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When Large Deformation Analysis Meets Large Deformation Phenomenon: Comparative Study and Improvement

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Abstract

Large deformation phenomena in rock engineering are commonly key bottlenecks impeding engineering progress. Correspondingly, large deformation analysis in rock mechanics has a widespread impact on mechanism understanding, prevention, and control guidance. Various large deformation schemes broadly categorized as hypoelastic- and hyperelastic-based models exist in the literature and software. Without an understanding of the capabilities and demerits of these schemes, the diversity in choices inadvertently leads to pitfalls, placing engineering endeavors at a disadvantage. In this work, we review and compare the most prevalent schemes (including PK2-Green, Jaumann, Green–Naghdi, Truesdell, and hyperelastic-based schemes) from perspective of rock mechanics, with an emphasis on their application in rock engineering, and further develop a hyperelastic-based large deformation scheme (marked as Cauchy-Ln scheme) based on Cauchy stress and Hencky logarithm strain. The proposed model realizes the separation of material nonlinearity and geometry nonlinearity. Several typical large deformation phenomena in rock engineering are studied. The relationships between large deformation phenomena and large deformation analyses are clarified under different circumstances. Especially under large strain conditions, the PK2-Green scheme is enfeebled, and the hypoelastic-based scheme should be used with caution. For rock material with an apparent pressure-sensitive effect, the proposed Cauchy-Ln scheme is superior.

Highlights

  • The connections between large deformation phenomenon and large deformation analysis are delineated.

  • Prevalent large deformation schemes are compared from a rock mechanics perspective with an emphasis on application in rock engineering.

  • A newly hyperelastic-based large deformation numerical model, enhancing compatibility with rocks, is developed.

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Data availability

All data and codes are available from the corresponding author upon reasonable request.

Abbreviations

\({\mathbb{a}}\) :

Spatial tangent modulus

\({\varvec{b}}^{{\text{e}}}\) :

Elastic left Cauchy–Green deformation tensor

\({\varvec{b}}\) :

Body force

\( {\mathbb{D}}^{{\text{e}}}\) :

Linear isotropic elasticity tensor

\( {\mathbb{D}}^{{{\text{ep}}}}\) :

Elastoplastic tangent modulus

\({\varvec{d}}\) :

Rate of deformation tensor

\( {\varvec{d}}^{{\text{e}}}\) :

Rate of elastic deformation tensor

\( {\varvec{d}}^{{\text{p}}}\) :

Rate of plastic deformation tensor

\(E\) :

Young’s modulus

\({\varvec{E}}\) :

Green strain tensor

\({\varvec{e}}\) :

Eulerian Hencky strain tensor

\({\varvec{F}}\) :

Deformation gradient tensor

\({\varvec{I}}\) :

Second-order identity tensor

\(J\) :

Jacobian of the deformation gradient

\({\varvec{l}}\) :

Velocity gradient tensor

\({\varvec{P}}\) :

First Piola–Kirchhoff stress tensor

\(q\) :

Hardening force

\({\varvec{R}}\) :

Rotation tensor

\({\varvec{S}}\) :

Second Piola–Kirchhoff stress tensor

\({\varvec{t}}\) :

Surface traction vector

\({\varvec{U}}\) :

Right stretch tensor

\({\varvec{u}}\) :

Displacement vector

\({\varvec{V}}\) :

Left stretch tensor

\({\varvec{w}}\) :

Spin tensor

x :

Position vector in current configuration

X :

Position vector in initial configuration

\(\gamma\) :

Plastic multiplier

\({\varvec{\varepsilon}}\) :

Infinitesimal strain tensor

\( {\varvec{\varepsilon}}^{{\text{e}}}\) :

Elastic strain tensor

\( {\varvec{\varepsilon}}^{{\text{p}}}\) :

Plastic strain tensor

\( \overline{\varepsilon }^{{\text{p}}}\) :

Effective plastic strain

\(\theta\) :

Frictional angle

\(\vartheta\) :

Dilatancy angle

\(\nu\) :

Poisson’s ratio

\({\varvec{\sigma}}\) :

Cauchy stress tensor

\({\varvec{\sigma}}^{\nabla }\) :

The specific stress rate of Cauchy stress

\({\varvec{\tau}}\) :

Kirchhoff stress tensor

\(\psi\) :

Strain density per unit reference volume

\({\Phi }\) :

Yield function

\({\Psi }\) :

Flow potential

\({\varvec{\varOmega}}\) :

Angular velocity tensor

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Acknowledgements

The authors would like to express their gratitude to the two anonymous reviewers and editors for their meticulous, professional, and insightful comments.

Funding

This work is supported in part by GHfund A (2023020122702), the China Postdoctoral Science Foundation (2023TQ0025), the Chinese Universities Scientific Fund (Z1090124057), and the National Natural Science Foundation of China (42050201, 41941018).

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Authors and Affiliations

  1. College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling, 712100, China

    Xiaofeng Cheng

  2. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China

    Xiaofeng Cheng & Chun’an Tang

  3. School of Civil and Resources Engineering, University of Science and Technology Beijing, Beijing, 100083, China

    Xianhui Feng

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  1. Xiaofeng Cheng
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  2. Chun’an Tang
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Correspondence to Xiaofeng Cheng.

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Cheng, X., Tang, C. & Feng, X. When Large Deformation Analysis Meets Large Deformation Phenomenon: Comparative Study and Improvement. Rock Mech Rock Eng (2024). https://doi.org/10.1007/s00603-024-03912-8

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