# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 3 Numerical Model of Viscoelastic Dampers  
The storage modulus \(G^{\prime}\) and the loss modulus \(G^{\prime\prime}\) control the shape of the hysteretic cycle of a VE material. The dependency of the VE material on the load frequency, ambient temperature, and self-heating is then reflected on these parameters. As an example, the dependency of the storage modulus and the loss modulus on strain rate and ambient temperature is shown in Figure 4 for the elastomer ISD-110 tested by [PERSON] et al. [8] for different load frequencies and with the control of the ambient temperature. The dependency of \(G^{\prime}\) and \(G^{\prime\prime}\) on strain rate is due to the molecular structure of elastomers: the faster the strain is applied, the less time molecular chains have to unroll. Figure 3: Vectorial representation of the complex modulus \(G^{*}\). Therefore, an increase in the strain rate determines an increase in the stiffness and dissipative capability, i.e., an increase in both the storage modulus and the loss modulus. Variations in the ambient temperature affect both the stiffness and dissipation capabilities of the elastomer. An increase in ambient temperature magnifies the rubbery behavior of the material, and both the storage modulus and the loss modulus decrease. Similarly, self-heating of the elastomer due to prolonged operating times (e.g., in the case of wind excitation) decreases the storage and loss moduli. In this paper, the strain rate dependency of the VE material is considered explicitly in nonlinear dynamic analyses through the VE material model introduced in this section. The effects of ambient temperature are taken into account by choosing experimental values of \(G^{\prime}\) and \(G^{\prime\prime}\) and \(G^{\prime\prime}\) and \(G^{\prime\prime}\) obtained for a selected value of the ambient temperature. The effects of the temperature variation (due to self-heating) on \(G^{\prime}\) and \(G^{\prime\prime}\) are neglected in this study. Indeed, the impact of self-heating on the mechanical properties of the elastomer in the case of short-duration events, like earthquakes, is considered negligible by some researchers [12, 19, 48]. Other researchers consider self-heating by means of the explicit calculation of the heat generated in the VE material during the loading history [27, 49]. The resulting temperature variation is then used to evaluate new values of the storage and loss moduli through appropriate shift functions [11, 49, 50] (i.e., functions that provide the values of \(G^{\prime}\) and \(G^{\prime\prime}\) depending on temperature and based on the time-temperature superposition principle). However, this process is not straightforward since it requires updating the material parameters during the numerical analysis. Moreover, this method requires the knowledge of additional properties of the selected VE material (e.g., specific heat capacity, time-temperature superposition functions) that are not always available in the literature. The most common approaches in the literature for the numerical representation of VE materials are linear models (also referred to as integer derivative models) [2, 7] and fractional derivative models [27, 50, 51]. Linear models are constituted by basic elements, i.e., springs and dashpots in various configurations. The simplest possible arrangements of one spring and one dashpot generate the well-known Kelvin-Voight model and Maxwell model, i.e., models with one spring and one dashpot connected in parallel and in series, respectively [51]. Even if these models are both able to reproduce the elliptical hysteretic cycle of the VE material by means of two parameters (i.e., the stiffness of the spring and the viscous constant of the dashpot), these simple models do not catch the strain rate dependency of the elastomer effectively. However, if more complex models are considered (i.e., with a larger number of parameters), the strain rate dependency of the elastomer may be effectively taken into account.