## 1 Introduction  
To date, one of the most widely adopted design approaches for highway bridges subject to static and dynamic conditions is the performance-based design concept [1, 2, 3]. As such, a designer needs better insight into structural requirements and responses in order to determine failure modes of each individual component of a bridge structure prior to the installation of metamaterials in practice. Common elastomeric bridge bearings, also called laminated rubber bearings (LRBs), are comprised of rubber layers and steel shims, resulting in a tradeoff between horizontal flexibility and high vertical stiffness. During dynamic loading, these bearings are vital components in a bridge system because they must alternately experience vertical forces and large horizontal displacements induced by the bridge deck. This leads to tensile (uplift) and compressive deformation to the LRBs, due to the rotation of a pier cap around a transverse axis [4]. In addition, these bearings are vulnerable to tensile loading, resulting in local failures in the interior of rubber as a well-known cavitation phenomenon if they are not bolted through top and bottom external plates [5, 6, 7, 8]. Unlike compression loading, the rubber is capable of easily sustaining large pressures without sustaining damage, as the tensile stresses are very low during the initial cavitation stage [6]. Under exceptional earthquakes, the horizontal displacement can range from 150 percent to more than 300 percent shear strain. Another failure of rubber bearings is rollover for dowelled bearings [9]. These limit states are not examined in this work. In our previous work [10], we reviewed numerous recent studies on the development of elastomeric bridge bearings using fibre reinforcement (fibre-reinforced rubber bearings, FRRBs), which exhibit improved mechanical properties [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22]. In contrast to LRBs, the fibre reinforcement used in FRRBs offers such benefits as light weight, lower stresses in bearings, and simple installation [10]. Nevertheless, the materials used in these bearings, whether reinforced by steel or fibre, can be better designed in order to obtain specific properties under expected loading conditions using metamaterials which cannot be found in nature [23; 24]. Their properties are obtained from the structural design rather than from their constituent materials. Recently, many researchers have investigated auxetic structures used for sandwich cores in a range of engineering applications. Auxetic structures possess the novel property of a negative Poisson's ratio (PR) [9; 25; 26]. Negative PR materials are well-known for their excellent of lateral contraction behaviour and resistance to expansion when exposed to compression and tension, respectively [27; 28; 29; 30; 31; 32; 33; 34]. While common natural materials, also referred to as positive PR materials, exhibit swell behaviour perpendicular to compressive loading and contraction behaviour under tensile loading, as given in Figure 1[25; 35], these auxetic materials provide a wide range of engineering advantages with superior mechanical properties such as shear resistance [9], higher fracture toughness, lower fatigue crack propagation [36; 37; 38; 39; 40; 41], vibration mitigation [42], indentation resistance [43], crashworthiness [44; 45], light weight [46], energy absorption [47; 48], etc. Therefore, auxetic structures are likely to be the ideal core for sandwich panels. A conventional sandwich panel consists of a lightweight core, for instance honeycomb [49; 50; 51; 52; 53; 54; 55], foam [56; 57; 58], etc. They are adhesively connected to two relatively thin facets which have high stiffness and strength. Numerous studies have been performed on sandwich panels with a two-dimensional auxetic metamaterial core to determine their vibration and bending behaviours [49; 50; 51; 52; 53; 54; 55]. On the other hand, there are only a few studies regarding sandwich panels with a three-dimensional auxetic lattice metamaterial core.