# Interfacial Shear Properties of Red Clay and Polyurethane with Different Densities
## 4 Model Validation
### Definition of Material Parameters  
The two-component polyurethane material is modeled using a linear elastic isotropic model, simplifying the polyurethane as a homogeneous material. By applying the strain  
Figure 5: The shear stress-shear displacement curves of the composite interface under different normal stresses: (**a**) polymer density of \(0.5\,\mathrm{g/cm^{3}}\); (**b**) polymer density of \(0.8\,\mathrm{g/cm^{3}}\). Figure 6: The relationship between shear strength and vertical stress. energy equivalence principle, this simplification enables the model to accurately reflect the structural response of the original complex system without requiring computationally intensive analyses; The bilinear cohesive strength model simulating the interface between polyurethane and red clay must ensure stable coupling between polyurethane and red clay. Complex constitutive models can easily lead to convergence issues; therefore, a linear elastic model is chosen to simulate the polyurethane material, studies [21; 40; 43; 44] have adopted this constitutive model for simulation experiments. The material parameters are listed in Table 3, where the elastic modulus of polyurethane at varying densities was derived from Equation (1) [37]. \[E=456.81\rho^{1.77}, \tag{1}\]  
where \(\rho\) represents the polyurethane density. The stress-strain relationship of red clay is based on the Mohr-Coulomb model built in ABAQUS [45], an extension of the classical Mohr-Coulomb yield rule. The Mohr-Coulomb yield function is adopted, incorporating isotropic hardening and softening of cohesion. The linear elastic model is based on the generalized Hooke's law, with the constitutive equation as follows:  
\[\sigma=D^{\text{el}}\varepsilon^{\text{el}}, \tag{2}\]  
In this equation: \(\sigma\) is the stress component vector; \(D^{\text{el}}\) is the elastic matrix; \(\varepsilon^{\text{el}}\) is the strain component vector. The yield strength of the Mohr-Coulomb model is defined as:  
\[\tau=c-\sigma\tan\varphi, \tag{3}\]  
where \(\tau\) is the shear strength, \(c\) is the soil cohesion, \(\sigma\) is the normal stress, and \(\varphi\) is the internal friction angle, which falls within the range of [0, 90\({}^{\circ}\)]. The parameter values are shown in Table 4.  
From the Mohr circle, the following relationship can be derived:  
\[\tau=s\cos\varphi, \tag{4}\]  
\[\sigma=\sigma_{m}+s\sin\varphi, \tag{5}\]  
Substituting \(\tau\) and \(\sigma\) into Equation (3), the Mohr-Coulomb criterion can be expressed as:  
\[s+\sigma_{m}\sin\varphi-c\cos\varphi=0, \tag{6}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Density** & **Elastic Modulus** & **Poisson** & **Angle of Internal Friction** & **Angle of Dilatancy** & **Plastic Strain** \\
**[g/cm\({}^{3}\)]** & **[kPa]** & **Ratio** & [\({}^{\circ}\)] & [\({}^{\circ}\)] & **[kPa]** \\ \hline
1.75 & 198 & 0.35 & 14.35 & 0.1 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material parameters of red clay. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Density [g/cm\({}^{3}\)]** & **Elastic Modulus [MPa]** & **Poisson’s Ratio** \\ \hline
0.2 & 128.5 & 0.2 \\
0.5 & 292.4 & 0.2 \\
0.8 & 487.3 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material Parameters of Polyurethane. Here, \(s=\frac{\sigma_{1}-\sigma_{2}}{2}\) is half the difference between the major and minor principal stresses, while \(\sigma_{m}=\frac{\sigma_{1}+\sigma_{2}}{2}\) is the average value of the major and minor principal stresses. The Mohr-Coulomb yield criterion assumes that failure is independent of the intermediate principal stress. While failure in typical geotechnical materials is often influenced by the intermediate principal stress, this effect is relatively small, making the Mohr-Coulomb criterion sufficiently accurate for most applications. Interface failure between polyurethane and red clay is modeled as type II delamination cracking, which is analyzed using the cohesive damage model [46].