# Empirical Models for the Viscoelastic Complex Modulus with an Application to Rubber Friction
## 2 Measured Viscoelastic Modulus  
When a sinusoidal force is applied to a purely elastic material, the stress and deformation occur concurrently, so that both quantities are in phase. On the contrary, when the same sinusoidal force is applied to a purely viscous fluid, the deformation will lag the stress by one-quarter cycle (\(\pi/2\) radians). This means that when the deformation is maximal, the force is minimal and vice versa. In the case of a viscoelastic material, the phase lag between the force and the deformation lies somewhere between zero and a quarter cycle. Mathematically, a sinusoidal strain can be written using [PERSON]'s formula, as shown in Equation (1), where \(\varepsilon_{0}\) is the amplitude of the strain and \(\omega\) is the frequency:  
\[\varepsilon(t)=\varepsilon_{0}e^{i\omega t} \tag{1}\]  
Similarly, the stress can be written as:  
\[\sigma(t)=\sigma_{0}e^{i(\omega t+\delta)} \tag{2}\]  
where \(\sigma_{0}\) is the amplitude of the stress and \(\delta\) is the phase angle. In Figure 2 the stress and strain phasors are shown as they rotate at a frequency \(\omega\), with the strain lagging behind the stress by an angle \(\delta\). Figure 1: Basic rheological models. \(k\) is the stiffness of the spring and \(\eta\) the viscosity of the dash-pot, representing the linear and viscous parts of the material, respectively. The following relationships can be deduced from Figure 2:  
\[\sigma^{\prime}_{0}=\sigma_{0}\cos(\delta) \tag{3}\]  
\[\sigma^{\prime\prime}_{0}=\sigma_{0}sin(\delta) \tag{4}\]  
With the above definitions, the dynamic or complex modulus will have a real and an imaginary part. The real or storage modulus is defined as the ratio between the real part of the stress and the strain:  
\[E^{\prime}=\sigma^{\prime}_{0}/\varepsilon_{0} \tag{5}\]  
The imaginary or loss modulus is defined as the ratio between the imaginary part of the stress and the strain:  
\[E^{\prime\prime}=\sigma^{\prime\prime}_{0}/\varepsilon_{0} \tag{6}\]  
Equations (1)-(6) lead to the broadly used complex or dynamic modulus formula [12]:  
\[E^{*}=E^{\prime}+iE^{\prime\prime} \tag{7}\]  
Finally, the tangent modulus, also known as the loss tangent is defined as:  
\[\tan\delta=E^{\prime\prime}/E^{\prime} \tag{8}\]  
By definition, the modulus of a material is considered as the overall resistance of the material to an applied deformation. Due to its viscoelastic nature, the rubber modulus is split into elastic (storage), \(E^{\prime}\), and viscous (loss), \(E\"\), components, denoting the ability of the material to store and dissipate energy as heat, respectively. The real and imaginary components of a viscoelastic modulus are collectively referred to as the material's complex modulus. Equally important is the loss tangent or \(\tan\delta\), which is defined as the ratio of energy loss to energy storage. The area near the peak value denotes high energy loss and is considered as the area where the tyre is designed to be operating most of the time. While spring-dashpot-based analytical models are useful for representing the physics of viscoelastic materials, the experimentally measured complex modulus as a function of excitation frequency is often used instead. In the case of rubber, the complex modulus is  
\[\sigma(t)=\sigma_{0}e^{i(\omega t+\delta)} \tag{2}\]0 A representation of the harmonic stress and strain. shown to be strongly linked to friction [13; 14]. Figure 3 shows the typical complex modulus of a rubber used in tyre manufacturing. Such moduli of rubber materials are obtained using oscillatory strain in tension or shear form, while monitoring the resulting stress [15]. When rubber is dynamically stretched and released, the returned energy is less than the energy put into the rubber in the first place. This viscoelastic effect can only be tested dynamically. Dynamic mechanical analysis (DMA) instruments measure the viscoelastic modulus in response to applied oscillating strain. The stress response to the deformation is recorded as a function of time or frequency.