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Fatigue life simulation of functionally graded materials under cyclic thermal load using XFEM
International Journal of Mechanical Sciences 82 (2014) 41–59
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Fatigue life simulation of functionally graded materials under cyclic thermal load using XFEM S. Bhattacharya, I.V. Singh n, B.K. Mishra Department of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India
art ic l e i nf o
a b s t r a c t
Article history: Received 28 January 2013 Accepted 3 March 2014 Available online 12 March 2014
In this work, the fatigue life of a functionally graded material (FGM) plate made of aluminum alloy and alumina (ceramic) is simulated under cyclic thermal load. The various discontinuities such as minor cracks, holes and inclusions of arbitrary size are randomly distributed in the plate. The fatigue lives of aluminum alloy, FGM and an equivalent composite (having the same composition as of FGM) plates are evaluated using Paris law in the presence of multiple discontinuities, and are compared with each other. & 2014 Elsevier Ltd. All rights reserved.
Keywords: FGM Thermal loading XFEM Fatigue crack growth Holes Inclusions
1. Introduction In recent years, a new class of graded composite materials more commonly known as functionally graded materials (FGMs) have been developed to meet the various needs of the engineering industry. A certain variation in the material properties of FGMs is achieved through local variation in composition or microstructure or both. The FGM has a remarkable capability to withstand bending as well as stretching, and is typically an inhomogeneous composite usually made from ceramic and metal. FGMs attain the multi-structural status due to their property gradation. By gradually varying the volume fraction of constituent materials, the material properties of FGMs exhibit a smooth and continuous change from one surface to another, thus eliminating interface problems and mitigating thermal stress concentrations. FGMs are processed in such a way that they possess continuous spatial variations in volume fractions of their constituents to generate a predetermined composition profile. These variations lead to the formation of a nonhomogeneous macrostructure with continuously varying mechanical and/or thermal properties in one or more than one direction. The microstructure of FGM is generally heterogeneous, and the dominant type of failure in FGM occurs from crack initiation and growth from flaws/inclusions. Thus, the study of the thermal fatigue life of FGMs becomes quite important in the presence of various flaws. Till date, some efforts have already been made to study and analyze the behavior of FGMs. Prabhakar and Tippur [14]
n
Corresponding author. Tel.: +91 1332 285888 (O); fax: +91 1332 285665. E-mail addresses: [email protected], [email protected] (I.V. Singh).
http://dx.doi.org/10.1016/j.ijmecsci.2014.03.005 0020-7403 & 2014 Elsevier Ltd. All rights reserved.
performed the static fracture analysis to calculate the crack tip stress field. Dolbow and Gosz and Rao and Rahman [5,16] analyzed the mixed-mode SIFs for 2-D orthotropic FGM using interaction energy contour integrals. Zhang et al. [23] analyzed the unidirectional and bidirectional FGM under mixed-mode loading using boundary integral equation method. Huang et al. [9,10] analyzed the multilayered model of FGM, and used Fourier transform approach for performing the fracture analysis of multilayered model. Zhang et al. [24] performed the elastostatic analysis of anti-plane cracks in unidirectional and bidirectional FGM using hypersingular boundary integral equation method. The analysis of FGMs under thermal loading has also been performed by Kokini et al. [12]. Rangaraj and Kokini [15] presented a methodology to estimate the crack growth resistance of functionally graded yttria stabilized zirconia bond coat alloy (made of nickel, cobalt, chromium, aluminum and yttria). Zhao et al. [25] evaluated the thermal shock resistance of Al2O3–TiC and Al2O3–(W, Ti)C functionally graded ceramic tool materials. Fazarinca et al. [6] estimated the suitability of the functionally graded materials subjected to thermal fatigue. Some studies on FGMs have also been carried out using FEM [21,18]. From the studies carried out so far, it was found that fatigue crack growth analysis of FGM has not been performed in the presence of multiple discontinuities under thermal loading. Moreover, the fatigue failure analyses of alloy/ceramic FGMs have not been explored much by any numerical scheme. As such, the traditional fracture based failure theories do not take into account the effect of small defects/discontinuities in the material for the life prediction. Therefore, in the present work, the fatigue crack growth analysis of alloy/ceramic FGMs, aluminum alloy and equivalent composite has been performed by XFEM in the
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S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
presence of multiple discontinuities (cracks, holes and inclusions) under thermal loading. A crack of an initial length is incorporated at the edge as well at the center of the FGM, pure aluminum alloy and an equivalent composite plate having the same composition as the FGM. The XFEM formulation for FGM is presented in Section 2. Section 3 depicts an overview of the method for calculating the SIF at the crack tip from a domain based interaction integral approach. Section 5 describes the physics of the FGM along with other relevant properties. The problem description and the discussion of the results obtained for various case of edge and center cracks is given in Section 6. Section 7 gives the conclusions derived from the present study.
2. XFEM formulation for FGM XFEM is a partition of the unity (PU) enriched finite element method which is particularly effective in handling strong as well as weak discontinuities in the domain. In this method, a crack is modeled by enrichment functions so a regular mesh is used for modeling the crack and crack growth without altering the original mesh. It also eliminates the need of singular elements for capturing the singularity at the crack tip (as in conventional FEM) as it makes use of asymptotic crack tip functions for this purpose.
The constitutive relations for the linear elastic material is given by Hook's law
rðuÞ ¼ DðxÞ : εðuÞ
ð4Þ
where, x is the vector of x and y-coordinates, DðxÞ is the constitutive matrix. 2.2. Variational formulation
Z Ω
A weak form of the equilibrium equation can be written as Z Z rðuÞ : εðvÞdΩ ¼ b vdΩ þ t vdΓ ð5Þ Ω
Γt
Substituting the constitutive relation rðuÞ ¼ DðxÞ : εðuÞ in the above equation, we obtain Z Z Z εðuÞ : DðxÞ : εðvÞdΩ ¼ b vdΩ þ t vdΓ ð6Þ Ω
Ω
Γt
_ _ The linear form, T , and the bilinear form, S, of the above equation are obtained as Z Z _ ð7Þ T ðvÞ ¼ b vdΩ þ t vdΩ Γ
Γt
2.1. Governing equations
x2
A two-dimensional domain (Ω) bounded by contour Γ divided into three parts i.e., Γ u , Γ t and Γ c with internal flaws like cracks, holes and inclusions is shown in Fig. 1. The displacement boundary conditions are imposed on Γ u , while tractions are applied on Γ t and traction free conditions are imposed on the crack surfaces Γ c . The equilibrium and boundary conditions for this problem may be described as ∇ r þ b ¼ 0 in Ω
Γo x1
ð1Þ
r U n^ ¼ t on Γ t
ð2aÞ
r U n^ ¼ 0 on Γ c
ð2bÞ
u ¼ u on Γ u
ð2cÞ
Ao
Fig. 2. Path independent closed contour around the crack tip.
where, r is the Cauchy stress tensor, u is the displacement field vector, b is the body force vector, t is the external traction vector ^ is the vector of unit outward normal. For small displaceand n ments, strain–displacement relation can be described as ε ¼ εðuÞ ¼ ∇s u
ð3Þ
y
where, ∇s is the symmetric part of the gradient operator.
y
100 %
t=t
Ω
Alloy
Γt
100 % Ceramic x
D
Γc Γu u=u
x Fig. 1. Domain with discontinuities.
L Fig. 3. Geometry of the FGM plate along with its dimensions.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
_ Sðu;vÞ ¼
Z Ω
εðuÞ : DðxÞ : εðvÞdΩ
ð8Þ
The quadratic energy functional can be written as 1_ _ ΠðuÞ ¼ Sðu;uÞ T ðuÞ 2
ð9Þ
On substitution of the relevant quantities, the expression for the same is obtained as Z Z Z 1 ΠðuÞ ¼ εðuÞ : DðxÞ : εðuÞdΩ b udλ t udλ ð10Þ 2 Ω Γ Γt
43
By using the trial and test functions and simplifying the above equation, the following set of discrete equations is obtained using the arbitrariness of nodal variations ½Kfdg ¼ ffg
ð11Þ
where, d is the vector of nodal unknowns, K and f are the global stiffness matrix and the external force vector respectively. The stiffness matrix and force vector are computed on element level, and are assembled into their global counterparts through usual finite element assembly procedure. 2.3. Displacement approximation
Table 1 Material properties of aluminum alloy and alumina ([2,4], Roylance, 2001 [19]). Material properties
Aluminum alloy Alumina
Elastic modulus E (GPa) Poisson's ratio, ν Coefficient of thermal expansion γ (/1C) Fracture toughness K IC (MPa√m) pffiffiffiffiffi Paris law parameter C in m=cycleðMPa mÞ m Paris law parameter, mðxÞ
70 0.33 25 10 6 29 10 12 3
100 % Alloy
300 0.213) 8.2 10 6 3.5 2.8 10 10 10
100 % Ceramic
a
For modeling cracks in XFEM [3,13], the approximation function takes the following form 2 3 6 7 n 4 6 j7 7 uh ðxÞ ¼ ∑ Ni ðxÞ6 u þ ΗðxÞa þ β ðxÞb ∑ j i7 6 i |fflfflffl{zfflfflffl}i i¼1 j¼1 4 5 i A nr |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
ð12aÞ
i A nA
where, i is the set of all nodes in the domain, N i ðxÞ is the element shape function associated with node i satisfying the partition of
100 % Alloy
a
D
D
D
2
2
100 % Ceramic
L
D
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 4. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
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S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
unity criterion, ui is the nodal displacement vector associated with the continuous part of the finite element solution, ai denotes the additional unknown degrees of freedom associated with the discontinuous Heaviside function ΗðxÞ, and is defined for those elements, which are completely cut by the crack to account for the j jump in the displacement field, bi is the additional degrees of freedom associated with those elements, which are partially cut by the crack, and accounts for stress singularity at the crack tip, n is the set of all nodes in the mesh, nr is the set of nodes belonging to those elements which are completely cut by the crack, and nA is the set of nodes belonging to those elements which are partially cut by the crack. For any node nr , Heaviside jump function, ΗðxÞ takes a constant value, and is equal to þ 1 on one side and 1 on other side of the crack. Thus, a shifted enrichment is used. If xi is the concerned node then Eq. (12a) can be written as 2 3 6 7 n 4 6 j7 7 u þ ½ΗðxÞ Hðx Þa þ ½β ðxÞ β ðx Þb uh ðxÞ ¼ ∑ N i ðxÞ6 ∑ i i j j i i7 6 i |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} i¼1 j¼1 4 5 i A nr |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
The sub-matrices and vectors that appear in the foregoing equations are given by Z ðΒri ÞT DΒsj dΩ where r; s ¼ u; a; b ð13bÞ Krs ij ¼ Ωe
Z
u
fi ¼
100 % Ceramic a
ð13aÞ
ð12bÞ
i A nA
100 % Alloy
In Eq. (12b), the difference between the values of the Heaviside function at the evaluation (Gauss) point and nodal point is taken to maintain the partition of unity. The tip enrichment is performed for those elements which contain the crack tip, and is achieved by adding additional crack tip functions. Using the approximation function defined in Eq. (12b), the e elemental matrices, Ke and f are obtained as 2 uu 3 K ij K ua K ub ij ij 6 7 n oT 6 au K aa K ab 7 u a b1 b2 b3 b4 e fi fi fi f ¼ fi fi fi and Keij ¼ 6 K ij ij ij 7 4 5 K bu K ba K bb ij ij ij
Ωe
Z Ν i bdΩ þ
Γt
Ν i tdΓ
ð13cÞ
100 % Alloy
100 % Ceramic a
D
D
D
2
2
D
L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0
0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 5. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
a
Z
fi ¼
b_ α
Z Ω
e
Ν i ðΗðxÞ Hðxi ÞÞbdΩ þ
Γt
Ν i ðΗðxÞ Hðxi ÞÞtdΓ
ð13dÞ
Z
fi ¼
Ωe
ðΝ i ðβ_α ðxÞ β_α ðxi ÞÞÞx 6 ¼40 ðΝ i ðβ_α ðxÞ β_α ðxi ÞÞÞy
3
0
ðΝ i ðβ_α ðxÞ β_α ðxi ÞÞÞy 7 5; ðΝ i ðβ_α ðxÞ β_α ðxi ÞÞÞx
Z
Γt
Ν i ðβ_α ðxÞ ðxi ÞÞtdΓ
where _ α ¼ 1; 2; 3; 4
ð13eÞ 3. Calculation of stress intensity factors
where, Ν i are finite element shape functions, and Βui , Βai , Βbi and are given by Ν i;x 60 u Βi ¼ 4 Ν i;y
0
_ α ¼ 1; 2; 3; 4; ð13hÞ
Ν i β_α ððxÞ ðxi ÞÞbdΩ
þ
2
2
_ Βbi α
45
_ Βbi α
3
Ν i;y 7 5
ð13f Þ
Ν i;x
2
3 ðΝ i ðΗðxÞ Hðxi ÞÞÞx 0 6 ðΝ i ðΗðxÞ Hðxi ÞÞÞy 7 Βai ¼ 4 0 5 ðΝ i ðΗðxÞ Hðxi ÞÞÞy ðΝ i ðΗðxÞ Hðxi ÞÞÞx
ð13gÞ
100 % Alloy
100 % Ceramic
a
In the present work, the domain based interaction integral approach [1,7,11,16,22] has been used for calculating the stress intensity factors for homogeneous and functionally graded materials under thermal load. For an elastic cracked FGM body as shown in Fig. 2, J-integral is defined as ∂u J ¼ ∮Γ0 sij i Wδ1j nj dΓ ð14Þ ∂x1 where, x ¼ ½x1 x2 T ½x yT , W is the strain energy density function and nj is the jth component of the outward unit vector normal to an arbitrary closed contour Γ 0 enclosing the area A0 .
