## 1 Introduction  
The study of understanding the physical properties and mechanical behaviors of granular media has always been a subject of interest [1; 2; 3; 4; 5; 6]. Investigating the propagation characteristics of elastic waves helps to better understand the mechanical response of granular materials to external loading. For example, for geotechnical materials, the wave velocity can be used as an important mechanical index to assess the small strain modulus [7] and anisotropy [8], and the elastic wave theory can be applied to engineering fields, such as foundation liquefaction, geological exploration, and structural seismic resistance [9]. Generally, there are two classes of method used in modeling the mechanical behaviors of granular media, called discrete element method and equivalent continuum method. In recent years, due to the rapid development of computer technology and numerical software, many breakthroughs have been achieved in understanding the static and dynamic properties of granular materials from a microscale viewpoint based on the discrete element method [10; 11; 12]. On the other hand, the continuous method is always a convenient and effective method to describe the mechanical properties of materials [13; 14; 15]. For granular materials, it is widely accepted that the modelling of granular media should incorporate the microscale information from particle level to capture more comprehensive microscale properties and hence model more complex mechanical behaviors. On this basis, many researchers have attempted to develop multiscale constitutive models considering the effect of microscale(particle) kinematics. For example, some micromechanics-based models considering particle kinematics and/or their high-order gradient terms are established [16; 17; 18; 19; 20]. As for wave propagation in granular media, many researchers have proved that the micromechanics-based models, such as the Cosserat model, micropolar model, high-gradient model, and other multiscale models, can describe well the propagational behaviors of plane waves and surface waves [21; 22; 23; 24]. Recently, when considering the local fluctuation of particle kinematics within granular media, [PERSON] et al. [25; 26; 27] developed a micromorphic model for granular materials based on micromechanical analysis and apply it to study the dispersion behaviors and bandgap features of elastic waves. The micromorphic model, also called the microstructure model, was firstly established by [PERSON] and [PERSON] [28; 29] for solids with microstructure. Based on the work of [PERSON] et al., [PERSON] et al. [30; 31] proposed a simplified micromorphic model considering only the first-order gradient of particle displacement and rotation and then also analyzed the propagation characteristics of plane waves. It is generally assumed that the granular materials are homogeneous and isotropic in these micromorphic (or microstructure) models. However, granular media, especially for geotechnical materials, show different degrees of anisotropy due to their depositional conditions or external loading effects. Furthermore, much research has shown that the fabric anisotropy has a significant impact on the macroscale and microscale mechanical properties of granular materials [32; 33; 34; 35]. In addition, the relative displacement and rotation at contacts are considered as basic independent variables in these models [25; 26; 27; 30; 31]. However, particle displacement and rotation are interrelated. It would lead to problem of double counting in the calculation of strain energy and kinematic energy in their models. Different from the existing micromorphic models mentioned above, in this paper, the particle translation and rotation are selected as basic independent variables in this model and the particle displacement at contact due to particle rotation is ignored for simplicity. In this way, the problems of double counting in the calculation of strain energy and kinematic energy would disappear because these two variables are naturally independent.