Acta Mechanica Sinica

, Volume 35, Issue 4, pp 786–798 | Cite as

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

  • Sheng-Qi YangEmail author
  • Bo Hu
  • Peng Xu
Research Paper


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.


Rock deformation Damage evolution Softening Numerical calculation 

List of symbols


Damage variable


Tensile damage


Shear damage


Elastic modulus at yield strengthening stage

E, Es

Young’s modulus


Initial value of elastic modulus

K, G

Bulk and shear modulus


Ratio of yield stress to peak strength


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


Hoek–Brown yield criterion


Mohr–Coulomb yield criterion


Dimensionless empirical constant


Shear yield plane


Tensile yield plane


Boundary plane between shear and tensile yield plane










Equivalent cohesion


Cohesion at peak strength point


Cohesion at residual strength stage


Angle of internal friction


Equivalent internal friction angle


Angle of internal friction at peak strength point


Angle of internal friction at residual strength stage


Dilation angle

σ1, σ3

Major and minor principal stresses


Uniaxial compressive strength of rock


The maximum confining pressure


Tensile strength


Yield stress


Peak stress


Tensile strength


Initial value of tensile strength


Equivalent shear plastic strain


Equivalent tensile plastic strain


Critical equivalent plastic strain of rock entering the residual deformation stage


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


Initial values of shear strength parameters


Residual values of shear strength parameters

σIe, σNe

Stress matrix of the unit before and after updating


Strain matrix of the unit


Stiffness matrix of the unit


Increment of shear plastic strain


Increment of tensile plastic strain


Increment of equivalent shear plastic strain


Increment of equivalent tensile plastic strain



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.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil EngineeringChina University of Mining and TechnologyXuzhouChina
  2. 2.China Construction Third Engineering Design Bureau Co, LtdWuhanChina

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