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Experimental and damage constitutive study of the stress-induced post-peak deformation and brittle–ductile behaviours of prismatic deeply buried marble

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Abstract

Unlike shallow rock, deeply buried rock presents significant brittle–ductile behaviour during the post-peak deformation process. This paper concerns the mechanical characteristics of stress-induced brittle–ductile behaviour of prismatic deeply buried marble. For this purpose, a series of triaxial compression tests were carried out on prismatic marble under different confining pressures, and then the strength, deformation and brittle–ductile characteristics of marble were analysed. A method for the two key parameters (β and Bd) affecting pre-peak hardening rate and post-peak damage rate to control the brittle–ductile characteristics was proposed, and the exponential functional relationships between the key parameters (β and Bd) and confining pressure were established. Thus, an elastoplastic damage constitutive relation was developed within the framework of thermodynamics, and it was implemented numerically based on a semi-implicit return mapping algorithm. In addition, sensitivity analysis of the key parameters β and Bd on the influence of pre- and post-peak deformation and brittle–ductile behaviour of rock were also investigated. Furthermore, the numerical simulation results predicted by the proposed model were also compared with the experimental results. It is found that the proposed model can capture the stress-induced brittle–ductile behaviour of deeply buried rock.

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Data availability

The authors confirm they have included a data availability statement in their main manuscript file. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Abbreviations

A 1,  B 1 and C 1 :

Model parameters associated with hardening function

A 2,  B 2 and C 2 :

Model parameters associated with damage evolution function

B d and β :

Damage rate parameter and hardening rate parameter, respectively

c, φ  and ϕ :

Cohesion, friction angle and dilatancy angle, respectively

C 0, C(ω) and C ep :

Elastic stiffness matrix of undamaged material, elastic stiffness matrix of damaged material and tangent stiffness matrix, respectively

d ε,  d ε e and d ε p :

Total strain increment tensor, elastic strain increment tensor and plastic strain increment tensor, respectively

dλ p :

Plastic increment parameter

d σ :

Stress increment tensor

e p :

Plastic deviatoric strain tensor

E 0 and \(\overline{E }\) :

Young’s modulus of undamaged and damaged material, respectively

f d, f p and g :

Damage evolution function, yield function and plastic potential function, respectively

h p :

Hardening damage function

\({h}_{\mathrm{p}}^{0}\) and \({h}_{\mathrm{p}}^{\mathrm{m}}\) :

Initial value of hardening damage function and final value of hardening damage function, respectively

I :

The fourth-order unit tensor

I BD :

Brittle–ductile index

J 2 :

The second invariant of the deviatoric stress

k(ω):

Bulk modulus of damaged rocks

M :

Post-peak modulus

p and q :

Mean stress and deviatoric stress, respectively

s :

Deviatoric stress tensor

\({Y}_{\mathrm{d}}^{\mathrm{p}}\) :

Damage force

α and κ :

Strength parameters of Drucker–Prager yield function

α g :

Strength parameters of plastic potential function

γ p :

Equivalent plastic shear strain

δ :

The second-order unit tensor

ε 1, ε 2 and ε 3 :

Maximum, intermediate and minimum principal strains, respectively

ε, ε e and ε p :

Total strain tensor, elastic strain tensor and plastic strain tensor, respectively

ε y, ε p and ε r :

Corresponding strain of initial yield strength, peak strength and residual strength, respectively

μ(ω):

Shear modulus of damaged rocks

σ 1,  σ 2 and σ 3 :

Maximum, intermediate and minimum principal stresses, respectively

σ y,  σ p and σ r :

Initial yield strength, peak strength and residual strength, respectively

ψ :

Helmholtz free energy function

ψ e and ψ p :

Elastic part and plastic part of Helmholtz free energy, respectively

ω and ω c :

Damage variable and the critical value of damage variable, respectively

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Acknowledgements

The authors sincerely thank Professor Xia-Ting Feng and his teams for their help in the experiment. And the authors sincerely thank reviewers for their help and attention.

Funding

The authors acknowledge the financial support received from the National Natural Science Foundation of China (Grant No. 52109119), the Guangxi Natural Science Foundation (Grant No. 2021GXNSFBA075030), the Guangxi Science and Technology Project (Grant No. Guike AD20325002), the Systematic Project of Guangxi Key Laboratory of Disaster Prevention and Engineering Safety (No. 2020ZDK007) and the Open Research Fund of Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University (Grant No. B210204004).

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Authors and Affiliations

  1. Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education, Guangxi Key Laboratory of Disaster Prevention and Engineering Safety, College of Civil Engineering and Architecture, Guangxi University, Nanning, Guangxi, 530004, China

    Zhi Zheng, Hao Su & Guoxiong Mei

  2. Key Laboratory of Ministry of Education On Safe Mining of Deep Metal Mines, Northeastern University, Shenyang, 110819, China

    Zhi Zheng

  3. Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing, 210098, China

    Zhi Zheng, Yajun Cao & Wei Wang

  4. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071, China

    Guangliang Feng & Quan Jiang

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  1. Zhi Zheng
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  2. Hao Su
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  3. Guoxiong Mei
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  4. Yajun Cao
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  5. Wei Wang
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Correspondence to Zhi Zheng.

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Highlights

• A series of triaxial compression experiments were performed on prismatic deeply buried marble under different confining pressures to investigate the influence of confining pressure on post-peak deformation and brittle–ductile characteristics.

• An elastoplastic damage mechanics model based on the framework of thermodynamics for rock hardening and softening was established, which can accurately simulate the marble experiment results under different confining pressures.

• A method for the two key parameters (β and Bd) affecting pre-peak hardening rate and post-peak damage rate to control the brittle–ductile characteristics of rock was proposed. The exponential functional relationships between the key parameters (β and Bd) and confining pressure were established.

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Zheng, Z., Su, H., Mei, G. et al. Experimental and damage constitutive study of the stress-induced post-peak deformation and brittle–ductile behaviours of prismatic deeply buried marble. Bull Eng Geol Environ 81, 427 (2022). https://doi.org/10.1007/s10064-022-02909-3

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