## 1 Introduction  
Elastic metamaterials are architecturally engineered structures that exhibit unusual properties to control the propagation of elastic waves through their structures [1, 2]. Elastic metamaterials can be composed of periodically arranged phononic crystals (PnCs) that control elastic waves at a specific frequency range where the incoming wave cannot propagate--named stop bands or band gaps. The observation of the band gap results from two possible mechanisms: Bragg scattering and localized resonances of elastic waves [3]. The observation of the Bragg scattering results from the interaction of elastic waves scattered by periodic unit cells. The localized resonance is elastic waves locally oscillated around the center of the inclusion structure, mainly depending on the inclusion structure itself. Therefore, by tuning the geometry, size, arrangement, and shape of the scatterers within the soft host, it could be possible to achieve band gaps with a wider frequency range by combining these two mechanisms [4]. This is useful for a variety of applications such as noise and vibration isolation [5, 6, 7, 8, 9], energy harvesting [10, 11, 12, 13], and acoustic and elastic filters and waveguides [14, 15, 16, 17]. PnCs are typically designed by placing a high-stiffness material inclusion in a low-stiffness (soft) matrix [18]. Previous studies have reported that multiple bandgaps can be realized using different shapes of inclusions within a soft matrix. For example, [PERSON] and [PERSON] achieved low-frequency band gaps using two- and three-component sphere inclusions [19]. [PERSON] et al. achieved complete multiple band gaps using two-dimensional phononic crystal slabs with periodic cross-like holes [20]. [PERSON] et al. achieved attenuation of broadband vibration using continuous carbon fiber-reinforced plastics with steel disks [21]. Elastic band gaps of PnCs could also be achieved by other different shapes of inclusions, such as comb-like inclusions [22], spindle shaped [23], quasi-Sierpinski carpet [24], and square/triangle shaped [25; 26]. Among these studies, the geometric effect of the inclusions on the bandwidth and central frequency of the band gap are discussed by changing the size, inclusion number, and filling fraction of the unit cell. However, the effect of irregular size and locations of inclusions within a unit cell are rarely discussed but will be included in this work. Another critical factor in determining the bandwidth and central frequencies of the bandgaps is the material properties of the inclusions and matrix used in a phononic crystal [27], which includes mass density ratio, the shear modulus ratio and Poisson's ratios. Several studies have shown that the central frequencies of the band gap can generally be increased by increasing Poisson's ratio of the inclusion/host material [28; 29; 30]. In addition, the increase in density and Young's modulus ratios increase the bandwidth of the band gaps [31]. Recently, tungsten carbide inclusion and epoxy matrix has been used in different elastic metamaterials for broadband elastic attenuation applications owing to their high Young's modulus ratios [32; 33]. For example, [PERSON] et al. have demonstrated that a phononic crystal composed of tungsten carbide and epoxy can achieve wider broadband phononic band gaps [34]. [PERSON] et al. achieved wide elastic bandgaps using the optimized structure of tungsten carbide spheres embraced in an epoxy matrix [35]. To realize a broad elastic bandgap in the low-frequency range, this work presents a numerical study of a periodic elastic composite composed of tungsten carbide within the epoxy host. The effects of square inclusion size within the unit cell (filling fraction), the stiffness of square inclusion, number of square inclusions, rotation angles of square inclusions, graded inclusion size, and irregular inclusion locations on their bandgap are studied systematically. A relationship between the resulting band gaps and the geometric dimensions is provided.