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An elasto-plastic and viscoplastic damage constitutive model for dilatancy and fracturing behavior of soft rock squeezing deformation

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Abstract

Soft rock squeezing deformation mainly consists of pre-peak damage-dilatancy and post-peak fracture-bulking at the excavation unloading instant, and creep-dilatancy caused by time-dependent damage and fracturing. Based on the classic elasto-plastic and Perzyna over-stress viscoplastic theories, as well as triaxial unloading confining pressure test and triaxial unloading creep test results, an elasto-plastic and viscoplastic damage constitutive model is established for the short- and long-term dilatancy and fracturing behavior of soft rock squeezing deformation. Firstly, the criteria for each deformation and failure stage are expressed as a linear function of confining pressure. Secondly, the total damage evolution equation considering time-dependent damage is proposed, including the initial damage produced at the excavation instant, in which the damage variable increases exponentially with the lateral strain, and creep damage. Thirdly, a transient five-stages elasto-plastic constitutive equation for the short-term deformation after excavation that comprised of elasticity, pre-peak damage-dilatancy, post-peak brittle-drop, linear strain-softening, and residual perfectly-plastic regimes is developed based on incremental elasto-plastic theory and the nonassociated flow rule. Fourthly, regarding the time-dependent properties of soft rock, based on the Perzyna viscoplastic over-stress theory, a viscoplastic damage model is set up to capture creep damage and dilatancy behavior. Viscoplastic strain is produced when the stress exceeds the initial static yield surface fs; the distance between the static yield surface fs and the dynamic yield surface fd determines the viscoplastic strain rate. Finally, the established constitutive model is numerically implemented and field applied to the -848 m belt conveyer haulage roadway of Huainan Panyidong Coal Mine. Laboratory test results and in-situ monitoring results validate the rationality of the established constitutive model. The presented model takes both the transient and time-dependent damage and fracturing into consideration.

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Abbreviations

σ 1 :

Axial stress

σ 3 :

Confining pressure

σ 03 :

Initial confining pressure

σ 1cd :

Damage-dilatancy stress

σ 1p :

Limited bearing capacity

σ 1l :

Linear strain-softening stress

σ t :

Tensile strength

σ ij :

Effective stress

η :

Internal softening variable

c :

Cohesion

φ :

Internal friction angle

c o :

Initial cohesion

φ o :

Initial internal friction angle

c r :

Residual cohesion

φ r :

Residual internal friction angle

η * cr :

Critical softening parameter corresponding to residual cohesion

η * φr :

Critical softening parameter corresponding to residual internal friction angle

a :

Fitting parameter of the yield criterion

l :

Fitting parameter of the yield criterion

ε e1 :

Axial elastic strain

ε e3 :

Lateral elastic strain

ε p1 :

Axial plastic strain

ε p3 :

Lateral plastic strain

ε p v :

Volumetric plastic strain

ε c3 :

Lateral crack strain

ε vp ij :

Viscoplastic strain

ε vp :

Equivalent viscoplastic strain

\({J_{2,{\varepsilon _{vp}}}}\) :

Second invariant of viscoplastic deviatoric strain

μ :

Poisson’s ratio

μ e :

Poisson’s ratio in the elastic regime

μ c :

Additional Poisson’s ratio

E :

Elastic modulus

E m :

Deformation modulus

D :

Damage tensor

ω :

Damage variable

D o(t=o):

Initial damage

D(σ/σ f):

Damage induced by stress changes

D c(t/t F):

Creep damage

t F :

Creep failure time

[C e]:

Elastic flexibility matrix

[D e]:

Elastic stiffness matrix

[D p]:

Plastic stiffness matrix

[D ep]:

Elasto-plastic stiffness matrix

dλ :

Plastic flow factor

λ t :

Tensile yield flow factor

A :

Hardening function

g i :

Plastic potential function in the strain space

G i :

Plastic potential function in the stress space

g vp :

Viscoplastic potential function

β :

Dilatancy parameter

ψ :

Dilatancy angle

B :

Softening modulus

f i :

Critical criterion in the i-th deformation or failure phase. When i=d, p, l, r, fi represents the yield criterion of the damage-dilatancy, post-peak brittle-drop, linear strain-softening, and residual strength stage, respectively

f vp :

Viscoplastic yield surface

f s :

Static yield surface

fd’:

Dynamic yield surface

γ :

Viscosity factor

F :

Function of the viscoplastic yield criterion

Φ(F):

Function of over-stress F

〈〉:

Macauley operator

θ :

Lode angle

γ o, m, n, Q :

Viscoplastic parameters

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Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant No. 52074258, Grant No. 41941018, Grant No. 51974289, and Grant No. 51874232), the Natural Science Basic Research Program of Shaanxi Province (Shaanxi Coal and Chemical Industry Group Co., Ltd. Joint Fund Project, Grant No. 2021JLM-06), and the open project of State Key Laboratory of Shield Machine and Boring Technology (Grant No. E01Z440101). Their support is gratefully acknowledged.

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Authors and Affiliations

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

    Xing Huang & Bin Liu

  2. Key Laboratory of Geotechnical and Structural Engineering Safety of Hubei Province, School of Civil Engineering, Wuhan University, Wuhan, 430072, China

    Quan-sheng Liu, Yin Bo & Quan-tai Zhang

  3. College of Energy Engineering, Xi’an University of Science and Technology, Xi’an, 710054, China

    Zi-wei Ding

Authors
  1. Xing Huang
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  2. Quan-sheng Liu
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  3. Yin Bo
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  4. Bin Liu
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  5. Zi-wei Ding
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  6. Quan-tai Zhang
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Correspondence to Bin Liu.

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Huang, X., Liu, Qs., Bo, Y. et al. An elasto-plastic and viscoplastic damage constitutive model for dilatancy and fracturing behavior of soft rock squeezing deformation. J. Mt. Sci. 19, 826–848 (2022). https://doi.org/10.1007/s11629-020-6530-4

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  • DOI: https://doi.org/10.1007/s11629-020-6530-4

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