# A Numerical Approach for the Efficient Concept Design of Laser-Based Hybrid Joints
## 3 Demonstration of the Approach
### Determination of CZM Parameters  
To determine the strength of the microstructure concepts, a cohesive zone model (CZM) for a specific microstructure was developed as presented in Section 2.2. The microstructure used is a cavity geometry resulting from six laser runs because these cavities show high strength and a distinctive undercut. Details regarding the corresponding dimensions of the cavities (depth, undercut, etc. ), laser setup (machine type, etc. ), and laser settings (laser power, etc.) can be found in ref. [18]. The principle for determining the CZM parameters is to determine the force-displacement curves of a single cavity using an FE model on the micro level (\"RVE micromodel\") and then calculate the CZM parameters from this data. After calibration of the CZM based on the RVE micromodel, the CZM can then be used in larger assemblies and can be applied to larger joining zones. In Table 4 the material parameters for polymer (PP-GF40) metal (1.4301) are shown, which were used to build the RVE micromodel. It has a width of 300 \(\upmu\)m, which is equivalent to a distance of 300 \(\upmu\)m between the cavities. The thickness of the RVE model is equal to the width. The plastic is given a plastic material behavior with fracture criterion. For the metal, a linear elastic material behavior was chosen since the stress in the metal is relatively low. A hard contact was used between the plastic and the metal, which completely prevents the metal and polymer surfaces from intersecting [46]. The micromodel was used to perform a tensile test (theta = 90\({}^{\circ}\)) and a shear test (theta = 0\({}^{\circ}\)) and to generate the force-displacement curve. The objective of the CZM is to represent the mechanical properties of the micromodel. To confirm the validity of the CZM beforehand on a simple small-scale example, a reference model (Figure 6b) with the same dimensions and boundary conditions as the micromodel was developed with a CZM and the force-displacement curves were compared. The maximum strength \(\sigma_{\text{max}}\), [PERSON]'s modulus E in tensile and shear directions, as well as the energy G (see also Figure 7) need to be defined for creating a CZM in an FE software. To model the trapezoidal CZM curve, the trapezoidal ratio h of the horizontal lines needs to be specified. The values (\(\sigma_{\text{max}}\), E, G, and h) were optimized so that the force-displacement curves from micromodel and CZM matched. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Steel (1.4301)** & **Plastic (PP/GF40)** \\ \hline Modulus (MPa) & 200,000 & 9200 \\ Density (g/cm\({}^{3}\)) & 7.9 & 1.2 \\ Poisson ratio & 0.3 & 0.35 \\ Tensile strength (MPa) & - & 185 \\ Strain at break (\%) & - & 2.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material properties of the metal and plastic component. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Parameter** & **Values** \\ \hline \multirow{4}{*}{Algorithm settings} & Population Size & 12 \\  & Number of Generations & 20 \\  & Crossover Probability & 0.9 \\  & Crossover Distribution Index & 10 \\  & Mutation Distribution Index & 20 \\  & Initialization Mode & Random \\ \hline \multirow{2}{*}{Design Variables} & stressThreshold & [0; 35] \\  & Distances d\({}_{1}\)–d\({}_{n}\) & [300 \(\upmu\)m; 1000 \(\upmu\)m] \\  & Number of clusters & 3 \\ \hline \multirow{2}{*}{Objectives} & Strength & \(\geq\)Benchmark \\  & Costs & minimize \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimization settings for NSGA-II, design variables, and objectives. The comparison of the curves after optimization is shown in Figure 12. In the curves of tensile and shear, the anisotropy can be clearly recognized. While the plastic in the neck broke off abruptly in the shear direction, it was pulled out gradually in the tensile direction, leading to the trapezoidal shape. The stiffness and strength in the shear direction were higher compared with the tensile direction. These investigations agree with the relevant literature.