Undrained expansion of a cylindrical cavity in clays with fabric anisotropy: theoretical solution

Abstract

This paper presents a novel, exact, semi-analytical solution for the quasi-static undrained expansion of a cylindrical cavity in soft soils with fabric anisotropy. This is the first theoretical solution of the undrained expansion of a cylindrical cavity under plane strain conditions for soft soils with anisotropic behaviour of plastic nature. The solution is rigorously developed in detail, introducing a new stress invariant to deal with the soil fabric. The semi-analytical solution requires numerical evaluation of a system of six first-order ordinary differential equations. The results agree with finite element analyses and show the influence of anisotropic plastic behaviour. The effective stresses at critical state are constant, and they may be analytically related to the undrained shear strength. The initial vertical cross-anisotropy caused by soil deposition changes towards a radial cross-anisotropy after cavity expansion. The analysis of the stress paths shows that proper modelling of anisotropic plastic behaviour involves modelling not only the initial fabric anisotropy but also its evolution with plastic straining.

Appendices

Appendix 1: derivatives

The partial derivatives used in the analytical solution are

$$\frac{{\partial f_{y} }}{{\partial \sigma_{i}^{'} }} = \frac{{p^{\prime}\left( {M^{2} - \alpha^{2} - {\bar{\eta }}^{2} } \right)}}{3} + \left( {3s_{i} - s_{r} \alpha_{r}^{d} - s_{\theta } \alpha_{\theta }^{d} - s_{z} \alpha_{z}^{d} } \right)\,{\text{for}}\, i = r,\theta ,z$$

where

$${\bar{\eta }} = \frac{{\bar{q}}}{p^{\prime}}$$

(58)

$$\bar{q} = \sqrt {\frac{3}{2}Q}$$

(59)

and

$$\frac{{\partial f_{y} }}{{\partial p_{m}^{'} }} = - p^{\prime}\left( {M^{2} - \alpha^{2} } \right)$$

(60)

$$\frac{{\partial p_{m}^{'} }}{{\partial \varepsilon_{v}^{p} }} = \frac{{\upsilon p^{\prime}}}{{\left( {\lambda - \kappa } \right)\left( {M^{2} - \alpha^{2} } \right)}}\left( {M^{2} - \alpha^{2} + {\bar{\eta }}^{2} } \right)$$

(61)

$$\frac{{\partial f_{y} }}{{\partial p^{\prime}}} = p^{\prime}\left( {M^{2} - \alpha^{2} - {\bar{\eta }}^{2} } \right) - 3\left( {s_{r} \alpha_{r}^{d} + s_{\theta } \alpha_{\theta }^{d} + s_{z} \alpha_{z}^{d} } \right)$$

(62)

$$\frac{{\partial f_{y} }}{{\partial \alpha_{i}^{d} }} = - 3s_{i} p^{\prime} + 3\alpha_{i}^{d} \frac{{\bar{q}^{2} }}{{M^{2} - \alpha^{2} }} \quad{\text{for}}\, i = r,\theta ,z$$

(63)

$$\frac{{\partial f_{y} }}{{\partial \sigma_{i}^{'d} }} = 3s_{i} \quad{\text{for}}\,i = r,\theta ,z$$

(64)

$$\frac{{\partial \alpha_{i}^{d} }}{{\partial \varepsilon_{v}^{p} }} = \omega \left( {\frac{{3\left( {\sigma_{i}^{'} - p^{\prime}} \right)}}{4p^{\prime}} - \alpha_{i}^{d} } \right)\quad{\text{for}}\,i = r,\theta ,z$$

(65)

$$\frac{{\partial \alpha_{i}^{d} }}{{\partial \varepsilon_{d}^{p} }} = \omega \omega_{d} \left( {\frac{{\left( {\sigma_{i}^{'} - p^{\prime}} \right)}}{3p^{\prime}} - \alpha_{i}^{d} } \right)\quad{\text{for}}\,i = r,\theta ,z$$

(66)

$$\sqrt {\frac{2}{3}\left\{ {\frac{{\partial f_{y} }}{{\partial\varvec{\sigma}^{'d} }}} \right\} \cdot \left\{ {\frac{{\partial f_{y} }}{{\partial\varvec{\sigma}^{'d} }}} \right\}} = 2\bar{q}$$

(67)

Appendix 2: elastic solution

The solution for the elastic total stresses, ($\sigma_{r}$, $\sigma_{\theta }$, $\sigma_{z}$), and the radial displacement ($u_{r}$) can be obtained imposing the assumption of total volumetric strain increment is zero under undrained deformation (for details see e.g. Yu [34])

$$\sigma_{r} = \sigma_{H} + \left( {\sigma_{p} - \sigma_{H} } \right)\left( {\frac{{r_{p} }}{r}} \right)^{2}$$

(68)

$$\sigma_{\theta } = \sigma_{H} + \left( {\sigma_{p} - \sigma_{H} } \right)\left( {\frac{{r_{p} }}{r}} \right)^{2}$$

(69)

$$\sigma_{z} = \sigma_{V}$$

(70)

$$u_{r} = \frac{{\sigma_{p} - \sigma_{H} }}{{2G_{0} }}\frac{{r_{p}^{2} }}{r}$$

(71)

where $\sigma_{p}$ is the total radial stress at the elasto/plastic boundary and $\sigma_{H}$ and $\sigma_{V}$ are the total horizontal and vertical stresses, respectively.