A phenomenological modelling of rocks based on the influence of damage initiation

  • Heng Zhao
  • Shuai Zhou
  • Ling ZhangEmail author
Original Article


It is recognized that the stress-induced damage impacts the progressive failure behavior of rocks. A phenomenological model for the compressive failure of rocks is thus presented in this study. The model addresses the progressive growth of damage that leads to the strength weakening on a macroscopic scale. Considering dramatic difference between uniaxial compression and tension strengths for rocks, the admitted Mises–Schleiche Drucker–Prager strength criterion is adopted to characterize the damage initiation. On this basis, a two-parameter Weibull-type probability function is used to define the strength distribution of representative volume elements, followed by the use of damage variable for addressing the accumulated probability of failure. The proposed damage variable essentially characterizes both the critical stress level of damage initiation and progressive damage evolution law. Detailed comparisons have been carried out between the predictions and experimental observations, and issues related to the damage evolution are particularly addressed. In addition, the results further validate the proposed model considering damage initiation.


Rocks Damage initiation MSDP criterion Loading function Strain-softening 

List of symbols

\({\sigma _{\text{1}}},{\sigma _{\text{2}}},{\sigma _{\text{3}}}\)

Major, intermediate, and minor apparent principal stresses, respectively

\(\sigma _{1}^{*},\sigma _{2}^{*},\sigma _{3}^{*}\)

Major, intermediate, and minor net principal stresses, respectively

\({\sigma _{{\text{1}},{\text{p}}}}\)

Maximum principal stress at the peak point

\({\varepsilon _{\text{1}}},{\varepsilon _{\text{2}}},{\varepsilon _{\text{3}}}\)

Major, intermediate, and minor apparent principal strains, respectively

\({\varepsilon _{{\text{1}},{\text{p}}}}\)

Maximum principal axial strain at the peak point


An empirical constant

\({\sigma _{\text{c}}}\)

Uniaxial compression strength

\({\sigma _{{\text{cd}}}}\)

An empirical constant

\({\sigma _{\text{t}}}\)

Uniaxial tension strength


Uniaxial compression strength

\(\left| {{T_0}} \right|\)

Uniaxial tension strength


Damage variable


Elasticity modulus of rocks


A fictitious loading function


Poisson’s ratio of rocks

\(\varphi ,c\)

Cohesion and angles of internal friction of rocks, respectively


Statistical parameters for Weibull distribution


Transition condition for the MSDP criterion


First invariant of the stress tensor


Second invariant of the deviatoric stress tensor


Third invariant of the deviatoric stress tensor


Material constants for Drucker–Prager criterion


Fitting constant


Fitting constant


Material constant



Particular values at peak point of stress–strain curves



Net stress



This work was fully supported by the National Natural Science Foundation of China under contract nos. 51608540 and 51678231, and the Basal Research Fund Support by Hunan University.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Geotechnical EngineeringHunan UniversityChangshaPeople’s Republic of China

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