## 2 General Approach to Homogenization  
It was shown in the study by [38] that the elastic properties of a phonon crystal are based on the basic physical ideas of [PERSON]'s law and [PERSON]'s second law. Using [PERSON]'s notation, the matrix-unified representation of these two laws is as follows:  
\[\left[ \begin{array}{cccc}0_{3}&\
abla_{3\times 6}\\ (\
abla_{3\times 6})^{\mathrm{T}}&0_{6}\end{array} \right]\mathbf{v}(\mathbf{r})= \Omega\ A(\mathbf{r})\mathbf{v}(\mathbf{r}), \tag{1}\]  
where:  
\[\
abla_{3\times 6}=\left( \begin{array}{cccc}\
abla_{1}&0&0&0&\
abla_{3}& \
abla_{2}\\ 0&\
abla_{2}&0&\
abla_{3}&0&\
abla_{1}\\ 0&0&\
abla_{3}&\
abla_{2}&\
abla_{1}&0\end{array} \right), \tag{2}\]  
\[[\mathbf{v}(\mathbf{r})]^{T}=(u_{1},u_{2},u_{3},\sigma_{1},\sigma_{2},\sigma_ {3},\sigma_{4},\sigma_{5},\sigma_{6}), \tag{3}\]  
\[\Omega=\left[ \begin{array}{cc}-\omega^{2}I_{3}&0_{3\times 6}\\ 0_{6\times 3}&I_{6}\end{array} \right]. \tag{4}\]  
In the above expressions, \(\omega\) is the frequency, \(I_{i}\) and \(0_{i\times j}\) are the unity and zero matrices, the order of which is indicated by the subscripts, and \([\mathbf{v}(\mathbf{r})]^{T}\) corresponds to the \(9\times 9\) column vector with the components of the displacement and stress vectors. The matrix \(A(\mathbf{r})\) of order \(9\times 9\) is a periodic function of the coordinate \(\mathbf{r}\) and is expressed in terms of the mass density \(\rho(\mathbf{r})\) and the \(6\times 6\) compliance tensor, \(S(\mathbf{r})\), so this can be cast as:  
\[A(\mathbf{r})=\left[ \begin{array}{cc}\rho(\mathbf{r})I_{3}&0_{3\times 6}\\ 0_{6\times 3}&S(\mathbf{r})\end{array} \right]. \tag{5}\]  
Using the previous definitions and following the homogenization methodology, based on the plane-wave expansion indicated in the study by [38], the effective tensors of the homogenized phononic crystal \(\overline{\rho}_{eff}\) and \(\overline{S}_{eff}\), as a function of the wave vector \(\mathbf{k}\) and the frequency \(\omega\), are defined by the following effective non-local response matrix:  
\[A_{eff}(\mathbf{k})=i\Omega^{-1}\{D^{-1}(\mathbf{k};0,0)\}^{-1}+i\Omega^{-1}\left[ \begin{array}{cc}0_{3}&K_{3\times 6}(\mathbf{k})\\ (K_{3\times 6}(\mathbf{k}))^{T}&0_{6}\end{array} \right]=\left[ \begin{array}{cc} \overline{\rho}_{eff}(\mathbf{k},\omega)&0_{3\times 6}\\ 0_{6\times 3}&\overline{S}_{eff}(\mathbf{k},\omega)\end{array} \right], \tag{6}\]  
where the matrix elements of \(K_{3\times 6}\) are:  
\[K_{3\times 6}(\mathbf{k})=\left( \begin{array}{cccc}k_{1}&0&0&0&k_{3}&k_{2}\\ 0&k_{2}&0&k_{3}&0&k_{1}\\ 0&0&k_{3}&k_{2}&k_{1}&0\end{array} \right), \tag{7}\]being \(D^{-1}(\mathbf{k};0,0)\), a matrix block of size \(9\times 9\) obtained from the inverse of the matrix of infinite size \(D^{-1}(\mathbf{k};\mathbf{G},\mathbf{G}^{{}^{\prime}})\), such that:  
\[D(\mathbf{k};\mathbf{G},\mathbf{G}^{{}^{\prime}})=-\left[ \begin{array}{cc}0_{3}&K_ {3\times 6}(\mathbf{k}+\mathbf{G})\\ \left(K_{3\times 6}(\mathbf{k}+\mathbf{G})\right)^{T}&0_{6}\end{array} \right]\delta_{ \mathbf{G},\mathbf{G}^{{}^{\prime}}}-i\Omega A\big{(}\mathbf{G}-\mathbf{G}^{ {}^{\prime}}\big{)}, \tag{8}\]  
with \(\mathbf{G}\) and \(A(\mathbf{G}-\mathbf{G}^{{}^{\prime}})\) as the reciprocal lattice vector and the Fourier coefficient of the matrix \(A(\mathbf{r})\), respectively. The long-wavelength regime corresponds to the lower part of the band structure, that is, when \(\mathbf{k}\to 0,\omega\to 0\). Applying this criterion to Equation (6), the \(A_{eff}(\mathbf{k}\to 0,\omega\to 0)\) matrix corresponding to the quasistatic limit takes the form:  
\[A_{eff}(\mathbf{k}\to 0)=i\lim_{\mathbf{k}\to 0}\Omega^{-1}\Big{\{}D^{-1}(\mathbf{k};0,0) \Big{\}}^{-1}=\left[ \begin{array}{cc}\overline{p}_{eff}&0_{3\times 6}\\ 0_{6\times 3}&\overline{s}_{eff}\end{array} \right].