end do

and accordingly, in the current configuration. With the procedure in [5] the 2D tangential vectors and finally the 2D deformation gradient can be computed that describes the membrane deformation.

#### *4.2. Returning Stress and Material Tangent*

The stress returned from a standard Traction Separation Law as a material routine for interfaces contains three components: normal direction ( *N*) and two shearing directions (*S*1 and *S*2)

$$\mathbf{s}^{\;}\;=\begin{pmatrix}\mathbf{s}\_{N} \\ \mathbf{s}\_{S1} \\ \mathbf{s}\_{S2}\end{pmatrix}\tag{7}$$

The corresponding tangent considers the changes with respect to normal and (two) shearing separations

$$d = \begin{pmatrix} \begin{array}{cccc} \partial \text{s}\_{N} / \partial \delta\_{N} & \partial \text{s}\_{N} / \partial \delta\_{S1} & \partial \text{s}\_{N} / \partial \delta\_{S2} \\ \partial \text{s}\_{S1} / \partial \delta\_{N} & \partial \text{s}\_{S1} / \partial \delta\_{S1} & \partial \text{s}\_{S1} / \partial \delta\_{S2} \\ \partial \text{s}\_{S2} / \partial \delta\_{N} & \partial \text{s}\_{S2} / \partial \delta\_{S1} & \partial \text{s}\_{S2} / \partial \delta\_{S2} \end{array} \end{pmatrix} \tag{8}$$

Taking all directions into account, two in plane stretching (*IP*1 and *IP*2) directions and the out of plane shearing direction (*OPS*) have to be added

$$\begin{array}{rcl} s &=& \begin{pmatrix} s\_N \\ & s\_{S1} \\ & & s\_{S2} \\ & & \\ & & s\_{IP1} \\ & & & \\ & & \text{sIP2} \\ & & & \\ & & \text{s\_{OPS}} \end{pmatrix} \end{array} \tag{9}$$


The resistance of a zero thickness surface regarding in plane stretching and out of plane shearing is assumed to be zero. This can be justified considering a very thin layer where the reaction force depends on the integral over the thickness. Without thickness, the reaction force becomes zero. Hence in-plane stretching and out-of-plane shearing are assumed not to contribute to the residual and thus Membrane Mode Enhanced Cohesive Zone Elements build their element residual solely based on the separation modes. The other directions are still considered for damage calculations but can be omitted in the stress and consequently also in the tangent

$$\begin{array}{rcl} s &=& \begin{pmatrix} s\_N \\ s\_{S1} \\ s\_{S2} \\ 0 \\ 0 \\ 0 \end{pmatrix} \end{array} \tag{11}$$

*Metals* **2020**, *10*, 1333


At this point, it can be noticed that all necessary stress components for Membrane Mode Enhanced Cohesive Zone Elements can be returned and processed in the standard Cohesive Zone Element manner. Though the influence of in plane modes in the tangent cannot be captured in this way what might have an influence on the convergence. In [5], failure is updated explicitly. For this reason, the influence of membrane mode deformations during one load step cancels out

$$d = \begin{pmatrix} \begin{array}{cccc} \partial \text{s} / \partial \delta \text{N} & \partial \text{s} / \partial \delta \text{S}\_{S1} & \partial \text{s} / \partial \delta \text{S}\_{S2} & 0 & 0 & 0\\ \partial \text{s}\_{S1} / \partial \delta \text{N} & \partial \text{s}\_{S1} / \partial \delta \text{S}\_{S1} & \partial \text{s}\_{S1} / \partial \delta \text{S}\_{S2} & 0 & 0 & 0\\ \partial \text{s}\_{S2} / \partial \delta \text{N} & \partial \text{s}\_{S2} / \partial \delta \text{S}\_{S1} & \partial \text{s}\_{S2} / \partial \delta \text{S}\_{S2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \tag{13}$$

and the tangent can also be processed in the standard Cohesive Zone Element manner.

#### *4.3. Thermal and Mechanical Parameters*

In [5], several simulations are carried out to fit parameters for the material model. The maximum allowable tensile and shear traction are determined to be *tt*,*max* = 365 MPa and *ts*,*max* = 300 MPa. The Lemaitre damage parameters are *s* = 1.0 and *S* = 0.34.

