Nonlinear solution for cylindrical cavity expansion in strain-softening geomaterials

Chao Li; Yingke Liu; Chuanfeng Fang; Qi Ge

doi:10.1515/jncds-2024-0083

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Nonlinear solution for cylindrical cavity expansion in strain-softening geomaterials

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Published/Copyright: January 23, 2026

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From the journal Journal of Nonlinear, Complex and Data Science Volume 27 Issue 1

Abstract

This study presents a nonlinear solution for cavity expansion on the basis of unified strength theory, and considers the influence of strain-softening and drained condition. By assuming that the small-strain in elastic region, and large-strain in softening and plastic flow region, instead of, the assumption of elastic region and plastic region were usually used in calculation. The initial stress may be anisotropic due to the influence of the initial consolidation of geomaterials. Besides, a drained cylindrical cavity expansion analysis definitely will be more suitable for the interpretation of in situ soil testing in geomaterials with very high permeability. In the end, some data are conducted to verify the suitability of this study. The research results provide a theoretical basis for the analysis of cavity expansion in geomaterials with very high permeability, and have a certain reference value for similar engineering designs.

Keywords: nonlinear solution; cylindrical cavity expansion; unified strength theory; strain-softening

Corresponding author: Yingke Liu, School of Safety Engineering, China University of Mining and Technology, Xuzhou, 221116, China, E-mail: ykliu@cumt.edu.cn

Funding source: the Science and Technology Planning Project of Jiangsu Province (BK20231079)

Award Identifier / Grant number: BK20231079

Funding source: the Foundation of State Key Laboratory of Mountain Bridge and Tunnel Engineering

Award Identifier / Grant number: SKLBT-2213

Funding source: the National Natural Science Foundation of China

Award Identifier / Grant number: U24B2044

Funding source: the Doctor of entrepreneurship and innovation in Jiangsu Province (JSSCBS20221497)

Award Identifier / Grant number: JSSCBS20221497

Acknowledgements

This work was supported by the National Natural Science Foundation of China (U24B2044), National Science and Technology Major Project(2025ZD1011001-4), the Doctor of Entrepreneurship and Innovation in Jiangsu Province (JSSCBS20221497), the Science and Technology Planning Project of Jiangsu Province (BK20231079), the Open Fund of the State Key Laboratory of Coalburst Theory and Prevention Technology National Mine Safety Administration (Preparatory) (Grant No. KFJJ-2025-0020), State Key Laboratory Cultivation Base for Gas Geology and Gas Control (Henan Polytechnic University) (WS2025B07).

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: This manuscript is an original work conducted by the undersigned authors and has not been published or submitted elsewhere for publication. All listed authors have made significant contributions to the research.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: No conflict of interest exits in the submission of this manuscript, and manuscript entitled “Nonlinear solution for cylindrical cavity expansion in strain-softening geomaterials” is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part.
Research funding: This work was supported by the National Natural Science Foundation of China (U24B2044), National Science and Technology Major Project(2025ZD1011001-4), the Doctor of Entrepreneurship and Innovation in Jiangsu Province (JSSCBS20221497), the Science and Technology Planning Project of Jiangsu Province (BK20231079), the Open Fund of the State Key Laboratory of Coalburst Theory and Prevention Technology National Mine Safety Administration (Preparatory) (Grant No. KFJJ-2025-0020), State Key Laboratory Cultivation Base for Gas Geology and Gas Control (Henan Polytechnic University) (WS2025B07).
Data availability: Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Appendix A. Derivation of the radius ratio (r _b/a)

By assuming that the continuous deformation around the cavity is geometrically self-similar according to Yu and Carter [40]. When the current radius has a small increment dr _b, and the corresponding displacement is du, du = dr = Vdr _b, V is the relative velocity [40], u is a function of the current radius r and the radius r _b, namely, u = u(r, r _b), r _b and r are two independent variables. According to Yu and Carter [40]; the differential equation can be expressed, in order to avoid excessive repetition, there is no detailed description of this differential equation. The detailed derivation of this differential equation can be referred to the paper of Yu and Carter [40];

(A.1) ∂ V ∂ r + ζ V β r = − ξ β ∂ σ r ∂ r b + V ∂ σ r ∂ r

where

(A.2) ξ = 1 M β − ν 1 − ν + 1 M R ( 1 − 2 ν ) + 2 ν − β ν 1 − ν M = E 1 − v 2 β = 1 + sin ⁡ ψ / 1 − sin ⁡ ψ

E is the Young’s modulus of the geomaterial.

Therefore,

(A.3) ∂ V ∂ r + P ( r ) V = Q ( r )

where

(A.4) P ( r ) = 1 β r − ξ q ( R − 1 ) R β r r b r R − 1 R

(A.5) Q ( r ) = − s r b r b r R − 1 R

(A.6) q = σ r b + 1 R − 1 σ 0

(A.7) s = ξ q R − 1 R β

According to Eqs. (5) and (7),

(A.8) V r = r b = 2 δ δ = σ r b − σ h 0 2 G

The Eq. (A.1) can also be obtained,

(A.9) V = exp − ξ q β r b r R − 1 R ∑ n = 0 ∞ H n r b r ( R − 1 ) ( 1 + n ) R − 1 + 2 δ ⁡ exp ξ q β − ∑ n = 0 ∞ H n r b r 1 β

where

(A.10) H n = 1 n ! ξ q β n R β s R + R β − β ( R − 1 ) ( 1 + n )

For the cavity wall r = a, V = da/dr _b, therefore,

(A.11) d a d r b = exp − ξ q β r b r R − 1 R ∑ n = 0 ∞ H n r b r ( R − 1 ) ( 1 + n ) R − 1 + 2 δ ⁡ exp ξ q β − ∑ n = 0 ∞ H n r b r 1 β

The geometrically similar can be written [40],

(A.12) d a d r b = a r b

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Received: 2024-07-10

Accepted: 2026-01-05

Published Online: 2026-01-23

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Keywords for this article

nonlinear solution; cylindrical cavity expansion; unified strength theory; strain-softening

Nonlinear solution for cylindrical cavity expansion in strain-softening geomaterials

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Appendix A. Derivation of the radius ratio (r b /a)

References

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Appendix A. Derivation of the radius ratio (r _b/a)