# Functionally Graded Plate Fracture Analysis Using the Field Boundary Element Method
## 4 Discussion and Conclusions
Moreover, in Table 1, the comparison of numerical SIF for different variation types of the elasticity evidenced that it was influenced by the modulus decrease or increase moving from the crack tip toward the interior of the plate. In particular, one can see that when the modulus decreased moving from the crack along the crack direction, that is perpendicular to the applied load, the SIF was greater than when the modulus increased with distance. This can be explained by the fact that a uniformly loaded plate response in terms of stress concentrated where the modulus increased. Hence, when the modulus increased toward the crack, the stress increased with a higher level than the homogeneous case. In particular, it can be seen that the KI for homogeneous plate was between the one for increasing modulus and the decreasing one. \[KI_{XC}\leq KI_{H}\leq KI_{XD} \tag{17}\]  
When the variability of the material property developed along the load direction the influence of the variability on the SIF was much less evident. The SIF decreased when the modulus increased with distance and increased when the modulus decreased but it remained almost equal to the homogeneous SIF. The variability of SIF on the crack length can be seen from the graphics in Figure 7. The figure highlights that the material variability shows that the curve representing the variability of the SIF on the crack length moved accordingly to the material stiffness at the tip. The shift of the curve was meaningful in the cases of x-variation of the modulus and negligible when the modulus varied along the y-direction. A final consideration suggested that the numerical calculation showed the feasibility of the method that allowed the application of integral equations to fracture mechanics even in the cases of heterogeneous materials; moreover, it was shown that the variability of the mechanical properties of the structure affected the value of the SIF. The qualitative behavior of the KI can be seen in Figure 8, Figure 9, and Figure 10 where the bars and the percentage variation (%V) of the KI were delimiting values of the homogeneous case provided the variation developed in the x-direction. Conversely, when the variation followed the y-direction, the behavior was quiet in comparison to the homogeneous one. Finally, some remarks on the comparison between FEM and FBEM should be addressed here. A comparison was made with the standard FEM program. The finite element calculation deals with piecewise constant elastic modulus at an element scale; no variability has been accounted for within the element. The actual variability of the material properties as well as the variability of the anisotropy class at the element scale should require iso-parametric or ad hoc shape functions for the elasticity description. Such iso-parametric or hetero-parametric modelling of the material properties requires ad hoc FEM coding together with an element-wise description of anisotropy directions oriented with respect to the local axes of the element. Consequently, the FEM modelling produces more mesh dependence than FBEM due to the dependence on the mesh of the material description too; such approximation does not affect FBEM modelling [37]. **Author Contributions:** Conceptualization, S.P., L.E., P.F., R.Z. and V.M. ; methodology, S.P., L.E., P.F., R.Z. and V.M. ; software, S.P., L.E., P.F., R.Z. and V.M. ; validation, S.P., L.E., P.F., R.Z. and V.M. ; formal analysis, S.P., L.E., P.F., R.Z. and V.M. ; investigation, S.P., L.E., P.F., R.Z. and V.M. ; resources, S.P., L.E., P.F., R.Z. and V.M. ; data curation, S.P., L.E., P.F., R.Z. and V.M. ; writing--original draft preparation, S.P., L.E., P.F., R.Z. and V.M. ; writing--review and editing, S.P., L.E., P.F., R.Z. and V.M. ; visualization, S.P., L.E., P.F., R.Z. and V.M. ; supervision, V.M. ; project administration, V.M. ; funding acquisition, V.M. All authors have read and agreed to the published version of the manuscript. **Funding:** The authors acknowledge the financial grant program VALERE: \"VAnviteLli pEr la RicErca\".