Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 86 (2014) 727 – 731
1st International Conference on Structural Integrity, ICONS-2014
Fatigue Crack Growth Simulations of FGM Plate under Cyclic Thermal Load by XFEM Somnath Bhattacharyaa* and Kamal Sharmab a
Department of Mechanical Engineering, NIT Raipur, Chhattisgarh 492010, India b
Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India E-mail ID:
[email protected]
Abstract In many practical applications, functionally graded materials (FGMs) are subjected to cyclic thermal loads which cause thermal fatigue failure of the components. Thus, the thermal fatigue needs to be considered in instrumenting the design of the components made from FGMs. In the present work, the numerical simulation of fatigue crack growth in three different materials namely a unidirectional FGM made of aluminum alloy and alumina (ceramic), an equivalent composite consisting of the same volume fractions of alloy and the ceramic as the FGM, and the aluminum alloy, is carried out using extended finite element method under thermal cyclic load. These simulations show that the presence of multiple flaws greatly influences the fatigue life of the plate. © © 2014 2014 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. Peer-review under responsibility of the Indira Gandhi Centre for Atomic Research Keywords: FGM, Thermal loading, Fatigue, XFEM, Multiple Discontinuities.
1. Introduction Over the years, multi-phase composite materials known as functionally graded materials (FGMs) have been developed in which the composition or microstructure or both is locally varied so that a certain variation of the local material properties is achieved. These variations lead to the formation of a non-homogeneous microstructure with continuously varying mechanical and/or thermal properties in one or more than one direction. These also eliminate interface problems and mitigate thermal stress concentrations. The ceramic phase of FGMs provides a good resistance to heat, while the metal phase provides a strong mechanical performance and thus reduces the possibility of catastrophic failure. FGMs find widespread applications in the field of engineering, and are now becoming increasingly popular in the aerospace and the computer circuit industries as they possess good capability to withstand high thermal gradients. A typical application of FGM is as thermal barrier coatings (TBCs), which are widely used in automotive and aircraft industries.
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Indira Gandhi Centre for Atomic Research doi:10.1016/j.proeng.2014.11.091
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In general, all structural components are subjected to thermo-mechanical cyclic load under service conditions. The fatigue life of these components is normally estimated without accounting for the effect of defects/discontinuities. Voids and micro-defects near the tip of a major crack enhance the effective SIF at the tip of the major crack, and the fatigue life is significantly influenced thereby. Hence, in the present study, the fatigue crack propagation in FGM, an equivalent composite and aluminum alloy is investigated in the presence of discontinuities/defects under cyclic thermal loading conditions using the extended finite element method (XFEM). XFEM eliminates the need of a conformal mesh for solving moving discontinuities, and yields more accurate results. 2. Displacement Approximation in XFEM The approximation of unknown displacement field in the presence of internal flaws such as cracks, holes and inclusions is given as [1]:
ª º 4 « » α u (x) = ¦ N i (x) « ui + [Ǿ ( x) − Ǿ ( x i ) ]a i + ¦ [βα (x) − βα (x i )]b i + ϕ ( x) c i + [ψ (x) −ψ ( x i )]d i » α =1 i =1 « » i∈nr i∈ni i∈n h i∈n A ¬« ¼» n
h
where,
(1)
u i = unknown nodal displacement vector; n = set of all nodes in the mesh; n r = set of nodes belonging to
n A = set of nodes belonging to those elements which are partially cut by the crack; H (x ) = discontinuous Heaviside function; a i =nodal enriched degrees of freedom those elements which are completely cut by the crack;
associated with Heaviside function;
βα (x)
α
= the asymptotic crack tip enrichment functions, b i = nodal enriched
degrees of freedom vector associated with crack tip enrichment
βα (x) ; ni
= set of nodes belonging to those
n h = set of nodes belonging to those elements which are cut by holes; c i = nodal enriched degrees of freedom associated with ϕ (x) , b i = nodal enriched degrees of freedom associated with ψ (x) .