100 % Ceramic
100 % Alloy
a
D
D
D 2
D 2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 6. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
46
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
Now, J can be converted into the equivalent domain form using divergence theorem, Z ∂u J¼ sij i Wδ1j dA ð15Þ ∂x1 A0 j The J-integral is path independent, where Γ 0 is a path starting from the lower crack face and terminating on the upper crack face and A0 is the enclosed area within Γ 0 . 1 m m In Eq. (15), W is defined as W ¼ 12sij εm ij ¼ 2C ijkl εkl εij , where, εm ¼ ε γΔTδ , γ is the coefficient of thermal expansion and ΔT ij ij ij is the temperature difference. Let q be a weight function such that its value is one at the crack tip, zero at the crack face, and arbitrary elsewhere. Now, Eq. (15) can be simplified as Z ∂ ∂u J¼ sij i Wδ1j q dA ð16Þ ∂x1 A0 ∂xj Substituting the value of W in Eq. (17), it can be written as Z ∂ ∂u 1 J¼ sij i sik εm δ ik 1j q dA ∂x1 2 A0 ∂xj
ð17Þ
100 % Ceramic
a
State 1 (actual state): sij εij ui J~ State 2 (auxiliary state): saij εaij uai
D
Ja
Defining the J-integral for both the states J~ ¼
Z
∂ A0 ∂xj
Z Ja ¼
∂ A0 ∂xj
sij
∂ui 1 sik εm δ q dA 1j ik ∂x1 2
saij
∂uai 1 a a sik εik δ1j q dA ∂x1 2
ð18Þ
ð19Þ
The J-integral for these two superimposed state can be defined as Z
For calculating the interaction integral for an elastic body, we consider two equilibrium states of the cracked body. State 1 is
100 % Alloy
taken as the actual state, state 2 is taken as an auxiliary state and superscript a denotes the parameters for the auxiliary state.
JT ¼
∂ A0 ∂xj
∂ui ∂uai 1 a ðsij þ saij Þ þ ðsik þ saik Þðεm ik þεik Þδ1j q dA 2 ∂x1 ∂x1 ð20Þ
J T ¼ J~ þ J a þ M 12
ð21Þ
100 % Alloy
a
D
100 % Ceramic
D
D
2
2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0
0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 7. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
where, Z M 12 ¼
∂ A0 ∂xj
∂ua ∂u 1 sij i þ saij i ðsik εaik þ saik εm ik Þδ1j q dA ∂x1 ∂x1 2
ð22Þ
where, the auxiliary field for the FGM is taken from [22] ! a a ∂ual 1 ∂uai ∂uj tip 1 ∂uk a a a a ; εij ¼ Sijkl ðxÞskl and εij a sij ¼ C ijkl þ þ 2 ∂xl ∂xk 2 ∂xj ∂xi
ð23Þ
Using sik εaik ¼ sik Sikpq ðxÞsapq ¼ εpq sapq ¼ saik εik , Eq. (22) can be written as Z ∂ua ∂ ∂u sij i þ saij i saik εm ð24Þ M 12 ¼ ik δ1j q dA ∂x1 ∂x1 A0 ∂xj Eq. (24) can be modified as Z M 12 ¼ A0
Z þ
A0
sij
∂uai ∂u ∂q þ saij i saik εm dA ik δ1j ∂x1 ∂x1 ∂xj
! m 2 ∂saij ∂ui ∂saik m ∂sij ∂uai ∂2 uai a ∂ ui a ∂εik þ sij þ þ sij ε sik qdA ∂xj ∂x1 ∂xj ∂x1 ∂xj ∂x1 ∂xj ∂x1 ∂x1 ik ∂x1
47
Using equilibrium equation and symmetry of the stress tensor, Eq. (25) is simplified as Z ∂ua ∂u ∂q M 12 ¼ sij i þ saij i saik εik δ1j dA ∂xj ∂x1 ∂x1 A0 ! ! Z a ∂saik 1 ∂ ∂uai ∂uj ∂ þ saij qdA sij þ ðεij εm Þ ε þ ij 2 ∂x1 ∂xj ∂xi ∂x1 ∂x1 ik A0 ð26Þ Using Eq. (24) and εm ij ¼ εij γΔTδij , Eq. (26) is further simplified as Z a ∂u ∂u ∂q M 12 ¼ sij i þ saij i saik εm dA ik δ1j ∂xj ∂x1 ∂x1 A0 Z ∂sakl ∂γ ∂ΔT a δij qdA ð27Þ sij ðStip S ðxÞÞ þ s ΔT þγ þ ijkl ij ijkl ∂x1 ∂x1 ∂x1 A0 For aluminum alloy and equivalent composite, Stip ¼ Sijkl ðxÞ. Now, ijkl Eq. (27) reduces to Z M 12 ¼ A0
sij
Z ∂uai ∂u ∂q ∂γ ∂ΔT δij qdA þ saij i saik εm dA þ saij ΔT þγ ik δ1j ∂x1 ∂x1 ∂xj ∂x ∂x 1 1 A0
ð25Þ
100 % Alloy
100 % Ceramic
a
ð28Þ
100 % Ceramic
100 % Alloy
a
D
D
D 2
D 2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0
0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 8. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
48
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
SIF at the crack tip can be computed using the above interaction integral as K I;II ¼
M 12 Ε nTip
ð29Þ
2ð1 ν2Tip Þ
where, Ε ntip is evaluated at the crack tip which is equal to Ε ntip ¼ Ε tip for plane stress and Ε ntip ¼ Ε tip =1 v2tip for plane strain.
direction are given as θc θc θc K Ieq ¼ K I cos 3 3K II cos 2 sin 2 2 2 0 θc ¼ 2 tan 1 @
KI
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 K 2I þ 8K 2II A 4K II
ð31Þ
From Eq. (30), the equivalent SIF can be found as θc θc θc ΔK Ieq ¼ ΔK I cos 3 3ΔK II cos 2 sin 2 2 2 4. Fatigue crack growth
For stable crack growth, Paris' law is given as
In this study, Paris law has been used to assess the fatigue life under cyclic thermal loading. The SIFs corresponding to the maximum and minimum load can be evaluated using the approach outlined in the previous section. At each crack tip, the direction of crack growth θc is determined using the maximum principal stress theory [20]. The crack is assumed to grow in a direction perpendicular to the maximum principal stress. According to this criterion, the equivalent mode-I SIF and crack growth
da ¼ CðxÞðΔK Ieq ÞmðxÞ dN
100 % Alloy
100 % Ceramic
a
ð30Þ
ð32Þ
ð33Þ
where, CðxÞ and mðxÞ are the functions of x. For implementation purpose, a value of crack growth, Δa is assumed and the corresponding failure cycles (ΔN) are calculated from Eq. (33). When more than one crack tip is present in the domain, Δa for the most dominant crack tip is assumed, corresponding ΔN is calculated and then the crack growth at the other
100 % Ceramic
100 % Alloy
D
a
D
D
D
2
2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 9. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
crack tips are calculated corresponding to this known value of ΔN. Finally, when the maximum value of K Ieq for any crack tip becomes more than the local value of the fracture toughness K IC then the simulation is stopped. At this point, the total number of cycles becomes the fatigue life of the material.
