# A Numerical Approach for the Efficient Concept Design of Laser-Based Hybrid Joints
## 2 Method and Models
### Cohesive Zone Model for Strength Calculation  
The strength of the joint is a crucial criterion for the decision for or against a joining technology and therefore needs to be considered in the product development of laser-based  
Figure 5: Decomposition of the stress vector S into an in-plane angle (phi) and out-of-plane angle (theta). points. Current research mainly focuses on the experimental determination of the joint strength, which investigates different material combinations, microstructures (e.g., patterns, number of runs), and laser production settings [20; 22; 34; 35; 36; 37]. Few approaches exist for the estimation of the joint strength using simulation models [16; 18; 21; 23]. However, only approach [16] is capable of considering the direct influence of the microstructure parameters on the strength. As mentioned above, both cost and strength can be adjusted via the microstructure parameters, which is why these parameters must be taken into account in the design process. In the approach, the cavities are modeled in a 2D shell finite-element model with full geometric detail with all its curves to represent the drop-shaped geometry (see Figure 6a). The artificial thickness (length of the cavity) of the shell elements was chosen to cover the whole metal surface width. The plastic had a plastic material behavior with fracture criterion. A hard contact condition was implemented between the plastic and the metal and friction was implemented as a tangential behavior. Thereby, the separation and the crack propagation could be modeled for one or multiple cavities. Using this approach, the strength could be predicted with an accuracy of \(<\)5%. So far, this approach could only be demonstrated for the two-dimensional case since the calculation time is high. This model was not able to determine the strength of the three-dimensional concepts generated with the method from Section 2.1. Therefore, this paper presents a new approach that enables the efficient estimation of the strength for the 3D-case by so-called cohesive zone models (CZM). The idea of the CZM approach is the homogenization of the effective mechanical properties of the bonding behavior between plastic and metal as shown in Figure 6b. CZM are well-known models to represent the mechanical behavior of boundary layers in joining zones. CZM have been successfully applied to many joints, such as adhesive bonds [38; 39]. CZM work as follows: CZM are modeled as bulk material. They do not represent a direct material, but describe the relation between traction (stress) and separation (displacement) of the surfaces [40; 41]. The relationship between traction and separation is described by the so-called traction-separation law. In the simplest case, a bilinear relationship is defined (see Figure 7a). Here \(\sigma_{\text{max}}\) denotes the point of maximum stress and damage initiation, respectively. The damage evolution describes the mechanical behavior after reaching \(\sigma_{\text{max}}\), which is just linear in the bilinear case. In order to describe the anisotropic material behavior of the technology also with CZM, different traction-separation behaviors can be defined in different directions (see Figure 7b). Mode-I marks the mechanical behavior for the normal load (theta = 90\({}^{\circ}\)) and Mode-II the pure shear load case (theta = 0\({}^{\circ}\)). Figure 6: (**a**) RVE model and (**b**) cohesive zone model approach for laser-based hybrid joints. Typically, the CZM parameters are identified phenomenologically, which means in a reverse-engineering approach where numerous experiments are required [40]. For this new approach, however, the parameters of the CZM interlayer shall be determined in a direct model-based approach. The relevant parameters for the traction separation law are therefore derived from a micromodel RVE, which represents exactly one cavity of the microstructure of interest (see e.g., Figure 6b). Using the RVE approach, the mechanical behavior of any microstructure can be quickly determined. The approach is demonstrated in Section 3.