elements which are cut by inclusions;
3. Evaluation of Stress Intensity Factors A domain based interaction integral approach is applied for calculating the stress intensity factors for homogeneous and functionally graded materials under thermal loading conditions [1-4]. Using this approach the modified interaction integral M 12 can be written as [1]:
§ ∂u a ∂u M 12 = ³ ¨¨ σ ij i + σ ija i − σ ika ε ikm δ1 j ∂x1 ∂x1 Ao ©
· ∂q ¸¸ dA + ¹ ∂x j
§ § ∂γ ∂σ kla ∂ΔT · ·¸ tip ¨ ¸ δ ij q dA σ S S (x ) − + σ ija ¨¨ ΔT + γ ij ijkl ijkl ³A ¨© ∂x1 ∂x1 ¸¹ ¸¹ © ∂x1 o
(
)
(2)
a represents the auxiliary state, superscript m denotes the mechanical component, q is a weight function which is one at the inner path Γ1 , zero at the outer path Γ2 , and arbitrary elsewhere, and γ is the
where, superscript
coefficient of thermal expansion. In case of aluminum alloy and the equivalent composite (homogeneous materials) tip
we substitute S ijkl
= S ijkl (x) in Eq. 2. The SIF is extracted from M 12 at the location of the crack tip SIF using the
above interaction integral as:
K I , II =
* M 12 ǼTip 2 2(1 − ν Tip )
(3)
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where,
* * * Ǽtip is evaluated at the crack tip which is equal to Ǽtip = Ǽtip for plane stress and Ǽtip =
Ǽtip 2 1 − vtip
for plane
strain. The generalized Paris’ law is used for stable crack propagation. 4. Problem Description, Results and Discussion A rectangular edge cracked plate (initial crack of length a = 20 mm) of dimensions L = 100 mm and
D = 200 mm made of graded material (with 100% aluminum alloy on the left side and 100% ceramic (alumina) on the right side. The gradation is done exponentially in the horizontal direction from left to right. An equivalent composite and the aluminum alloy plate are also taken for the purpose of analysis. The equivalent composite have the same volume fractions of the ceramic and the alloy phases as used in FGM. The material properties of the aluminum alloy and alumina used in FGM are given in [1]. A uniform mesh consisting of nine node quadrilateral elements and made of 117 equally distributed nodes in horizontal and 235 equally distributed nodes in vertical direction is used for the simulations with a total of 6786 elements in the domain. For each case, the boundary conditions are depicted in the figures. Thermal fatigue loading is given to the plate by inducing a temperature difference ΔT = 23.86 oC in cycles. The edge crack is given a constant increment of Δa =
a = 2 mm to 10
evaluate the fatigue failure life of aluminum alloy, FGM and the equivalent composite in the presence of a major discontinuity. Moreover, other discontinuities like holes, inclusions and minor cracks and their combinations are also incorporated in the plate in addition to the major center crack. The material properties of the alloy and the ceramic are listed in Table 1. The material properties of the inclusions are taken as Ǽ = 20 GPa and ν = 0.2 in the simulations. Fig. 1 and Fig. 2 show a plate with a major edge crack of length a = 20 mm at its left and right edges respectively along with 36 minor cracks, 15 holes and 15 inclusions. The minor cracks, holes and 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 0o to 60o. The radii of the holes and inclusions vary randomly from 3 to 4.5 mm. In case of FGM, the major edge crack either initiates from the aluminum alloy rich side or ceramic rich side. The crack propagates under the impact of cyclic thermal load. The plots of the fatigue life for aluminum alloy, FGM and equivalent composite without the minor defects/discontinuities are as shown in Fig. 3 whereas the plots of their fatigue life with the minor discontinuities present are shown in Fig. 4. The fatigue life of the materials with and without the minor discontinuities is tabulated in Table 2. From the results presented in table and figures, it can be concluded that the fatigue life of the plate is greatly influenced by the presence of discontinuities. Table 1: Material properties of aluminum alloy and alumina [1] Material Properties
Aluminum alloy
Alumina
Elastic modulus E (GPa)
70
300
Poisson’s ratio, ν Coefficient of thermal expansion
0.33
γ (/°C)
25×10
Fracture toughness K IC (MPa ¥m) Paris law parameter C in m/cycle( MPa m ) Paris law parameter, m(x )
−m
0.21 -6
8.2 × 10-6
29
3.5
10-12
2.8 x 10-10
3
10
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100 % Alloy
100 % Ceramic
100 % Alloy
100 % Ceramic
a
a
D
D
D 2
D 2
L
L
Fig. 1: Plate with an edge crack on the alloy rich side under thermal loading
Fig. 2: Plate with an edge crack on the ceramic rich side under thermal loading
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 4
x 10
Fig. 3: A plot of crack extension with number of cycles without minor discontinuities
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Somnath Bhattacharya and Kamal Sharma / Procedia Engineering 86 (2014) 727 – 731 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 4
x 10
Fig. 4: A plot of crack extension with number of cycles with minor discontinuities Table 2. Fatigue life of the materials
FGM (Crack on alloy rich side) FGM (Crack on ceramic rich side) Equivalent Composite Aluminum Alloy
Without minor discontinuities
With minor discontinuities
21642
19740
Reduction in fatigue life (%) 6.03
12264
9751
36.15
14773 22705
12885 21730
11.78 5.42
5. Conclusions In this study, the fatigue crack growth simulations have been performed for FGM, equivalent composite and aluminum alloy under cyclic thermal load in the presence of a combination of holes/voids, inclusions and minor cracks using XFEM. It is observed that the fatigue life of the materials is reduced considerably when these discontinuities are simultaneously present in the domain. In addition, the FGM fails much earlier when a crack is present on the ceramic rich side as compared to the alloy rich side as the former is much weaker in fracture as compared to the alloy. The equivalent composite exhibits a moderate fatigue life while the aluminum alloy has the maximum fatigue life among these materials. 6. References [1] S. Bhattacharya, I.V. Singh, B.K. Mishra, T.Q. Bui, Fatigue crack growth simulations of interfacial cracks in bi-layered FGMs using XFEM, Computational Mechanics, 52 (2013) 799-814. [2] J.H. Kim, and G.H. Paulino, Consistent formulations of the interaction integral method for fracture of functionally graded materials, Journal of Applied Mechanics, 72 (2005) 351–364. [3] K.C. Amit, and J.H. Kim, Interaction integrals for thermal fracture of functionally graded materials, Engineering Fracture Mechanics, 75 (2008) 2542–2565. [4] L. Guo, F. Guo, H. Yu, and L. Zhang, An interaction energy integral method for nonhomogeneous materials with interfaces under thermal loading, International Journal of Solids Structures, 49 (2012) 355– 365.
