## 2 Materials and Methods
### Elastic Constants for Parametric Study
#### 2.2.3 Shear Modulus \(G^{\prime}\) and Saint-Venant Approximation  
The shear modulus \(G^{\prime}\) is the most laborious parameter to determine from the five independent elastic parameters of a transversely isotropic material [29]. The parameter \(G^{\prime}\) may be determined directly in UCS tests on samples with inclined isotropy planes from the conduction of strain measurements at 45\({}^{\circ}\) to the cylinder axis [10; 25]. Besides the direct determination, \(G^{\prime}\) may likewise be obtained from the solution of the system of stress-strain equations arising from the transformed [PERSON]'s law [2; 14; 36]. This requires the execution of multiple tests and measurements on commonly three samples with isotropy plane inclinations of \(\beta=0^{\circ}\), \(\beta=90^{\circ}\) and \(\beta\
eq 0^{\circ},90^{\circ}\).  
Given the increased effort for the determination of the shear modulus \(G^{\prime}\), in many practical applications, this parameter is approximated by \(G^{\prime}_{sv}\) acc. to Equation (5).  
\[G^{\prime}\approx G^{\prime}_{sv}=\frac{EE^{\prime}}{E(1+2\
u^{\prime})+E^{ \prime}} \tag{5}\]  
This relationship, in the literature often referred to as the Saint-Venant approximation of the shear modulus, originates from the early studies of [PERSON] [37] and is indicated here in the form of [PERSON] [11]. Although this approximation is frequently described as being purely empirical [2; 38], it still relates to a theoretical background, as shown by [PERSON] et al. [16]. Through rotating the compliance matrix \(S^{\prime}\), as well as stresses and strains from the local coordinate system \(x^{\prime}y^{\prime}z^{\prime}\) by an angle \(\beta\) around the \(y\)-axis (see Figure 4), with the \(y\)-axis oriented parallel to the isotropy plane's strike direction, the apparent shear modulus \(G^{\prime}_{\beta}\cong G_{xz}\) within the coordinate system \(xyz\) is related to the shear stress \(\tau_{xz}\) and shear strain \(\gamma_{xz}\) by \(\tau_{xz}=G^{\prime}_{\beta}\,\gamma_{xz}\) with  
\[\frac{1}{G^{\prime}_{\beta}}=\underbrace{\left(\frac{1}{E}+\frac{1+2\
u^{ \prime}}{E^{\prime}}\right)}_{G^{\prime}_{sv}}\sin^{2}(2\beta)+\frac{1}{G^{ \prime}}\cos^{2}(2\beta). \tag{6}\]  
From Equation (6), it is obvious that, at angles \(\beta=45^{\circ}+i\cdot 90^{\circ}\), with \(i\) corresponding to any integer number, the apparent shear modulus \(G^{\prime}_{\beta}\) equals the value from the approximation \(G^{\prime}_{sv}\) due to the vanishing cosine term. This justifies the adoption of Equation (5) for simplification purposes, with deviations in the true to the approximated value of the shear modulus often  
Figure 4: (**a**) Illustration of a quadratic cutout from a transversely isotropic solid oriented in the \(xyz\) coordinate system and subjected to shear stress \(\tau_{xz}\). (**b**) Polar plot of \(G^{\prime}_{\beta}/G^{\prime}_{sv}\) dependent on polar angle \(\beta\) and for varying values of the shear modulus \(G^{\prime}\) assumed with 80, 100 and 120 % of \(G^{\prime}_{sv}\) equal to unity (based on [PERSON] et al. [16]).  
reported within a range smaller than 20% [16]. Emphasis should be given that this statement is only valid for rocks with low to moderate degrees of anisotropy, i.e., \(E/E^{\prime}<2\).  
For the parametric analytical study in this research, the shear modulus was taken as \(G^{\prime}_{\text{sp}}\) acc. to Equation (5). To further investigate the influence of varying values for this elastic parameter on the developing strains, an additional value of \(G^{\prime}_{\text{sp},Ent=2}\) computed at a ratio \(E/E^{\prime}=2.0\) was additionally chosen for combinations with [PERSON]'s modulus ratios smaller than 2. Numerical studies were conducted with values for the transverse shear modulus \(G^{\prime}\) of 0.8-\(G^{\prime}_{\text{sp}^{\prime}}\), \(G^{\prime}_{\text{sp}^{\prime}}\) and 1.2-\(G^{\prime}_{\text{sp}}\) considering commonly observed deviations from the approximated value \(G^{\prime}_{\text{sp}}\).