# A Numerical Approach for the Efficient Concept Design of Laser-Based Hybrid Joints
## 3 Demonstration of the Approach
### Concept Development  
The geometry of the single-lap-joint (SLJ) specimen is shown in Figure 9. The overlapped length (joining zone) is 12.5 \(\times\) 25.0 mm. The dimensions of the specimen and the dimensions of the joining zone were not changed during optimization. Only the micro  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Capital costs C\({}_{\text{C}}\) & 100,000 \(\upxi\) \\ Write-off time t\({}_{\text{w}}\) & 8 years \\ Relative maintenance costs C\({}_{\text{m}}\) & 20\% p.a. \\ Footprint F & 5 m\({}^{2}\) \\ Building costs C\({}_{\text{F}}\) & 3.5 \(\upxi\)/m\({}^{2}\) p.M. \\ Running hours t\({}_{\text{r}}\) & 2000 h p.a. \\ Energy costs C\({}_{\text{E}}\) & 0.405 \(\upxi\)/h \\ \hline \hline \end{tabular}
\end{table}
Table 1: Values for the laser cost calculation (reproduced from ref. [16] with kind permission of Trans Tech Publications).  
structure on the metal surface was modified. The metal component was fully clamped at its end. A displacement constraint with U1 = 0.15 mm was applied to the plastic surface. For the polymer part, glass-fiber-reinforced polypropylene was used, (PP-GF40) and for the metal part, regular stainless steel was used (1.4301).  
In Figure 10, the results from applying the design method to the demonstration example are shown. The von Mises stress (e), in-plane angle phi (b), and out-of-plane angle theta (a) distributions that are derived from the interlayer between the polymer and metal are shown. It can be observed that the stress distribution on the joining zone varied significantly. Low-stressed areas can be excluded for further consideration. The result for an example of stressThreshold = 0.1 is shown in Figure 10f.  
From the stress tensor, the angles theta and phi were calculated. Especially for theta, highly varying angles between shear (\(\sim\)0\({}^{\circ}\)), tensile (\(\sim\)-20\({}^{\circ}\)), and compressive (\(\sim\)-30\({}^{\circ}\)) load emerged.  
The clusters were derived using the von Mises stress and the theta angles.  
Pre-tests with phi angles of 0\({}^{\circ}\), 45\({}^{\circ}\), and 90\({}^{\circ}\) showed no difference in the strength for parallel linear cavities. Thus, the orientation of the microstructure within the plane does not affect the strength. Cross-section images show that the cavities are irregular in the direction of the grooves due to variations in the manufacturing process. This means that the cross-sections change in the direction of the grooves, causing additional undercuts. Thus, the phi angle was not considered for the optimization.  
The resulting clusters are shown for two, three, and four clusters for a value of stressThreshold = 0.1 in Figure 10. For two clusters, a large inner cluster and an outer cluster were formed, which, however, were not connected. A similar behavior resulted for three clusters, whereby the third cluster was strongly oriented according to the highly stressed area on the left edge of the surface. Pre-investigations showed that the cluster number of three for the given example had a good trade-off between manufacturability and granularity. For two clusters, the standard deviation of theta and von Mises was high, while for more than three clusters the areas became small and scattered so that no cavities could be placed and the concepts could not be manufactured. Thus, the cluster number of three was set for the following investigations. However, for new components the number of clusters shall be determined again.  
Figure 9: Dimensions and boundary conditions of the single-lap-joint specimen.