A Numerical Evaluation of SIFs of 2-D Functionally Graded Materials by Enriched Natural Element Method
A Numerical Evaluation of SIFs of 2-D Functionally Graded Materials by Enriched Natural Element...
Cho, Jin-Rae
2019-09-01 00:00:00
applied sciences Article A Numerical Evaluation of SIFs of 2-D Functionally Graded Materials by Enriched Natural Element Method Jin-Rae Cho Department of Naval Architecture and Ocean Engineering, Hongik University, Jochiwon, Sejong 30016, Korea; jrcho@hongik.ac.kr; Tel.: +82-44-860-2546 Received: 29 July 2019; Accepted: 25 August 2019; Published: 1 September 2019 Featured Application: Prediction of crack propagation of functionally graded materials (FGMs). Abstract: This paper presents the numerical prediction of stress intensity factors (SIFs) of 2-D inhomogeneous functionally graded materials (FGMs) by an enriched Petrov-Galerkin natural element method (PG-NEM). The overall trial displacement field was approximated in terms of Laplace interpolation functions, and the crack tip one was enhanced by the crack-tip singular displacement field. The overall stress and strain distributions, which were obtained by PG-NEM, were smoothened (1,2) and improved by the stress recovery. The modified interaction integral M was employed to evaluate the stress intensity factors of FGMs with spatially varying elastic moduli. The proposed method was validated through the representative numerical examples and the eectiveness was justified by comparing the numerical results with the reference solutions. Keywords: enhanced PG-NEM; functionally graded material (FGM); stress intensity factor (SIF); modified interaction integral 1. Introduction In the late 1980s, a new material concept called functionally graded material (FGM) was proposed to resolve the inherent problem of traditional lamination-type composites [1]. The sharp material discontinuity across the layer interface causes the stress concentration, which may trigger the layer delamination. This crucial stress concentration can be significantly minimized by inserting a graded layer between two distinct homogeneous material layers [2,3]. This is because the material discontinuity completely disappears according to the material composition gradation, where the constituent particles of two base materials are mixed up in a random microstructure within a graded layer to maximize the desired performance [4–6]. Naturally, a functionally graded material is an inhomogeneous material, with spatially non-uniform material properties characterized by continuity and functionality. In addition to the suppression of stress concentration, the material concept of FGM rapidly and continuously spread throughout engineering and fields [7–10]. Early research eorts were concentrated on material characterization, fabrication, modeling, and analysis [1,11,12]. This was because the mechanical behaviors of FGMs are governed by the geometric dimensions and orientation, microstructure, and volume fractions of constituent particles. Recently, however, the concern toward the crack problems has increased because the structural failure of FGMs is dominated by micro-cracking [7,13,14]. In this regard, an accurate numerical prediction of stress intensity factors and the crack propagation has been an essential research subject [15,16]. For these subjects, an accurate reproduction of the singularity in the near-tip stress field in highly heterogeneous media becomes a key task [17–20]. Appl. Sci. 2019, 9, 3581; doi:10.3390/app9173581 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 3581 2 of 16 To evaluate the stress intensity factors of FGMs with cracks, one can consider the use of the well-known J or M integral methods. However, these conventional indirect integral methods cannot reflect the spatially varying material properties of FGMs. The studies on the fracture mechanics of inhomogeneous bodies were initiated in the 1970~80s by assuming the spatially varying elastic modulus as an exponential function [17,21]. The standard integral methods were modified and/or refined to reflect the spatial