# A Numerical Approach for the Efficient Concept Design of Laser-Based Hybrid Joints
## 3 Demonstration of the Approach
### Determination of CZM Parameters
The curves of the micromodel and CZM matched well, so the CZM can represent the mechanical behavior of the micromodel. The CZM does not require complex contact calculations, plastic material behavior, or friction and thus has a significantly lower cost of simulation. Furthermore, the implementation of the CZM can be performed easily on any arbitrary components. The critical time step for explicit finite element analysis is proportional to the smallest element mesh size [47]. The CZM does not require extremely small mesh sizes as the micromodel does, and is therefore significantly faster. The final parameters of the CZM for tensile and shear directions are shown in Table 5.  
To demonstrate the accuracy of the CZM approach for larger joining zones with multiple cavities, the CZM was applied to the single-lap-joint specimen (see Figure 9). A microstructure with parallel cavities of 300 \(\upmu\)m distance was chosen for comparison and the force at failure for this configuration was compared to experiments and the micromodel. For these settings, there were multiple cavities on the joining zone as shown in Figure 13a. Figure 13b shows the force at failure. The error between the experiment, micromodel, and CZM was small (-3%) and within the standard deviation of the experiment. The validity and applicability of the CZM approach for laser-based plastic-metal joints can therefore be confirmed. Figure 12: Comparison of force–displacement curves for the micromodel and the cohesive-zone model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(\sigma_{\text{max,shear}}\) & 38.55 & MPa \\ \(\sigma_{\text{max,tensile}}\) & 18.88 & MPa \\ \(\text{E}_{\text{shear}}\) & 9200 & MPa \\ \(\text{E}_{\text{tensile}}\) & 1750 & MPa \\ \(\text{R}_{\text{shear}}\) & 0.1 & - \\ \(\text{R}_{\text{tensile}}\) & 0.6 & - \\ \(\text{G}_{\text{shear}}\) & 0.6 & ] \\ \(\text{G}_{\text{tensile}}\) & 0.4 & ] \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters for the cohesive-zone model (distance = 300 \(\upmu\)m). The parameters shown in Table 5 were determined for a reference distance of 300 \(\upmu\)m. However, for the optimization, the distance in each cluster is a continuous design variable and can take any decimal numbers between 300 \(\upmu\)m and 1000 \(\upmu\)m. Therefore, the parameters of the CZM are always calculated for each specific distance by scaling the values in relation to the parameters for a distance of 300 \(\upmu\)m. In Figure 14 CZMs are shown, in the three-dimensional space, as examples for distances of 300 \(\upmu\)m, 600 \(\upmu\)m, and 900 \(\upmu\)m. As expected, the highest strength could be achieved in the shear direction and the lowest strength in the tensile direction. At the same time, the strength decreased with increasing distance. However, the separations in the tensile direction were lower those in the shear direction. It should be noted that the results in Figure 14 only represent the mechanical behavior of one single, small CZM element. The mechanical properties of the entire joining zone in an assembly cannot directly be derived or extrapolated from these CZM properties. Figure 14: 3D illustration of the cohesive-zone models for different distances. Figure 13: Comparison of the estimated force at failure for the experiment, micromodel, and CZM (**b**) for a single-lap joint with distance = 300 \(\upmu\)m (**a**).