# A Unified Nonlinear Elastic Model for Rock Material
## 3 A New Unified Nonlinear Elastic Model  
This section presents a new unified nonlinear elastic model for rock material based on [PERSON]'s law. [PERSON]'s law can be used to describe the linear relationship between the strain and stress of linear elastic material under compression. When the compression direction is positive, [PERSON]'s law is expressed as  
\[\sigma=E\varepsilon \tag{9}\]  
\[d\sigma=E\varepsilon \tag{10}\]  
\[\varepsilon=-\frac{L-L_{0}}{L_{0}} \tag{11}\]  
\[d\varepsilon=-\frac{dL}{L_{0}} \tag{12}\]  
where \(E\), \(\varepsilon\) and \(L\) are the elastic modulus, strain and length of the material under stress \(\sigma\), respectively, and \(L_{0}\) is the initial length of the material. The elastic modulus of the linear elastic material remains unchanged during deformation, as shown in Figure 3a. The \(E_{ni}\) parameter is introduced, which represents the initial elastic modulus of the material, and we have the following:  
\[E=E_{ni} \tag{13}\]  
Figure 2: Linear and nonlinear elastic deformation of rock adapted from Liu. Adapted from [30]. Substituting Equations (12) and (13) into (10) yields the following:  
\[d\sigma=-E\frac{dL}{L_{0}}=-E_{ni}\frac{dL}{L_{0}} \tag{14}\]  
If the elastic modulus of the material is not a fixed value, but is related to the material length during deformation, its stress-strain curve is nonlinear, as shown in Figure 3b. The elastic modulus during deformation follows the following formula:  
\[E=E_{ni}\bigg{(}\frac{L_{0}}{L}\bigg{)}^{m}\quad(m\geq 0) \tag{15}\]  
where \(m\) is a constant. The nonlinear elastic curve has obvious differences compared to the linear elastic curve. In the nonlinear elastic curve, the closer to the maximum strain, the faster the curve rises, and more stress is required for further deformation of the material. As shown by Equation (15), the elastic modulus of the material decreases with increasing length during the deformation process. Substituting Equations (12) and (15) into (10) yields the following:  
\[d\sigma=-E_{ni}L_{0}^{m-1}\frac{dL}{L^{m}} \tag{16}\]  
The length \(L\) of the material under stress \(\sigma\) can be obtained by integrating Equation (16). The \(m\) value is a special case when it is equal to 1, and the derivation method is different from that when it is not equal to 1. When \(m\) is equal to 1  
\[L=L_{0}\exp\bigg{(}-\frac{\sigma}{E_{n}}\bigg{)} \tag{17}\]  
When \(m\) is not equal to 1  
\[d\sigma=E\varepsilon \tag{10}\]0  
After substituting Equations (17) and (18) into (11) and simplifying the formula, we have the following:  
When \(m\) is equal to 1  
\[d\sigma=E\varepsilon \tag{10}\]1  
\[d\sigma=E\varepsilon \tag{10}\]2 Linear and nonlinear elastic curves (**a**-Linear elastic; **b**-Nonlinear elastic). When \(m\) is not equal to 1  
\[d\sigma=E\varepsilon \tag{10}\]3  
The new unified nonlinear elastic model is composed of Equations (19) and (20). Special forms of the formula exist when \(m\) is equal to 0, 1 and 2. When \(m\) is equal to 0, the unified nonlinear elastic model can be simplified into a linear elastic model:  
\[d\sigma=E\varepsilon \tag{10}\]4  
When \(m\) is equal to 1, the unified nonlinear elastic model is Equation (19), it is the same as Equation (8) of the two-part [PERSON]'s model, and it is the same as Equation (5) of the simple exponential model under the following conditions:  
\[\left\{ \begin{array}{c}E_{ni}=K_{ni}d_{\max}\\ \varepsilon=d_{n}/d_{\max}\end{array} \right. \tag{22}\]  
When \(m\) is equal to 2, Equation (20) can be simplified as  
\[\sigma=E_{ni}\frac{1}{1/\varepsilon-1} \tag{23}\]  
It is the same as Equation (3) of the BB model when Equation (22) is established. Note that the stress displacement coordinate system is often used for the nonlinear deformation of joints, and if it is changed to the stress-strain coordinate system that is the same as the rock, Equation (22) will be established.