5. Variation in the properties of FGM
49
The rule of mixtures is applied to estimate the local elastic modulus of the FGM using the local volume fraction of ceramic. Thus, the volume fractions of ceramic and aluminum alloy in the FGM are given as V FGM ceramic ðxÞ ¼
EðxÞ Ealloy Ealloy eαx Ealloy ¼ Eceramic Ealloy Eceramic Ealloy
FGM V FGM alloy ðxÞ ¼ 1 V ceramic ðxÞ:
The FGM plate is composed of aluminum alloy and ceramic (alumina) as shown in Fig. 3. The variation in the composition of FGM is taken along x-direction. From the figure, it is seen that the FGM plate has 100% aluminum alloy at x ¼ 0 and 100% ceramic at x ¼ L. The material properties of the aluminum alloy and alumina used in FGM are given in Table 1. An exponential variation in the elastic modulus for FGM is taken as EðxÞ ¼ Ealloy eαx
ð35bÞ
The Poisson's Ratio for the FGM can be calculated using Halpin– Tsai equation [8] νðxÞ ¼
FGM νalloy V FGM alloy ðxÞE ceramic þ νceramic V ceramic ðxÞE alloy FGM V FGM alloy ðxÞE ceramic þ V ceramic ðxÞE alloy
ð34Þ
100 % Alloy
100 % Ceramic
a
ð36Þ
The rule of mixtures is applied again to estimate the coefficient of thermal expansion for the FGM as FGM γðxÞ ¼ γ alloy V FGM alloy ðxÞ þ γ ceramic V ceramic ðxÞ
where, α is given as 1 E α ¼ ln ceramic L Ealloy
ð35aÞ
ð37Þ
The fracture toughness of the FGM can be expressed as a function of the volume fraction of the ceramic [17] using the
100 % Ceramic
100 % Alloy
D
a
D
D
D
2
2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 10. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
50
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
following formula K IC ðxÞ ¼
K alloy þ K ceramic IC IC 2
þ
K alloy K ceramic IC IC 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 V FGM V FGM ceramic ðxÞ ceramic ðxÞÞ
ð38Þ The Paris law parameters vary in the similar manner as the elastic modulus, and are given by the following equations 1 C ð39Þ CðxÞ ¼ C alloy eϑx ; where ϑ ¼ ln ceramic L C alloy mðxÞ ¼ malloy eςx
where ς ¼
1 mceramic ln L malloy
ð40Þ
The equivalent composite consists of the same amount of the alloy and the ceramic as in the FGM. The fatigue life of the FGM plate is compared with that of the equivalent composite. Hence, the volume fraction of the ceramic in the equivalent composite can be calculated as Z 1 L FGM V composite ¼ V ðxÞdx ð41Þ ceramic L 0 ceramic
100 % Alloy
100 % Ceramic a
where, L is the length of the plate., which is taken as L ¼ 100 mm in the present simulations. Hence, we obtain V composite ceramic ¼ 38:3% and V composite ¼ 61:72%. Now, using the rule of mixtures for the alloy equivalent composite, the elastic modulus is obtained as þEceramic V composite Ecomposite ¼ Ealloy V composite ceramic alloy
ð42Þ
Using the Halpin–Tsai equation, Poisson's ratio is calculated as νcomposite ¼
νalloy V composite Eceramic þ νceramic V composite ceramic E alloy alloy V composite Eceramic þ V composite ceramic E alloy alloy
ð43Þ
The coefficient of thermal expansion for the equivalent composite is estimated by the rule of mixtures as þ γ ceramic V composite γ composite ¼ γ alloy V composite ceramic alloy
ð44Þ
The fracture toughness of equivalent composite is estimated in the same manner as for FGM ¼ K composite IC
K alloy þ K ceramic K alloy K ceramic IC IC IC þ IC 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
1 V composite ceramic
V composite ceramic Þ
100 % Alloy
100 % Ceramic a
D
D
ð45Þ
D
D
2
2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 11. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
nodes in x-direction and 235 equally distributed nodes in y-direction is used for the simulations. For each case, the boundary conditions are shown in the respective figures. The gradation in material property is taken along x-direction where x varies from x ¼ 0 to x ¼ 100 mm. In all cases, an initial crack of length a ¼ 20 mm is assumed at the edge as well as at the center of the plate. The dimensions of the plate are taken same for FGM, composite and pure aluminum alloy. For simulating the thermal fatigue, a cyclic thermal load is applied equivalent to a stress of Δs ¼ 70 MPa. The temperature difference (ΔT) obtained using the properties of the equivalent composite Δs ¼ Ecomposite γ composite ΔT is found to be ΔT ¼ 23:86 o C for inducing thermal fatigue. An a increment of Δa ¼ 10 ¼ 2 mm is taken to evaluate the fatigue failure life of aluminum alloy, FGM and equivalent composite in the presence of a major discontinuity. The other discontinuities like holes, inclusions and minor cracks and their combinations are
For the equivalent composite, the location x in the FGM can be E found using the relation x ¼ 1α ln composite , where α is defined in Ealloy Eq. (34). The Paris law parameters of the equivalent composite are assumed to be same as that of the FGM at x ¼ x . Thus, C composite ¼ C alloy eϑx
ð46aÞ
mcomposite ¼ malloy eςx
ð46bÞ
51
6. Problem description, results and discussions A rectangular plate of dimensions L ¼ 100 mm and D ¼ 200 mm of graded material having 100% aluminum alloy on the left side and 100% ceramic (alumina) on the right side is taken for the analysis. A uniform mesh consisting of 117 equally distributed
100 % Alloy
100 % Ceramic
a
D
D 2 L
25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
2
2.5
No. of Cycles
3
3.5
4
4.5 x 10
4
Fig. 12. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
52
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
22,705 cycles before failure. Moreover, it is observed that when a crack initiates from the ceramic (alumina) side, it fails much earlier (only 12,264 cycles) than when the same is initiated from the aluminum alloy side. In addition to the major crack, 30 randomly distributed holes are added in the domain above and below the major crack as shown in Fig. 5a and b respectively. The sizes of the holes vary randomly from 3 mm to 4.5 mm. These simulations reveal that the number of cycles to failure in case of aluminum alloy is about 20,326 cycles; whereas the average fatigue life of FGM with a crack on the alloy rich side and ceramic rich side is found to be 17,741 cycles and 9607 cycles respectively. The fatigue life of the equivalent composite is found to be 12,352 cycles. Due to the presence of holes, the fatigue life of the aluminum alloy goes down by about 10.48% whereas the fatigue life of FGM with crack on the ceramic rich side is reduced by 21.67%. Even when the crack is on
added in the plate in addition to the major crack. The material properties of the inclusions are taken as Ε ¼ 20 GPa and ν ¼ 0:20 in all the simulations. 6.1. Plate with an edge crack and multiple discontinuities Fig. 4a and b shows a plate with a major edge crack of length a ¼ 20 mm on the left and right edges of the plate respectively. The simulations are performed for aluminum alloy, equivalent composite and FGM cracked plate in the presence of multiple discontinuities. In case of FGM, the major edge crack either initiates from the aluminum alloy rich side or ceramic rich side. The plots of the fatigue life for aluminum alloy, FGM and equivalent composite are as shown in Fig. 4c. From this analysis, it is found that the equivalent composite survives 14,773 cycles before it fails while the FGM withstands 21,642 cycles and aluminum undergoes
100 % Alloy
100 % Ceramic a
D
D 2 L
25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
2
2.5
No. of Cycles
3
3.5
4
4.5 x 10
4
Fig. 13. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
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53
10,049 cycles respectively. Thus, the fatigue life of the aluminum alloy plate reduced by 1.15% due to the presence of inclusions; whereas the fatigue life of FGM decreased by 2.92% and 18.06% when the crack is lying on the alloy and ceramic rich sides respectively. The fatigue life of the composite is reduced by 6.91%. Further, it is noticed that the effect of holes on the fatigue life is more severe as compared to the same number of inclusions. In the subsequent study, 54 minor cracks are randomly added in the plate above and below the major crack with their inclination randomly varying from 01 to 601 as shown in Fig. 7a and b respectively. In this study it is observed that the number of cycles to failure in case of aluminum alloy is about 21,642 cycles whereas in the case of FGM, the fatigue life is found nearly 20,023 and 11,024 cycles when the crack lies on the alloy and ceramic rich sides respectively. The fatigue life of composite is found to be
the alloy rich side, the fatigue life of FGM plate is still reduced by about 18.03%. For the composite, the fatigue life decreases by 16.39%. Thus, it is evident that the fatigue life of FGM and composite is influenced to some extent due to the presence of holes. The plots of crack extension with the number of cycles for various materials (aluminum alloy, equivalent composite, FGM) for this case are shown in Fig. 5c. Fig. 6a and b shows the major edge crack along with 30 randomly located inclusions. The sizes of the inclusions vary randomly from 3 mm to 4.5 mm as in case of holes. The plots of crack extension with the number of cycles are shown in Fig. 6c for all the materials. On the basis of these simulations, it is found that the fatigue lives of the aluminum alloy and composite are found to be 22,444 and 13,752 cycles; whereas the life of FGM with crack on the alloy and ceramic rich side is found to be 21,009 cycles and
100 % Alloy
100 % Ceramic a
D
D 2 L
25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
2
2.5
No. of Cycles
3
3.5
4
4.5 x 10
4
Fig. 14. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
54
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crack extension with the number of cycles are shown in Fig. 8c. In this case, it is found that the average number of cycles to failure for the aluminum alloy and the composite are found to 21,708 and 13,886 cycles respectively. The life of the FGM plate with crack on the alloy and ceramic rich sides is found to be 20,723 cycles and 11,005 cycles respectively. Thus, it is observed that the life of the aluminum alloy is decreased by 4.39%; whereas it is decreased by 4.25% and 10.27% for the FGM with crack on the alloy and ceramic side respectively. The life of the composite is reduced by 6% due to the presence of minor cracks and holes. When the 23 holes/voids are replaced by 23 randomly located inclusions (radii of the inclusions also vary randomly from 3 to 4.5 mm) in the previous simulation along with the randomly located minor cracks in the domain as shown in Fig. 9a and b the number of cycles to failure in case of aluminum alloy is about
12,828 cycles. Thus, it is observed that the life of the aluminum alloy plate goes down by about 4.68%; whereas the life of the FGM plate goes down by 7.48% and 10.11% respectively depending upon whether the crack is on the alloy rich side or the ceramic rich side. The life of the composite goes down by 13.17% due to the presence of minor cracks. The plots for crack extension with the number of cycles are shown in Fig. 7c. Now, 42 minor cracks and 23 holes/voids are added in the plate containing the major edge crack made from aluminum alloy, FGM and the composite as shown in Fig. 8a and b respectively with the locations of the minor cracks and holes are randomly distributed in the domain above and below the major crack. The sizes of the minor cracks vary randomly from 3.5 mm to 4.5 mm. These cracks are randomly oriented at angles ranging from 01 to 601. The radii of the holes also vary randomly from 3 to 4.5 mm. The plots of
100 % Alloy
100 % Ceramic D
a D 2 L
25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
2
2.5
No. of Cycles
3
3.5
4
4.5 x 10
4
Fig. 15. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
55
respectively for left and right edge cracks. The plots of crack extension with the number of cycles for each material are shown in Fig. 10c. From these plots, it is observed that the number of cycles to failure in case of aluminum alloy is found to be 21,995 cycles; whereas in the case of FGM with crack on the alloy rich and ceramic rich sides, fatigue life is found to be 20,314 and 7791 cycles respectively. The failure life of the cracked composite plate is found to be 11,145 cycles. Thus, due to the presence of holes and inclusions, the life of the aluminum alloy goes down by about 3.13%; whereas the life of the FGM with the crack on the alloy and ceramic sides is reduced nearly by 6.14% and 36.47% respectively. The life of the equivalent composite is reduced by 24.56%. Finally 36 minor cracks, 15 holes and 15 inclusions are taken in the domain along with the major left and right edge cracks as shown in Fig. 11a and b respectively. The minor cracks, holes and
20,999 cycles. In FGM with crack on the alloy and ceramic rich sides are about 20,672 and 9773 cycles respectively. The failure life for composite is found to be 11,633 cycles. Thus, it is observed that the life of the aluminum alloy goes down by about 7.51%; whereas the fatigue life of FGM goes down by 4.48% and 2031% respectively depending on whether the crack is on the alloy side or the ceramic side. The fatigue life of the equivalent composite is reduced nearly by 21.25% due to the presence of minor cracks and inclusions. The plots of crack extension with the number of cycles are shown in Fig. 9c. In the next study, 20 holes and 20 inclusions are incorporated in the domain in addition to the major edge crack. The holes and inclusions are randomly distributed in the plate above and below the major crack. The radii of both the holes and inclusions vary randomly from 3 mm to 4.5 mm as depicted in Fig. 10a and b
100 % Alloy
100 % Ceramic
a
D
D 2 L
25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0
0
0.5
1
1.5
2
2.5
No. of Cycles
3
3.5
4
4.5 x 10
4
Fig. 16. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
56
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reduced by 11.78%. The plots of crack extension with the number of cycles are shown in Fig. 11c. From all these simulations, it is evident that the FGM shows the least fatigue life as compared to other materials. In FGM, when the major crack is present on the right edge i.e. on the ceramic rich side, the fatigue life diminishes to a great extent as compared to the case when the crack is present on the left edge i.e. on the alloy rich side. The equivalent composite shows a moderate life whereby the alloy shows the maximum fatigue life among three materials
