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Application of EFGM and XFEM for Fatigue Crack growth Analysis of Functionally Graded Materials
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 173 (2017) 1231 – 1238
11th International Symposium on Plasticity and Impact Mechanics, Implast 2016
Application of EFGM and XFEM for Fatigue Crack growth Analysis of Functionally Graded Materials Mohit Pant2,Kamal Sharma1*,Somnath Bhattacharya3 *1 Bhabha Atomic Reseacrh Centre, Mumbai, India Department of Mechancial Engineering, NIT, Hamirpur, India 3 Department of Mechancial Engineering, NIT,Raipur, India
2
Abstract The present work investigates the fatigue life of a functionally graded material (FGM) made of aluminum alloy and alumina (ceramic) under cyclic mix mode loading. Element free Galerkin Method (EFGM) and extended finite element method (XFEM) are employed to simulate and compare the fatigue crack growth. The fatigue lives of aluminum alloy, FGM and an equivalent composite (having the same composition as of FGM) are compared for a major edge crack in a rectangular domain. © Published by Elsevier Ltd. This © 2017 2016The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of organizing committee of Implast 2016. Peer-review under responsibility of the organizing committee of Implast 2016 Keywords: FGM; XFEM; Fatigue Crack Growth; Discontinuities
1. Introduction Composite materials in which the composition or microstructure or both are locally varied so that a certain variation of the local material properties is achieved are defined as functionally graded materials (FGMs). FGMs possess nonhomogeneous macrostructure with continuously varying mechanical and/or thermal properties in one or more than one direction. The fatigue life of these components is normally estimated without accounting for the effect of defects/discontinuities. Fatigue life is significantly affected by presence of voids and micro-defects near the tip of a major crack and further enhances the effective SIF at the tip of the major crack. The severity of failure is more when a structure is subjected to mixed mode loading as compared to mode-I loading. Moreover, the crack growth under
* Corresponding author. Tel.: +91-9820771897 E-mail address: [email protected]/[email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of Implast 2016
doi:10.1016/j.proeng.2016.12.135
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Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238
mixed mode loading may not be in a self-similar direction adherence issues like crack growth direction also become important. In view of this, the fatigue analysis of FGM under mixed mode loading becomes quite important. In past, few studies on mixed mode fatigue crack growth have been carried out for different materials. Some efforts have already been made to study and analyze the behavior of FGMs. Pandey and Patel [1] studied the crack growth in a panel with an inclined crack subjected to biaxial fatigue loading. They analyzed the 7075-T6 aluminum alloy, and predicted the effect of crack angle on the fatigue life of the component. Qian and Fatemi [2] reviewed the various criteria and parameters for the prediction of mixed mode crack growth directions and rates. Prabhakar and Tippur [3] performed the static fracture analysis to calculate the crack tip stress field. Dolbow and Gosz [4] and Rao and Rahman [5, 6] analyzed the mixed mode SIF for 2-D orthotropic FGM using interaction energy contour integrals. Kim and Paulino [7] presented a path-independent J -integral in conjunction with the finite element method for mode-I and mixed-mode crack problems in orthotropic functionally graded materials having exponential and linear variations in material properties. Pavlou et al. [8] proposed a new methodology to simulate the fatigue crack trajectories under in-plane cyclic loading. They used a new factor related to the accumulated elastic strain energy within a circular core around the crack tip for predicting the mixed-mode fatigue crack path and fatigue crack growth rate. Zhang et al. [9] analyzed the unidirectional and bidirectional FGM under mixed mode loading using boundary integral equation method. Huang et al. [10,11] analyzed the multilayered model of FGM, and used Fourier transform approach for performing the fracture analysis of multilayered model. 