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Undrained cylindrical cavity expansion in modified cam-clay soil: A semi-analytical solution considering biaxial in-situ stresses

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Abstract

This paper presents a semi-analytical solution for undrained cylindrical cavity expansion in modified Cam-clay soils under biaxial in-situ stresses based on the small strain and large strain assumptions in the elastic and plastic regions, respectively. The cylindrical cavity expansion problem is treated as a boundary value problem and formulated by solving a system of first-order differential equations with four stress components as basic unknown variables. The validity of the proposed solution is examined by comparing with the existing solution, where the in-situ stress is uniform. Extensive parametric studies are conducted to investigate the effects of biaxial in-situ stresses on the stress distribution, the expansion process and the shape and size of elastic-plastic boundary. The results indicate that the elastic-plastic boundary is in the shape of an ellipse due to the shear stress induced by biaxial in-situ stresses. The major axis of the elliptical elastic-plastic boundary is in accordance with the direction of the larger in-situ stress. The present solution is free of the limitation that the cavity expands under the uniform in-situ stress, therefore it is expected to provide a more general method for practical geotechnical and petroleum problems.

Introduction

The cavity expansion theory belongs to one of the fundamental boundary value problems and has obtained wide applications in both geotechnical and petroleum engineering, since many practical problems in these two engineering fields can be theoretically modelled as a cavity expansion or contraction with slight simplifications and assumptions on real conditions. Such examples include the interpretation of pressuremeter and cone penetration tests (Houlsby and Withers, 1988, Yu, 1996), the prediction on load carrying capacity of driven piles (Li et al., 2016, Li et al., 2017, Gong et al., 2017, Li and Gong, 2019) as well as the simulation of tunneling excavation (Yu and Rowe, 1999, Mo and Yu, 2017).

Since it was initially proposed by Lamé (1852), numerous research efforts have been devoted to enriching the cavity expansion theory. Among them, most focus on employing sophisticated constitutive models and failure criteria to provide more realistic simulations for soil behaviours during cavity expansion. The elastic-perfectly plastic model, as the pioneer among the adopted constitutive models, was widely used to describe the stress–strain relationship of soils during cavity expansion in the early years. For example, Hill (1950) derived a solution to cavity expansion based on the Tresca criterion; Vesic (1972) proposed a general solution to both cylindrical and spherical cavity expansions in infinite Mohr-Coulomb soils; Carter et al. (1986) later presented a closed-form pressure-expansion relationship for the spherical and cylindrical cavities in an ideal, cohesive frictional soil on the basis of the infinitesimal deformation theory; Yu and Carter (2002) further developed the rigorous closed-form solution for the cavity expanding from zero initial radius by means of utilizing the Mohr-Coulomb criterion to describe the soil behavior. However, unlike metallic materials, the mechanical behaviours of soils are much more difficult to be determined, because of the complex geological activities on soils such as erosion, consolidation, sedimentation, etc. Therefore, some failure criteria, like the Tresca criterion that is appropriate for steel, are not ideal to be applied to model cavity expansion problems in soils. Besides, the abovementioned solutions all simplified the soils as ideal elastic-perfectly solids and hence are incapable of characterizing the strain-hardening/softening behaviour of soils, which however is a basic property of soils.

For the purpose of modelling the elastoplastic behavior of soils more accurately, Roscoe et al. (1963) initially and innovatively proposed the original Cam-clay model; and then the modified Cam-clay (MCC) model (Roscoe and Burland, 1968), which is well-known for its capability of capturing the volumetric hardening behavior of soils as well as the effects of stress history, was proposed. Since Carter et al. (1979) firstly employed the Cam-clay model to investigate the expansion process of a cylindrical cavity, many other researchers (e.g. Li et al., 2016, Li et al., 2017, Collins and Stimpson, 1994, Collins and Yu, 1996, Chen and Abousleiman, 2012, Su and Yang, 2019, Yang et al., 2020, Yang et al., 2020) have also employed the MCC model or the constitutive models developed based on the MCC model to make contributions to the development of cavity expansion theory. Although the existing MCC model-based solutions can more realistically simulate the soil behaviours in the cylindrical cavity expansion process, almost all of them make the assumption that the cylindrical cavity expands under the uniform radial pressure in the horizontal plane, which generally contradicts the realistic in-situ stresses under some certain conditions, especially for tunneling excavation, horizontal wellbore drilling and underground structures buried at a relatively shallow depth in soils (Zou et al., 2018, Zou et al., 2019). Other typical example includes the interpretation of pressuremeter tests under anisotropic stress in the testing plane. Under this kind of in-situ stresses, the stress distribution around an expanding cavity is significantly different from that with the isotropic in-situ stress. Hence, it is meaningful and necessary to propose a solution for the cylindrical cavity expansion under biaxial stresses. In recent, Zhou et al., 2014, Zhou et al., 2016) and Zhuang and Yu (2018) have tried to solve the cavity expansion problem biaxial stresses and investigate its corresponding effects on the stress distribution and the shape of elastic–plastic boundary, however the soil behaviours are simulated by Tresca or Mohr-Coulomb criterion in their jobs. Hence, their solutions are unable to simulate the cavity expansion in soils that show the strain-hardening/softening behaviour. In another word, to the best of authors’ knowledge, the undrained solution to cylindrical cavity expansion in the modified Cam-clay soil under biaxial in-situ stresses is still unavailable.