The thermal conductivity of the joining zone considered as bulk material within an InTEx element is investigated in [9]. For undamaged (*D* = 0) or fully damaged (*D* = 1) material, the conductivity is

κ(*<sup>D</sup>* = 0) = 68 WmK , κ(*<sup>D</sup>* = 1) = 0.01 WmK . (14)

This can be converted to a heat transfer coefficient *h*

$$h(D=0) = \text{ } 6800000 \frac{\text{W}}{\text{m}^2 \text{K}} \quad , \quad h(D=1) = 1000 \frac{\text{W}}{\text{m}^2 \text{K}} \quad , \tag{15}$$

with the corresponding bulk material thickness of *t* = 10 μ*<sup>m</sup>*. It is assumed that *h* linearly depends on *D*

$$h(D) = \left(6800000 \left(1 - D\right) + 1000 \, D\right) \frac{\text{W}}{\text{m}^2 \text{K}} \quad . \tag{16}$$

#### **5. Simulation of a Transverse Link**

The transverse link is an advanced demonstrator component for Tailored Forming. All processes and also the final geometry are currently under development. The raw hybrid part is produced by extrusion; a forging process and afterwards machining is used to generate the final geometry. According to the current state, the raw part is an L formed steel profile (32 mm × 32 mm, width = 7 mm) that is filled with aluminium (Figure 7a). The final geometry is flattened, contains an indentation in the middle and has three holes for mounting (Figure 7b).

**Figure 7.** Transverse link; raw (**a**) and final geometry (**b**).

The forging process is simulated here to test Membrane Mode Enhanced Cohesive Zone Elements in a new load case especially with the implementation in MSC Marc. The simulations might also give an idea of the impact of forming on this specific part, though the validity is limited as no material data for this certain joining zone is available; the parameters in Section 4.3 stem from a di fferent process. For the bulk materials temperatures of Θ*st* = 750 ◦C and Θ*al* = 480 ◦C and yield stresses of <sup>σ</sup>*y*,*st* = 270 MPa and <sup>σ</sup>*y*,*al* = 40 MPa [10] are assumed.

The component has a symmetry in thickness direction that is utilised during the simulations. In a first step, the forming tool is modelled by subtracting the final geometry (except for the holes) from a solid block. To prevent burr formation, the tool is extended in forming direction, see Figure 8; practically this can be realised in a closed die process. Simulations show that the indentation in the middle induces a severe loading of the joining zone; the slope at the edge of the indentation is large and coincides with the joining zone. This causes failure of the joining zone in a very early stage. Figure 9a shows a damage contour plot of the slightly formed component. Nodal averaging is activated to visualise the damage also in the bulk elements around. Even if damage ranges from 0 to 1, here 0.5 is used as upper limit as interface elements contribute a damage and bulk elements do not. In the case of a tetraeder mesh, or also if the corner here would be damaged from both sides, the averaging result can only be utilised to indicate that damage occurs. The extent is influenced as the share of damage contributing elements varies. Figure 9b only contains the interface elements, nodal averaging is deactivated and the damage can be seen directly on the interface elements using a scale from 0 to 1.

**Figure 8.** Forming tool.

**Figure 9.** Forming process using the original geometry; whole geometry (**a**) and joining zone (**b**): joining zone damage.

Next, the simplified geometry in Figure 10 is used that does not contain the indentation in the middle. The results strongly depend on the starting position of the specimen in the form. In the case depicted in Figure 11a, a crack arises after the aluminium gets in contact with the radius of the form. With a slightly different position (1 mm shift in x and y direction), three positions ge<sup>t</sup> cracks, though later (Figure 11b). The inner corner is strongly shear loaded. On the backside (Figure 12) two shear induced cracks arise that are both close to the stiffening between mounting 1 and 2 (Figure 7b). The strong dependency on the position of the specimen in the form necessitates a holder or special geometry of the form that allows reliable and repeatable positioning. To further reduce loading, the form is again simplified by dropping the stiffening.

**Figure 10.** Simplified geometry.

**Figure 11.** Forming process using a simplified geometry with two different initial positions (**a**) and (**b**) of the specimen in the form: joining zone damage.

**Figure 12.** Forming process using a simplified geometry (backside of Figure 11b): joining zone damage.

The process can now be executed further. Though filling the edge around mounting 1 (Figure 7b) causes severe di fficulties. The stronger steel and a thin portion of the weaker aluminium enter the groove and come into contact on both sides (Figure 13a). Further forming induces compression, the aluminium starts yielding normal to the compression direction and decohesion is caused by shearing (Figure 13b). The decohesion induces a reduced heat flow that results in a temperature jump, see Figure 14. Using an adapted temperature range (Figure 14b) the e ffect can be noticed better compared to the plot with the whole temperature range (Figure 14a).

**Figure 13.** Forming process using a further simplified geometry; before (**a**) and after failure initiation (**b**): Joining zone damage.

**Figure 14.** Forming process using a further simplified geometry; whole (**a**) and adapted temperature range (**b**): temperature distribution.

Using a tetraeder mesh (tetraeder for the bulk, triangle or wedge like elements for the interface) instead of a hexaeder mesh (hexaeder for the bulk, quadrilaterial or hexaeder like elements for the interface) leads to very similar results. Figure 15 shows the same increment as Figure 13b and the specimen exhibits failure in the same position.

**Figure 15.** Forming process using tetraeder elements: joining zone damage.
