# 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
Inserting this basis into the motion (42) yields  
\[\vec{x}=\vec{\chi}_{\text{R}}(\vec{X},t)=\bar{r}(Z)\cos(DZ)\vec{E}_{R}(\Phi)+\bar {r}(Z)\sin(DZ)\vec{E}_{\Phi}(\Phi)+\hat{z}(Z)\vec{E}_{Z}. \tag{44}\]  
The in-plane deformation gradient (1) requires the tangent vectors  
\[\vec{a}_{1}=\frac{\partial\vec{x}}{\partial\Phi}=\hat{a}_{R}(Z) \vec{E}_{R}(\Phi)+\hat{a}_{\Phi}(Z)\vec{E}_{\Phi}(\Phi) \tag{45}\] \[\vec{a}_{2}=\frac{\partial\vec{x}}{\partial Z}=\hat{b}_{R}(Z) \vec{E}_{R}(\Phi)+\hat{b}_{\Phi}(Z)\vec{E}_{\Phi}(\Phi)+\hat{b}_{Z}(Z)\vec{E}_ {Z} \tag{46}\]  
with  
\[\hat{a}_{R}(Z)=-\bar{r}(Z)\sin(DZ),\qquad\qquad\qquad\hat{a}_{\Phi }(Z)=\bar{r}(Z)\cos(DZ), \tag{47}\] \[\hat{b}_{R}(Z)=\bar{r}^{\prime}(Z)\cos(DZ)-D\bar{r}(Z)\sin(DZ), \quad\hat{b}_{\Phi}(Z)=\bar{r}^{\prime}(Z)\sin(DZ)+D\bar{r}(Z)\cos(DZ), \tag{48}\]  
and \(\hat{b}_{Z}(Z)=\hat{z}^{\prime}(Z)\). When calculating the gradient vectors \(\vec{A}^{\alpha}\), the tangent vectors  
\[\vec{A}_{1}=\frac{\partial\vec{X}}{\partial\Phi}=\hat{R}(Z)\vec{E}_{\Phi}( \Phi),\quad\vec{A}_{2}=\frac{\partial\vec{X}}{\partial Z}=\hat{R}^{\prime}(Z) \vec{E}_{R}(\Phi)+\vec{E}_{Z} \tag{49}\]  
are required. Here, the property \(\vec{E}^{\prime}_{R}(\Phi)=\vec{E}_{\Phi}(\Phi)\) is exploited (the prime means the derivative with respect to the argument of the function). The inverse of the matrix of covariant metric coefficients  
\[[A_{\alpha\beta}]= \begin{bmatrix}\hat{R}^{2}(Z)&\\ &1+\hat{R}^{\prime 2}(Z)\end{bmatrix}  \tag{50}\]  
i.e.,  
\[[A^{\alpha\beta}]= \begin{bmatrix}1/\hat{R}^{2}(Z)&\\ &1/(1+\hat{R}^{\prime 2}(Z))\end{bmatrix}  \tag{51}\]  
yields together with \(\vec{A}^{\alpha}=A^{\alpha\beta}\vec{A}_{\beta}\) the contravariant vectors  
\[\vec{A}^{1}=\frac{1}{\hat{R}(Z)}\vec{E}_{\Phi},\quad\vec{A}^{2}=\frac{\hat{ R}^{\prime}(Z)}{1+\hat{R}^{\prime 2}(Z)}\vec{E}_{R}(\Phi)+\frac{1}{1+\hat{R}^{ \prime 2}(Z)}\vec{E}_{Z}. \tag{52}\]  
In the following, the abbreviations  
\[\hat{A}_{\Phi}(Z)=\frac{1}{\hat{R}(Z)},\quad\hat{B}_{R}(Z)=\frac{\hat{R}^{ \prime}(Z)}{1+\hat{R}^{\prime 2}(Z)},\quad\hat{B}_{Z}(Z)=\frac{1}{1+\hat{R}^{ \prime 2}(Z)} \tag{53}\]  
are used. Evaluating the sum of dyadic products (1) leads to the in-plane deformation gradient  
\[\hat{\mathbf{F}}= \begin{bmatrix}b_{R}B_{R}&a_{R}A_{\Phi}&b_{R}B_{Z}\\ b_{\Phi}B_{R}&a_{\Phi}A_{\Phi}&b_{\Phi}B_{Z}\\ b_{Z}B_{R}&0&b_{Z}B_{Z}\end{bmatrix} \vec{E}_{I}\otimes\vec{E}_{I}, \tag{54}\]  
\(I,J=R,\Phi,Z\), and the in-plane right Cauchy-Green tensor  
\[\hat{\mathbf{C}}=\hat{\mathbf{F}}^{T}\hat{\mathbf{F}}= \begin{bmatrix}B_{R}^{2} (b_{R}^{2}+b_{\Phi}^{2}+b_{\Phi}^{2})&A_{\Phi}B_{R}(a_{R}b_{R}+a_{\Phi}b_{\Phi })&B_{R}B_{Z}(b_{R}^{2}+b_{R}^{2}+b_{Z}^{2})\\ A_{\Phi}B_{R}(a_{R}b_{R}+a_{\Phi}b_{\Phi})&A_{\Phi}^{2}(a_{\Phi}^{2}+a_{\Phi}^{2} )&A_{\Phi}B_{Z}(a_{R}b_{R}+a_{\Phi}b_{\Phi})\\ B_{R}B_{Z}(b_{R}^{2}+b_{\Phi}^{2}+b_{Z}^{2})&A_{\Phi}B_{Z}(a_{R}b_{R}+a_{\Phi}b_{ \Phi})&B_{Z}^{2}(b_{R}^{2}+b_{\Phi}^{2}+b_{Z}^{2})\end{bmatrix} \vec{E}_{I} \otimes\vec{E}_{I}. \tag{55}\]Since the tensor is singular, one eigenvalue is zero. Evaluating Equation (9) with the (full) invariants  
\[\mathrm{I}_{\hat{\mathbf{C}}} =\mathrm{tr}\,\hat{\mathbf{C}} =A_{\Phi}^{2}(a_{R}^{2}+a_{\Phi}^{2})+(B_{R}^{2}+B_{Z}^{2})(b_{R}^ {2}+b_{\Phi}^{2}+b_{Z}^{2}), \tag{56}\] \[\mathrm{II}_{\hat{\mathbf{C}}} =\frac{1}{2}\Big{(}(\mathrm{tr}\,\hat{\mathbf{C}})^{2}-\mathrm{ tr}\,\hat{\mathbf{C}}^{2}\Big{)} =A_{\Phi}^{2}(B_{R}^{2}+B_{Z}^{2})\Big{(}(a_{R}b_{\Phi}-a_{\Phi}b_{ R})^{2}+(a_{R}^{2}+a_{\Phi}^{2})b_{Z}^{2}\Big{)}, \tag{57}\]  
are now depend on the axial coordinate \(Z\), i.e., the principal stretches are  
\[\lambda_{1,2}(Z)=\sqrt{\mu_{1,2}(Z)}\text{ with }\mu_{1,2}=\frac{\mathrm{I}_{ \hat{\mathbf{C}}}}{2}\pm\sqrt{\frac{\mathrm{I}_{\hat{\mathbf{C}}}^{2}}{4}- \mathrm{II}_{\hat{\mathbf{C}}}}. \tag{58}\]  
(full means that the basis is not in the tangential plane so that the trace has to be built over the entire tensor coefficients).