# Accuracy of Dynamic Modulus Models of Asphalt Mixtures Containing Reclaimed Asphalt (RA)
## 1 Introduction  
The dynamic modulus (\(|E^{*}|\)) is a fundamental material property used to characterize the stiffness and elastic response of viscoelastic materials, such as hot mix asphalt (HMA) [1]. It is an essential parameter in the design and analysis of road pavements as it quantifies a material's ability to resist deformation under dynamic (cyclic) loading conditions, commonly experienced by road surfaces due to vehicle traffic and temperature variations. The \(|E^{*}|\) is typically determined through laboratory testing and is defined as the ratio of dynamic stress (\(\sigma\)) to dynamic strain (\(\epsilon\)) experienced by a material under sinusoidal loading (see Figure 1) conditions at various temperatures [2]. Mathematically, it is represented as in Equation (1). Figure 1: Sinusoidal stress (\(\sigma\)) and strain (\(\epsilon\)) waveform in time (\(\sigma\) is the stress, \(\sigma_{0}\) is the amplitude of the stress, \(\varepsilon\) is the strain, \(\varepsilon_{0}\) is the amplitude of the strain, \(\omega\) is the angular frequency of the oscillation, t is the time variable, and \(\phi\) is the phase angle). Under the noted conditions--sinusoidal loading at various temperatures--the asphalt mix behaves mainly in a linear viscoelastic manner, meaning that the material's response to sinusoidal stress is sinusoidal, with a phase lag (\(\Phi\)) between strain and stress (deformation and applied stress). Given this, the dynamic modulus (\(|E^{*}|\)) is calculated. \(|E^{*}|\) has a real part (\(E_{1}\)) and an imaginary part (\(E_{2}\)), as shown in Figure 2, providing essential information about the elastic, viscous, and linear viscoelastic behaviour of an asphalt mixture over a wide range of temperatures and loading frequencies. Various test methods exist to proceed with the calculation of this modulus, with loading type varying depending on the specific test protocol. The load can be applied either in compression (axial loading) in the case of an indirect tension dynamic test or in bending in the case of 2- and 4-point bending tests. In this research, the 4-point bending test (according to EN 12697-26 Annex B) was performed, the dynamic modulus was calculated, and the master curve was then generated [3]. However, dynamic tests, including the 4-point bending test, are known to be lengthy and costly. Therefore, many researchers have explored methods for predicting the modulus, with a view to obtaining an order of magnitude that can be used in pavement design [4; 5; 6; 7; 8]. Investigations have established that there are three main methods of prediction: empirical prediction equations (based on regression analyses), meso-mechanical prediction methods (based on composite material theory), and numerical simulation (based on computer technology) [9]. In the past few years, research focusing on predicting the dynamic modulus of asphalt mixtures containing RA has increased significantly. These studies utilize various techniques, such as empirical, mechanistic, and machine learning-based, as well as hybrid models, to improve the accuracy of predictions of RA-based asphalt mixtures. To name a few, [PERSON] et al. [9] presented an n-layer-inclusions meso-mechanical model to predict the \(|E^{*}|\) of reclaimed asphalt materials. They recommended dividing asphalt layers into N = 50 layers for better prediction since the growth rate of reclaimed asphalt (predicted \(|E^{*}|\)) decreased significantly and became closer and closer after N = 50. [PERSON] et al. [10] evaluated the viscoelastic properties of RA-blended asphalt binders and studied the Witczak model's ability to estimate the dynamic modulus master curves of asphalt mixtures containing reclaimed asphalt. They concluded that the Witczak model is capable of predicting \(E^{*}\) master curves of these types of mixtures. Finally, [PERSON] et al.