SLOWNESS SURFACES AND ENERGY FOCUSING PATTERNS OF AUXETIC CUBIC MEDIA

Abstract

Abstract: We study properties of slowness surfaces and energy focusing patterns of cubic elastic media. We restricted ourselves to the region of the stability triangle where Poisson’s ratio σ P of the specimen stretched in the [001] direction and measured for [100] is negative, i.e. we consider all cubic auxetic materials. We study properties of surfaces and energy focusing patterns for all elastic auxetic media characterized by σ P = !1/3. 1. INTRODUCTION Recently there is considerable interest in auxetics – elastic materials with negative Poisson’s ratio (cf. [1-3]). Generally the calculation of Poisson’s ratio is complicated for directions oblique to the crystal axes [4]. Therefore, in our previous papers [5, 6] we restricted ourselves to Poisson’s ratio for [001] stretch measured for [100] lateral direction and established the region of the stability triangle (ST) (cf. [6]) where the mentioned Poisson’s ratio is negative. We shall underline that the characteristics of particular region of auxeticity depends on the choice of the stretch direction n and the direction m of measurement [7]. To indicate this dependence we shall use notation nm-auxeticity region of ST. The present paper is devoted to the study of geometrical properties of slowness surfaces of long wavelength acoustic phonons [8, 9] in the [001][100]-auxeticity region (for simplicity we shall use a short notation zx-Poisson’s ratio and zx-auxeticity region). We also obtained the energy focusing patterns [8, 9] which are images of these surfaces under the mapping induced by phonon focusing, which is mathematically known as the Gauss maps (cf. [10]). These maps are obtained in time-of-flight experiments with ballistic phonon beams [8, 9]. 2. CHOICE OF ELASTICITY PARAMETERS 2.1. Partition of the stability triangle Within the framework of theory of elastic media, in a chosen Cartesian coordinate system, the phase c and group velocities v, as well as polarization vectors e of long wavelength acous-