non-uniformity of material properties by subsequent investigators. The most works were made by utilizing the finite element method [15,22–25]. However, since late 1990s, the employment of meshless methods to crack problems of inhomogeneous bodies has been actively advanced, particularly for the computation of SIFs by the modified M integral method [26,27]. Here, the extension of the natural element method (NEM) is worth noting, even though it was restricted to 2-D homogenous material [28]. The natural element method was introduced to overcome the demerits of conventional meshless methods [29], the diculty in enforcing the displacement constraint and the numerical integration. The Laplace interpolation functions in NEM strictly obey the Kronecker delta property so that the imposition of displacement constraint becomes easy. In addition, Delaunay triangles, which were produced during the process for introducing Laplace interpolation functions, also serve as a background mesh for the numerical integration. In particular, PG-NEM can further improve the numerical integration accuracy by maintaining the consistency between the Delaunay triangle and the integration region [30]. Although Laplace interpolation functions provide the high smoothness of C -continuity, there is still room for further improvement in capturing the high stress singularity at the crack tip. In this context, this paper introduces an enriched PG-NEM to explore whether and how much the enrichment of interpolation function increases the prediction reliability of stress intensity factors for FGMs. The validity of enrichment was reported for homogeneous materials [31,32], but it was rarely reported for inhomogeneous materials. The trial function is enriched by the asymptotic displacement fields of mode I and II, and the approximated overall stress field is enhanced by the patch recovery technique. The proposed method was validated through the illustrative numerical examples and its eectiveness was quantitatively evaluated. 2. 2-D Inhomogeneous Cracked Bodies 2.1. Linear Elasticity of 2-D Cracked Bodies Figure 1 represents a 2-D linearly elastic isotropic inhomogeneous material with an edge crack which is contained within a bounded domain W 2 < with the boundary @W = G [ G [ G . D N c Here, G and G indicate the displacement and force boundary regions, and G = G [ G denotes the D N c c c traction-free crack surface. As a representative non-homogeneous material, FGMs are characterized by the spatially varying elastic modulus E and Poisson’s ratio over the bounded domain W. For the mathematical description purpose, two Cartesian co-ordinate systems are employed, x, y for the 0 0 2-D linear elasticity problem and x , y for the SIF evaluation of crack respectively. Assuming the crack surface is traction-free and neglecting the body force b, then the displacement field u(x) in the Cartesian coordinate system x, y is subjected to the equilibrium equations r = 0 in W (1) with the displacement constraint u = uˆ on G (2) and the force boundary condition t on G n = (3) 0 on G c Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 15 t on Γ Appl. Sci. 2019, 9, 3581 N 3 of 16 σ ⋅ n = (3) 0 on Γ c Here, are the Cauchy stresses, n the outward unit vector normal to @W, and t the contour traction. ∂Ω Here, are the Cauchy stresses, the outward unit vector normal to , and t the σ n When the displacement and strains are assumed to be small, the Cauchy strain " is constituted to the contour traction. When the displacement and strains are assumed to be small, the Cauchy strain Cauchy stress via the (3 2) gradient-like operator L such that is constituted to the Cauchy stress σ via the () 3 × 2 gradient-like operator such that " = "(u) = Lu (4) ε = ε() u = Lu (4) Letting D be the constitutive tensor, the stresses and strains are constituted by Letting D be the constitutive tensor, the stresses and strains are constituted by σ = D : ε (5) = D : " (5) Note that the displacement, strains, and stresses are calculated based on the co-ordinate system Note that the displacement, strains, and stresses are calculated based on the co-ordinate system {x′,y′} {} x, y and transformed into the co-ordinate system . 