# Functionally Graded Plate Fracture Analysis Using the Field Boundary Element Method
###### Abstract  
This paper describes the Field Boundary Element Method (FBEM) applied to the fracture analysis of a 2D rectangular plate made of Functionally Graded Material (FGM) to calculate Mode I Stress Intensity Factor (SIF). The case study of this Field Boundary Element Method is the transversely isotropic plane plate. Its material presents an exponential variation of the elasticity tensor depending on a scalar function of position, i.e., the elastic tensor results from multiplying a scalar function by a constant taken as a reference. Several examples using a parametric representation of the structural response show the suitability of the method that constitutes a Stress Intensity Factor evaluation of Functionally Graded Materials plane plates even in the case of more complex geometries. Functionally Graded Materials (FGM); Field Boundary Element Method (FBEM); interface; Stress Intensity Factor (SIF) +
Footnote †: journal: Applied Sciences  
**Citation**: [PERSON], [PERSON]; [PERSON], [PERSON]; [PERSON], [PERSON]; [PERSON], [PERSON]; [PERSON], [PERSON] Functionally Graded Plate Fracture Analysis Using the Field Boundary Element Method. As previously mentioned, the increasing interest in cracks in Functionally Graded Materials (FGM) gave rise to several methods such as the Weight Function approach [8; 9], Energy Release Method [10], and others. The present work aimed to apply the boundary element method to heterogeneous material fractures. The BEM applied to fracture mechanics is split into two different approaches. The first, the dual boundary element method, is based on two dual integral equations, the first based on the displacement Somigliana identity and the second obtained through the projection on the boundary of the gradient of the first one resulting in a traction identity [11]. The reason for such a model is based on the fact that the displacement equation should be collocated on both sides of the crack that produced the linear dependence of the equations. To avoid the singularity, the two different primary and dual equations can be placed on each edge of the fracture line. The disadvantage of this approach is the presence of hypersingular integrals. The second strategy, to avoid the dual approach, uses a multizone formulation where the structure is divided into parts along the crack line so that different displacement integral equations are formulated on the separate regions and coupling is obtained using boundary conditions at the common points and on the crack edges [12]. The formulation proposed in the present work is based on the displacement integral equation specified for FGMs. The fracture model is borrowed from the multizone approach; in particular, the proposed examples deal with symmetrical structures where the crack belongs to one edge. A multizone approach for FGM has been proposed in [13]; this formulation can be extended to fracture mechanics with no complication. A simple approach has been chosen here to focus on the relevant aspects of the SIF calculation. The calculation used an enriched formulation of the boundary element method with field terms that account for the variation of material elasticity [13; 14; 15; 16; 17]. Using the formulation, the application of Elastic Fracture Mechanics (EFM) is shown to calculate the Stress Intensity Factor under Mode I crack propagation. The calculations followed the traditional formulation for EFM where it is supposed that around the crack tip the elasticity of the material is constant within the small region of stress concentration. The interface, the joints, or the surfaces, have great importance in graded material because the stress is usually limited to these locations. In these cases, the specific zone is the so-called \"graded zone\" and if the volume fractions of the constituent materials change by a certain smooth function, the material is defined as a functionally graded material, FGM [18; 19; 20]. Furthermore, several benchmarks and reports of models using Linear Elastic Fracture Mechanics (LEFM) are presented in [21; 22].