PSI - Issue 58

P.C. Sidharth et al. / Procedia Structural Integrity 58 (2024) 115–121 P.C. Sidharth and B.N. Rao/ Structural Integrity Procedia 00 (2019) 000–000

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Fracture is a fundamental consideration in the performance and reliability of engineering materials, and FGMs are no exception. However, traditional fracture mechanics approaches, suitable for homogeneous materials, encounter limitations when dealing with the spatially varying material properties inherent in FGMs. Consequently, there is a growing need for innovative computational techniques capable of effectively capturing the complex interplay between microstructure, material gradients, and fracture processes within these materials. In this regard, the present paper explores the utilization of exponential finite element shape functions as a key element in phase-field modeling for the simulation of fracture in functionally graded materials. This approach offers a robust and versatile framework to address intricate fracture phenomena while seamlessly accommodating the heterogeneous nature of FGMs. By representing the fracture process as a diffuse interface within a continuous field variable and leveraging exponential finite element shape functions, we can gain valuable insights into critical aspects such as crack initiation, propagation, and branching within FGMs. Through an in-depth examination of the fundamental principles behind phase-field modeling with exponential finite element shape functions, we emphasize the benefits and unique capabilities of this approach in the context of FGMs. Drawing upon recent research and practical case studies, we demonstrate how this methodology contributes to a more profound understanding of fracture mechanisms in FGMs and empowers the optimization of these advanced materials across various engineering applications. By leveraging exponential finite element shape functions within the phase-field modeling framework, our paper seeks to bridge the gap between computational mechanics and the intricate material properties of FGMs. This novel approach not only promises to advance the understanding of fracture behavior in functionally graded materials but also facilitates their design and engineering for enhanced structural integrity and reliability. 1.1. Functionally graded materials In our current study, we have focused on two-dimensional functionally graded material (FGM) plates, where the properties vary along the length of the plate. However, it is important to note that in typical applications like thermal barriers and protective coatings, the property gradient occurs along the thickness direction of the plate, which we have not considered in this study. This aspect is reserved for future research and exploration. For our analysis, we have employed the Voigt rule of mixtures to homogenize the graded properties of the FGM plate at material points. Consequently, the spatial distribution of these varying material properties can be expressed in terms of the spatial coordinates as follows: (1) In this context, the symbol "P" represents a specific material property whose variation is mathematically described by equation 1. This property can include parameters such as Young's modulus (E), Poisson's ratio (ν), or critical energy release rate (G c ). The term "V x/y " refers to the material volume fraction of one of the two constituents that make up the functionally graded (FG) material. The subscripts "x" and "y" are indicative of the chosen direction of gradation. To further clarify, we define "p" as the volume fraction exponent or the material power law index, while "l x " and "l y " represent the dimensions of the plate in the x and y directions, respectively. The expressions for volume fractions along the x and y coordinates can be expressed as follows: / ( ) 1 ( 2 1) x y P x P P P V   

p

p

  

  

  

  

1 2

1 2

x

y

(2)

,

V

V

 

  

x

y

l

l

x

y

1.2. Phase field modeling of fracture Within phase-field models, the depiction of cracks relies on the utilization of a phase-field parameter. This parameter, denoted by a variable spanning from 0 to 1, serves as a representation of the transition between uncracked and cracked material phases. Consequently, the system's total energy can be conceptualized as a summation of the strain energy contained within the material's bulk and the energy dissipation stemming from the