## 1 Introduction  
In recent years, the design and fabrication of metamaterials have demonstrated exotic effects and phenomena for all types of travelling waves, from electromagnetic [1] to mechanical [2]. The terminology of metamaterial was first used to describe a class of structured materials (or composites) for which wave effects arise as a collective manifestation of locally resonant constituent units [3]. The resonant frequency of a metamaterial's subunit depends only on its inertia and restoring force, meaning that the incident wavelength may be several orders of magnitude greater than the physical dimension of the constituent subunits. This sub-wavelength property is characteristic of all metamaterials; therefore, it is now widely accepted that the definition has been broadened to include all sub-wavelength systems that support exotic wave effects not found in nature. Examples for mechanical metamaterials and phononic crystals include negative refraction [4], superlensing [5], filtering [6], and cloaking [7]. The nature of metamaterial resonant subunits affects the wave propagation effects, but the most commonly used class of metamaterials is characterised by periodic repetitions of unit cells. Analytical techniques use Bloch-Floquet theory to construct dispersion diagrams, from which stop and pass bands are easily identifiable, as well as special features such as Dirac points and standing modes [8]. Structured elastic Kirchhoff-Love plates, comprising periodic distributions of scatterers and also referred to as platonic crystals [9], have attracted great interest in the last fifteen years. These two-dimensional elastic plate analogues of photonic crystals feature many of the anisotropic effects typically observed in photonics, such as ultra-refraction, Dirac-like cones and topological insulator, and edge mode properties for flexural [10; 11; 12; 13; 14] waves. Similar effects have also been observed for in-plane elastic waves [15; 16; 17]. This article considers a two-dimensional locally resonant metamaterial for in-plane elastic waves: a plain-strain, homogeneous medium is patterned with a doubly periodic array of cross-shaped perforations. Previous studies related to similar systems include the articles by [PERSON] et al. [18] and [PERSON] et al. [19]. The focus of [18] was a polyvinyl chloride phononic plate and the identification and location of complete band gaps for a square lattice of crosslike holes using numerical methods and experimental validations. Very good agreement between the predicted band gap frequencies and the experimental results was found. In order to perform ultrasonic pitch-catch experimental tests, the plate was machined to produce hollow and rounded cross-cylinder inclusions [18], with a tolerance of 0.1 mm for each cross (with characteristic length 18 mm) within a \(20\times 20\) mm\({}^{2}\) unit cell. For incident frequencies up to 50 kHz, the corresponding wavelengths are of the order 18 mm, i.e., of the same order as the inclusions. As stated by [18], the study was motivated by the work of [20], in which an extensive numerical investigation indicated that cross-shaped holes could generate multiple and complete band gaps within this lower frequency range. A subsequent study [19] investigated finite width slabs of a metamaterial patterned with cross-shaped holes, using a semianalytical solution of the relaxed micromorphic model [21; 22; 23] to identify wave scattering properties. Solutions were compared with frequency-domain finite element simulations and the reflection and transmission coefficients (and band gap locations) showed excellent agreement over a range of frequencies and angles of incidence. The dimensions of this material were significantly smaller than those considered by [18], with the area of the unit cell being 400 times smaller (\(1\times 1\) mm\({}^{2}\) compared with \(20\times 20\) mm\({}^{2}\)). Consequently, frequencies were considerably larger with band gaps covering a range from 100 kHz up to 2.5 MHz.