2. XFEM FORMULATION The XFEM is a partition of unity (PU) enriched finite element method. It has proved to be a competent mathematical tool for the analysis of problems involving discontinuities due to the local enrichment of the approximation space. The enrichment is done through the PU concept. The method is particularly useful for the approximation of the solutions with prominent non-smooth characteristics in small parts of the computational domain like near discontinuities and singularities. A planar domain ( : ) bounded by contour * divided into three parts i.e., , * t and *c with internal flaws is shown in Figure 1. The displacement boundary conditions are imposed on *u , while tractions are applied on * t and traction free condition are imposed on crack surfaces *c . The equilibrium and boundary conditions for this problem may be described as t
:
x2
x1 *u
*t
*
u
Figure 1: Domain along with essential and natural boundary conditions
.σ b 0 in : σ. nˆ t on * t
(1)
0 on *c
(3)
σ .nˆ
(2)
(4) u u on *u where, σ is the Cauchy stress tensor, u is the displacement field vector, b is the body force vector per unit volume,
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Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238
t is the external traction vector and nˆ is the unit outward normal vector. For small displacements, straindisplacement relation can be described as ε
s u
ε(u)
(5)
where, s is the symmetric part of the gradient operator. The constitutive relations for the linear elastic FGM under consideration is given by Hooke's law
σ(u)
D(x) : ε(u)
(6)
where, x is the vector of x and y -coordinates, D(x) is the constitutive matrix, which can be written for plane strain condition as ª º «1 Q (x) Q (x) » 0 « » E ( x) (7) D ( x) 0 « Q ( x ) 1 Q ( x) » ^1 2Q (x)`^1 Q (x)` « » 1 2Q (x) 0 « 0 » 2 ¬ ¼ 3. EFGM FORMULATION In EFGM, the field variable u is approximated by moving least square approximation function u h ( x) which is given by [12, 13, 14] m
u h ( x)
¦ p (x)a (x) j
pT (x) a(x)
j
(8)
j 1
where, p(x) is a vector of basis functions, a(x) are unknown coefficients, and m is the number of terms in the basis. The unknown coefficients a(x) are obtained by minimizing a weighted least square sum of the difference between local approximation, u h ( x ) and field function nodal parameters u I . The weighted least square sum L(x) can be written in the following quadratic form: n
L(x)
¦ w(x x )[p
T
I
(x)a(x) uI ]2
(9)
I 1
where, u I is the nodal parameter (primary variable, displacement) associated with node I at x
x I . However,
u I are not the nodal values of u h (x x I ) since u h ( x ) is an approximant and not an interpolant as in the case of finite element method. By setting wL / wa =0, following set of linear equation is obtained: (10) A(x)a(x) B(x) u n
where,
¦II (x) y
I
y
I 1
B(x)
^w(x x1 ) p(x1 ), w(x x2 ) p(x2 ),......................., w(x xn ) p(xn )`
By substituting Eq. (10) in Eq. (8), the approximation function is obtained as: n
u h (x)
¦Φ (x) u I
(11)
I
I 1
where, I (x) I
m
1 ¦ p j (x)( A (x)B(x)) jI j
pT A1B I
The linear consistency requirements for the shape function I (x) are given as [15, 16, 17]: I
1234
Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238 n
¦II (x)
n
n
¦II (x) x
1,
I
I 1
¦II (x) y
x
I
y
(12)
I 1
I 1
The derivatives of MLS shape function are computed as:
ΦI ,x (x)
(pT A 1B I ) ,x pT ,x A1 BI pT (A1 ),x BI pT A1 BI ,x
where, B I , x (x)
(13)
dw (x x I )p(x I ) dx
and A 1, x is computed by
A1,x A1A,x A1 where, n dw A, x ¦ ( x x I ) p ( x I ) pT ( x I ) I 1 dx
(14)
