Homogenization theories provide models that simplify the constitutive relations of hetero- geneous media while retaining their macroscopic features. These theories have shown how the governing ﬁelds can be macroscopically coupled, even if they are microscopically independent. A prominent example is the Willis theory which predicted the strain-momentum coupling in elastodynamic metamaterials. Recently, a theory that is based on the Green’s function method predicted analogous electro-momentum coupling in piezoelectric metamaterials. Here, we de- velop a simpler scheme for ﬁbrous piezoelectric composites undergoing antiplane shear waves. We employ a source-driven approach that delivers a unique set of effective properties for arbi- trary frequency-wavevector pairs. We numerically show how the resultant homogenized model recovers exactly the dispersion of free waves in the composite. We also compute the effective properties in the long-wavelength limit and off the dispersion curves, and show that the resul- tant model satisfy causality, reciprocity and energy conservation. By contrast, we show how equivalent models that neglect the electromomentum coupling violate these physical laws. 1 Introduction Homogenization theories provide a simpliﬁed description of the physics of composites using a ﬁctitious homogeneous medium that behaves the same in some appropriate average sense [5, 13, 20, 37, 46, 65, 67]. The simpliﬁed description shows that the behavior of certain composites can be essentially different from the behavior of their comprising constituents, in which case they are called metamaterials [8, 12, 14, 15, 21, 28, 30, 59, 61, 62, 68, 71]. 1 Central to the ﬁeld of metamaterials in elastodynamics is the theory of Willis [72, 73, 74, 78], which was developed at the level of generality of random media, and relies on ensemble averaging to handle the randomness. Notably, the theory also applies when periodicity is present as a particular case. The theory shows that the effective response of a material point is nonlocal in space (i.e., depends on the state of other points) and time (i.e., depends on previous states). The theory further shows that the effective velocity may induce an effective stress, and that the effective strain may induce an effective momentum. These nonlocal couplings—which are absent in conventional materials—are now termed the Willis couplings. To date, the theory is still being reﬁned and analyzed by Willis [77, 79, 80] and others [33, 35, 41, 45, 47, 49, 50, 52, 54, 56, 57, 64, 69]. These theoretical studies are accompanied with experiments that measure and exploit the Willis coupling for applicational purposes [9, 26, 29, 32, 34–36, 38, 42, 53]. As mentioned, the ﬁrst developments of the theory were for elastodynamics. Owing to the similarity in the wave equations of different physical media, the applicability of the Willis theory extends beyond elastodynamics, e.g., to acoustics, ﬂuid dynamics and electromagnetism [43, 54, 75]. In the latter case, the theory predicts that the electric displacement is coupled with the magnetic ﬁeld, and that the magnetic induction is coupled with the electric ﬁeld [75]. Thereby, Willis theory agrees with what is already known in the photonics community as bianisotropic materials, described by the so-called magnetoelectric tensor [7, 23, 36, 57]. Most recently, Pernas-Salomón and Shmuel [49] have extended the approach of Willis to de- scribe the dynamics of elastic composites that exhibit a coupling with thermal1-, electric- or mag- netic ﬁelds at the microscopic scale. The generalized theory for piezoelectric constituents revealed additional couplings of Willis type between the electric displacement and the velocity, and between the momentum and the electric ﬁeld. This result not only introduces new parameters to the design space of metamaterials, but also establishes a different route to manipulate elastic waves using non- mechanical stimuli. Indeed, the recent one-dimensional study by Pernas-Salomón et al. [51] has shown how the electromomentum coupling modiﬁes the effective wave velocity and impedance, and generates asymmetry in the phase angle in addition to the asymmetry that the Willis coupling generates. Pernas-Salomón et al. [51] further showed how this asymmetry can be switched on and off by changing the electrical circuit boundary conditions. The formulae of Willis [74, 75, 79] and Pernas-Salomón and Shmuel [49] are based on the Green’s function of the relevant medium. However, the Green’s function cannot be determined for most problems, and to obtain explicit expressions for the effective properties in the general case there is a need for other methods. In the elastic periodic case, it proved useful to use the plane wave expansion method [24, 58] as part of the averaging process [47, 48, 52]. Here, we extend this approach to obtain explicit expressions for the effective properties of periodic piezoelectric com- 1See also the work of Torrent et al. [70] on thermal materials. 