## 2 Materials and Methods
### Deriving New Quantities from Full-Field Receptances
#### 2.2.2 Estimating the Stress FRFs  
The _Stress FRF tensor_\(\mathbf{H}_{\mathbf{\sigma}}(x,y,\omega)\in\mathbb{C}\) can be evaluated only indirectly from the _total Strain FRFs_ in Equation (3), once a functional relation \(\mathcal{G}_{cm}()\) is used as the constitutive model13 of the specific material of the component. Note that the components of the _Stress  
Figure 3: Examples of SLDV-based _von Mises equivalent stress FRF maps_ from optical techniques in Equation (7) from shaker 1: at 121 Hz in **(a)**, at 127 Hz in **(b)**, at 246 Hz in **(c)**, at 250 Hz in **(d)**, at 279 Hz in **(e)**, at 284 Hz in **(f)**.  
FRF tensor_\(\mathbf{H}_{\sigma}(x,y,\omega)\) are expressed in [1/m\({}^{2}\)] and can be here obtained from Equation (B.2) as:  
\[\mathbf{H}_{\sigma}(x,y,\omega)=\mathcal{G}_{cm}(\mathbf{H}_{e}(x,y,\omega))=\mathcal{G} _{cm}(\mathcal{D}_{e}(\mathbf{H}_{d}(x,y,\omega)))\in\mathbb{C}. \tag{5}\]  
The same approach can be followed with any specific material (anisotropic and locally linearised included), by means of the deployment of the proper functional relation \(\mathcal{G}_{cm}()\).  
Once the _experiment-based Stress FRF tensor_\(\mathbf{H}_{\sigma}(x,y,\omega)\) in Equation (5) is diagonalised, the _Principal Stress FRF maps_\(\mathbf{H}_{\mathbf{P}\sigma}(x,y,\omega)\)14\(\in\mathbb{C}\) are obtained, therefore, from the _full-field_ _receptances_\(\mathbf{H}_{d}(x,y,\omega)\), simply as:  
Footnote 14: \(\mathbf{H}_{\sigma}(x,y,\omega)\), \(\mathbf{H}_{\mathbf{P}\sigma}(x,y,\omega)\), \(\mathbf{H}_{\mathbf{P}\sigma}(x,y,\omega)\) are the _First-_, the _Second-_, and the _Third-Principal Stress FRF eigenvalue_ maps. In this plane application, \(\mathbf{H}_{\mathbf{P}\sigma}(x,y,\omega)\) was very small compared to the others.  
\[\mathbf{H}_{\mathbf{P}\sigma}(x,y,\omega)=diag[\mathbf{H}_{\mathbf{P}\sigma},\mathbf{H}_{\mathbf{P} \sigma},\mathbf{H}_{\mathbf{P}\sigma}](x,y,\omega)\in\mathbb{C}. \tag{6}\]  
By following the general multi-axes von Mises Equivalent Stress15 definition, a rearrangement of the _Principal Stress FRF maps_\(\mathbf{H}_{\mathbf{P}\sigma}(x,y,\omega)\) of Equation (6), according to the functional \(\mathcal{F}_{\sigma M}()\) defined in Equation (C.2), can bring the _von Mises Equivalent Stress_\(FR\)s \(\mathbf{H}_{\sigma M}(x,y,\omega)\in\mathbb{C}\), which depend upon only the specific constitutive model \(\mathcal{G}_{cm}()\) and  
Footnote 15: Specifically, see [85; 86; 87] and Appendix C for all the details.  
Figure 4: Examples of SLDV-based _von Mises equivalent stress FRF maps_ from optical techniques in Equation (7) from shaker 1: at 333 Hz in (**g**), at 336 Hz in (**h**), at 496 Hz in (**i**), at 752 Hz in (**l**), at 755 Hz in (**m**), at 991 Hz in (**n**).  
upon the _full-field receptances_\(\mathbf{H_{d}}(x,y,\omega)\), but remain independent from the forcing, while in linearity:  
\[\mathbf{H_{\sigma_{vM}}}(x,y,\omega)=\mathcal{F}_{vM}(\mathbf{H_{p\sigma}}(x,y,\omega)). \tag{7}\]  
Examples of the _von Mises Equivalent Stress FRF_\(\mathbf{H_{\sigma_{vM}}}(x,y,\omega)\) maps in Equation (7) are given in Figures 3,4-5,6, or of single dof graphs in Figure 7, as later recalled in Section 3.1.