4. CALCULATION OF STRESS INTENSITY FACTORS FOR FGM The domain based interaction integral approach has been widely used for calculating the stress intensity factors for homogeneous, bi-material and functionally graded materials under thermal and mechanical loads [6, 7, 18, 19]. For an elastic body subjected to mechanical load (Figure 1), the J-integral is given by § wu · (15) J ³ ¨ V ij i W G1 j ¸ q n j d * wx1 ¹ * © o
where, x [ x1
x2 ] T { [ x
y ] T , W is the strain energy density function and n j is the jth component of the
outward unit vector normal to an arbitrary closed contour * enclosing the area A (Figure 2). q stands for a weight function chosen such that it has a value of unity at the crack tip, zero along the boundary of the domain and arbitrary elsewhere
A
*
x2
x1
Figure 2: Path
* surrounding a crack with an enclosed area A
5. PROBLEM DESCRIPTION, RESULTS AND DISCUSSIONS A rectangular FGM plate of dimensions L 100 mm and D 200 mm of graded material (with 100% aluminum alloy on the left side and 100% ceramic (alumina) on the right side) is considered for the purpose of analysis. A uniform mesh consisting of 117 equally distributed nodes in x -direction and 235 equally distributed nodes in y direction are used for both EFGM and XFEM simulations. Nine node quadrilateral elements are used with a total of 6786 elements in the domain, for XEFM. Four point Gauss quadrature is used for numerical integration of EFG
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Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238
equations. For each case, the boundary conditions are depicted in the figures. The gradation in material property is taken along horizontal direction i.e. x-direction where x varies from x 0 to x 100 mm. In all the cases, an initial crack of a 20 mm length is assumed at the edge of the plate. The dimensions of the plate are taken same as that of FGM for the study of composite and pure aluminium alloy. A cyclic tensile load varying from
V
max
70 MPa to V min
0 MPa is applied in all the simulations. An increment of 'a
a 10
2 mm is taken to
evaluate the fatigue failure life of aluminum alloy, FGM and equivalent composite. 5.1 Plate with an Edge Crack Figures 3a and 3b show the plates with an edge crack on the alloy rich and ceramic rich sides respectively along with the relevant boundary conditions. The problem has been analyzed under plane strain condition using a uniform mesh of 117 by 235 nodes for both EFGM and XFEM. A crack extension vs number of cycles curve have been plotted for FGM, Equivalent composite and aluminium alloy as shown in Figure 3c. The fatigue failure cycles have been tabulated in table 1. Both, the plot and table depicts that EFGM results are in good agreement with the XFEM results. It is also observed that when the crack initiates from the ceramic (alumina) side, it fails much earlier than when the crack initiates from the alloy rich side. Table 1: (No. of Cycles) S. Fatigue Life Simulation Technique No 1 EFGM 2
Crack on Alloy Rich Side 15561
Crack on Ceramic Rich Side 4872
Crack on Equivalent Composite 7885
Crack on Aluminum Allloy 19145
15559
4874
7881
19141
XFEM
'V
'V
100% Ceramic
100% Alloy
100 % Ceramic
100 % Alloy
a
a
D
D D 2
D 2
L Figure 3a: Plate with an edge crack on the alloy rich side under mode-I
L Figure 3b: Plate with an edge crack on the ceramic rich side under mode-I
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Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238 22 FGM (crack on alloy rich side)-XFEM FGM (crack on ceramic rich side)-XFEM Equivalent composite-XFEM Aluminum Alloy-XFEM EFGM Data
20 18
Crack Extension (mm)
16 14 12 10 8 6 4 2 0
0
0.2
0.4
0.6
0.8 1 1.2 No. of Cycles
1.4
1.6
1.8
2 4
x 10
Figure 3c: A plot of crack extension with number of cycles
