## 3 Mathematical Derivation
### Nonlocal Kernel and Operator
There is an explicit relation between lattice vectors in reciprocal and direct lattices \((\overset{\rightarrow}{l}_{i}\bullet\overset{\rightarrow}{m}_{k})=\delta_{ ik}\), where, \(\delta_{ik}\) is the Kronecker \((\delta)\) symbol. Hence, the similar potential is true for a wave vector \(\overset{\rightarrow}{k^{\prime}}=\overset{\rightarrow}{k}+\overset{ \rightarrow}{S}\), where \(\overset{\rightarrow}{S}\) is a translation vector \((\overset{\rightarrow}{S}=\xi_{1}\overset{\rightarrow}{m}_{1}+\xi_{2} \overset{\rightarrow}{m}_{2}+\xi_{3}\overset{\rightarrow}{m}_{3})\) in reciprocal lattice system. There is a specific periodic relationship between the frequency \((\omega)\) and \(\overset{\rightarrow}{k}\) to enforce the lattice scale relationships [40]. This relationship must satisfy the boundary conditions at the boundaries of the Brillouin zones [40], specifically at the boundaries of the first Brillouin zone. The functional relationship presented by [PERSON] from the lattice dynamics study can be further approximated [41]. The approximated atomic dispersion relation can be written as  
\[\frac{\omega_{i}(\overset{\rightarrow}{k})}{c_{i}\sqrt{(k_{1}^{2}+k_{2}^{2}+k_{3 }^{2})}}=\frac{1}{\left\{1+e_{0}{}^{2}\ell^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}) \right\}} \tag{1}\]  
where, \(i\) is indicate waves modes (show, fast shear and longitudinal), \(\varepsilon_{ls}=e_{0}\ell\) is the intrinsic length scale factor, and \(\ell\) is intrinsic length scale such as lattice dimension, granular distance and so on [41]. The operator \(L\) can be written as  
\[L=(1-\ell_{ls}^{2}\
abla^{2}) \tag{2}\]  
where, \(\
abla^{2}\) is a Laplace operator in three dimensions. The suitable Kernel function can be written as [42]  
\[\kappa(|\mathbf{y}-\mathbf{x}|)=\frac{1}{4\pi}\frac{1}{c_{p}^{2}-c_{s}^{2}} \frac{1}{|(\mathbf{y}-\mathbf{x})|}\left\{\exp\bigl{(}-|(\mathbf{y}-\mathbf{x })|/c_{p}\bigr{)}-\exp(-|(\mathbf{y}-\mathbf{x})|/c_{s})\right\} \tag{3}\]  
where, \(c_{p}^{2}=\frac{(e_{0}\ell)^{2}}{2}\)  
[PERSON] et al. recommended the use of bi-Helmholtz type operator to circumvent the problem of satisfying the boundary condition [42]. According to [PERSON] et al. [42], the atomic dispersion relation can be written as  
\[\frac{\omega_{i}(\overset{\rightarrow}{k})}{c_{i}\sqrt{(k_{1}^{2}+k_{2}^{2}+k _{3}^{2})}}=\frac{1}{\left\{1+e_{0}{}^{2}\ell^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{ 2})+f_{0}{}^{4}\ell^{4}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})^{2}\right\}} \tag{4}\]  
where, \(f_{0}\ell\) is the higher order intrinsic length scale factor. The operator which can explicitly catch the dispersion equation at intrinsic length scale has been derived by several researchers [41,43,44] and adopted in the present study. The suitable operator \(L\) can be written as  
\[L=(1-e_{0}{}^{2}\ell^{2}\
abla^{2}) \tag{5}\]  
where, \(\
abla^{2}\) is a Laplace operator in three dimensions. The Kernel function can be written as [42]  
\[\kappa(|\mathbf{y}-\mathbf{x}|)=\frac{1}{4\pi}\frac{1}{c_{p}^{2}-c_{s}^{2}} \frac{1}{|(\mathbf{y}-\mathbf{x})|}\left\{\exp\bigl{(}-|(\mathbf{y}-\mathbf{x })|/c_{p}\bigr{)}-\exp(-|(\mathbf{y}-\mathbf{x})|/c_{s})\right\} \tag{6}\]  
where, \(c_{p,s}^{2}=\frac{(e_{0}\ell)^{2}}{2}\biggl{(}1\pm\sqrt{1-4\frac{(f_{0}\ell)^ {4}}{(e_{0}\ell)^{4}}}\biggr{)}\)