime}_{0}$$ :

Initial values of $$p$$ and $$p^{\prime }$$

$$U$$, $$U_{0}$$, $$\Delta U$$ :

Total, initial ambient, excess pore pressures

$$\left. {\Delta U} \right|_{r = a}$$, $$\left. {\Delta U} \right|_{r = b}$$ :

Excess pore pressures at $$r = a$$ and at $$r = b$$

$$\left. {\Delta U} \right|_{r = a}$$, $$\left. {\Delta U} \right|_{r = b}$$ :

Excess pore pressures at $$r = c$$ and at $$r = r_{\text{cs}}$$

$$\sigma^{\prime}_{\text{r}}$$, $$\sigma^{\prime}_{\theta }$$ :

$$\sigma_{\text{r}}$$, $$\sigma_{\theta }$$ :

$$\varepsilon_{\text{r}}$$, $$\varepsilon_{\theta }$$ :

$$\delta$$, $$\gamma$$ :

Volumetric and shear strains

$$a_{0}$$, $$a$$; $$b_{0}$$, $$b$$; $$c_{0}$$, $$c$$ :

Initial and current radii of the inner cavity wall, the outer cavity wall, the elastic–plastic boundary

$$r_{\text{cs}}$$ :

$$p^{\prime}_{a}$$, $$q_{a}$$ :

Mean effective and shear stresses at $$r = a$$

$$p^{\prime}_{b}$$, $$q_{b}$$ :

Mean effective and shear stresses at $$r = b$$

$$\gamma_{a}$$, $$\gamma_{b}$$ :

Shear strains at $$r = a$$ and at $$r = b$$

$$\gamma_{\text{ep}}$$, $$q_{\text{ep}}$$ :

Shear strain and shear stress at the state just enters plastic yielding

$$K$$, $$G$$ :

Instantaneous bulk and shear moduli with initial values of $$K_{0}$$ and $$G_{0}$$

$$M$$ :

The slope of the CSL in the $$p^{\prime}$$$$q$$ space

$$\lambda$$ :

Slope of the normally compression line

$$\varGamma$$ :

The value of $$v$$ on the CSL at $$p^{\prime} = 1\,{\text{kPa}}$$

$$v$$, $$\mu$$ :

Specific volume and Poisson’s ratio of soil

$$\kappa$$ :

Slope of the swelling line

$$\varLambda$$ :

Plastic volumetric strain ratio, equals $${{\left( {\lambda - \kappa } \right)} \mathord{\left/ {\vphantom {{\left( {\lambda - \kappa } \right)} \lambda }} \right. \kern-0pt} \lambda }$$

$$R_{0}$$ :

Isotropic over-consolidation ratio is defined as $$p^{\prime}_{y0} /p^{\prime}_{0}$$

n, $$r^{*}$$ :

Stress-state coefficient and spacing ratio in CASM

$$p^{\prime}_{y}$$, $$p^{\prime}_{y0}$$ :

Preconsolidation pressure and its initial value

$$s_{\text{u}}$$ :

Undrained shear strength of soil

$$\eta$$, $$\eta_{\text{ep}}$$ :

Stress ratio and its value at the elastic–plastic boundary

$$\varphi_{{\rm cs}}$$ :

Critical-state friction angle, Hvorslev friction angle

$$\varphi_{\text{tc}}$$ :

Critical-state friction angle under triaxial compression and plane strain

$$\Delta V/V_{0}$$ :

Cavity volumetric strain

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## Acknowledgements

The authors would like to acknowledge the Open Research Fund of the State Key Laboratory for Geomechanics and Deep Underground Engineering China University of Mining and Technology (SKLGDUEK1802) and the International Mobility Fund from the University of Leeds. The first author also acknowledges the support of the ‘Taishan’ Scholar Program of Shandong Province, China (No. tsqn201909016) and the ‘Qilu’ Scholar Program of Shandong University.

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Correspondence to Pin-Qiang Mo.