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Study on the damage-softening constitutive model of rock and experimental verification

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Abstract

A damage-softening model is presented to describe the stress–strain curve of rock. By comparing the Hoek–Brown (H–B) and Mohr–Coulomb (M–C) yield criterion, the equivalent M–C yield criterion is selected as the strength criterion in this model. To better characterize the rock damage and failure processes with considering the relationship between damage and deformation, the concept of yield stress ratio is introduced to describe the yield-strengthening deformation before rock peak stress. Damage events are described by two cumulative damage evolution laws. The evolution equations of tensile and shear damage are presented based on the equivalent plastic strains, and the maximum value between tensile and shear damage represents the total damage for rock. Considering that rock cannot bear tensile load after tensile failure but still has a certain shear strength, its tensile and shear strengths are small after shear failure. The elastic modulus is affected by tensile damage, whereas the angle of internal friction, the cohesion, and dilation angles are influenced by shear damage. The proposed damage-softening model describes the strain softening, brittle stress drop, and residual strength of rock after peak stress, and finally the model is implemented in FLAC3D. Comparing the test and the numerical calculation results, the damage-softening model better describes the total stress–strain curve of rock.

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Abbreviations

D :

Damage variable

D t :

Tensile damage

D s :

Shear damage

E′:

Elastic modulus at yield strengthening stage

E, E s :

Young’s modulus

E 0 :

Initial value of elastic modulus

K, G :

Bulk and shear modulus

k 1 :

Ratio of yield stress to peak strength

k 2 :

Ratio of elastic modulus at yield strengthening stage to Young’s modulus

H–B:

Hoek–Brown yield criterion

M–C:

Mohr–Coulomb yield criterion

m i :

Dimensionless empirical constant

f s :

Shear yield plane

g t :

Tensile yield plane

h :

Boundary plane between shear and tensile yield plane

T:

Tensile

T–S:

Tensile–shear

C–S:

Compressive–shear

c :

Cohesion

c′:

Equivalent cohesion

c p :

Cohesion at peak strength point

c r :

Cohesion at residual strength stage

φ :

Angle of internal friction

φ′:

Equivalent internal friction angle

φ p :

Angle of internal friction at peak strength point

φ r :

Angle of internal friction at residual strength stage

ψ :

Dilation angle

σ 1, σ 3 :

Major and minor principal stresses

σ c :

Uniaxial compressive strength of rock

σ 3max :

The maximum confining pressure

σ t :

Tensile strength

σ yield :

Yield stress

σ peak :

Peak stress

σ t :

Tensile strength

σ t0 :

Initial value of tensile strength

ε ps :

Equivalent shear plastic strain

ε pt :

Equivalent tensile plastic strain

ε psL :

Critical equivalent plastic strain of rock entering the residual deformation stage

ω :

Shear strength parameter [i.e., fraction angle (φ), cohesion (c), and dilation angle (ψ)]

ω p :

Initial values of shear strength parameters

ω r :

Residual values of shear strength parameters

σ e I , σ e N :

Stress matrix of the unit before and after updating

∆ε I :

Strain matrix of the unit

D :

Stiffness matrix of the unit

\(\Delta\varvec{\varepsilon}_{ij}^{\text{ps}}\) :

Increment of shear plastic strain

\(\Delta\varvec{\varepsilon}_{ij}^{\text{pt}}\) :

Increment of tensile plastic strain

ΔK s :

Increment of equivalent shear plastic strain

ΔK t :

Increment of equivalent tensile plastic strain

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grants 51734009 & 51179189), the Fifth “333” Project of Jiangsu Province (2016) and the China Postdoctoral Science Foundation (Grant 2018M642360). The authors would like to express their sincere gratitude to the editor and two anonymous reviewers for their valuable comments which have greatly improved this paper.

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Authors and Affiliations

  1. State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, 221116, China

    Sheng-Qi Yang & Bo Hu

  2. China Construction Third Engineering Design Bureau Co, Ltd, Wuhan, 430064, China

    Peng Xu

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  1. Sheng-Qi Yang
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Correspondence to Sheng-Qi Yang.

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Yang, SQ., Hu, B. & Xu, P. Study on the damage-softening constitutive model of rock and experimental verification. Acta Mech. Sin. 35, 786–798 (2019). https://doi.org/10.1007/s10409-018-00833-y

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