# Viscoelastic Strains of Palaeozoic Shales under the Burger's Model Description
## 4 Theoretical Basics of Rock Rheology in Triaxial Test Conditions for the Burger's Model  
The presented analyses refer to the behavior of shale samples in a complex stress state, therefore the description of rock behavior should be linked with relationships in a triaxial stress field. Assuming isotropy of the medium, the stress and strain tensors may be represented by their axiator and deviator. The relationships of linear elasticity take in this case the following form:  
\[S_{ij}=2G\ e_{ij} \tag{3}\]  
\[\sigma_{m}=3K\varepsilon_{m} \tag{4}\]  
where  
\(\sigma_{m}\) is axiator of the stress tensor \(\sigma_{m}=(\sigma_{1}+\sigma_{2}+\sigma_{3})/3=\sigma_{\rm{c}kt}\),  
\(\varepsilon_{m}\) is axiator of the strain tensor \(\varepsilon_{m}=(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3})/3= \varepsilon_{\rm{c}kt}\),  
\(S_{ij}\) is deviatoric stress tensor,  
\(\varepsilon_{ij}\) is deviatoric strain tensor. Referring to the model of the linear theory of elasticity, its generalization to the differential form of the constitutive equations of linear viscoelastic media in a spatial state of stress has the following form [38; 41]:  
\[P_{1}\{S_{ij}(t)\}=2Q_{1}\{e_{ij}(t)\} \tag{5}\]  
\[P_{2}\{\sigma_{m}(t)\}=Q_{2}\{e_{m}(t)\} \tag{6}\]  
\(P_{1}\), \(Q_{1}\), \(P_{2}\), \(Q_{2}\) are linear differential operators, which are different for the law of shear change and different for the law of volume change. In an extended notation, the equations attain the following form:  
\[\bigg{(}p_{0}^{\prime}+p_{1}^{\prime}\ \frac{\partial}{\partial t}+p_{2}^{ \prime}\ \frac{\partial^{2}}{\partial t^{2}}+\ldots+p_{a}^{\prime}\ \frac{\partial^{a}}{\partial t^{a}}\bigg{)}S_{ij}(t)=2 \bigg{(}q_{0}^{\prime}+q_{1}^{\prime}\ \frac{\partial}{\partial t}+q_{2}^{\prime}\ \frac{\partial^{2}}{\partial t^{2}}+\ldots+p_{b}^{ \prime}\ \frac{\partial^{b}}{\partial t^{b}}\bigg{)}e_{ij}(t) \tag{7}\]  
\[\bigg{(}p_{0}^{\prime\prime}+p_{1}^{\prime\prime}\ \frac{\partial}{\partial t}+p_{2}^{ \prime\prime}\ \frac{\partial^{2}}{\partial t^{2}}+\ldots+p_{a}^{\prime\prime}\ \frac{\partial^{a}}{\partial t^{a}}\bigg{)}\sigma_{m}(t)= \bigg{(}q_{0}^{\prime\prime}+q_{1}^{\prime\prime}\ \frac{\partial}{\partial t}+q_{2}^{\prime\prime}\ \frac{\partial^{2}}{\partial t^{2}}+\ldots+p_{b}^{ \prime\prime}\ \frac{\partial^{b}}{\partial t^{b}}\bigg{)}e_{m}(t) \tag{8}\]  
They are linear differential equations, with constant coefficients \(pi\) and \(qi\). Using the Laplace transformation, the equations can be noted with sub-division into axiator and deviator changes in the following form [41]:  
\[S_{ij}+\bigg{(}\frac{\eta_{s}^{M}}{G^{M}}+\frac{\eta_{s}^{M}+\eta_{s}^{K}}{G^ {K}}\bigg{)}\dot{S}_{ij}+\frac{\eta_{s}^{M}\eta_{s}^{K}}{G^{M}G^{K}}\ddot{S}_ {ij}=2\eta_{s}^{M}\dot{e}_{ij}+\frac{2\eta_{s}^{M}\eta_{s}^{K}}{G^{K}}\ddot{e} _{ij} \tag{9}\]  
\[\sigma_{m}+\bigg{(}\frac{\eta_{m}^{M}}{K^{M}}+\frac{\eta_{m}^{M}+\eta_{m}^{K} }{K^{K}}\bigg{)}\dot{\sigma}_{m}+\frac{\eta_{m}^{M}\eta_{m}^{K}}{K^{M}K^{K}} \ddot{\sigma}_{m}=\eta_{m}^{M}\dot{e}_{\rm{c}rol}+\frac{\eta_{m}^{M}\eta_{m}^ {K}}{K^{K}}\ddot{e}_{\rm{c}rol} \tag{10}\]  
where\(\dot{\sigma}\) and \(\ddot{\sigma}\) are the first and second derivative of the average stress, respectively, \(\dot{e}_{vol}\) and \(\ddot{e}_{vol}\) are the first and second derivatives of the volumetric strain, respectively, \(\dot{S}_{ij}\) and \(\ddot{S}_{ij}\) are the first and second derivatives of deviatoric stress tensor, respectively, \(\dot{e}_{\ddot{y}}\) and \(\ddot{e}_{\ddot{y}}\) are the first and second derivatives of the deviatoric strain tensor, respectively.