inclusions are randomly distributed in the domain above and below the major crack. The sizes of the minor cracks vary randomly from 3.5 mm to 4.5 mm. These cracks are randomly oriented in between 01 and 601. The radii of the holes and inclusions vary randomly from 3 to 4.5 mm. In this case it is seen that the number of cycles to failure in case of aluminum alloy is found to be 21,730 cycles; whereas in case of FGM with crack on the alloy and ceramic rich sides, the fatigue life is found to be 19,740 and 9751 cycles respectively. The fatigue life of the composite plate is found to be 12,885 cycles. On the basis of these simulations, it is observed that due to the presence of minor cracks, holes and inclusions, the life of the alloy is nearly reduced by 5.42%; whereas the fatigue life of the FGM with crack on the alloy and ceramic rich sides is reduced by 6.03% and 36.15% respectively. The fatigue life of the equivalent composite is
6.2. Plate with a center crack and multiple discontinuities Fig. 12a shows a plate with a center crack subjected to cyclic thermal load along with the boundary conditions. The plots of the
100 % Ceramic
100 % Alloy a
D
D 2 L
25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0
0
0.5
1
1.5
2
2.5
No. of Cycles
3
3.5
4
4.5 x 10
4
Fig. 17. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
57
aluminum alloy, FGM and composite plates is reduced by about 8.91%, 25.32% and 22.71% respectively. In the next study, a major crack is taken at the center of the domain along with 30 inclusions as shown in Fig. 14a. These inclusions are randomly distributed in the plate above and below the major crack with their size varying randomly from 3 mm to 4.5 mm as for holes. The plots of crack extension with the number of fatigue cycles are shown in Fig. 14b for FGM, aluminum alloy and equivalent composite. These plots show that the number of cycles to failure for aluminum alloy, FGM and equivalent composite are around to be 40,520, 15,821 and 21,111 cycles respectively. The FGM shows the minimum life, and the equivalent composite shows a moderate life; whereas the alloy has the maximum
fatigue life with crack extension are shown in Fig. 12b. These plots show that the equivalent composite survives 28,264 cycles before it fails while the FGM withstands 25,530 cycles, and aluminum alloy undergoes 41,589 cycles before failure. Thirty holes are taken next along with the major center crack as shown in Fig. 13a. The holes are randomly distributed in the plate above and below the major crack. The size of the holes varies randomly from 3 mm to 4.5 mm. The plots of crack extension with the number of cycles are shown in Fig. 13b. These simulations show that the number of cycles causing failure for aluminum alloy and FGM are found to be 37,882 cycles and 19,065 cycles respectively; whereas the life is found to be 21,844 cycles for the composite. Thus, due to the presence of holes, the life of the
100 % Alloy
100 % Ceramic a D
D 2 L
25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0
0
0.5
1
1.5
2
2.5
No. of Cycles
3
3.5
4
4.5 x 10
4
Fig. 18. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
58
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14.92% and 12.09% respectively due to the presence of minor cracks. Again, 42 minor cracks and 23 holes are taken along with the major crack as shown in Fig. 16a. These minor cracks and holes are randomly distributed in the plate above and below the major crack. The minor cracks are randomly oriented in between 01 and 601. The size of the minor cracks also varies randomly from 3.5 mm to 4.5 mm. The radii of the holes randomly vary from 3 to 4.5 mm. A cyclic thermal load is applied to the plate. The plots of crack extension with the number of cycles for aluminum alloy, FGM and composite are shown in Fig. 16b. It is observed that the number of cycles to failure in case of aluminum alloy, FGM and equivalent composite are found nearly 36,868 cycles, 18,001 cycles and 20,509 cycles respectively. The drop in the fatigue life due to
fatigue life among these materials. Thus, due to the presence of inclusions, the life of the aluminum alloy, FGM and equivalent composite plate is reduced by about 2.57%, 38.03% and 25.31% respectively. Next a major crack is taken along with 54 minor cracks (Fig. 15a). These minor cracks are randomly distributed in the domain above and below the major crack. The size of the minor cracks varies randomly from 3.5 mm to 4.5 mm and their inclination varies from 01 to 601. A cyclic thermal load is applied at the plate. The plots of crack extension with the number of cycles are shown in Fig. 15b. It is found that the failure cycles for aluminum alloy, FGM and equivalent composite are about 39,710, 21,721 and 24,848 cycles respectively. Thus it can be seen that the life of the aluminum alloy, FGM and composite is reduced by about 4.52%,
100 % Ceramic
100 % Alloy a
D D 2 L
25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
2
2.5
No. of Cycles
3
3.5
4
4.5 x 10
4
Fig. 19. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
the presence of minor cracks and holes is about 11.35%, 29.49% and 27.44% for aluminum alloy, FGM and the equivalent composite respectively. In the next simulation, 23 inclusions along with 42 minor cracks are taken with the major center crack as shown in Fig. 17a. These minor cracks and inclusions are randomly distributed in the plate above and below the major crack. This distribution of discontinuities remains the same for aluminum alloy, composite and FGM. The size of the minor cracks varies randomly from 3.5 mm to 4.5 mm, and the orientation of these cracks is randomly selected in between 01 and 601. The radii of the inclusions vary randomly from 3 to 4.5 mm. The fatigue life of aluminum alloy, composite and FGM is evaluated under the same load and geometric conditions. The plots of the crack extension with the number of cycles are shown in Fig. 17b for all these materials. These simulations show that the number of fatigue cycles in case of aluminum alloy, FGM and the composite is found to be 39,793, 21,814 and 23,556 cycles respectively. Thus, it has been observed that the life of the aluminum alloy plate is reduced nearly by 4.32%; whereas the fatigue life of the FGM and equivalent composite plate is reduced by 14.56% and 16.66% respectively. Next, a major center crack along with 20 holes and 20 inclusions are considered as shown in Fig. 18a. These holes and inclusions are randomly incorporated in the plate of aluminum alloy, FGM and composite above and below the major crack. The radii of the holes and inclusions vary randomly from 3 to 4.5 mm. The fatigue life of aluminum alloy, FGM and equivalent composite are shown in Fig. 18b. These simulations show that the number of cycles to failure in case of aluminum alloy, FGM and equivalent composite is found to be 38,005, 18,715 and 21,135 cycles respectively. Thus, it can be stated that the life of the aluminum alloy, FGM and composite plate is reduced by 8.62%, 26.69% and 25.22% respectively. Finally, a major crack is taken at the center of the domain along with 36 minor crack, 15 holes and 15 inclusions as shown in Fig. 19a. The minor cracks of arbitrary size and random orientation are randomly located in each plate of aluminum alloy, composite and FGM above and below the major crack. The size of the minor cracks randomly varies from 3.5 mm to 4.5 mm, and their orientation varies from 01 to 601. In addition to these minor cracks, holes and inclusions of random size (3–4.5 mm) are also added in the plate above and below the major crack. The fatigue life of aluminum alloy, equivalent composite and FGM plates are shown in Fig. 19b. From these simulations, it is found that the number of fatigue failure cycles in case of aluminum alloy, FGM and composite are found to 38,232, 17,722, and 20,878 cycles respectively. Thus, the fatigue life of aluminum alloy, FGM and the composite plate is reduced by 8.07%, 30.58% and 26.13% due to the presence of minor cracks, inclusions and holes. The present simulations show that the FGM shows the least thermal fatigue life as compared to other materials; whereas the equivalent composite shows a moderate life and alloy show the maximum fatigue life among these materials. From these studies it is also observed that the crack advances at much faster rate towards the ceramic side in comparison to aluminum alloy side in FGM. The effect of holes is the most pronounced whereby the minor cracks bear the least effect on the fatigue life of the materials. Also when the minor discontinuities are present in combination the fatigue life is greatly affected. 7. Conclusions In the present work, the fatigue crack growth simulations of cracked plate made of FGM, equivalent composite and aluminum
59
alloy are performed under cyclic thermal load using XFEM. The major cracks are assumed at the edge as well as at the center of the plate. The discontinuities such as holes/voids, inclusions and minor cracks are also added in the plate in addition to the major crack. On the basis of the present simulations, it is found that the holes/voids in the domain greatly influence the fatigue life of the material when they present singly or in combination with other discontinuities (minor cracks and inclusions). The fatigue life of aluminum alloy, FGM and equivalent composite is found minimum when all discontinuities are simultaneously present in the domain. The minor cracks have least effect on the fatigue life of the materials. The life of the aluminum alloy is found maximum; whereas the equivalent composite exhibits greater fatigue life as compared to the FGM. References [1] Amit KC, Kim JH. Interaction integrals for thermal fracture of functionally graded materials. Eng Fract Mech 2008;75:2542–65. [2] Arola D, Huang MP, Sultan MB. The failure of amalgam dental restorations due to cyclic fatigue crack growth. J Mater Sci: Mater Med 1999;10:319–27. [3] Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 1999;45:601–20. [4] Bhattacharya S, Singh IV, Mishra BK, Bui TQ. Fatigue Crack Growth Simulations of Interfacial Cracks in Bi-layered FGMs using XFEM. Computational Mechanics 2013;52:799–814. [5] Dolbow JE, Gosz M. On the computation of mixed-mode stress intensity factors in functionally graded materials. Int J Solids Struct 2002;39:2557–74. [6] Fazarinca M, Muhičb T, Šaleja A, Bombača D, Fajfara P, Terčelja M, et al. Thermal fatigue testing of bulk functionally graded materials. Procedia Eng 2011;10:692–7. [7] Guo L, Guo F, Yu H, Zhang L. An interaction energy integral method for nonhomogeneous materials with interfaces under thermal loading. Int J Solids Struct 2012;49:355–65. [8] Hsieh CL, Tuang WH. Poisson's ratio of two phase composites. Mater Sci Eng A 2005;396:202–5. [9] Huang GY, Wang YS, Dietmar G. Fracture analysis of functionally graded coatings: plane deformation. Eur J Mech A/Solids 2003;22:535–44. [10] Huang GY, Wang YS, Yu SW. A new model for fracture analysis of functionally graded coatings under plane deformation. Mech Mater 2005;37:507–16. [11] Kim JH, Paulino GH. Consistent formulations of the interaction integral method for fracture of functionally graded materials. J Appl Mech 2005;72:351–64. [12] Kokini K, DeJongea J, Rangaraja S, Beardsleyb B. Thermal shock of functionally graded thermal barrier coatings with similar thermal resistance. Surf Coat Technol 2002;154:223–31. [13] Moes N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. Int J Numer Methods Eng 1999;46:131–50. [14] Prabhakar RM, Tippur HV. Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient. Int J Solids Struct 2000;37:5353–70. [15] Rangaraj S, Kokini K. Estimating the fracture resistance of functionally graded thermal barrier coatings from thermal shock tests. Surf Coat Technol 2003;173:201–12. [16] Rao BN, Rahman S. An interaction integral method for analysis of cracks in orthotropic functionally graded materials. Comput Mech 2003;32:40–51. [17] Raveendran KV, Verma AP, Rao CVSK. Effective fracture toughness of composites. Int J Fract 1991;47:R63–5. [18] Sabuncuoglu B, Dag S, Yildirim B. Three dimensional computational analysis of fatigue crack propagation in functionally graded materials. Comput Mater Sci 2012;52:246–52. [19] Roylance D. Fatigue. Department of Materials Science and Engineering, Cambridge: Massachusetts Institute of Technology; 2001. [20] Sukumar N, Prevost J. Modeling quasi-static crack growth with the extended finite element method Part I: computer implementation. Int J Solids Struct 2003;40:7513–37. [21] Xu FM, Zhu SJ, Zhao J, Qi M, Wang FG, Li SX, et al. Effect of stress ratio on fatigue crack propagation in a functionally graded metal matrix composite. Compos Sci Technol 2004;64:1795–803. [22] Yu H, Wu L, Guo L, He Q, Du S. Interaction integral method for the interfacial fracture problems of two non-homogeneous materials. Mech Mater 2010;42:435–50. [23] Zhang C, Sladek J, Sladek V. Crack analysis in unidirectionally and bidirectionally functionally graded materials. Int J Fract 2004;129:385–406. [24] Zhang C, Sladek J, Sladek V. Antiplane crack analysis of a functionally graded material by a BIEM. Comput Mater Sci 2005;32:611–9. [25] Zhao J, Ai X, Deng J, Wang J. Thermal shock behaviors of functionally graded ceramic tool materials. J Eur Ceram Soc 2004;24:847–54.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Fatigue life simulation of functionally graded materials under cyclic thermal load using XFEM S. Bhattacharya, I.V. Singh n, B.K. Mishra Department of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India
art ic l e i nf o
a b s t r a c t
Article history: Received 28 January 2013 Accepted 3 March 2014 Available online 12 March 2014
In this work, the fatigue life of a functionally graded material (FGM) plate made of aluminum alloy and alumina (ceramic) is simulated under cyclic thermal load. The various discontinuities such as minor cracks, holes and inclusions of arbitrary size are randomly distributed in the plate. The fatigue lives of aluminum alloy, FGM and an equivalent composite (having the same composition as of FGM) plates are evaluated using Paris law in the presence of multiple discontinuities, and are compared with each other. & 2014 Elsevier Ltd. All rights reserved.
