# Acoustic Applications of a Foamed Geopolymeric-Architected Metamaterial
## 2 Materials and Methods
### Methods
#### 2.3.3 Dynamic Mechanical Analysis Test Procedures and Theoretical Bases  
The dynamic mechanical analyses have been run on a Mettler Toledo Dynamic Mechanical Analyzer (DMA-SDTA 1+) operating in shear force (max 0.5 N) and displacement (max 10 \(\upmu\)m) control modes. The oscillatory shearing viscoelastic properties of the geopolymer were monitored in a frequency sweep from 10\({}^{-3}\) to 10\({}^{3}\) Hz. The dynamic mechanical characterisation discerns and isolates the viscous and elastic components of the system's mechanical response to an external oscillatory solicitation (Figure 3a). Two identical size specimens, 10 \(\times\) 10 \(\times\) 5 mm, as reported in Figure 4, have been symmetrically sandwiched in the sample assembly reported in Figure 3b (with a 5 mm thickness gap) between three circular steel plates of 20 mm diameter. The two stationary outer parts of the sample assembly are connected with a force sensor (the white part of the sample holder in Figure 3b) while the central disk moves at a controlled sinusoidal strain of steady amplitude (maximum displacement \(\Delta\)x = 10 \(\upmu\)m) and oscillation frequency f (or expressed as the angular frequency of \(\varpi\) = f 6.28 rad/s):  
\[\gamma(t)=\gamma_{0}\sin\varpi\ t. \tag{4}\]Figure 4: Morphology of the foamed geopolymers specimens 10 \(\times\) 10 \(\times\) 5 mm used in the DMA tests (Magnification \(\times\) 10). Figure 3: Viscoelasticity principles and analysis tools: (**a**) Shear behaviours in purely viscous liquids and purely elastic solid; (**b**) DMA shear test tool and viscoelastic behaviour measurements; (**c**) Vector representation of complex shear modulus and storage and loss components relationships. Due to the intrinsic characteristics of this test, we may be able to exploit the results in terms of viscous and elastic components derived from the experimentally measured complex shear moduli (enclosing both energy dissipative and storage contributions) according to the mechanical response time delay (\(\delta\)) of the material to an imposed oscillatory deformation. The force sensor detects the counter forces to keep the side plates in position:  
\[\mathrm{F/A=\tau(t)=\tau_{0}\sin\left(\varpi\ t+\delta\right)}. \tag{5}\]  
The recorded shear stress \(\tau(t)\), once correlated with the applied sinusoidal shear rate \(\mathrm{d}\mathrm{y}(t)/\mathrm{d}t\) of (Equation (1)), measures the apparent shear modulus \(\mathrm{G^{*}}\) containing both the elastic and viscous responses of the sample:  
\[\mathrm{G^{*}=\tau(t)/\mathrm{d}\mathrm{y}(t)/\mathrm{d}t}. \tag{6}\]  
The vectorial representation of the apparent modulus with a real and an imaginary part is used to account for the two components and separate them using the delay time \(\delta\):  
\[\mathbf{G^{*}=G^{\prime}+iG^{\prime\prime}}, \tag{7}\]  
the delay time \(\delta\), defined in Figure 3b, is the phase angle shift indicating the direction of the complex modulus (previously named _apparent modulus_) \(\mathbf{G^{*}}\) (blue vector in Figure 3c). At the same time, the two components are the vectors \(\mathbf{G^{\prime}}\) and \(\mathbf{G^{\prime\prime}}\) (green and red, respectively) with magnitude:  
\[\mathrm{G^{\prime}=G^{*}\sin(\delta)\qquad Storage\ modulus,} \tag{8}\]  
\[\mathrm{G^{\prime\prime}=G^{*}\cos(\delta)\qquad\qquad Loss\ modulus.} \tag{9}\]  
The direction of the elastic vector (storage modulus), which belongs to the purely elastic behaviour of the tested medium, is parallel to the direction of the applied deformation (the horizontal axis in Figure 3c). Conversely, the direction of the viscous vector (loss modulus) is orthogonal to that of the applied deformation (vertical axis in Figure 3c). Phase shift angles ranging from \(0^{\circ}\) (for purely elastic behaviour) to \(90^{\circ}\) (for purely viscous behaviour) indicate the relative predominance of the material elastic-storage (\(G^{\prime}\)) and viscous-dissipative (\(G^{\prime\prime}\)) varying responses of the material under mechanical deformation.