5.2 Edge crack under Mode II Loading Next we considered plates with an edge crack on the alloy rich and ceramic rich sides respectively (Figures 4a and 4b) along with the relevant boundary conditions. The dimensions of the plate, nodal density, and boundary conditions are considered same as that of FGM for the study of composite and pure aluminum alloy. An initial crack of a 20 mm length is assumed at the both at edge and centre of the plate. A cyclic tensile load varying from V max 70 MPa to V min 0 MPa is applied in all the simulations. In addition to the mode-I cyclic tensile load, a shear fatigue load of
W
max
10 MPa and W
life. An increment of 'a
a 10
min
0 MPa is applied to study the effect of mixed mode load on the fatigue
2 mm is taken to evaluate the fatigue failure life of aluminum alloy, FGM and
equivalent composite. Both EFGM and XFEM predict nearly same values of fatigue life as can be clearly seen from Table. 2 and Figure 4c. It is evident that the fatigue life reduces considerably in case of mixed mode mechanical load as compared to that of pure mode-I load. Table 2: (No. of Cycles) S. No 1
Fatigue Life Simulation Technique EFGM
Crack on Alloy Rich Side 12185
2
XFEM
12181
Crack on Ceramic Crack on quivalent Crack on Auminum Allloy Rich Side Composite 4575 6008 14893 4573
6006
14890
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Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238
'V
'V
'W
'W
100 % Alloy
100 % Ceramic
a
100 % Ceramic
100 % Alloy
a D
D
D 2
D 2 L
L
Figure 4a: Plate with an edge crack on the alloy rich side under mixed mode loading
Figure 4b: Plate with an edge crack on the ceramic rich side under mixed mode loading
21
FGM Crack on alloy Side EFGM Data FGM Crack on ceramic side EFGM Data Equivalent Composit XFEM data EFGM Data Aluminua Alloy XFEM EFGM Data
Crack Extension (mm)
18
15 12
9 6
3
0 0
3000
6000
9000
12000
15000
18000
No of Cycle
Figure 4c: A plot of crack extension with number of cycles
6. CONCLUSIONS In the present work, fatigue crack growth simulations have been performed for FGM, equivalent composite and aluminium alloy by EFGM and XFEM under mixed mode cyclic loading. On the basis of the present simulations, it is found that the spatial location of crack in a FGM plays a major role in determining the fatigue life of component. The results also reveal that the fatigue life reduces considerably in case of mixed mode loading as compared to that of pure mode-I loading. The simplicity and effectiveness of the proposed criterion shows its potential to simulate realistic problems of fracture in functionally graded materials using EFGM.
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REFERENCES [1] R.K. Pandey and A.B. Patel, Mixed-mode fatigue crack growth under biaxial loading, International Journal of Fatigue, 6 (1984) 119–123. [2] J. Qian and A.Fatemi, Mixed mode fatigue crack growth: A literature survey, Engineering Fracture Mechanics, 55(1996) 969-990. [3] R.M. Prabhakar and H.V. Tippur, Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient, International Journal of Solids and Structures, 37(2000) 5353–5370. [4] J.E. Dolbow and M. Gosz, On the computation of mixed-mode stress intensity factors in functionally graded materials, International Journal of Solids and Structures,39(2002)2557–2574. [5] B.N. Rao and S. Rahman, Meshfree analysis of cracks in isotropic functionally graded materials, Engineering Fracture Mechanics, 70(2003a)1−27. [6] B.N. Rao and S. Rahman, An interaction integral method for analysis of cracks in orthotropic functionally graded materials, Computational Mechanics, 32(2003b)40–51. [7] J.H. Kim and G.H. Paulino, Mixed-mode J-integral formulation and implementation using graded elements for fracture analysis of nonhomogeneous orthotropic materials, Mechanics of Materials, 35(2003)107–128. [8] D.G. Pavlou, G.N. Labeas, N.V. Vlachakis and, F.G. Pavlou, Fatigue crack propagation trajectories under mixed-mode cyclic loading, Engineering Structures, 25(2003) 869–875. [9] C. Zhang, J. Sladek, and V. Sladek, Crack analysis in unidirectionally and bidirectionally functionally graded materials, International Journal of Fracture,129(2004)385–406. [10] G.Y. Huang, Y.S. Wang and G. Dietmar, Fracture analysis of functionally