Keywords: FGM Thermal loading XFEM Fatigue crack growth Holes Inclusions
1. Introduction In recent years, a new class of graded composite materials more commonly known as functionally graded materials (FGMs) have been developed to meet the various needs of the engineering industry. A certain variation in the material properties of FGMs is achieved through local variation in composition or microstructure or both. The FGM has a remarkable capability to withstand bending as well as stretching, and is typically an inhomogeneous composite usually made from ceramic and metal. FGMs attain the multi-structural status due to their property gradation. By gradually varying the volume fraction of constituent materials, the material properties of FGMs exhibit a smooth and continuous change from one surface to another, thus eliminating interface problems and mitigating thermal stress concentrations. FGMs are processed in such a way that they possess continuous spatial variations in volume fractions of their constituents to generate a predetermined composition profile. These variations lead to the formation of a nonhomogeneous macrostructure with continuously varying mechanical and/or thermal properties in one or more than one direction. The microstructure of FGM is generally heterogeneous, and the dominant type of failure in FGM occurs from crack initiation and growth from flaws/inclusions. Thus, the study of the thermal fatigue life of FGMs becomes quite important in the presence of various flaws. Till date, some efforts have already been made to study and analyze the behavior of FGMs. Prabhakar and Tippur [14]
n
Corresponding author. Tel.: +91 1332 285888 (O); fax: +91 1332 285665. E-mail addresses: [email protected], [email protected] (I.V. Singh).
http://dx.doi.org/10.1016/j.ijmecsci.2014.03.005 0020-7403 & 2014 Elsevier Ltd. All rights reserved.
performed the static fracture analysis to calculate the crack tip stress field. Dolbow and Gosz and Rao and Rahman [5,16] analyzed the mixed-mode SIFs for 2-D orthotropic FGM using interaction energy contour integrals. Zhang et al. [23] analyzed the unidirectional and bidirectional FGM under mixed-mode loading using boundary integral equation method. Huang et al. [9,10] analyzed the multilayered model of FGM, and used Fourier transform approach for performing the fracture analysis of multilayered model. Zhang et al. [24] performed the elastostatic analysis of anti-plane cracks in unidirectional and bidirectional FGM using hypersingular boundary integral equation method. The analysis of FGMs under thermal loading has also been performed by Kokini et al. [12]. Rangaraj and Kokini [15] presented a methodology to estimate the crack growth resistance of functionally graded yttria stabilized zirconia bond coat alloy (made of nickel, cobalt, chromium, aluminum and yttria). Zhao et al. [25] evaluated the thermal shock resistance of Al2O3–TiC and Al2O3–(W, Ti)C functionally graded ceramic tool materials. Fazarinca et al. [6] estimated the suitability of the functionally graded materials subjected to thermal fatigue. Some studies on FGMs have also been carried out using FEM [21,18]. From the studies carried out so far, it was found that fatigue crack growth analysis of FGM has not been performed in the presence of multiple discontinuities under thermal loading. Moreover, the fatigue failure analyses of alloy/ceramic FGMs have not been explored much by any numerical scheme. As such, the traditional fracture based failure theories do not take into account the effect of small defects/discontinuities in the material for the life prediction. Therefore, in the present work, the fatigue crack growth analysis of alloy/ceramic FGMs, aluminum alloy and equivalent composite has been performed by XFEM in the
42
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
presence of multiple discontinuities (cracks, holes and inclusions) under thermal loading. A crack of an initial length is incorporated at the edge as well at the center of the FGM, pure aluminum alloy and an equivalent composite plate having the same composition as the FGM. The XFEM formulation for FGM is presented in Section 2. Section 3 depicts an overview of the method for calculating the SIF at the crack tip from a domain based interaction integral approach. Section 5 describes the physics of the FGM along with other relevant properties. The problem description and the discussion of the results obtained for various case of edge and center cracks is given in Section 6. Section 7 gives the conclusions derived from the present study.
2. XFEM formulation for FGM XFEM is a partition of the unity (PU) enriched finite element method which is particularly effective in handling strong as well as weak discontinuities in the domain. In this method, a crack is modeled by enrichment functions so a regular mesh is used for modeling the crack and crack growth without altering the original mesh. It also eliminates the need of singular elements for capturing the singularity at the crack tip (as in conventional FEM) as it makes use of asymptotic crack tip functions for this purpose.
The constitutive relations for the linear elastic material is given by Hook's law
rðuÞ ¼ DðxÞ : εðuÞ
ð4Þ
where, x is the vector of x and y-coordinates, DðxÞ is the constitutive matrix. 2.2. Variational formulation
Z Ω
A weak form of the equilibrium equation can be written as Z Z rðuÞ : εðvÞdΩ ¼ b vdΩ þ t vdΓ ð5Þ Ω
Γt
Substituting the constitutive relation rðuÞ ¼ DðxÞ : εðuÞ in the above equation, we obtain Z Z Z εðuÞ : DðxÞ : εðvÞdΩ ¼ b vdΩ þ t vdΓ ð6Þ Ω
Ω
Γt
_ _ The linear form, T , and the bilinear form, S, of the above equation are obtained as Z Z _ ð7Þ T ðvÞ ¼ b vdΩ þ t vdΩ Γ
Γt
2.1. Governing equations
x2
A two-dimensional domain (Ω) bounded by contour Γ divided into three parts i.e., Γ u , Γ t and Γ c with internal flaws like cracks, holes and inclusions is shown in Fig. 1. The displacement boundary conditions are imposed on Γ u , while tractions are applied on Γ t and traction free conditions are imposed on the crack surfaces Γ c . The equilibrium and boundary conditions for this problem may be described as ∇ r þ b ¼ 0 in Ω
Γo x1
ð1Þ
r U n^ ¼ t on Γ t
ð2aÞ
r U n^ ¼ 0 on Γ c
ð2bÞ
u ¼ u on Γ u
ð2cÞ
Ao
Fig. 2. Path independent closed contour around the crack tip.
where, r is the Cauchy stress tensor, u is the displacement field vector, b is the body force vector, t is the external traction vector ^ is the vector of unit outward normal. For small displaceand n ments, strain–displacement relation can be described as ε ¼ εðuÞ ¼ ∇s u
ð3Þ
y
where, ∇s is the symmetric part of the gradient operator.
y
100 %
t=t
Ω
Alloy
Γt
100 % Ceramic x
D
Γc Γu u=u
x Fig. 1. Domain with discontinuities.
L Fig. 3. Geometry of the FGM plate along with its dimensions.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
_ Sðu;vÞ ¼
Z Ω
εðuÞ : DðxÞ : εðvÞdΩ
ð8Þ
The quadratic energy functional can be written as 1_ _ ΠðuÞ ¼ Sðu;uÞ T ðuÞ 2
ð9Þ
On substitution of the relevant quantities, the expression for the same is obtained as Z Z Z 1 ΠðuÞ ¼ εðuÞ : DðxÞ : εðuÞdΩ b udλ t udλ ð10Þ 2 Ω Γ Γt
43
By using the trial and test functions and simplifying the above equation, the following set of discrete equations is obtained using the arbitrariness of nodal variations ½Kfdg ¼ ffg
ð11Þ
where, d is the vector of nodal unknowns, K and f are the global stiffness matrix and the external force vector respectively. The stiffness matrix and force vector are computed on element level, and are assembled into their global counterparts through usual finite element assembly procedure. 2.3. Displacement approximation
Table 1 Material properties of aluminum alloy and alumina ([2,4], Roylance, 2001 [19]). Material properties
Aluminum alloy Alumina
Elastic modulus E (GPa) Poisson's ratio, ν Coefficient of thermal expansion γ (/1C) Fracture toughness K IC (MPa√m) pffiffiffiffiffi Paris law parameter C in m=cycleðMPa mÞ m Paris law parameter, mðxÞ
70 0.33 25 10 6 29 10 12 3
100 % Alloy
300 0.213) 8.2 10 6 3.5 2.8 10 10 10
100 % Ceramic
a
For modeling cracks in XFEM [3,13], the approximation function takes the following form 2 3 6 7 n 4 6 j7 7 uh ðxÞ ¼ ∑ Ni ðxÞ6 u þ ΗðxÞa þ β ðxÞb ∑ j i7 6 i |fflfflffl{zfflfflffl}i i¼1 j¼1 4 5 i A nr |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
ð12aÞ
i A nA
where, i is the set of all nodes in the domain, N i ðxÞ is the element shape function associated with node i satisfying the partition of
100 % Alloy
a
D
D
D
2
2
100 % Ceramic
L
D
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 4. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
44
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
unity criterion, ui is the nodal displacement vector associated with the continuous part of the finite element solution, ai denotes the additional unknown degrees of freedom associated with the discontinuous Heaviside function ΗðxÞ, and is defined for those elements, which are completely cut by the crack to account for the j jump in the displacement field, bi is the additional degrees of freedom associated with those elements, which are partially cut by the crack, and accounts for stress singularity at the crack tip, n is the set of all nodes in the mesh, nr is the set of nodes belonging to those elements which are completely cut by the crack, and nA is the set of nodes belonging to those elements which are partially cut by the crack. For any node nr , Heaviside jump function, ΗðxÞ takes a constant value, and is equal to þ 1 on one side and 1 on other side of the crack. Thus, a shifted enrichment is used. If xi is the concerned node then Eq. (12a) can be written as 2 3 6 7 n 4 6 j7 7 u þ ½ΗðxÞ Hðx Þa þ ½β ðxÞ β ðx Þb uh ðxÞ ¼ ∑ N i ðxÞ6 ∑ i i j j i i7 6 i |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} i¼1 j¼1 4 5 i A nr |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
The sub-matrices and vectors that appear in the foregoing equations are given by Z ðΒri ÞT DΒsj dΩ where r; s ¼ u; a; b ð13bÞ Krs ij ¼ Ωe
Z
u
fi ¼
100 % Ceramic a
ð13aÞ
ð12bÞ
i A nA
100 % Alloy
In Eq. (12b), the difference between the values of the Heaviside function at the evaluation (Gauss) point and nodal point is taken to maintain the partition of unity. The tip enrichment is performed for those elements which contain the crack tip, and is achieved by adding additional crack tip functions. Using the approximation function defined in Eq. (12b), the e elemental matrices, Ke and f are obtained as 2 uu 3 K ij K ua K ub ij ij 6 7 n oT 6 au K aa K ab 7 u a b1 b2 b3 b4 e fi fi fi f ¼ fi fi fi and Keij ¼ 6 K ij ij ij 7 4 5 K bu K ba K bb ij ij ij
Ωe
Z Ν i bdΩ þ
Γt
Ν i tdΓ
ð13cÞ
100 % Alloy
100 % Ceramic a
D
D
D
2
2
D
L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0
0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 5. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
a
Z
fi ¼
b_ α
Z Ω
e
Ν i ðΗðxÞ Hðxi ÞÞbdΩ þ
Γt
Ν i ðΗðxÞ Hðxi ÞÞtdΓ
ð13dÞ
Z
fi ¼
Ωe
ðΝ i ðβ_α ðxÞ β_α ðxi ÞÞÞx 6 ¼40 ðΝ i ðβ_α ðxÞ β_α ðxi ÞÞÞy
3
0
ðΝ i ðβ_α ðxÞ β_α ðxi ÞÞÞy 7 5; ðΝ i ðβ_α ðxÞ β_α ðxi ÞÞÞx
Z
Γt
Ν i ðβ_α ðxÞ ðxi ÞÞtdΓ
where _ α ¼ 1; 2; 3; 4
ð13eÞ 3. Calculation of stress intensity factors
where, Ν i are finite element shape functions, and Βui , Βai , Βbi and are given by Ν i;x 60 u Βi ¼ 4 Ν i;y
0
_ α ¼ 1; 2; 3; 4; ð13hÞ
Ν i β_α ððxÞ ðxi ÞÞbdΩ
þ
2
2
_ Βbi α
45
_ Βbi α
3
Ν i;y 7 5
ð13f Þ
Ν i;x
2
3 ðΝ i ðΗðxÞ Hðxi ÞÞÞx 0 6 ðΝ i ðΗðxÞ Hðxi ÞÞÞy 7 Βai ¼ 4 0 5 ðΝ i ðΗðxÞ Hðxi ÞÞÞy ðΝ i ðΗðxÞ Hðxi ÞÞÞx
ð13gÞ
100 % Alloy
100 % Ceramic
a
In the present work, the domain based interaction integral approach [1,7,11,16,22] has been used for calculating the stress intensity factors for homogeneous and functionally graded materials under thermal load. For an elastic cracked FGM body as shown in Fig. 2, J-integral is defined as ∂u J ¼ ∮Γ0 sij i Wδ1j nj dΓ ð14Þ ∂x1 where, x ¼ ½x1 x2 T ½x yT , W is the strain energy density function and nj is the jth component of the outward unit vector normal to an arbitrary closed contour Γ 0 enclosing the area A0 .
