# Performance Study of Black Shale Modified Soil for Road Use Based on Eshelby-Mori-Tanaka Theory
## 5 Strength Model of Black Shale Modified Soil  
Establishing a mechanical model for material strength based on existing experimental data and previous theoretical foundations is crucial. It not only helps systematically summarize and integrate existing experimental results but also provides a more extensive and comprehensive understanding of material performance. In this study, cement  
Figure 14: Picture of unconfined compression test 21.”I. Contact Stage” refers to the stage when the specimen is fully in contact with the test machine and device; “IV. Failure Stage” refers to the stage when the specimen is obviously damaged. Figure 13: Unconfined compression test of test piece Y7S30. soil (cement + clay) is considered as the matrix, and black shale fragments are treated as the inclusion material, forming a two-phase composite material (Figure 15). Using the Eshelby-Mori-Tanaka equivalent method, a mechanical model is established that predicts the strength of black shale modified soil based on three indicators: soil strength, gravel content, and cement content. According to [PERSON] et al. 's study [23] based on the Eshelby-Mori-Tanaka equivalent method for the equivalent elastic modulus of sandy gravel soil, the relationship between the equivalent elastic modulus of sandy gravel soil and the matrix elastic modulus can be expressed as:  
\[E=E_{0}(av^{3}+bv^{2}+cv+1), \tag{3}\]  
In the formula, \(E\) is the elastic modulus of sandy gravel soil, \(E_{0}\) is the matrix elastic modulus, \(v\) is the volume fraction of the inclusion material, and \(a^{\prime},b^{\prime},c^{\prime}\) are constants. As per the literature [34], it is known that \(\sigma\propto E\) and \(m\propto v\), which implies:  
\[[\sigma]=[\sigma_{0}](a^{\prime}m^{3}+b^{\prime}m^{2}+c^{\prime}m+1), \tag{4}\]  
where \([\sigma]\) is the composite material strength, \([\sigma_{0}]\) is the matrix strength, \(a^{\prime},b^{\prime},c^{\prime}\) are constants related to the strength of the inclusion material, and \(m\) is the mass fraction of the inclusion material. The transformed equation for \([\sigma]\) in Equation (4) is given by  
\[\frac{[\sigma]}{[\sigma_{0}]}=a^{\prime}m^{3}+b^{\prime}m^{2}+c^{\prime}m+1, \tag{5}\]  
Through numerical fitting, as shown in Figure 16, we know that, \(a^{\prime}=2.255\), \(b^{\prime}=-2.468\), \(c^{\prime}=-0.321\), correlation coefficient \(r=0.955\), and  
\[\frac{[\sigma]}{[\sigma_{0}]}=2.255m^{3}-2.468m^{2}-0.321m+1, \tag{6}\]  
Figure 16: Test result fitting curve. Figure 15: Schematic diagram of composite material formation process. As \([\sigma_{0}]=[\sigma_{\text{cement-soil}}]\), then,  
\[[\sigma]=[\sigma_{\text{cement-soil}}]\left(2.255m^{3}-2.468m^{2}-0.321m+1\right), \tag{7}\]  
Substituting \(\gamma=19.51\) and Equation (2) into Equation (7), we can obtain  
\[[\sigma]=[\sigma_{\text{soil}}](1+19.51S)\left(2.255m^{3}-2.468m^{2}-0.321\,m+ 1\right), \tag{8}\]  
where \(S\) is the cement content of the cement-soil. From Equation (8), it is known that replacing the soil with higher strength or increasing the cement content can improve the composite material strength. Taking the partial derivative of M in Equation (8), it is obtained that  
Figure 13:0  
Setting \(3\times 2.255m^{2}-2\,\times\,2.468\,m-0.321\) to 0 yields \(m_{1}=-0.06\), \(m_{2}=0.79\). As \(m\in[0,1]\), then when \(0<m<0.79\) and \([\sigma]^{\prime}_{m}<0\), \([\sigma]\) is monotonically decreasing when \(m\in[0,0.79]\), and when \(0.79<m<1\), \([\sigma]\) is monotonically increasing when \(m\in[0.79,1]\). The composite material strength \([\sigma]\) shows a decreasing trend followed by an increase with the increase in the black shale content. However, due to the lack of bonding force between the black shale particles, when the shale content is too high, the overall strength is low. Therefore, the formula is more accurate when the shale content is less than 79%. When the shale content exceeds 79%, the closer it gets to 100%, the larger the error.