# Full-Field Strain Determination for Additively Manufactured Parts Using Radial Basis Functions
## 3 Evaluation of Strain Determination Using RBFs
### Verification Examples of WAAM-Produced Cylindrical Components
#### 3.1.2. Tension and Torsion of a Tube with Rough Surface  
With WAAM components that have already been manufactured, the actual objective is that the surfaces, see Figure 5 (right), as an example, are not treated any more. This poses further challenges to image correlation methods, since--although classical paints and spray-on speckle patterns can be used--the surfaces have a high degree of roughness. The strains on the surfaces, on the other hand, are difficult to evaluate because the interpolations smooth out the local features. To investigate this, it is possible to use a verification example of the following type: a cylindrical specimen with a rough surface in the initial configuration is considered  
\[\vec{X}(\Theta^{1},\Theta^{2})=\vec{X}(\Phi,Z)=\hat{R}(Z)\vec{E}_{R}(\Phi)+Z \vec{E}_{Z}, \tag{38}\]  
where \((\Theta^{1},\Theta^{2})=(\Phi,Z)\) are the curvilinear coordinates, and \(\vec{E}_{R}(\Phi)=\cos\Phi\vec{e}_{x}+\sin\Phi\vec{e}_{y}\), and \(\vec{E}_{\Phi}(\Phi)=-\sin\Phi\vec{e}_{x}+\cos\Phi\vec{e}_{y}\), the radial and circumferential directions. First, a smoothed step-function for the varying radius describing the welding bead stacks is applied,  
Figure 4: Principal directions \(\vec{u}_{\alpha}\), \(\alpha=1,2\), of the in-plane right stretch tensor \(\hat{\mathbf{U}}\) for the torsional example (\(\Delta d=0.25\,\text{mm}\) for inverse multi-quadrics with monomials \(c_{I}R_{0}=0.3\)). \[\tilde{R}(Z) =R_{0}+\frac{D_{S}}{2}\sum_{k=0}^{n_{\text{ws}}}(\tanh(c(Z-(Z_{k}+ \delta)))-\tanh(c(Z-(Z_{k+1}-\delta)))), \tag{39}\] \[Z_{k} =Z_{0}+kH. \tag{40}\]  
\(R_{0}\) is the mean radius of the tube, \(D_{S}\) controls the notch depth between two stacks of welding beads, and \(n_{\text{ws}}\) defines the number of stacks over the specimen's height. Here, one can only describe a pause by the welding robot, which has to stop after a certain number of welding beads to guarantee sufficient heat transport (The welding robot must be stopped after a few cycles of material application, as poor heat transfer causes the material to become too viscous and the specimens tend to melt away. The stopping time is necessary so that solidification is guaranteed). These welding layers build a welding stack of height \(H\). Here, \(\delta\ll H\) is chosen for a certain notch width between the step functions describing the stacks, and \(c\) (dimension: 1/length) serves to sharpen the notch, see Figure 6.  
The individual welding layers (welding beads) are then described with the aid of a superposition of sinusoidal functions,  
\[\hat{R}(Z)=\hat{R}(Z)+\frac{D_{W}}{2}\bigg{(}1+\sin\bigg{(}\frac{2\pi n_{ \text{wb}}}{H}Z\bigg{)}\bigg{)}, \tag{41}\]  
where \(n_{\text{wb}}\) represents the number of welding beads within one stack of height \(H\). Next, the deformation  
\[\vec{x}=\vec{\chi}_{\text{R}}(\vec{X},t)=\hat{r}(R)\vec{e}_{r}(\varphi)+\hat{ z}(Z)\vec{e}_{z}, \tag{42}\]  
is assumed, where the radial motion depends only on \(Z,\vec{r}(Z)=\hat{r}(\hat{R}(Z))\), see Equation (41). Furthermore, \(\varphi=\Phi+DZ\) holds for torsional problems, see Equation (32), which leads to  
\[ \begin{split}\vec{e}_{r}(\Phi,Z)&=\cos(DZ)\vec{E}_{ \text{R}}(\Phi)+\sin(DZ)\vec{E}_{\Phi}(\Phi)\\ \vec{e}_{\varphi}(\Phi,Z)&=-\sin(DZ)\vec{E}_{R}( \Phi)+\cos(DZ)\vec{E}_{\Phi}(\Phi)\\ \vec{e}_{z}&=\vec{E}_{Z}.\end{split}  \tag{43}\]  
Figure 5: Specimens made using WAAM process. (**left**) Specim without any outer surface treatment, (**right**) specimen, where the outer surface is machined off. Figure 6: Geometrical quantities of verification example for rough surface.