100 % Ceramic
100 % Alloy
a
D
D
D 2
D 2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 6. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
46
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
Now, J can be converted into the equivalent domain form using divergence theorem, Z ∂u J¼ sij i Wδ1j dA ð15Þ ∂x1 A0 j The J-integral is path independent, where Γ 0 is a path starting from the lower crack face and terminating on the upper crack face and A0 is the enclosed area within Γ 0 . 1 m m In Eq. (15), W is defined as W ¼ 12sij εm ij ¼ 2C ijkl εkl εij , where, εm ¼ ε γΔTδ , γ is the coefficient of thermal expansion and ΔT ij ij ij is the temperature difference. Let q be a weight function such that its value is one at the crack tip, zero at the crack face, and arbitrary elsewhere. Now, Eq. (15) can be simplified as Z ∂ ∂u J¼ sij i Wδ1j q dA ð16Þ ∂x1 A0 ∂xj Substituting the value of W in Eq. (17), it can be written as Z ∂ ∂u 1 J¼ sij i sik εm δ ik 1j q dA ∂x1 2 A0 ∂xj
ð17Þ
100 % Ceramic
a
State 1 (actual state): sij εij ui J~ State 2 (auxiliary state): saij εaij uai
D
Ja
Defining the J-integral for both the states J~ ¼
Z
∂ A0 ∂xj
Z Ja ¼
∂ A0 ∂xj
sij
∂ui 1 sik εm δ q dA 1j ik ∂x1 2
saij
∂uai 1 a a sik εik δ1j q dA ∂x1 2
ð18Þ
ð19Þ
The J-integral for these two superimposed state can be defined as Z
For calculating the interaction integral for an elastic body, we consider two equilibrium states of the cracked body. State 1 is
100 % Alloy
taken as the actual state, state 2 is taken as an auxiliary state and superscript a denotes the parameters for the auxiliary state.
JT ¼
∂ A0 ∂xj
∂ui ∂uai 1 a ðsij þ saij Þ þ ðsik þ saik Þðεm ik þεik Þδ1j q dA 2 ∂x1 ∂x1 ð20Þ
J T ¼ J~ þ J a þ M 12
ð21Þ
100 % Alloy
a
D
100 % Ceramic
D
D
2
2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0
0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 7. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
where, Z M 12 ¼
∂ A0 ∂xj
∂ua ∂u 1 sij i þ saij i ðsik εaik þ saik εm ik Þδ1j q dA ∂x1 ∂x1 2
ð22Þ
where, the auxiliary field for the FGM is taken from [22] ! a a ∂ual 1 ∂uai ∂uj tip 1 ∂uk a a a a ; εij ¼ Sijkl ðxÞskl and εij a sij ¼ C ijkl þ þ 2 ∂xl ∂xk 2 ∂xj ∂xi
ð23Þ
Using sik εaik ¼ sik Sikpq ðxÞsapq ¼ εpq sapq ¼ saik εik , Eq. (22) can be written as Z ∂ua ∂ ∂u sij i þ saij i saik εm ð24Þ M 12 ¼ ik δ1j q dA ∂x1 ∂x1 A0 ∂xj Eq. (24) can be modified as Z M 12 ¼ A0
Z þ
A0
sij
∂uai ∂u ∂q þ saij i saik εm dA ik δ1j ∂x1 ∂x1 ∂xj
! m 2 ∂saij ∂ui ∂saik m ∂sij ∂uai ∂2 uai a ∂ ui a ∂εik þ sij þ þ sij ε sik qdA ∂xj ∂x1 ∂xj ∂x1 ∂xj ∂x1 ∂xj ∂x1 ∂x1 ik ∂x1
47
Using equilibrium equation and symmetry of the stress tensor, Eq. (25) is simplified as Z ∂ua ∂u ∂q M 12 ¼ sij i þ saij i saik εik δ1j dA ∂xj ∂x1 ∂x1 A0 ! ! Z a ∂saik 1 ∂ ∂uai ∂uj ∂ þ saij qdA sij þ ðεij εm Þ ε þ ij 2 ∂x1 ∂xj ∂xi ∂x1 ∂x1 ik A0 ð26Þ Using Eq. (24) and εm ij ¼ εij γΔTδij , Eq. (26) is further simplified as Z a ∂u ∂u ∂q M 12 ¼ sij i þ saij i saik εm dA ik δ1j ∂xj ∂x1 ∂x1 A0 Z ∂sakl ∂γ ∂ΔT a δij qdA ð27Þ sij ðStip S ðxÞÞ þ s ΔT þγ þ ijkl ij ijkl ∂x1 ∂x1 ∂x1 A0 For aluminum alloy and equivalent composite, Stip ¼ Sijkl ðxÞ. Now, ijkl Eq. (27) reduces to Z M 12 ¼ A0
sij
Z ∂uai ∂u ∂q ∂γ ∂ΔT δij qdA þ saij i saik εm dA þ saij ΔT þγ ik δ1j ∂x1 ∂x1 ∂xj ∂x ∂x 1 1 A0
ð25Þ
100 % Alloy
100 % Ceramic
a
ð28Þ
100 % Ceramic
100 % Alloy
a
D
D
D 2
D 2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0
0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 8. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
48
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
SIF at the crack tip can be computed using the above interaction integral as K I;II ¼
M 12 Ε nTip
ð29Þ
2ð1 ν2Tip Þ
where, Ε ntip is evaluated at the crack tip which is equal to Ε ntip ¼ Ε tip for plane stress and Ε ntip ¼ Ε tip =1 v2tip for plane strain.
direction are given as θc θc θc K Ieq ¼ K I cos 3 3K II cos 2 sin 2 2 2 0 θc ¼ 2 tan 1 @
KI
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 K 2I þ 8K 2II A 4K II
ð31Þ
From Eq. (30), the equivalent SIF can be found as θc θc θc ΔK Ieq ¼ ΔK I cos 3 3ΔK II cos 2 sin 2 2 2 4. Fatigue crack growth
For stable crack growth, Paris' law is given as
In this study, Paris law has been used to assess the fatigue life under cyclic thermal loading. The SIFs corresponding to the maximum and minimum load can be evaluated using the approach outlined in the previous section. At each crack tip, the direction of crack growth θc is determined using the maximum principal stress theory [20]. The crack is assumed to grow in a direction perpendicular to the maximum principal stress. According to this criterion, the equivalent mode-I SIF and crack growth
da ¼ CðxÞðΔK Ieq ÞmðxÞ dN
100 % Alloy
100 % Ceramic
a
ð30Þ
ð32Þ
ð33Þ
where, CðxÞ and mðxÞ are the functions of x. For implementation purpose, a value of crack growth, Δa is assumed and the corresponding failure cycles (ΔN) are calculated from Eq. (33). When more than one crack tip is present in the domain, Δa for the most dominant crack tip is assumed, corresponding ΔN is calculated and then the crack growth at the other
100 % Ceramic
100 % Alloy
D
a
D
D
D
2
2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 9. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
crack tips are calculated corresponding to this known value of ΔN. Finally, when the maximum value of K Ieq for any crack tip becomes more than the local value of the fracture toughness K IC then the simulation is stopped. At this point, the total number of cycles becomes the fatigue life of the material.
5. Variation in the properties of FGM
49
The rule of mixtures is applied to estimate the local elastic modulus of the FGM using the local volume fraction of ceramic. Thus, the volume fractions of ceramic and aluminum alloy in the FGM are given as V FGM ceramic ðxÞ ¼
EðxÞ Ealloy Ealloy eαx Ealloy ¼ Eceramic Ealloy Eceramic Ealloy
FGM V FGM alloy ðxÞ ¼ 1 V ceramic ðxÞ:
The FGM plate is composed of aluminum alloy and ceramic (alumina) as shown in Fig. 3. The variation in the composition of FGM is taken along x-direction. From the figure, it is seen that the FGM plate has 100% aluminum alloy at x ¼ 0 and 100% ceramic at x ¼ L. The material properties of the aluminum alloy and alumina used in FGM are given in Table 1. An exponential variation in the elastic modulus for FGM is taken as EðxÞ ¼ Ealloy eαx
ð35bÞ
The Poisson's Ratio for the FGM can be calculated using Halpin– Tsai equation [8] νðxÞ ¼
FGM νalloy V FGM alloy ðxÞE ceramic þ νceramic V ceramic ðxÞE alloy FGM V FGM alloy ðxÞE ceramic þ V ceramic ðxÞE alloy
ð34Þ
100 % Alloy
100 % Ceramic
a
ð36Þ
The rule of mixtures is applied again to estimate the coefficient of thermal expansion for the FGM as FGM γðxÞ ¼ γ alloy V FGM alloy ðxÞ þ γ ceramic V ceramic ðxÞ
where, α is given as 1 E α ¼ ln ceramic L Ealloy
ð35aÞ
ð37Þ
The fracture toughness of the FGM can be expressed as a function of the volume fraction of the ceramic [17] using the
100 % Ceramic
100 % Alloy
D
a
D
D
D
2
2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 10. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
50
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
following formula K IC ðxÞ ¼
K alloy þ K ceramic IC IC 2
þ
K alloy K ceramic IC IC 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 V FGM V FGM ceramic ðxÞ ceramic ðxÞÞ
ð38Þ The Paris law parameters vary in the similar manner as the elastic modulus, and are given by the following equations 1 C ð39Þ CðxÞ ¼ C alloy eϑx ; where ϑ ¼ ln ceramic L C alloy mðxÞ ¼ malloy eςx
where ς ¼
1 mceramic ln L malloy
ð40Þ
The equivalent composite consists of the same amount of the alloy and the ceramic as in the FGM. The fatigue life of the FGM plate is compared with that of the equivalent composite. Hence, the volume fraction of the ceramic in the equivalent composite can be calculated as Z 1 L FGM V composite ¼ V ðxÞdx ð41Þ ceramic L 0 ceramic
100 % Alloy
100 % Ceramic a
where, L is the length of the plate., which is taken as L ¼ 100 mm in the present simulations. Hence, we obtain V composite ceramic ¼ 38:3% and V composite ¼ 61:72%. Now, using the rule of mixtures for the alloy equivalent composite, the elastic modulus is obtained as þEceramic V composite Ecomposite ¼ Ealloy V composite ceramic alloy
ð42Þ
Using the Halpin–Tsai equation, Poisson's ratio is calculated as νcomposite ¼
νalloy V composite Eceramic þ νceramic V composite ceramic E alloy alloy V composite Eceramic þ V composite ceramic E alloy alloy
ð43Þ
The coefficient of thermal expansion for the equivalent composite is estimated by the rule of mixtures as þ γ ceramic V composite γ composite ¼ γ alloy V composite ceramic alloy
ð44Þ
The fracture toughness of equivalent composite is estimated in the same manner as for FGM ¼ K composite IC
K alloy þ K ceramic K alloy K ceramic IC IC IC þ IC 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
1 V composite ceramic
V composite ceramic Þ
100 % Alloy
100 % Ceramic a
D
D
ð45Þ
D
D
2
2 L
L
25 FGM (crack on alloy rich side) FGM (crack on ceramic rich side) Equivalent composite Aluminum alloy
Crack Extension (mm)
20
15
10
5
0 0
0.5
1
1.5
No. of Cycles
2
2.5 x 10
4
Fig. 11. (a) Plate with an edge crack on the alloy rich side under thermal loading, (b) plate with an edge crack on the ceramic rich side under thermal loading, and (c) a plot of crack extension with number of cycles.
