# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 2 Dynamic Characteristics of Viscoelastic Materials  
The response of a viscoelastic solid material to a shear deformation history \(\gamma(t)\) is intermediate between the response of an elastic solid material and a viscous fluid. If \(\gamma(t)\) is a harmonic function with amplitude \(\gamma_{0}\) and pulse \(\omega\), the steady-state stress response \(\tau(t)\) of the VE material is lagged by a phase angle \(\delta\) ranging from \(0^{\circ}\) (elastic behavior) to \(90^{\circ}\) (viscous behavior), i.e.,  
\[\gamma(t)=\gamma_{0}\sin\omega t \tag{1}\]  
\[\tau(t)=\tau_{0}\sin(\omega t+\delta) \tag{2}\]  
The steady-state stress-strain hysteretic diagram described by Equations (1) and (2) has the shape of an ellipse centered on the origin and inclined with respect to the \(\gamma\) axe. The inclination of the ellipse depends on the elastic component of the response, while the enclosed area depends on the viscous component, as shown in Figure 2.  
The hysteretic diagram of the VE material can be described by a pseudo-elastic law, i.e.,  
\[\tau(t)=G^{*}\gamma(t) \tag{3}\]  
where \(G^{*}\) is called the \"dynamic modulus\". This quantity accounts for both the elastic and viscous components of the response, whose contribution to the total response is a function of the phase angle \(\delta\). Equations (1) and (2) can be expressed in the frequency domain as follows:  
\[\gamma(i\omega)=\gamma_{0}e^{i\omega t} \tag{4}\]  
\[\tau(i\omega)=\tau_{0}e^{i(\omega t+\delta)} \tag{5}\]  
Figure 1: (**a**) Schematic representation of a braced bay and (**b**) particular of the VED. Figure 2: (**a**) Mechanical response to cyclic deformation and (**b**) stress–strain diagrams for viscoelastic materials. The dynamic modulus is a complex number given by the following relation  
\[G^{*}=\frac{\tau_{0}e^{i(\omega t+\delta)}}{\gamma_{0}e^{i\omega t}}=\frac{\tau_{ 0}}{\gamma_{0}}e^{i\delta} \tag{6}\]  
According to complex notation, the dynamic modulus can also be expressed in polar coordinates  
\[G^{*}=\frac{\tau_{0}}{\gamma_{0}}(\cos\delta+i\sin\delta) \tag{7}\]  
or, equivalently, in Cartesian coordinates  
\[\tau(t)=\tau_{0}\sin(\omega t+\delta) \tag{2}\]0  
where \(G^{\prime}\) is called the \"shear storage modulus\" and represents the elastic component of the material behavior, whereas \(G^{\prime\prime}\) is called the \"shear loss modulus\" and represents the viscous components of the material behavior [13; 35]. By equating Equations (7) and (8), \(G^{\prime}\) and \(G^{\prime\prime}\) can be obtained  
\[\tau(t)=\tau_{0}\sin(\omega t+\delta) \tag{2}\]1  
\[\tau(t)=\tau_{0}\sin(\omega t+\delta) \tag{2}\]2  
According to Equations (7) and (8), the dynamic modulus \(G^{*}\) can be represented as a vector of the complex plane whose angle with the axis of the real numbers is the phase angle \(\delta\), as shown in \[\tau(t)=\tau_{0}\sin(\omega t+\delta) \tag{2}\]3 The intensity of \(G^{*}\) is called the \"complex modulus\" and is given as follows:  
\[\tau(t)=\tau_{0}\sin(\omega t+\delta) \tag{2}\]4  
This mathematical representation is consistent with the actual behavior of VE materials. In fact, when the phase angle tends to \(0^{\circ}\) (elastic behavior), the loss modulus tends to zero. Vice versa, when the phase angle tends to \(90^{\circ}\) (viscous behavior), the storage modulus tends to zero. The ratio between the loss modulus and the storage modulus is called the \"loss factor\" \(\eta\) and represents an indicator of the damping capability of an elastomer [47]  
\[\tau(t)=\tau_{0}\sin(\omega t+\delta) \tag{2}\]5  
Finally, the intensity of the dynamic modulus can also be expressed as a function of \(G^{\prime}\) and \(G^{\prime}\)  
\[\tau(t)=\tau_{0}\sin(\omega t+\delta) \tag{2}\]6  
The complex modulus \(|G^{*}|\), the storage modulus \(G^{\prime}\), the loss modulus \(G^{\prime\prime}\), and the loss factor \(\eta\) are the parameters that will be used in the next sections to characterize the elastomer.