graded coatings: plane deformation, European Journal of Mechanics, A/Solids, 22 (2003)535–544. [11] G.Y. Huang, Y.S. Wang and S.W. Yu, A new model for fracture analysis of functionally graded coatings under plane deformation, Mechanics of Materials, 37(2005)507–516. [12] T. Belytschko, Y.Y. Lu and L. Gu, Element-free Galerkin methods. International Journal for Numerical Methods in Engineering 37(1994)229–256. [13] Sharma, K., Bhasin, V., Singh, I.V., Mishra, B.K, Effect of Parameter in EFG for 2-D Analysis, International Journal of Engineering Science and Technology, 2(10), (2010), p. 5838. [14] Sharma, K., Bhasin, V., Singh, I.V., Mishra, B.K., Parametric Study of Element Free Galerkin for Crack Analysis, JP Journal of Solid and Structures, 4(2), (2010), p. 85. [15] T. Belytschko, Y. Krongauz, M. Fleming and D. Organ, Smoothing and accelerated computations in element-free Galerkin method. Journal of Computational and Applied Mathematics 74(1996) 111–126. [16] Sharma, K., Singh, I.V., Mishra, B.K., Shedbale, A.S., The Effect of Inhomogeneities on Edge Crack: A Numerical Study using XFEM, International Journal for Computational Methods in Engineering Science & Mechanics, 14(6), (2013), p. 505. [17] Sharma, K., Singh, I.V., Mishra, B.K., Bhasin, V., Vaze, K.K., Parameter and Interaction Study of Edge Crack Problem using Meshfree Mehtod, International Journal of Modeling, Simulation, and Scientific Computing, (2012), 3(4). [18] Sharma, K., Application of XFEM for Evaluating fracture Integrity of the DMW, LAP Lambert Academic Publishing, ISBN-13: 978-3-65992072-1, 2016. [19] Sharma, K. , Simulation of EPFM Problems in Functionally Graded Material with XFEM, LAP Lambert Academic Publishing, ISBN-13: 978-3-659-92285-5, 2016.
ScienceDirect Procedia Engineering 173 (2017) 1231 – 1238
11th International Symposium on Plasticity and Impact Mechanics, Implast 2016
Application of EFGM and XFEM for Fatigue Crack growth Analysis of Functionally Graded Materials Mohit Pant2,Kamal Sharma1*,Somnath Bhattacharya3 *1 Bhabha Atomic Reseacrh Centre, Mumbai, India Department of Mechancial Engineering, NIT, Hamirpur, India 3 Department of Mechancial Engineering, NIT,Raipur, India
2
Abstract The present work investigates the fatigue life of a functionally graded material (FGM) made of aluminum alloy and alumina (ceramic) under cyclic mix mode loading. Element free Galerkin Method (EFGM) and extended finite element method (XFEM) are employed to simulate and compare the fatigue crack growth. The fatigue lives of aluminum alloy, FGM and an equivalent composite (having the same composition as of FGM) are compared for a major edge crack in a rectangular domain. © Published by Elsevier Ltd. This © 2017 2016The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of organizing committee of Implast 2016. Peer-review under responsibility of the organizing committee of Implast 2016 Keywords: FGM; XFEM; Fatigue Crack Growth; Discontinuities
1. Introduction Composite materials in which the composition or microstructure or both are locally varied so that a certain variation of the local material properties is achieved are defined as functionally graded materials (FGMs). FGMs possess nonhomogeneous macrostructure with continuously varying mechanical and/or thermal properties in one or more than one direction. The fatigue life of these components is normally estimated without accounting for the effect of defects/discontinuities. Fatigue life is significantly affected by presence of voids and micro-defects near the tip of a major crack and further enhances the effective SIF at the tip of the major crack. The severity of failure is more when a structure is subjected to mixed mode loading as compared to mode-I loading. Moreover, the crack growth under
* Corresponding author. Tel.: +91-9820771897 E-mail address: [email protected]/[email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of Implast 2016
doi:10.1016/j.proeng.2016.12.135
