# Interfacial Shear Properties of Red Clay and Polyurethane with Different Densities
## 4 Model Validation
### Definition of Material Parameters
The concept of a cohesive zone was first proposed by [PERSON] [47] and [PERSON] [48], and represents a simplified elastic-plastic model that uses damage variables and damage evolution criteria to model the interface constitutive relationships. In ABAQUS, two types of cohesive zone unit models exist. This study uses cohesive elements to model the interface between polyurethane and red clay, neglecting delamination behavior at the bonding point. The contact interface is set as a complete contact, with cohesive elements for modeling, the cohesive zone model was employed in studies [21; 44]. The constitutive model of the cohesive force unit adopts a bilinear traction-separation law, which is described by the following Figure 7:  
\[t=\left\{ \begin{array}{l}t_{n}\\ t_{s}\\ t_{t}\end{array} \right\}=\left[ \begin{array}{l}K_{nn}&K_{ns}&K_{nt}\\ K_{ns}&K_{ss}&K_{st}\\ K_{nt}&K_{st}&K_{tt}\end{array} \right]\left\{ \begin{array}{l}\delta_{n}\\ \delta_{s}\\ \delta_{t}\end{array} \right\}=K\delta, \tag{7}\]  
Here, \(t\) is the nominal interface stress, \(\delta\) is the interface displacement, and \(K\) is the stiffness matrix of the contact interface. The subscripts \(n\), \(s\) and \(t\) correspond to the normal, first shear, and second shear displacement components, respectively. In the bilinear cohesive zone model, the cohesive element's initial failure criterion represents the material's initial damage failure, while the damage evolution criterion governs crack propagation. According to the design of the direct shear test, the interface normal is subjected only to compressive stress, and shear force is applied in a single direction on the interface plane. Thus, the damage strength is controlled by the shear strength. When the stress reaches its maximum value, the interface's bearing capacity  
Figure 7: Bilinear traction separation response constitutive model. decreases, initiating damage. Therefore, the maximum nominal stress criterion is used to determine interface damage, as shown in Equation (8):  
\[\max\left\{\frac{\langle\sigma_{n}\rangle}{\sigma_{n}^{\max},\sigma_{s}^{\max}, \frac{\sigma_{t}}{\sigma_{t}^{\max}}}\right\}=1, \tag{8}\]  
where \(\sigma_{n}\), \(\sigma_{s}\)\(\sigma_{t}\) and represent the nominal stress in the normal, first shear, and second shear directions, respectively, and \(\sigma_{n}^{\max}\), \(\sigma_{s}^{\max}\) and \(\sigma_{t}^{\max}\) are the peak nominal stresses in these directions. Since the composite specimen is axisymmetric, the interface strength in each shear direction is assumed to be equivalent. For simplicity, the model assumes homogeneity of interface strength, i.e., \(\sigma_{n}=\sigma_{s}=\sigma_{t}\). During the interface damage evolution stage, the displacement corresponding to the shear strength represents the initial separation displacement of the interface damage. The displacement at complete interface failure corresponds to the complete failure separation displacement. The energy consumed during the failure process is associated with the interfacial shear failure energy, which corresponds to the slope of the elastic stage in the shear stress-displacement curve, the peak stress, and the area under the curve. These parameters are derived from the stress-displacement curve of the test results, as shown in Table 5.