Loading and unloading of a thick-walled cylinder of critical-state soils: large strain analysis with applications


Thick-walled cylinder (TWC) tests are widely used to obtain soil properties and investigate wellbore instability problems in laboratory-controlled conditions. This paper presents analytical cavity expansion and contraction solutions for modelling undrained TWC tests under three typical loading and unloading programs. Both cylindrical and spherical cavities in critical-state soils with a finite radial extent subjected to monotonic loading or unloading under undrained conditions are considered. The solutions are developed in terms of finite strain formulations, and the procedure is applicable to any isotropically hardening materials. Parametric studies show the boundary effect may significantly affect the cavity expansion/contraction response. A limit outer-to-inner diameter ratio of the soil sample exists, beyond which the boundary effect becomes negligible. The limit ratio varies with the cavity geometry, soil stress history (OCR) and cavity deformation level. For undrained TWC tests, a diameter ratio over 20 should normally be adequate to remove the possible boundary effect. Predicted expansion and contraction curves by the new solutions are compared with published data of TWC tests in the literature, and good agreement is shown in each loading/unloading program. This indicates that the boundary effect, which greatly limits the application of conventional cavity expansion/contraction solutions into TWC problems, is successfully captured by the present solutions. The solutions can also serve as valuable benchmark for verifying various numerical methods involving critical-state plasticity models.

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\(p_{\text{a}}\), \(p_{\text{in}}\), \(p_{\text{out}}\) :

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 \(r = r_{\text{cs}}\)

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

Effective radial and circumferential stresses

\(\sigma_{\text{r}}\), \(\sigma_{\theta }\) :

Total radial and circumferential stresses

\(\varepsilon_{\text{r}}\), \(\varepsilon_{\theta }\) :

Radial and circumferential strains

\(\delta\), \(\gamma\) :

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

\(r_{\text{cs}}\) :

Radius of the plastic-critical-state boundary

\(p^{\prime}_{a}\), \(q_{a}\) :

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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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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Zhuang, P., Yu, H., Mooney, S.J. et al. Loading and unloading of a thick-walled cylinder of critical-state soils: large strain analysis with applications. Acta Geotech. (2020). https://doi.org/10.1007/s11440-020-00994-w

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  • Boundary effect
  • Cavity contraction
  • Cavity expansion
  • Critical-state soil
  • Thick-walled cylinder tests