# A Strain Hardening and Softening Constitutive Model for Hard Brittle Rocks
## 4 Establishment and Verification of the Constitutive Model
### Verification  
To verify the rationality of the strain hardening and softening constitutive model for hard brittle rocks under different confining pressures, triaxial compression tests on granite and diabase are simulated using FLAC\({}^{3\text{D}}\) software. Because the numerical model carries a uniform load, the number of elements in the numerical model has little influence on the simulation results [44], and only a single-grid model (1 m\({}^{3}\)) is established, as shown in Figure 9. The boundary conditions are as follows:  
Figure 8: Flow chart of the secondary development of the constitutive model.  
(i) The displacement at the bottom boundary is restricted when the model runs;  
(ii) A stress \(\sigma_{x}=\sigma_{y}\) is applied to the surface of the model to simulate the confining pressure;  
(iii) A constant displacement velocity \(u_{z}\) (\(10^{-8}\) m/step) is applied to the upper surface of the model to simulate axial displacement loading.  
Numerical triaxial compression tests are performed on Beishan granite [26] under six confining pressures (\(\sigma_{x}\) = 0.2 MPa, 1.0 MPa, 3.5 MPa, 10 MPa, 20 MPa and 40 MPa) and on diabase [37] under four confining pressures (\(\sigma_{x}\) = 5 MPa, 10 MPa, 20 MPa and 40 MPa). The complete stress-strain curves of the two hard brittle rocks under different confining pressures are shown in Figure 10.  
Figure 10: Comparison of the test curves and model curves: (**a**) Beishan granite [26]; (**b**) Danjiangkou diabase [37].  
Figure 9: Numerical model of the triaxial compression test in FLAC\({}^{\text{3D}}\).  
Figure 10 illustrates that the simulated results agree well with the test data. The initial yield stress, peak stress, peak strain and residual stress values at different confining pressures, especially at 0.2 MPa, 1.0 MPa, 20 MPa and 40 MPa for granite and 5 MPa, 10 MPa and 40 MPa for diabase are satisfactorily captured by the strain hardening and softening model, which demonstrates that the proposed constitutive model can describe the nonlinear behaviour of rock. The constitutive model can adequately capture the transition from brittle failure to plastic failure under high confining pressures and well describe the confining pressure-dependent characteristics of hard brittle rocks, which verifies the rationality and correctness of the strain hardening and softening constitutive model for hard brittle rocks under different confining pressures. It should be noted that the simulated peak stress for granite at 10 MPa in Figure 10a differs significantly from the experimental peak stress. This occurs due to the deviation in fitting the peak stress of each confining pressure with the Mohr-Coulomb linear yield criterion. In addition, the discrete experimental data may result in deviations when fitting the width parameters \(\xi_{c}\) and \(\xi_{\varphi}\) with the discrete data of the confining pressures, so the post-peak softening behaviours at a few confining pressure values (e.g., granite at 3.5 MPa, diabase at 20 MPa) are inconsistent with the true experimental curve.