0 0 x, y and transformed into the co-ordinate system x , y . Figure 1. An inhomogeneous isotropic body with an edge crack. Figure 1. An inhomogeneous isotropic body with an edge crack. For a homogeneous cracked body, the energy release rate per unit crack propagation along the For a homogeneous cracked body, the energy release rate per unit crack propagation along the x axis can be estimated by the path-independent J integral defined by x′ − axis can be estimated by the path-independent J − integral defined by @u ∂u J = W n ds (6) 1 j i j i j ′ J = Wδ − σ @x n ds (6) G 1 1 j ij j ∂x 1 0 0 0 0 using the indicial notation (i.e., x = x and x = y ), the Dirac delta function , and the strain-energy 1 1 j density using the Win= di cia "l/ nota 2 =ti"on D(i.e., " /2′. Her ′e, an G d indicates ′ ′an ), the Di arbitrary rac del closed ta functi path, on which , and the strain- surrounds the x = x x = y i j i jkl kl 1j 1 2 crack tip counterclockwise. As shown in Figure 2, it is expanded to C = G + G + G + G in order to c c ε D ε / 2 W = σ ⋅ε / 2 = Γ energy density . Here, indicates an arbitrary closed path, which ij ijkl kl generate a grayed donut-type region A, in which a smooth function q(x)(0 q 1) is introduced to surrounds the crack tip counterclockwise. As shown in Figure 2, it is expanded to recast the integral into an equivalent domain form [33]. The function q, called by weighting function, − + becomes unity on G, zero on G , and arbitrary value between 0 and 1 within the grayed donut region A. C = Γ + Γ + Γ + Γ in order to generate a grayed donut-type region A, in which a smooth c c o Then, the above Equation (6) can be expanded as following function is introduced to recast the integral into an equivalent domain form [33]. The q() x (0 ≤ q ≤ 1) ! ! Z Z @q @u @ @u i i q Γ Γ function , called by weighting function, becomes unity on , zero on , and arbitrary value J = W dA + W qdA (7) i j 1 j i j 1 j 0 0 0 0 @x @x @x @x 1 j j 1 A A between 0 and 1 within the grayed donut region A. Then, the above Equation (6) can be expanded as following 0 0 according to the divergence theorem, together with n = n on G in C. By further expanding the second term on the right hand side, Equation (7) becomes ∂u ∂q ∂ ∂u i i J = σ −Wδ dA + σ −Wδ qdA (7) ij 1 j ij 1 j Z ! Z ! A A ∂x′ ∂x′ ∂x′ 2 ∂x′ @ @" @D @q 1 j j 1 @u i j @u @ u i j 1 i jkl i i i J = W dA + + " " qdA (8) i j 1 j i j i j i j kl 0 0 0 0 0 0 0 0 @x @x @x @x @x @x @x 2 @x 1 j j 1 j 1 1 1 A A according to the divergence theorem, together with n′ = −n′ on in C . By further expanding the second term on the right hand side, Equation (7) becomes Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 15 ∂σ ∂ε ∂D ∂u ∂q ∂u ∂ u 1 ij ij ijkl i i i J = σ − Wδ dA + + σ − σ − ε ε qdA (8) ij 1 j ij ij ij kl A A ′ ′ ′ ′ ′ ′ ′ ′ ∂x ∂x ∂x ∂x ∂x ∂x ∂x 2 ∂x 1 j j 1 j 1 1 1 Appl. Sci. 2019, 9, 3581 4 of 16 But, the second integral on the right-hand side of Equation (8) becomes zero according to the But, the second integral on the right-hand side of Equation (8) becomes zero according to equilibrium (1), compatibility (4), and the material uniformity. Therefore, the J − integral for the equilibrium (1), compatibility (4), and the material uniformity. Therefore, the J integral for homogeneous materials becomes the area integral defined by homogeneous materials becomes the area integral defined by ∂ u ∂q i ! J = σ −Wδ dA (9) @q ij@u 1 j i ′ ′ ∂x ∂x J = W dA (9) i j 1 1 j j 0 0 @x @x A 1 j But, for non-homogeneous materials, the last material derivation term within the second But, for non-homogeneous materials, the last material derivation term within the second integrand of Equation (8) does not become zero. Therefore, Equation (8) ends up with a more general integrand ~ of Equation (8) does not become zero. Therefore, Equation (8) ends up with a more J − integral [34], which is given by general J integral [34], which is given by ∂D ∂u ∂q 1 ~ ijkl Z ! Z J = σ −Wδ dA − ε ε qdA @D @q (10) ij@u 1 j 1 ij i jkl kl i e A A J = ∂x′