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Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238
mixed mode loading may not be in a self-similar direction adherence issues like crack growth direction also become important. In view of this, the fatigue analysis of FGM under mixed mode loading becomes quite important. In past, few studies on mixed mode fatigue crack growth have been carried out for different materials. Some efforts have already been made to study and analyze the behavior of FGMs. Pandey and Patel [1] studied the crack growth in a panel with an inclined crack subjected to biaxial fatigue loading. They analyzed the 7075-T6 aluminum alloy, and predicted the effect of crack angle on the fatigue life of the component. Qian and Fatemi [2] reviewed the various criteria and parameters for the prediction of mixed mode crack growth directions and rates. Prabhakar and Tippur [3] performed the static fracture analysis to calculate the crack tip stress field. Dolbow and Gosz [4] and Rao and Rahman [5, 6] analyzed the mixed mode SIF for 2-D orthotropic FGM using interaction energy contour integrals. Kim and Paulino [7] presented a path-independent J -integral in conjunction with the finite element method for mode-I and mixed-mode crack problems in orthotropic functionally graded materials having exponential and linear variations in material properties. Pavlou et al. [8] proposed a new methodology to simulate the fatigue crack trajectories under in-plane cyclic loading. They used a new factor related to the accumulated elastic strain energy within a circular core around the crack tip for predicting the mixed-mode fatigue crack path and fatigue crack growth rate. Zhang et al. [9] analyzed the unidirectional and bidirectional FGM under mixed mode loading using boundary integral equation method. Huang et al. [10,11] analyzed the multilayered model of FGM, and used Fourier transform approach for performing the fracture analysis of multilayered model. 2. XFEM FORMULATION The XFEM is a partition of unity (PU) enriched finite element method. It has proved to be a competent mathematical tool for the analysis of problems involving discontinuities due to the local enrichment of the approximation space. The enrichment is done through the PU concept. The method is particularly useful for the approximation of the solutions with prominent non-smooth characteristics in small parts of the computational domain like near discontinuities and singularities. A planar domain ( : ) bounded by contour * divided into three parts i.e., , * t and *c with internal flaws is shown in Figure 1. The displacement boundary conditions are imposed on *u , while tractions are applied on * t and traction free condition are imposed on crack surfaces *c . The equilibrium and boundary conditions for this problem may be described as t
:
x2
x1 *u
*t
*
u
Figure 1: Domain along with essential and natural boundary conditions
.σ b 0 in : σ. nˆ t on * t
(1)
0 on *c
(3)
σ .nˆ
(2)
(4) u u on *u where, σ is the Cauchy stress tensor, u is the displacement field vector, b is the body force vector per unit volume,
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Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238
t is the external traction vector and nˆ is the unit outward normal vector. For small displacements, straindisplacement relation can be described as ε
s u
ε(u)
(5)
where, s is the symmetric part of the gradient operator. The constitutive relations for the linear elastic FGM under consideration is given by Hooke's law
σ(u)
D(x) : ε(u)
(6)
where, x is the vector of x and y -coordinates, D(x) is the constitutive matrix, which can be written for plane strain condition as ª º «1 Q (x) Q (x) » 0 « » E ( x) (7) D ( x) 0 « Q ( x ) 1 Q ( x) » ^1 2Q (x)`^1 Q (x)` « » 1 2Q (x) 0 « 0 » 2 ¬ ¼ 3. EFGM FORMULATION In EFGM, the field variable u is approximated by moving least square approximation function u h ( x) which is given by [12, 13, 14] m
u h ( x)
¦ p (x)a (x) j
pT (x) a(x)
j
(8)
j 1
