# A Unified Nonlinear Elastic Model for Rock Material
## 7 Summary and Conclusions  
In this paper, the initial elastic modulus \(E_{ni}\) and modulus change rate \(m\) are introduced in order to establish a new unified nonlinear elastic model for the nonlinear elastic deformation of intact rocks. Based on the RMT experimental system, a large number of uniaxial compression tests for dolomite, granite, limestone and sandstone were performed, and their nonlinear deformation stress-strain curves were employed to fit the unified nonlinear elastic model. The main outcomes are summarized as follows:  
(1) The nonlinear elastic characteristic is obvious in the initial compression stage of the intact rock, which is mainly attributable to the porosity and/or microfractures present depending on the different rock types.  
(2) In the unified nonlinear elastic model, the physical meanings of two key parameters, namely, the \(E_{ni}\) representing the initial elastic modulus and the \(m\) representing its change rate, were clearly defined. \(E_{ni}\) determines the slope in the initial stage of deformation, and \(m\) determines the degree of nonlinearity in the middle and later stages. The unified nonlinear elastic model not only covers the existing nonlinear elastic models, such as the simple exponential model (\(m=1\)) and the BB model (\(m=2\)) of the joint and the two-part Hooke's model (\(m=1\)) of the rock, but is also able to provide some new models.  
(3) The unified nonlinear elastic model was able to describe the nonlinear deformation elastic characteristics of the intact rocks in their initial compression stage rather well. This was justified as the data fit with the experimental results.  
(4) The \(E_{ni}\) values have a good linear correlation with the uniaxial compressive strength (_UCS_) values of the intact rocks. Based on this observation, a new empirical formula for the prediction of the _UCS_ value using the \(E_{ni}\) value was proposed.  
(5) The unified nonlinear elastic model can also be employed to describe the nonlinear elastic deformation behavior of the jointed rock mass, but the range of the \(m\) values may be different from that of the intact rocks, which needs further investigation. For example, we could divide the jointed rock into two distinct portions (intact rock and joints) and analyze their deformation individually. Afterwards, they could be combined according to the principles of \"strain superimposition\" [40] or \"energy equivalence\" [41].  
Conceptualization, C.C. and S.C.; methodology, C.C.; software, H.L.; validation, Y.Z., H.L. and Y.W.; formal analysis, H.L.; investigation, Y.W.; data curation, Y.Z.; writing--original draft preparation, C.C.; writing--review and editing, S.C.; funding acquisition, C.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China under Grant 42277186 and 51927815.  
Not applicable.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Type of Rock**} & \multicolumn{2}{c}{**Linear Fitting y = ax + b**} & \multirow{2}{*}{**Coefficients of**} \\ \cline{2-3} \cline{5-5}  & **a** & **b** & **Determination R2** \\ \hline Dolomite & 8.92 & 15.74 & 0.950 \\ Granite & 10.44 & \(-\)6.52 & 0.978 \\ Limestone & 10.94 & 0.87 & 0.988 \\ Sandstone & 9.29 & 7.38 & 0.915 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Linear relationships between _UCS_ and \(E_{ni}\) of different rock types.