# Full-Field Strain Determination for Additively Manufactured Parts Using Radial Basis Functions
## 2 Surface Strain Determination
### Strain Computation in Surfaces  
One difficulty in connection with surface strain measurement is to obtain strains within a curvilinear surface. Thus, the surfaces in the initial configuration \(\vec{X}=\hat{\vec{X}}(\Theta^{1},\Theta^{2})\) and in the current configuration \(\vec{x}=\hat{\vec{X}}(\Theta^{1},\Theta^{2},t)\) have to be approximated. We assume that both functions depend on the same surface parameters \(\Theta^{\alpha}\), \(\alpha=1,2\) (convective coordinates). Let \(\vec{a}_{\alpha}=\partial\hat{\vec{X}}/\partial\Theta^{\alpha}\) be the tangent vectors to the surface description in the current configuration, and \(\vec{A}_{\alpha}=\partial\hat{\vec{X}}/\partial\Theta^{\alpha}\) those in the initial state. Obviously, the functions \(\hat{\vec{X}}(\Theta^{1},\Theta^{2})\) and \(\hat{\vec{x}}(\Theta^{1},\Theta^{2},t)\) must be specified. In a first step, it is assumed that both vectorial functions are known, in Section 2.2 they will be specified by RBFs. The in-plane deformation gradient  
\[\hat{\vec{\mathbf{r}}}=\vec{a}_{\alpha}\otimes\vec{A}^{\alpha}, \tag{1}\]  
(Greek letters that occur twice as indices imply the sum of 1 to 2, see [4]) requires--apart from the tangent vectors \(\vec{a}_{\alpha}\)--the contravariant (gradient) vectors \(\vec{A}^{\alpha}=A^{\alpha\beta}\vec{A}_{\beta}\), \(\beta=1,2\). The matrix of contravariant metric coefficients \(A^{\alpha\beta}=\vec{A}^{\alpha}\cdot\vec{A}^{\beta}\) is obtained by the inverse of the matrix of covariant metric coefficients \(A_{\alpha\beta}=\vec{A}_{\alpha}\cdot\vec{A}_{\beta}\),  
\[\left[A^{\alpha\beta}\right]=\left[A_{\alpha\beta}\right]^{-1}, \tag{2}\]  
see, for example [18]. It should be remembered that \(\vec{a}_{\alpha}\in\mathbb{V}^{3}\) holds even though there are only two tangent vectors (in each configuration). The in-plane right Cauchy-Green tensor  
\[\hat{\vec{\mathbf{C}}}=\hat{\vec{\mathbf{F}}}^{T}\hat{\vec{\mathbf{F}}}=a_{ \alpha\beta}\vec{A}^{\alpha}\otimes\vec{A}^{\beta}, \tag{3}\]  
in which \(a_{\alpha\beta}=\vec{a}_{\alpha}\cdot\vec{a}_{\beta}\) define the covariant metric coefficients in the current configuration, has to be changed to the mixed-variant formulation  
\[\hat{\vec{\mathbf{C}}}=\underbrace{a_{\alpha\beta}A^{\alpha\gamma}}_{\vec{C} ^{\gamma}_{\beta}}\vec{A}_{\gamma}\otimes\vec{A}^{\beta}=\hat{C}^{\gamma}_{ \beta}\vec{A}_{\gamma}\otimes\vec{A}^{\beta} \tag{4}\]  
to arrive at the eigenvalue problem  
\[\left((\hat{C}^{\gamma}_{\beta}-\mu\delta^{\gamma}_{\beta})\vec{A}_{\gamma} \otimes\vec{A}^{\beta})(q^{\alpha}\vec{A}_{\alpha})=[\hat{C}^{\gamma}_{\alpha }-\mu\delta^{\gamma}_{\alpha}]q^{\alpha}\vec{A}_{\gamma}=\vec{0}, \tag{5}\]  
remember \(\vec{A}^{\gamma}\cdot\vec{A}_{\alpha}=\delta^{\gamma}_{\alpha}\) holds, with the Kronecker-symbol \(\delta^{\gamma}_{\alpha}\) (\(\delta^{\gamma}_{\alpha}=1\) for \(\gamma=\alpha\), \(\delta^{\gamma}_{\alpha}=0\) for \(\gamma\
eq\alpha\)). For linearly independent tangent vectors \(\vec{A}_{\gamma}\), the coefficients must vanish (eigenvalue problem) leading to \[\det[\hat{C}^{\gamma}_{\ \beta}-\mu\delta^{\gamma}_{\ \beta}] =\left| \begin{matrix}\hat{C}^{1}_{\ 1}-\mu&\hat{C}^{1}_{\ 2}\\ \hat{C}^{2}_{\ 1}&\hat{C}^{2}_{\ 2}-\mu\end{matrix}