where, p(x) is a vector of basis functions, a(x) are unknown coefficients, and m is the number of terms in the basis. The unknown coefficients a(x) are obtained by minimizing a weighted least square sum of the difference between local approximation, u h ( x ) and field function nodal parameters u I . The weighted least square sum L(x) can be written in the following quadratic form: n
L(x)
¦ w(x x )[p
T
I
(x)a(x) uI ]2
(9)
I 1
where, u I is the nodal parameter (primary variable, displacement) associated with node I at x
x I . However,
u I are not the nodal values of u h (x x I ) since u h ( x ) is an approximant and not an interpolant as in the case of finite element method. By setting wL / wa =0, following set of linear equation is obtained: (10) A(x)a(x) B(x) u n
where,
¦II (x) y
I
y
I 1
B(x)
^w(x x1 ) p(x1 ), w(x x2 ) p(x2 ),......................., w(x xn ) p(xn )`
By substituting Eq. (10) in Eq. (8), the approximation function is obtained as: n
u h (x)
¦Φ (x) u I
(11)
I
I 1
where, I (x) I
m
1 ¦ p j (x)( A (x)B(x)) jI j
pT A1B I
The linear consistency requirements for the shape function I (x) are given as [15, 16, 17]: I
1234
Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238 n
¦II (x)
n
n
¦II (x) x
1,
I
I 1
¦II (x) y
x
I
y
(12)
I 1
I 1
The derivatives of MLS shape function are computed as:
ΦI ,x (x)
(pT A 1B I ) ,x pT ,x A1 BI pT (A1 ),x BI pT A1 BI ,x
where, B I , x (x)
(13)
dw (x x I )p(x I ) dx
and A 1, x is computed by
A1,x A1A,x A1 where, n dw A, x ¦ ( x x I ) p ( x I ) pT ( x I ) I 1 dx
(14)
4. CALCULATION OF STRESS INTENSITY FACTORS FOR FGM The domain based interaction integral approach has been widely used for calculating the stress intensity factors for homogeneous, bi-material and functionally graded materials under thermal and mechanical loads [6, 7, 18, 19]. For an elastic body subjected to mechanical load (Figure 1), the J-integral is given by § wu · (15) J ³ ¨ V ij i W G1 j ¸ q n j d * wx1 ¹ * © o
where, x [ x1
x2 ] T { [ x
y ] T , W is the strain energy density function and n j is the jth component of the
outward unit vector normal to an arbitrary closed contour * enclosing the area A (Figure 2). q stands for a weight function chosen such that it has a value of unity at the crack tip, zero along the boundary of the domain and arbitrary elsewhere
A
*
x2
x1
Figure 2: Path
* surrounding a crack with an enclosed area A
5. PROBLEM DESCRIPTION, RESULTS AND DISCUSSIONS A rectangular FGM plate of dimensions L 100 mm and D 200 mm of graded material (with 100% aluminum alloy on the left side and 100% ceramic (alumina) on the right side) is considered for the purpose of analysis. A uniform mesh consisting of 117 equally distributed nodes in x -direction and 235 equally distributed nodes in y direction are used for both EFGM and XFEM simulations. Nine node quadrilateral elements are used with a total of 6786 elements in the domain, for XEFM. Four point Gauss quadrature is used for numerical integration of EFG
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Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238
equations. For each case, the boundary conditions are depicted in the figures. The gradation in material property is taken along horizontal direction i.e. x-direction where x varies from x 0 to x 100 mm. In all the cases, an initial crack of a 20 mm length is assumed at the edge of the plate. The dimensions of the plate are taken same as that of FGM for the study of composite and pure aluminium alloy. A cyclic tensile load varying from
V
max
70 MPa to V min
0 MPa is applied in all the simulations. An increment of 'a
a 10
2 mm is taken to