S. Bhattacharya et al. / International Journal of Mechanical Sciences 82 (2014) 41–59
nodes in x-direction and 235 equally distributed nodes in y-direction is used for the simulations. For each case, the boundary conditions are shown in the respective figures. The gradation in material property is taken along x-direction where x varies from x ¼ 0 to x ¼ 100 mm. In all cases, an initial crack of length a ¼ 20 mm is assumed at the edge as well as at the center of the plate. The dimensions of the plate are taken same for FGM, composite and pure aluminum alloy. For simulating the thermal fatigue, a cyclic thermal load is applied equivalent to a stress of Δs ¼ 70 MPa. The temperature difference (ΔT) obtained using the properties of the equivalent composite Δs ¼ Ecomposite γ composite ΔT is found to be ΔT ¼ 23:86 o C for inducing thermal fatigue. An a increment of Δa ¼ 10 ¼ 2 mm is taken to evaluate the fatigue failure life of aluminum alloy, FGM and equivalent composite in the presence of a major discontinuity. The other discontinuities like holes, inclusions and minor cracks and their combinations are
For the equivalent composite, the location x in the FGM can be E found using the relation x ¼ 1α ln composite , where α is defined in Ealloy Eq. (34). The Paris law parameters of the equivalent composite are assumed to be same as that of the FGM at x ¼ x . Thus, C composite ¼ C alloy eϑx
ð46aÞ
mcomposite ¼ malloy eςx
ð46bÞ
51
6. Problem description, results and discussions A rectangular plate of dimensions L ¼ 100 mm and D ¼ 200 mm of graded material having 100% aluminum alloy on the left side and 100% ceramic (alumina) on the right side is taken for the analysis. A uniform mesh consisting of 117 equally distributed
100 % Alloy
100 % Ceramic
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Fig. 12. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
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22,705 cycles before failure. Moreover, it is observed that when a crack initiates from the ceramic (alumina) side, it fails much earlier (only 12,264 cycles) than when the same is initiated from the aluminum alloy side. In addition to the major crack, 30 randomly distributed holes are added in the domain above and below the major crack as shown in Fig. 5a and b respectively. The sizes of the holes vary randomly from 3 mm to 4.5 mm. These simulations reveal that the number of cycles to failure in case of aluminum alloy is about 20,326 cycles; whereas the average fatigue life of FGM with a crack on the alloy rich side and ceramic rich side is found to be 17,741 cycles and 9607 cycles respectively. The fatigue life of the equivalent composite is found to be 12,352 cycles. Due to the presence of holes, the fatigue life of the aluminum alloy goes down by about 10.48% whereas the fatigue life of FGM with crack on the ceramic rich side is reduced by 21.67%. Even when the crack is on
added in the plate in addition to the major crack. The material properties of the inclusions are taken as Ε ¼ 20 GPa and ν ¼ 0:20 in all the simulations. 6.1. Plate with an edge crack and multiple discontinuities Fig. 4a and b shows a plate with a major edge crack of length a ¼ 20 mm on the left and right edges of the plate respectively. The simulations are performed for aluminum alloy, equivalent composite and FGM cracked plate in the presence of multiple discontinuities. In case of FGM, the major edge crack either initiates from the aluminum alloy rich side or ceramic rich side. The plots of the fatigue life for aluminum alloy, FGM and equivalent composite are as shown in Fig. 4c. From this analysis, it is found that the equivalent composite survives 14,773 cycles before it fails while the FGM withstands 21,642 cycles and aluminum undergoes
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Fig. 13. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
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10,049 cycles respectively. Thus, the fatigue life of the aluminum alloy plate reduced by 1.15% due to the presence of inclusions; whereas the fatigue life of FGM decreased by 2.92% and 18.06% when the crack is lying on the alloy and ceramic rich sides respectively. The fatigue life of the composite is reduced by 6.91%. Further, it is noticed that the effect of holes on the fatigue life is more severe as compared to the same number of inclusions. In the subsequent study, 54 minor cracks are randomly added in the plate above and below the major crack with their inclination randomly varying from 01 to 601 as shown in Fig. 7a and b respectively. In this study it is observed that the number of cycles to failure in case of aluminum alloy is about 21,642 cycles whereas in the case of FGM, the fatigue life is found nearly 20,023 and 11,024 cycles when the crack lies on the alloy and ceramic rich sides respectively. The fatigue life of composite is found to be
the alloy rich side, the fatigue life of FGM plate is still reduced by about 18.03%. For the composite, the fatigue life decreases by 16.39%. Thus, it is evident that the fatigue life of FGM and composite is influenced to some extent due to the presence of holes. The plots of crack extension with the number of cycles for various materials (aluminum alloy, equivalent composite, FGM) for this case are shown in Fig. 5c. Fig. 6a and b shows the major edge crack along with 30 randomly located inclusions. The sizes of the inclusions vary randomly from 3 mm to 4.5 mm as in case of holes. The plots of crack extension with the number of cycles are shown in Fig. 6c for all the materials. On the basis of these simulations, it is found that the fatigue lives of the aluminum alloy and composite are found to be 22,444 and 13,752 cycles; whereas the life of FGM with crack on the alloy and ceramic rich side is found to be 21,009 cycles and
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D 2 L
25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
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Fig. 14. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
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crack extension with the number of cycles are shown in Fig. 8c. In this case, it is found that the average number of cycles to failure for the aluminum alloy and the composite are found to 21,708 and 13,886 cycles respectively. The life of the FGM plate with crack on the alloy and ceramic rich sides is found to be 20,723 cycles and 11,005 cycles respectively. Thus, it is observed that the life of the aluminum alloy is decreased by 4.39%; whereas it is decreased by 4.25% and 10.27% for the FGM with crack on the alloy and ceramic side respectively. The life of the composite is reduced by 6% due to the presence of minor cracks and holes. When the 23 holes/voids are replaced by 23 randomly located inclusions (radii of the inclusions also vary randomly from 3 to 4.5 mm) in the previous simulation along with the randomly located minor cracks in the domain as shown in Fig. 9a and b the number of cycles to failure in case of aluminum alloy is about
12,828 cycles. Thus, it is observed that the life of the aluminum alloy plate goes down by about 4.68%; whereas the life of the FGM plate goes down by 7.48% and 10.11% respectively depending upon whether the crack is on the alloy rich side or the ceramic rich side. The life of the composite goes down by 13.17% due to the presence of minor cracks. The plots for crack extension with the number of cycles are shown in Fig. 7c. Now, 42 minor cracks and 23 holes/voids are added in the plate containing the major edge crack made from aluminum alloy, FGM and the composite as shown in Fig. 8a and b respectively with the locations of the minor cracks and holes are randomly distributed in the domain above and below the major crack. The sizes of the minor cracks vary randomly from 3.5 mm to 4.5 mm. These cracks are randomly oriented at angles ranging from 01 to 601. The radii of the holes also vary randomly from 3 to 4.5 mm. The plots of
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25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
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Fig. 15. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
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respectively for left and right edge cracks. The plots of crack extension with the number of cycles for each material are shown in Fig. 10c. From these plots, it is observed that the number of cycles to failure in case of aluminum alloy is found to be 21,995 cycles; whereas in the case of FGM with crack on the alloy rich and ceramic rich sides, fatigue life is found to be 20,314 and 7791 cycles respectively. The failure life of the cracked composite plate is found to be 11,145 cycles. Thus, due to the presence of holes and inclusions, the life of the aluminum alloy goes down by about 3.13%; whereas the life of the FGM with the crack on the alloy and ceramic sides is reduced nearly by 6.14% and 36.47% respectively. The life of the equivalent composite is reduced by 24.56%. Finally 36 minor cracks, 15 holes and 15 inclusions are taken in the domain along with the major left and right edge cracks as shown in Fig. 11a and b respectively. The minor cracks, holes and
20,999 cycles. In FGM with crack on the alloy and ceramic rich sides are about 20,672 and 9773 cycles respectively. The failure life for composite is found to be 11,633 cycles. Thus, it is observed that the life of the aluminum alloy goes down by about 7.51%; whereas the fatigue life of FGM goes down by 4.48% and 2031% respectively depending on whether the crack is on the alloy side or the ceramic side. The fatigue life of the equivalent composite is reduced nearly by 21.25% due to the presence of minor cracks and inclusions. The plots of crack extension with the number of cycles are shown in Fig. 9c. In the next study, 20 holes and 20 inclusions are incorporated in the domain in addition to the major edge crack. The holes and inclusions are randomly distributed in the plate above and below the major crack. The radii of both the holes and inclusions vary randomly from 3 mm to 4.5 mm as depicted in Fig. 10a and b
100 % Alloy
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25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
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Fig. 16. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
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reduced by 11.78%. The plots of crack extension with the number of cycles are shown in Fig. 11c. From all these simulations, it is evident that the FGM shows the least fatigue life as compared to other materials. In FGM, when the major crack is present on the right edge i.e. on the ceramic rich side, the fatigue life diminishes to a great extent as compared to the case when the crack is present on the left edge i.e. on the alloy rich side. The equivalent composite shows a moderate life whereby the alloy shows the maximum fatigue life among three materials
inclusions are randomly distributed in the domain above and below the major crack. The sizes of the minor cracks vary randomly from 3.5 mm to 4.5 mm. These cracks are randomly oriented in between 01 and 601. The radii of the holes and inclusions vary randomly from 3 to 4.5 mm. In this case it is seen that the number of cycles to failure in case of aluminum alloy is found to be 21,730 cycles; whereas in case of FGM with crack on the alloy and ceramic rich sides, the fatigue life is found to be 19,740 and 9751 cycles respectively. The fatigue life of the composite plate is found to be 12,885 cycles. On the basis of these simulations, it is observed that due to the presence of minor cracks, holes and inclusions, the life of the alloy is nearly reduced by 5.42%; whereas the fatigue life of the FGM with crack on the alloy and ceramic rich sides is reduced by 6.03% and 36.15% respectively. The fatigue life of the equivalent composite is
6.2. Plate with a center crack and multiple discontinuities Fig. 12a shows a plate with a center crack subjected to cyclic thermal load along with the boundary conditions. The plots of the
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Crack Extension (mm)
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Fig. 17. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