2 posites that undergo antiplane shear waves of Bloch-Floquet type. While this is the simplest setting that admits tensor-valued properties, spatial nonlocality and the number of properties that govern the medium render the problem very rich, as we show in the sequel. Its extension to problems that are more general is conceptually straightforward, however signiﬁcantly more complicated from a technically perspective, and not pursued here. The method in this work shares two key elements with the schemes of Willis [75] and Pernas- Salomón and Shmuel [49]. First, our deﬁnition for effective waves relies on the ensemble aver- aging of the microscopic ﬁelds, which reduces to volume averaging over the periodic part of the microscopic Bloch-Floquet waves. Second, we account for body forces and volume charge in our governing equations, and further introduce an eigenstrain in the microscopic constitutive relations. The main motivation for introducing these sources is mathematical: while the form they have in our formulation is challenging to access experimentally, if at all possible, this form delivers unique set of effective properties that is applicable for general frequency-wavevector pairs, not only for those that are related by the normal modes of the composite. (At the very basic level, without these sources the effective state variables become algebraically dependent; for an in-depth discussion on the issue of uniqueness, the reader is referred to Refs. [40, 45, 50, 79].) Importantly, the effec- tive description that results from this approach satisﬁes basic physical laws, in contrast with other homogenization schemes, as we discuss in more detail later. This type of source-driven homoge- nization is rooted in the works of Fietz and Shvets [16, 17], Alù [4] and Willis [75]. The scheme is applied for two types of mixtures of PMMA matrix with PZT4 inclusions. The two mixtures differ by the shape of the inclusions: one mixture considers circular inclusions and the other considers inclusions without axial symmetry. The dispersion relations that the effective properties predict for free waves (i.e., the normal modes) are shown to be in perfect agreement with those of the original composites. We also apply our homogenization scheme in the long wavelength limit, where the medium can be approximated using local effective properties2, and outside the dispersion curves for arbitrary frequency-wavevector pairs. These calculations show that the effective properties that our homogenization scheme provides respect the mathematical conditions that follow from reciprocity, energy conservation and causality [3, 25, 41, 50, 57, 64, 66]; the homogenized description should respect these conditions, since the components of the composite satisfy them. (Recent works actually seek to break these restrictions [44] using active components and digital control devices [9–11, 53, 81]. With such electronic circuits, researchers were able to engineer the injected energy and all the effective properties independently, thereby generating nonreciprocal, nonconservative response.) As mentioned, in the absence of sources it is possible to deﬁne alternative homogenized mod- 2Milton et al. [36] were the ﬁrst to introduce equations with local cross-coupling terms in elastodynamics, hence these equations can referred to as the Milton-Briane-Willis equations [39]. 3 els. By construction, these models may reproduce certain features in the macroscopic response of the composite, e.g., recover the dispersion curves, while neglecting the electromomentum cou- pling. We adopt the terminology in Refs. [4, 57] and refer to these models as equivalent models with equivalent properties. We calculate the equivalent properties on the dispersion curves and in the long wavelength limit, and show that they violate reciprocity and energy conservation. This observation supports our source-driven homogenization scheme, and speciﬁcally the need for the electromomentum tensor in the homogenized model to be physically admissible. It is in direct analogy with the observations of Alù [3] and Sieck et al. [57], that the bianisotropic and Willis tensors are needed in order to obtain physical effective properties of electromagnetic- and acoustic composites, respectively. Our results are presented as follows. Sec. 2 summarizes the governing equations and introduces our deﬁnition of effective ﬁelds in periodic piezoelectric media. Sec. 3 develops the source-driven homogenization scheme for antiplane shear waves of Bloch-Floquet type in two-dimensional piezo- electric composites, using the plane wave expansion method. A numerical study is carried out in Sec. 4, followed by a summary of our main results and insights in Sec. 5. 2 Governing equations and effective ﬁelds The state of a piezoelectric medium is determined from the equations of motion in conjunction with the quasi-electrostatic approximation of Maxwell’s equations. This approximation is valid when at the considered frequencies, the mechanical wavelengths are much shorter than the electromagnetic wavelengths. Accordingly, the equations are Ñsss+f=˙p;ÑD=q;ÑE=000; (1) where fis the body force density, sssis the Cauchy stress, pis the linear momentum, Dis the electric displacement, qis the charge density and Eis the electric ﬁeld. It is convenient to identically satisfy Eq. (1) 3using an electric potential f, such that E=Ñf. The state variables of the medium are related through the constitutive relations 0 B@sss D p1 CA=0 B@C BT0 BA0 0 0 r1 CA0 B@Ñuhhh Ñf ˙u1 CA; (2) where uis the displacement ﬁeld and hhhis an inelastic strain, the purpose of which is discussed in detail in Refs. [49, 50, 75, 79]; the symbols r;A;B;Cdenote the mass density and the dielec- tric, piezoelectric and elasticity tensors, which are functions of the position xfor a heterogeneous 4 medium. In the sequel, we refer to the quantities in the right- and left column vectors as kinematic- and kinetic variables, respectively. The objective of homogenization theories is to replace the description of the heterogeneous medium by a ﬁctitious homogeneous medium that effectively be- haves the same in some appropriate sense. This is carried out by ﬁrst deﬁning average effective ﬁelds, and then determining the effective constitutive relations between them. Here, we focus on time-harmonic Bloch-Floquet waves in periodic media, and adopt the deﬁnition of Willis [75], Alù [4] and the