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

Nallathamby Sivasithamparam; Jorge Castro

doi:10.1007/s11440-017-0587-4

Acta Geotechnica

June 2018, Volume 13, Issue 3, pp 729–746 | Cite as

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

Authors
Authors and affiliations

Nallathamby Sivasithamparam
Jorge CastroEmail author

Research Paper

First Online: 13 September 2017

Received: 20 December 2016

Accepted: 27 July 2017

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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.

Keywords

Anisotropy Clays Fabric of soils Plasticity Stress path Theoretical analysis

List of symbols

a: Radius of the cylindrical cavity
$c_{u}$: Undrained shear strength
$c_{{u,{\text{TX}}}}$: Undrained shear strength for triaxial compression conditions
$c_{{u,{\text{PS}}}}$: Undrained shear strength for plane strain conditions
d: Incremental operator
D: Elastic stiffness matrix
E: Young’s modulus
e: Void ratio
e_M: Void ratio at critical state
f_y: Function of the yield surface
G: Shear modulus
K_0NC: Coefficient of lateral earth pressure at rest in normally consolidated conditions
$K_{0}$: Coefficient of lateral earth pressure at rest
$M$: Slope of the critical state line
$p^{\prime}$: Mean effective stress
$p^{\prime}_{m}$: Preconsolidation pressure
$q$: Deviatoric stress
$\overline{q}$: Invariant for anisotropic models. Radius of the yield surface in π-plane
Q: Invariant for anisotropic models: $Q = \frac{2}{3}\bar{q}^{2}$
R: Isotropic overconsolidation ratio
s: Deviatoric stress
u: Pore pressure
u_r: Radial displacement
$\upsilon$: Specific volume
${\varvec{\upalpha}}$: Fabric tensor
$\alpha$: Inclination of the yield surface
${\varvec{\upalpha}}_{{\mathbf{d}}}$: Deviatoric fabric tensor
Λ: Plastic multiplier
ε: Strain scalar
ε: Strain tensor
${\text{d}}\varepsilon_{v}$: Change in volumetric strain ${\text{d}}\varepsilon_{v} = {\text{d}}\varepsilon_{r} + {\text{d}}\varepsilon_{\theta } + {\text{d}}\varepsilon_{z }$
${\text{d}}\varepsilon_{d}$: Change in deviatoric strain ${\text{d}}\varepsilon_{d} = \sqrt {\frac{2}{3}\left\{ {{\text{d}}\varvec{\varepsilon}_{d} } \right\}^{T} \cdot \left\{ {{\text{d}}\varvec{\varepsilon}_{d} } \right\}}$
η: Stress ratio: η = q/$p^{\prime}$ or ${\varvec{\upeta}} = {\varvec{\upsigma}}_{{\mathbf{d}}} /p^{\prime}$ (tensor)
θ: Lode’s angle: $\theta = { \tan }^{ - 1} \left[ {\frac{1}{\sqrt 3 }\left( {2\frac{{\sigma_{2}^{'} - \sigma_{3}^{'} }}{{\sigma_{1}^{'} - \sigma_{3}^{'} }} - 1} \right)} \right]$
$\kappa$: Slope of swelling line from $\upsilon - \ln p^{\prime}$ space
$\lambda$: Slope of post-yield compression line from $\upsilon - \ln p^{\prime}$ space
$\nu$: Poisson’s ratio
$\sigma$,$\sigma '$: Total and effective stresses
$\sigma_{a}$: Internal cavity pressure
$\sigma_{p}$: Total radial stress at the elastic/plastic boundary
$\varvec{\sigma}_{\varvec{d}}^{\varvec{'}}$: deviatoric stress tensor
ϕ: Friction angle
ω, ω_d: Absolute and relative effectiveness of rotational hardening

Abbreviations

CS: Critical state
CSL: Critical state line
ESP: Effective stress path
FEM: Finite element method
MCC: Modified Cam clay
OCR: Overconsolidation ratio
RH: Rotational hardening
YS: Yield surface

Subscripts/superscripts

0: Initial
d, v: Deviatoric, volumetric
H, V: Horizontal, vertical
i: Any of the axis components r, θ, z
p: Plastic
r, θ, z: Cylindrical coordinates

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Notes

Acknowledgements

The research was initiated as part of GEO-INSTALL (Modelling Installation Effects in Geotechnical Engineering, PIAP-GA-2009-230638) and CREEP (Creep of Geomaterials, PIAP-GA-2011-286397) projects supported by the European Community through the programme Marie Curie Industry-Academia Partnerships and Pathways (IAPP) under the 7th Framework Programme.

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.

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Authors and Affiliations

Nallathamby Sivasithamparam
- 1
Jorge Castro
- 2
Email authorView author's OrcID profile

1.Computational Geomechanics DivisionNorwegian Geotechnical InstituteOsloNorway
2.Group of Geotechnical Engineering, Department of Ground Engineering and Materials ScienceUniversity of CantabriaSantanderSpain

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

Abstract

Keywords

List of symbols

Abbreviations

Subscripts/superscripts

Notes

Acknowledgements

References

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