# Elastic-Plastic Threshold and Rational Unloading Level of Rocks
## 1 Introduction
### Evolution Analysis of Deformation Parameter  
This research focused on the stress state of rock specimens during the loading process under various hydrostatic pressures. Based on the test data, Figure 4 depicts the stress-strain curves of siltstone specimens in the conventional triaxial compression test at a confining pressure of 10 MPa, wherein \(\varepsilon_{1}(\varepsilon_{3},\varepsilon_{v})\) is the axial (lateral, volumetric) strain of siltstone specimens which occurred under the load. Figure 1: Typical rock specimens. Figure 2: MTS 815.02 Electro-hydraulic servo-controlled rock mechanics testing system. Elastic modulus and Poisson's ratio are important mechanical parameters of rocks, whose changes are closely associated with rock deformation damage and the elastic-plastic transition. Assuming that rock specimens under three-dimensional stress follow the generalized Hooke's law, then they are consistent with the condition listed below:  
\[\left\{ \begin{array}{l}\varepsilon_{1}=\frac{1}{E}[\sigma_{1}-\overline{\mu}( \sigma_{2}+\sigma_{3})]\\ \varepsilon_{2}=\frac{1}{E}[\sigma_{2}-\overline{\mu}(\sigma_{1}+\sigma_{3})] \\ \varepsilon_{3}=\frac{1}{E}[\sigma_{3}-\overline{\mu}(\sigma_{1}+\sigma_{2})] \end{array} \right. \tag{1}\]  
where \(\varepsilon_{1}(\varepsilon_{2},\varepsilon_{3})\) is the axial strain (lateral strain), \(\sigma_{1}(\sigma_{2},\sigma_{3})\) is the axial pressure (confining pressure), \(\overline{E}\) is nominal elastic modulus, and \(\overline{\mu}\) is nominal Poisson's ratio. Figure 4: Stress–strain curves of siltstone specimens at confining pressure of 10 MPa. Figure 3: Stress path and loading method used in the conventional triaxial compression test. Under the conventional triaxial compression, \(\sigma_{2}=\sigma_{3}\), Equations of nominal elastic modulus \(\overline{E}\) and nominal Poisson's ratio \(\overline{\mu}\) for the deformation parameter of rocks [28; 29] are formulated below by substituting \(\sigma_{2}=\sigma_{3}\), into Equation (1), that is:  
\[\left\{ \begin{array}{l}\overline{E}=\frac{\sigma_{1}-2\overline{\mu}\sigma_{ 3}}{\sigma_{1}}\\ \overline{\mu}=\frac{\frac{\sigma_{3}}{\sigma_{1}}\sigma_{1}-\sigma_{3}}{ \left(2\frac{\sigma_{3}}{\sigma_{1}}-1\right)\sigma_{3}-\sigma_{1}}\end{array} \right. \tag{2}\]  
Under a fixed confining pressure, the change in deformation parameter is only related to axial pressure. To analyze the effects of axial pressure change, we introduce the concept of deformation parameter-axial pressure increment ratio ([PERSON] et al. 2012), namely the ratio of nominal elastic modulus (nominal Poisson's ratio) increment to the axial pressure increment, which is expressed as:  
\[\left\{ \begin{array}{l}\frac{\Delta\overline{E_{t+1}}}{\Delta\sigma_{1(t+1) }}=\frac{\overline{E_{t+1}}-\overline{E_{t}}}{\sigma_{1(t+1)}-\sigma_{1(t)}} \\ \frac{\Delta\overline{\mu_{t+1}}}{\Delta\sigma_{1(t+1)}}=\frac{\overline{\mu_ {t+1}}-\overline{\mu_{t}}}{\sigma_{1(t+1)}-\sigma_{1(t)}}\end{array} \right. \tag{3}\]  
where \(\Delta\overline{E_{t+1}}\) is the increment of nominal elastic modulus at time \((t+1)\), \(\overline{E_{t+1}},\overline{E_{t}}\) is nominal elastic modulus at time \((t+1)\) (and \(t\)), \(\Delta\overline{\mu_{t+1}}\) is the increment of nominal [PERSON]'s ratio at time \((t+1)\), \(\overline{\mu_{t+1}},\overline{\mu_{t}}\) is nominal [PERSON]'s ratio at time \((t+1)\) (and \(t\)), \(\Delta\sigma_{1(t+1)}\) is the increment of axial pressure at \((t+1)\), and \(\sigma_{1(t+1)}\Big{(}\sigma_{1(t)}\Big{)}\) is the axial pressure at time \((t+1)\) (and \(t\)). The strain-axial pressure increment ratio is to describe a physical quantity, indicating the change rate of strain as a function of axial pressure. This ratio can sufficiently reflect the effects of axial pressure change on the deformation damage of rocks during loading, as well as the response rate of the internal structure to the external macro mechanical state change.