In order to build the postpeak strain softening model of rock, the evolution laws of rock parameters
It is well known that the rock material is one type of heterogeneous brittle material with complex mechanical properties, which is significantly different from the metal material. In terms of postpeak strain softening characteristics, it is impossible to describe by the classical strength theory [
In order to study the postpeak strain softening mechanical properties of rock, the mechanical models described above are based on the Mohr–Coulomb criterion. However, the extent of rock damage was not studied in their studies. It is seldom to establish the postpeak strain softening model based on the Hoek–Brown criterion. Tan et al. [
Figure
Complete stressstrain curves of Hawkesbury sandstone specimens under different confining stresses [
Test curves and idealized stressstrain curves.
In the postpeak deformation stage, the development of strain softening is induced by the change of the rock strength parameters [
According to the Hoek–Brown criterion [
There are usually two methods for selecting strain softening parameters. The first one is based on the increment method:
In another method, the strain softening parameter is considered as a function of the internal variable. The major principal strain
As the rock parameters
The relationship between postpeak strength parameters and strain softening parameter, namely, the evolution of strength parameters, can be obtained by the laboratory experiment and numerical simulation. For simplicity, the mentioned relationship is often assumed as the form of a piecewise linear function [
Variation of the strength parameter
According to (
By substituting (
Before calculation by (
Cohesion
Postpeak strain softening simplified model stress states under the same unloading path.
Mohr diagram for the tested specimens at the same plastic shear strain
Uniaxial compressive strength of rock mass may be expressed as [
Uniaxial tensile strength of rock mass may be written as [
By combining with the Mohr–Coulomb criterion, cohesion
By combining (
Equations (
By combining (
Substituting (
After substituting the strength parameters
Equations (
In terms of several groups of
In order to verify the developed postpeak strain softening model, the triaxial compression test results of the Hawkesbury sandstone [
Data of the Hawkesbury sandstone in triaxial compression tests.
Confining pressure (MPa)  At the peak  At the starting point of the residual stage  








2  5.5  5.66  31.9  6.3  0.79  32.7 
5  6.5  6.46  27.4  8.0  1.74  28.7 
8  7.7  10.28  19.2  9.0  4.89  21.5 
12  8.8  12.19  14.2  10.0  7.05  15.3 
18  12.3  14.17  9.6  13  8.61  9.9 
Substituting the data in Table
Rock strength parameters
Confining pressure (MPa)  At the peak  At the starting point of the residual stage 
 






 
2  5.5  1.99  0.46  6.3  0.29  0.01  48.00 
5  6.5  1.65  0.50  8.0  0.49  0.04  13.09 
8  7.7  1.42  0.93  9.0  0.98  0.33  4.06 
12  8.8  1.09  0.99  10.0  0.70  0.38  2.87 
18  12.3  0.77  1.25  13.0  0.48  0.47  — 
It can be observed from Table
Relationships between the minimum and maximum principal stresses for equivalent Hoek–Brown and Mohr–Coulomb criteria [
In terms of the upper limit of the maximum confining pressure
As indicated in Tables
Relationship of
Relationship of
Substituting (
It can be found from Table
Within a certain range of confining pressure and increasing confining pressure at the peak,
By substituting the corresponding values of
Comparison between Hawkesbury sandstone triaxial compression test data and numerical simulation curves.
In order to verify the analytical results, the same analytical method was carried out with laboratory results of the representative rock types: Indiana limestone [
Rock strength parameters
Rock type  Confining pressure (MPa) 








Indiana limestone  2  3.4  7.67  0.47  5.5  1.17  0.03  16.94 
4  4.2  3.81  0.71  6.3  1.25  0.05  16.60  
8  5.0  2.86  0.86  8.1  2.15  0.19  4.79  
12  5.9  2.70  0.98  8.0  2.25  0.40  2.52  
Tennessee marble  3  2.4  4.1  0.84  3.2  2.54  0.12  7.00 
7  2.8  4.0  0.85  3.8  2.50  0.15  5.67  
14  3.2  3.9  0.92  4.9  2.42  0.22  4.18  
21  3.5  3.2  1.00  6.0  1.80  0.31  3.23  
Hegang granite  2  4.2  18.83  0.37  5.6  2.36  0.02  24.18 
5  5.5  14.44  0.44  8.8  2.45  0.02  19.82  
10  6.7  13.50  0.64  12.6  2.53  0.09  7.29  
15  7.5  11.99  0.98  20.0  2.95  0.15  6.63 
Comparison between Indiana limestone triaxial compression test data and numerical simulation curves.
Comparison between Tennessee marble triaxial compression test data and numerical simulation curves.
As can be seen from Figures
Comparison between Hegang granite triaxial compression test data and numerical simulation curves.
For better understanding of the postpeak softening characteristics and the degree of fragmentation and the degree of brittleness of rock, the brittleness index is proposed for
Relationships between
As the confining pressure decreases,
A reasonable postpeak strain softening model plays an important role in the analysis of rock behaviour. Based on the nonlinear Hoek–Brown criterion, a postpeak strain softening model was proposed in this paper, and the conclusions are listed below:
Based on the Hoek–Brown criterion, the maximum principal strain
By theoretical derivation, the analytical relation of parameters
The laboratory test results of the Hawkesbury sandstone show that with the increase of confining pressure, the rock hardness in peak strength becomes smaller. However, the degree of rock fragmentation in peak strength becomes lower within the range of confining pressure from 0 to
The analytical results of the developed strain softening model have been verified with laboratory results of the Hawkesbury sandstone, Indiana limestone, Tennessee marble, and Hegang granite. The results of the analytical method and laboratory tests are in good agreement, and it is concluded that the proposed model can predict the postpeak mechanical properties of the rock accurately.
Different types of rock may have different values of
The authors declare that they have no conflicts of interest.
The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (51174228 and 11772358).