Journal of Mechanical Science and Technology 33 (1) (2019) 299~306 www.springerlink.com/content/1738-494x(Print)/1976-3824(Online) DOI 10.1007/s12206-018-1229-y Evaluation of stress intensity factors in functionally graded materials by natural element method † Jin-Rae Cho * Department of Naval Architecture and Ocean Engineering, Hongik University, Sejong 30016, Korea (Manuscript Received August 28, 2018; Revised September 25, 2018; Accepted October 10, 2018) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract This paper is concerned with the numerical evaluation of the stress intensity factors (SIFs) of 2-D isotropic functionally graded materi- als (FGMs) by the natural element method (more exactly, Petrov-Galerkin NEM). The spatial variation of elastic modulus in inhomoge- neous FGMs is reflected into the modified interaction integral ( ) 1,2 M % . The local NEM grid near the crack tip is refined, and the strain and stress fields that were directly approximated by PG-NEM were enhanced and smoothened by the patch recovery technique. Numeri- cal examples with the exponentially varying elastic modulus are taken to illustrate the proposed method. The stress intensity factors are parametrically evaluated with respect to the exponent index in the elastic modulus and the crack length, and those were compared with the other reported results. It has been justified from the numerical results that the present method successfully and accurately evaluates the stress intensity factors of 2-D inhomogeneous functionally graded materials. Keywords: Functionally graded materials (FGM); Stress intensity factor (SIF); Modified interaction integral; Near-tip grid refinement; Petrov-Galerkin natural element method (PG-NEM) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction A functionally graded material (FGM) was introduced in the late 1980s to overcome the demerits of traditional layered heat-resisting composite materials [1]. The thermal stress con- centration at the layer interface, which was the most signifi- cant problem of layered heat-resisting composites, could be successfully reduced by inserting a graded layer between two different homogeneous material layers [2]. Here, a graded layer is meant by an inhomogeneous material layer in which the constituent particles of two homogeneous material layers are mixed up in a complex microstructure pattern [3]. Besides the reduction of thermal stress concentration at the layer inter- face by removing the sharp material discontinuity, the thermo- mechanical performance of FGM could be maximized by appropriately tailoring the volume fraction distribution of constituent particles within the graded layer [4]. In the beginning, most research efforts focused on the mate- rial characterization, thermo-mechanical analysis and fabrica- tion [5, 6]. But later, other mechanical behaviors, such as bending and buckling, vibration and fracture, were evaluated in order to expand the potential application fields of FGMs [7- 10]. The thermo-mechanical behaviors of FGMs are influ- enced by the geometry, dimension, orientation and micro- structure of constituent particles as well as the external loading and constraint. In particular, the structural failure of FGMs is dominated by micro-cracking because the microstructure is highly heterogeneous [11]. In this context, the computation of stress intensity factors and the crack propagation simulation have been an important research subject [12-14], in order to tailor FGMs which can be protected from the crack-driven failure. One can employ traditional J - or interaction integrals for homogeneous materials. But, these standard integral methods provide path-dependent SIFs when applied to inhomogeneous materials. It is because the material properties vary point by point in inhomogeneous materials, but these standard integral methods cannot account for this spatial variation of material properties [15]. The studies on the crack problems for inho- mogeneous materials were initiated in the 1960~70s by as- suming the spatial variation of the elastic modulus as an expo- nential function [16, 17]. Eishen [18] numerically investigated cracks in inhomogeneous materials by the finite element method, while Gu et al. [15] presented a simplified method for calculating the crack-tip field of FGMs using the equivalent domain integral technique. Anlas et al. [19] numerically evaluated SIFs in FGMs by discretizing the material property variation and by assigning different homogeneous elastic properties to each element. Kim and Paulino [13] evaluated * Corresponding author. Tel.: +82 44 860 2546, Fax.: +82 41 862 0940 E-mail address: jrcho@hongik.ac.kr † Recommended by Associate Editor Heung Soo Kim © KSME & Springer 2019