# 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
#### 3.1.1 Tension and Torsion of a Cylindrical Tube  
To study the accuracy of the new approach using RBFs, the analytic deformation of tension and torsion is compared to the numerical results. For more details on the analytical equations and their derivation, refer to [23; 24; 25; 26], and with an application to DIC, to [4]. Here, the motion  
\[\vec{x}=\vec{x}_{\mathrm{R}}(\vec{X},t)=\hat{r}(R)\vec{e}_{r}(\Phi,Z)+\hat{z}( Z)\vec{e}_{z} \tag{31}\]  
is considered, with  
\[\hat{r}(R)=\lambda^{-1/2}R,\quad\hat{\varphi}(\Phi,Z)=\Phi+DZ,\quad\hat{z}(Z)= \lambda Z, \tag{32}\]  
and  
\[\vec{e}_{r}=\cos\varphi\vec{e}_{x}+\sin\varphi\vec{e}_{y},\quad\vec{e}_{\varphi }=-\sin\varphi\vec{e}_{x}+\cos\varphi\vec{e}_{y}. \tag{33}\]  
\(\lambda=L/L_{0}\) defines the axial stretch, where \(L\) is the current length and \(L_{0}\) the initial length of the cylinder. \(D=\alpha/L_{0}\) symbolizes the twist, where \(\alpha\) represents the torsion angle. The motion (31) yields the principal stretches  
\[\lambda_{1}=\lambda^{-1/2},\quad\lambda_{2,3}=\sqrt{\frac{1+\lambda^{3}+(DR)^ {2}\pm\sqrt{(1+\lambda^{3}+(DR)^{2})^{2}-4\lambda^{3}}}{2\lambda}}, \tag{34}\]representing the eigenvalues of the right stretch tensor \(\mathbf{U}\) of the polar decomposition \(\mathbf{F}=\mathbf{R}\mathbf{U}\), where \(\mathbf{F}(\vec{X},t)=\operatorname{Grad}\vec{\chi}_{\mathrm{R}}(\vec{X},t)\) defines the deformation gradient, and \(\mathbf{R}\) with \(\det\mathbf{R}=+1\), \(\mathbf{R}^{T}=\mathbf{R}^{-1}\), the rotation tensor. In the following example, a fixed radius \(R=15\,\mathrm{mm}\), \(L_{0}=80\,\mathrm{mm}\), \(\lambda=1.2\), and a torsional angle of \(\alpha=20^{\circ}\) are assumed, so that \(D=\alpha/L_{0}=20^{\circ}\pi/180^{\circ}/80\,\mathrm{mm}\approx 0.44\, \mathrm{mm}^{-1}\). See Figure 1 for the points on the surface in the reference and the current configuration. The analytical solution yields the stretches  
\[\lambda_{1}=0.91287,\quad\lambda_{2}=1.2035,\quad\lambda_{3}=0.91022. \tag{35}\]  
We are interested in the in-plane stretches \(\lambda_{2}\) and \(\lambda_{3}\), which can be compared to the DIC-results. \(\lambda_{1}\) defines the out-of-plane component, which cannot be measured. The maximum principal stretch is \(\lambda_{2}\) (\((+)\)-sign in Equation (34)\({}_{2}\)). For the numerically generated DIC-data, a center region (\(\Phi\in[-45^{\circ},+45^{\circ}]\), \(Z\in[20\,\mathrm{mm},60\,\mathrm{mm}]\)) is assumed. Then, equidistant points are generated using Equation (31) for constant \(R\) with a distance \(\Delta d\) in each \(\Phi\)- and \(Z\)-direction. In a first investigation, the pointwise relative error of the maximum principal stretch of the RBF-approach relative to the analytical (exact) stretch \(\lambda_{\mathrm{ex}}=\lambda_{2}\) in Equation (35)  
\[e_{\lambda_{k}}=\frac{\left|\lambda_{k}^{\mathrm{DIC}}-\lambda_{\mathrm{ex}} \right|}{\lambda_{\mathrm{ex}}}\times 100, \tag{36}\]  
is defined. Here, \(\Delta d=0.15\,\mathrm{mm}\) is chosen. Further, the Wendland function \(\hat{m}_{W}(\rho)\) combined with the linear polynomial (monomials), see Table 1, with \(R_{0}=1.5\,\mathrm{mm}\) is applied to visualize the boundary effects, see Figure 2.  
Figure 1: Tension–torsion example (smooth surface): generated (pattern) points on the surface in the reference (green) and calculated positions in the current configuration (dark red). Although the points are fulfilled exactly, the derivatives--here represented by the stretches--show larger errors at the boundary of the evaluation region, which is one of the weaknesses of the RBF-method that were found in the scope of the study. This will be considered in the following investigation. In the following figures, the mean value of all errors is calculated  
\[e_{m}=\frac{1}{n^{\text{DIC}}}\sum_{k=1}^{n^{\text{DIC}}}e_{\lambda_{k}}, \tag{37}\]  
in a region of interest. \(n^{\text{DIC}}\) is the number of data points considered.