# Complex-Eigenfrequency Band Structure of Viscoelastic Phononic Crystals
## 1 Introduction  
Phononic crystals for elastic waves are classical analogs to crystal lattices for phonons [1; 2]. They are described with continuum mechanics and can also be viewed as periodic composites. At the frequencies involved, that go from audible sound (a few Hz), to ultrasound (up to a few tens of GHz), it is usual to neglect material losses in a first approximation. In any solid material, however, material loss is present and ultimately increases with frequency. The Kelvin-Voigt viscoelastic model for isotropic solid materials, that turns [PERSON]'s modulus \(E\) into a complex and dispersive number \(E+\mu\omega\eta\) with \(\omega\) being the angular frequency and \(\eta\) the viscosity, applies well at ultrasonic frequencies. In crystals such as silicon or quartz, material loss can often be modeled by adding an imaginary part to the elastic tensor, in a first approximation proportional to the frequency [3; 4]. As a result, the elastic tensor \(c_{ijkl}\) becomes the complex valued tensor \(c_{ijkl}+\imath\omega\eta_{ijkl}\), with \(\eta_{ijkl}\) the phonon viscosity tensor. Sonic crystals for acoustic waves also contain viscous fluids that can be described by a dynamical viscosity leading to a bulk modulus whose imaginary part increases proportionally to frequency. This applies to common fluids supporting sound propagation, such as air and water.  
In the literature on phononic crystals, having a reliable measure of the effect of material loss on wave propagation has been a standing salient problem, with obvious practical implications [5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. Since the band structure gives the dispersion relation of Bloch waves, i.e., of the eigenmodes of phononic crystals, it is natural to try to generalize it to complex quantities describing attenuation inspace or damping in time. The complex band structure gives the complex-valued Bloch wavevector as a function of real frequency [7; 15; 16]. It is in essence well suited to the description of the attenuation in space of monochromatic waves originating from a source of finite extent excited at a given frequency. There are other situations where the wavevector can be imposed, such as with laser generated ultrasound, but waves will be damped as time goes by after the excitation process. Enhanced energy damping in metamaterials has also been proposed [17; 18]. It is the purpose of this paper to propose a complex-eigenfrequency band structure method that is suited for viscoelastic materials. In the following, an augmented complex eigenvalue problem is defined using an auxilliary field technique, giving direct damping information for every Bloch wave in the band structure. Examples are given for a sonic crystal of rigid scatterers in air, for steel-epoxy phononic crystals, and for a holey silicon phononic crystal. It is found that numerical results follow closely the perturbation theory in Ref. [11], according to which the damping of a given Bloch wave can be estimated at any frequency from a volume average of the viscoelastic constituents weighted by the Bloch wave distribution. In particular, the complex-eigenfrequency band structure \(\omega(k)\) does not capture any spatial propagation information but provides one with information complementary to the complex band structure \(k(\omega)\).