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aluminum alloy, FGM and composite plates is reduced by about 8.91%, 25.32% and 22.71% respectively. In the next study, a major crack is taken at the center of the domain along with 30 inclusions as shown in Fig. 14a. These inclusions are randomly distributed in the plate above and below the major crack with their size varying randomly from 3 mm to 4.5 mm as for holes. The plots of crack extension with the number of fatigue cycles are shown in Fig. 14b for FGM, aluminum alloy and equivalent composite. These plots show that the number of cycles to failure for aluminum alloy, FGM and equivalent composite are around to be 40,520, 15,821 and 21,111 cycles respectively. The FGM shows the minimum life, and the equivalent composite shows a moderate life; whereas the alloy has the maximum
fatigue life with crack extension are shown in Fig. 12b. These plots show that the equivalent composite survives 28,264 cycles before it fails while the FGM withstands 25,530 cycles, and aluminum alloy undergoes 41,589 cycles before failure. Thirty holes are taken next along with the major center crack as shown in Fig. 13a. The holes are randomly distributed in the plate above and below the major crack. The size of the holes varies randomly from 3 mm to 4.5 mm. The plots of crack extension with the number of cycles are shown in Fig. 13b. These simulations show that the number of cycles causing failure for aluminum alloy and FGM are found to be 37,882 cycles and 19,065 cycles respectively; whereas the life is found to be 21,844 cycles for the composite. Thus, due to the presence of holes, the life of the
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25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
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Fig. 18. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
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14.92% and 12.09% respectively due to the presence of minor cracks. Again, 42 minor cracks and 23 holes are taken along with the major crack as shown in Fig. 16a. These minor cracks and holes are randomly distributed in the plate above and below the major crack. The minor cracks are randomly oriented in between 01 and 601. The size of the minor cracks also varies randomly from 3.5 mm to 4.5 mm. The radii of the holes randomly vary from 3 to 4.5 mm. A cyclic thermal load is applied to the plate. The plots of crack extension with the number of cycles for aluminum alloy, FGM and composite are shown in Fig. 16b. It is observed that the number of cycles to failure in case of aluminum alloy, FGM and equivalent composite are found nearly 36,868 cycles, 18,001 cycles and 20,509 cycles respectively. The drop in the fatigue life due to
fatigue life among these materials. Thus, due to the presence of inclusions, the life of the aluminum alloy, FGM and equivalent composite plate is reduced by about 2.57%, 38.03% and 25.31% respectively. Next a major crack is taken along with 54 minor cracks (Fig. 15a). These minor cracks are randomly distributed in the domain above and below the major crack. The size of the minor cracks varies randomly from 3.5 mm to 4.5 mm and their inclination varies from 01 to 601. A cyclic thermal load is applied at the plate. The plots of crack extension with the number of cycles are shown in Fig. 15b. It is found that the failure cycles for aluminum alloy, FGM and equivalent composite are about 39,710, 21,721 and 24,848 cycles respectively. Thus it can be seen that the life of the aluminum alloy, FGM and composite is reduced by about 4.52%,
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25 FGM (left crack tip) FGM (right crack tip) Equivalent composite Aluminum alloy
Crack Extension (mm)
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Fig. 19. (a) Plate with a center crack under thermal loading and (b) a plot of crack extension with number of cycles.
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the presence of minor cracks and holes is about 11.35%, 29.49% and 27.44% for aluminum alloy, FGM and the equivalent composite respectively. In the next simulation, 23 inclusions along with 42 minor cracks are taken with the major center crack as shown in Fig. 17a. These minor cracks and inclusions are randomly distributed in the plate above and below the major crack. This distribution of discontinuities remains the same for aluminum alloy, composite and FGM. The size of the minor cracks varies randomly from 3.5 mm to 4.5 mm, and the orientation of these cracks is randomly selected in between 01 and 601. The radii of the inclusions vary randomly from 3 to 4.5 mm. The fatigue life of aluminum alloy, composite and FGM is evaluated under the same load and geometric conditions. The plots of the crack extension with the number of cycles are shown in Fig. 17b for all these materials. These simulations show that the number of fatigue cycles in case of aluminum alloy, FGM and the composite is found to be 39,793, 21,814 and 23,556 cycles respectively. Thus, it has been observed that the life of the aluminum alloy plate is reduced nearly by 4.32%; whereas the fatigue life of the FGM and equivalent composite plate is reduced by 14.56% and 16.66% respectively. Next, a major center crack along with 20 holes and 20 inclusions are considered as shown in Fig. 18a. These holes and inclusions are randomly incorporated in the plate of aluminum alloy, FGM and composite above and below the major crack. The radii of the holes and inclusions vary randomly from 3 to 4.5 mm. The fatigue life of aluminum alloy, FGM and equivalent composite are shown in Fig. 18b. These simulations show that the number of cycles to failure in case of aluminum alloy, FGM and equivalent composite is found to be 38,005, 18,715 and 21,135 cycles respectively. Thus, it can be stated that the life of the aluminum alloy, FGM and composite plate is reduced by 8.62%, 26.69% and 25.22% respectively. Finally, a major crack is taken at the center of the domain along with 36 minor crack, 15 holes and 15 inclusions as shown in Fig. 19a. The minor cracks of arbitrary size and random orientation are randomly located in each plate of aluminum alloy, composite and FGM above and below the major crack. The size of the minor cracks randomly varies from 3.5 mm to 4.5 mm, and their orientation varies from 01 to 601. In addition to these minor cracks, holes and inclusions of random size (3–4.5 mm) are also added in the plate above and below the major crack. The fatigue life of aluminum alloy, equivalent composite and FGM plates are shown in Fig. 19b. From these simulations, it is found that the number of fatigue failure cycles in case of aluminum alloy, FGM and composite are found to 38,232, 17,722, and 20,878 cycles respectively. Thus, the fatigue life of aluminum alloy, FGM and the composite plate is reduced by 8.07%, 30.58% and 26.13% due to the presence of minor cracks, inclusions and holes. The present simulations show that the FGM shows the least thermal fatigue life as compared to other materials; whereas the equivalent composite shows a moderate life and alloy show the maximum fatigue life among these materials. From these studies it is also observed that the crack advances at much faster rate towards the ceramic side in comparison to aluminum alloy side in FGM. The effect of holes is the most pronounced whereby the minor cracks bear the least effect on the fatigue life of the materials. Also when the minor discontinuities are present in combination the fatigue life is greatly affected. 7. Conclusions In the present work, the fatigue crack growth simulations of cracked plate made of FGM, equivalent composite and aluminum
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alloy are performed under cyclic thermal load using XFEM. The major cracks are assumed at the edge as well as at the center of the plate. The discontinuities such as holes/voids, inclusions and minor cracks are also added in the plate in addition to the major crack. On the basis of the present simulations, it is found that the holes/voids in the domain greatly influence the fatigue life of the material when they present singly or in combination with other discontinuities (minor cracks and inclusions). The fatigue life of aluminum alloy, FGM and equivalent composite is found minimum when all discontinuities are simultaneously present in the domain. The minor cracks have least effect on the fatigue life of the materials. The life of the aluminum alloy is found maximum; whereas the equivalent composite exhibits greater fatigue life as compared to the FGM. References [1] Amit KC, Kim JH. Interaction integrals for thermal fracture of functionally graded materials. Eng Fract Mech 2008;75:2542–65. [2] Arola D, Huang MP, Sultan MB. The failure of amalgam dental restorations due to cyclic fatigue crack growth. J Mater Sci: Mater Med 1999;10:319–27. [3] Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 1999;45:601–20. [4] Bhattacharya S, Singh IV, Mishra BK, Bui TQ. Fatigue Crack Growth Simulations of Interfacial Cracks in Bi-layered FGMs using XFEM. Computational Mechanics 2013;52:799–814. [5] Dolbow JE, Gosz M. On the computation of mixed-mode stress intensity factors in functionally graded materials. Int J Solids Struct 2002;39:2557–74. [6] Fazarinca M, Muhičb T, Šaleja A, Bombača D, Fajfara P, Terčelja M, et al. Thermal fatigue testing of bulk functionally graded materials. Procedia Eng 2011;10:692–7. [7] Guo L, Guo F, Yu H, Zhang L. An interaction energy integral method for nonhomogeneous materials with interfaces under thermal loading. Int J Solids Struct 2012;49:355–65. [8] Hsieh CL, Tuang WH. Poisson's ratio of two phase composites. Mater Sci Eng A 2005;396:202–5. [9] Huang GY, Wang YS, Dietmar G. Fracture analysis of functionally graded coatings: plane deformation. Eur J Mech A/Solids 2003;22:535–44. [10] Huang GY, Wang YS, Yu SW. A new model for fracture analysis of functionally graded coatings under plane deformation. Mech Mater 2005;37:507–16. [11] Kim JH, Paulino GH. Consistent formulations of the interaction integral method for fracture of functionally graded materials. J Appl Mech 2005;72:351–64. [12] Kokini K, DeJongea J, Rangaraja S, Beardsleyb B. Thermal shock of functionally graded thermal barrier coatings with similar thermal resistance. Surf Coat Technol 2002;154:223–31. [13] Moes N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. Int J Numer Methods Eng 1999;46:131–50. [14] Prabhakar RM, Tippur HV. Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient. Int J Solids Struct 2000;37:5353–70. [15] Rangaraj S, Kokini K. Estimating the fracture resistance of functionally graded thermal barrier coatings from thermal shock tests. Surf Coat Technol 2003;173:201–12. [16] Rao BN, Rahman S. An interaction integral method for analysis of cracks in orthotropic functionally graded materials. Comput Mech 2003;32:40–51. [17] Raveendran KV, Verma AP, Rao CVSK. Effective fracture toughness of composites. Int J Fract 1991;47:R63–5. [18] Sabuncuoglu B, Dag S, Yildirim B. Three dimensional computational analysis of fatigue crack propagation in functionally graded materials. Comput Mater Sci 2012;52:246–52. [19] Roylance D. Fatigue. Department of Materials Science and Engineering, Cambridge: Massachusetts Institute of Technology; 2001. [20] Sukumar N, Prevost J. Modeling quasi-static crack growth with the extended finite element method Part I: computer implementation. Int J Solids Struct 2003;40:7513–37. [21] Xu FM, Zhu SJ, Zhao J, Qi M, Wang FG, Li SX, et al. Effect of stress ratio on fatigue crack propagation in a functionally graded metal matrix composite. Compos Sci Technol 2004;64:1795–803. [22] Yu H, Wu L, Guo L, He Q, Du S. Interaction integral method for the interfacial fracture problems of two non-homogeneous materials. Mech Mater 2010;42:435–50. [23] Zhang C, Sladek J, Sladek V. Crack analysis in unidirectionally and bidirectionally functionally graded materials. Int J Fract 2004;129:385–406. [24] Zhang C, Sladek J, Sladek V. Antiplane crack analysis of a functionally graded material by a BIEM. Comput Mater Sci 2005;32:611–9. [25] Zhao J, Ai X, Deng J, Wang J. Thermal shock behaviors of functionally graded ceramic tool materials. J Eur Ceram Soc 2004;24:847–54.