## 3 Mathematical Derivation  
### Nonlocal Kernel and Operator  
In this study, we utilize the change in characteristics of surface acoustic wave propagation in the piezoelectric crystal, namely the time of flight (TOF) and its spectral signature, to quantify the precursor to damage. Damage initiation and crack evolution in the material is a multi-scale phenomenon. Interaction of the ultrasonic wave with the micro-crack helps in the detection and quantification of the precursor to damage in the material. The damage diagnosis and prognosis in the piezoelectric ceramics rely on the information of angular dispersion relationship, that is, directional changes in quasi-longitudinal and quasi-shear wave velocity profile. Any perturbation in the directional wave velocity is an important aspect that indicates the \"incubation of damage\" at the intrinsic length scale. Here, we can now relate the perturbation in wave velocity with the microscale damages/flaws. The study incorporates the nonlocal elastic theory in the framework of micro continuum mechanics and derives the relationship between angular dispersion and nonlocal parameters via [PERSON]'s solution [37]. [PERSON], and [PERSON] have explained the mathematical framework of the development of the wave dispersion relationship and angular dispersion [37; 38]. The study is efficient in predicting the nonlocal parameter by satisfying the boundary conditions in the first Brillouin zone. Eigenvectors and Eigenvalues from the solution of [PERSON]'s equation will be useful to understand the behavioral changes in the solution due to the incorporation of nonlocal parameters [11]. In this work, our aim is to evaluate the change in TOF of the SAW's in piezoelectric ceramics to quantify the precursor to damage. There are two scenarios to utilize the formulation proposed in nonlocal [PERSON]'s solution for quantification of the damage precursor. In the first case, if ab initio dynamics studies are performed on the damaged state of the material and prior information of nonlocal parameters is available then the perturbed velocity profile can be calculated from the nonlocal [PERSON]'s solution. In the second case, a dispersion curve is plotted for the varied nonlocal parameters using non-local field theory. The experimental wave velocity and the central frequency are overlaid on the dispersion plots to evaluate suitable nonlocal parameters. The nonlocal parameters will provide a parametric understanding of internal damage occurring even before the crack nucleates. The nonlocal parameter can be used in elegant prognostic theories to predict crack nucleation based on the present damaged state at the intrinsic length scale. The above concept is mathematically formulated and key aspects are discussed below. [PERSON] and [PERSON] have employed the similar theoretical model to quantify the precursor to damage in woven carbon composite structure damage under influence of high cycle fatigue loading [39]. Let us assume a position vector of a point in the direct lattice \(\overset{\rightarrow}{r}=\alpha_{1}\overset{\rightarrow}{l}_{1}+\alpha_{2} \overset{\rightarrow}{l}_{2}+\alpha_{3}\overset{\rightarrow}{l}_{3}\) where, \(\overset{\rightarrow}{l}_{1}\), \(\overset{\rightarrow}{l}_{2}\) and \(\overset{\rightarrow}{l}_{3}\) are the basis vectors of the lattice [40]. The wave potential of a wave propagating through the lattice can be observed only at the lattice points \(\overset{\rightarrow}{r}\), and the potential can be written as \(\phi=Ae^{\{i(k_{1}\alpha_{1}+k_{2}\alpha_{2}+k_{3}\alpha_{3})\}}\). The function \(\varphi\) is periodic in \(\overset{\rightarrow}{k}=k_{1}\overset{\rightarrow}{m}_{1}+k_{2}\overset{ \rightarrow}{m}_{2}+k_{3}\overset{\rightarrow}{m}_{3}\), where \(\overset{\rightarrow}{k}\) is the wave vector in reciprocal lattice and \(\overset{\rightarrow}{m}_{1},\overset{\rightarrow}{m}_{2},\overset{ \rightarrow}{m}_{3}\) are the basis vectors in reciprocal lattice system.