evaluate the fatigue failure life of aluminum alloy, FGM and equivalent composite. 5.1 Plate with an Edge Crack Figures 3a and 3b show the plates with an edge crack on the alloy rich and ceramic rich sides respectively along with the relevant boundary conditions. The problem has been analyzed under plane strain condition using a uniform mesh of 117 by 235 nodes for both EFGM and XFEM. A crack extension vs number of cycles curve have been plotted for FGM, Equivalent composite and aluminium alloy as shown in Figure 3c. The fatigue failure cycles have been tabulated in table 1. Both, the plot and table depicts that EFGM results are in good agreement with the XFEM results. It is also observed that when the crack initiates from the ceramic (alumina) side, it fails much earlier than when the crack initiates from the alloy rich side. Table 1: (No. of Cycles) S. Fatigue Life Simulation Technique No 1 EFGM 2
Crack on Alloy Rich Side 15561
Crack on Ceramic Rich Side 4872
Crack on Equivalent Composite 7885
Crack on Aluminum Allloy 19145
15559
4874
7881
19141
XFEM
'V
'V
100% Ceramic
100% Alloy
100 % Ceramic
100 % Alloy
a
a
D
D D 2
D 2
L Figure 3a: Plate with an edge crack on the alloy rich side under mode-I
L Figure 3b: Plate with an edge crack on the ceramic rich side under mode-I
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Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238 22 FGM (crack on alloy rich side)-XFEM FGM (crack on ceramic rich side)-XFEM Equivalent composite-XFEM Aluminum Alloy-XFEM EFGM Data
20 18
Crack Extension (mm)
16 14 12 10 8 6 4 2 0
0
0.2
0.4
0.6
0.8 1 1.2 No. of Cycles
1.4
1.6
1.8
2 4
x 10
Figure 3c: A plot of crack extension with number of cycles
5.2 Edge crack under Mode II Loading Next we considered plates with an edge crack on the alloy rich and ceramic rich sides respectively (Figures 4a and 4b) along with the relevant boundary conditions. The dimensions of the plate, nodal density, and boundary conditions are considered same as that of FGM for the study of composite and pure aluminum alloy. An initial crack of a 20 mm length is assumed at the both at edge and centre of the plate. A cyclic tensile load varying from V max 70 MPa to V min 0 MPa is applied in all the simulations. In addition to the mode-I cyclic tensile load, a shear fatigue load of
W
max
10 MPa and W
life. An increment of 'a
a 10
min
0 MPa is applied to study the effect of mixed mode load on the fatigue
2 mm is taken to evaluate the fatigue failure life of aluminum alloy, FGM and
equivalent composite. Both EFGM and XFEM predict nearly same values of fatigue life as can be clearly seen from Table. 2 and Figure 4c. It is evident that the fatigue life reduces considerably in case of mixed mode mechanical load as compared to that of pure mode-I load. Table 2: (No. of Cycles) S. No 1
Fatigue Life Simulation Technique EFGM
Crack on Alloy Rich Side 12185
2
XFEM
12181
Crack on Ceramic Crack on quivalent Crack on Auminum Allloy Rich Side Composite 4575 6008 14893 4573
6006
14890
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Mohit Pant et al. / Procedia Engineering 173 (2017) 1231 – 1238
'V
'V
'W
'W
100 % Alloy
100 % Ceramic
a
100 % Ceramic
100 % Alloy
a D
D
D 2
D 2 L
L
Figure 4a: Plate with an edge crack on the alloy rich side under mixed mode loading
Figure 4b: Plate with an edge crack on the ceramic rich side under mixed mode loading
21
FGM Crack on alloy Side EFGM Data FGM Crack on ceramic side EFGM Data Equivalent Composit XFEM data EFGM Data Aluminua Alloy XFEM EFGM Data
Crack Extension (mm)
18
15 12
9 6
3
0 0
3000
6000
9000
12000
15000
18000
No of Cycle
Figure 4c: A plot of crack extension with number of cycles
6. CONCLUSIONS In the present work, fatigue crack growth simulations have been performed for FGM, equivalent composite and aluminium alloy by EFGM and XFEM under mixed mode cyclic loading. On the basis of the present simulations, it is found that the spatial location of crack in a FGM plays a major role in determining the fatigue life of component. The results also reveal that the fatigue life reduces considerably in case of mixed mode loading as compared to that of pure mode-I loading. The simplicity and effectiveness of the proposed criterion shows its potential to simulate realistic problems of fracture in functionally graded materials using EFGM.
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