**Appendix A**

The stress balance in Figure 1 is based on shear strength and is expressed as follows:

$$
\sigma\_1 \cos \theta - \sigma\_3 \sin \theta = \mu\_{\text{laminar}} (\sigma\_3 \cos \theta + \sigma\_1 \sin \theta) + C\_{\text{laminar}} \tag{A1}
$$

where *f* = μ*lamina* × (<sup>σ</sup>3cosθ + <sup>σ</sup>1sinθ) and represents the physical and frictional interactions between the lamina and the surrounding rock, as shown in Figure 1. *Clamina* represents the chemical cementation between the lamina and the surrounding rock, which is treated as a constant parameter in this simulation.

To establish an equation in contrast to the Mohr–Coulomb failure criterion (Equation (A2)), Equation (A3) is achieved when Equation (A1) is transformed into (A2).

$$\sigma\_1 = 2S\_0 \frac{\cos \varphi}{1 - \sin \varphi} + \frac{1 + \sin \varphi}{1 - \sin \varphi} \sigma\_3 \tag{A2}$$

$$\sigma\_1 = \frac{\mathcal{C}\_{\text{laminar}}}{\cos \theta - \mu\_{\text{laminar}} \sin \theta} + \frac{\sin \theta + \mu\_{\text{laminar}} \cos \theta}{\cos \theta - \mu\_{\text{laminar}} \sin \theta} \sigma\_3 \tag{A3}$$

It is possible to compare Equation (A2) with (A3) when the lamina-induced fractures are considered a special product emerging when shear fractures slide along the lamina surface. Thus, an equivalent relationship between the constant term and the coefficient term of the variables σ1 and σ3 can be expressed as Equations (A4) and (A5):

$$\frac{C\_{\text{lamium}}}{\cos \theta - \mu\_{\text{lamium}} \sin \theta} = 2S\_0 \frac{\cos \varphi}{1 - \sin \varphi} \tag{A4}$$

$$\frac{1}{\cos\Theta - \mu\_{\text{laminar}}\sin\Theta} = \frac{1 + \sin\varphi}{1 - \sin\varphi} \tag{A5}$$

Then, ϕ and *S*0 can be expressed as Equations (A6) and (A7):

$$\varphi = \arcsin \frac{\mu\_{\text{laminar}} \cos \theta + \sin \theta}{(\mu\_{\text{laminar}} + 1) \cos \theta + (1 - \mu\_{\text{laminar}}) \sin \theta} \tag{A6}$$

$$S\_0 = \frac{(1 - \sin \varphi)C\_{\text{lauvun}}}{2 \cos \varphi (\cos \theta - \mu\_{\text{lauvun}} \sin \theta)}\tag{A7}$$

Thus, *S*0 can be further expressed as Equation (A8) when Equation (A6) is put into Equation (A7).

$$S\_0 = \frac{1}{2} \cdot \frac{\frac{\cos \theta - \mu\_{\text{lamba}} \sin \theta}{(\mu\_{\text{lamba}} + 1)\cos \theta + (1 - \mu\_{\text{lamba}})\sin \theta}}{\cos \arcsin \frac{\mu\_{\text{lamba}} \cos \theta + \sin \theta}{(\mu\_{\text{lamba}} + 1)\cos \theta + (1 - \mu\_{\text{lamba}})\sin \theta}} \cdot \frac{C\_{\text{lamba}}}{\cos \theta - \mu\_{\text{lamba}} \sin \theta} \tag{A8}$$