To present the semi-analytical solution, this paper employs the MCC model and formulate the undrained cylindrical cavity expansion under biaxial in-situ stresses as an initial boundary value problem with four stress components as the basic unknowns. The obtained results are compared with the well-established solutions, where the in-situ stresses are uniform, to examine the validity of the proposed formulation. Extensive parametric studies are carried out to explore the influence of biaxial in-situ stresses on the distribution of the stress components, the excess pore pressure as well as the expansion procedure. In addition, the variations of the size and the shape of elastic-plastic boundary are investigated in detail. This work, free of the conventional assumption that the radial pressure is uniform, therefore, is expected to have wider potential applications in real geotechnical and petroleum engineering problems.

Section snippets

Definition and assumptions

The cylindrical cavity expansion is generally modelled as a plane strain cavity expanding in an infinite, saturated and homogenous soil, the specific mechanical model of which is shown in Fig. 1. It can be seen that the cylindrical cavity owns an initial radius, a0, and is subjected to the in-situ stresses, which are composed of the horizontal biaxial stresses equal to σx0 and σy0 and the vertical stress equal to σz0. Unlike the traditional cylindrical cavity expansion under the axisymmetric

Elastic solution

The elastic solution of cylindrical cavity expansion under the isotopic in-situ stress has been given by Yu (2000) in detail, based on the theory of elasticity and the small strain theory. The solutions of the three stresses in the radial, circumferential and vertical directions and the radial displacement of undrained cylindrical cavity expansion under the biaxial in-situ stresses can be readily obtained with the simple modification of those presented by Yu (2000), which are written in terms

Validation and parametric studies

The parameters used for calculation are summarized in Table 1, where Cases A and C represent the cylindrical cavity expansions under the biaxial stresses while Case B stands for the cylindrical cavity expansion under the uniform far-field stress. Three different values of overconsolidation ratio, i.e.,OCR=1.2,3,10, which represent the slightly, moderately and heavily overconsolidated soils, respectively, are adopted to investigate the effects of stress history. The effects of biaxial in-situ

Discussion on potential limitations of proposed method

The limitation of the proposed method lies in the basic assumption that there is no circumferential displacement induced during the expansion process and the radial displacement is only the function of the radial position, r. Note that this is a widely adopted assumption in the existing analytical solutions of the cavity expansion problem involving biaxial in-situ stresses or shear stress (Zhou et al., 2014, Zhou et al., 2016, Zhuang and Yu, 2018). If the soil particles are allowed to undergo

Conclusions

This paper has derived a semi-analytical solution for undrained cylindrical cavity expansion in modified Cam-clay soils under biaxial in-situ stresses, based on the small strain assumption in the elastic region and the large strain assumption in the plastic region. Unlike conventional cylindrical cavity expansions under the uniform radial stress, in which no shear stress exists, the biaxial in-situ stresses will induce the shear stress in the soil around the cylindrical cavity during expansion

CRediT authorship contribution statement

Weibing Gong: Software, Validation, Methodology, Conceptualization, Data curation. Changyi Yang: Writing - original draft, Methodology. Jingpei Li: Supervision. Lichao Xu: Reviewing and Editing, Revising.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was financially supported by the National Natural Science Foundation of China (Grant No. 41772290).

References (29)

  • H.S. Yu et al.

    Plasticity solutions for soil behaviour around contracting cavities and tunnels

    Int. J. Numer. Anal. Meth. Geomech.

    (1999)
  • P.Q. Mo et al.

    Undrained cavity-contraction analysis for prediction of soil behavior around tunnels

    Int. J. Geomech.

    (2017)
  • Lamé G. Lecons sur la Theorie Mathematique de l'Elasticite des Corps Solides. Bachelier, Paris; 1852. (in...
  • R. Hill

    The mathematical theory of plasticity

    (1950)
  • Cited by (0)

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