Available online at www.sciencedirect.com
Engineering Fracture Mechanics 75 (2008) 2542–2565
www.elsevier.com/locate/engfracmech
Interaction integrals for thermal fracture of functionally
graded materials
Amit KC, Jeong-Ho Kim
*
Department of Civil and Environmental Engineering, University of Connecticut, 261 Glenbrook Rd. U-2037, Storrs, CT 06269, USA
Received 14 December 2006; received in revised form 22 May 2007; accepted 22 July 2007
Available online 31 July 2007
Abstract
This paper addresses finite element evaluation of the non-singular T-stress and mixed-mode stress intensity factors in
functionally graded materials (FGMs) under steady-state thermal loads by means of interaction integral. Interaction integral provides an accurate and efficient numerical framework in evaluating these fracture parameters in FGMs under thermal as well as mechanical loads. We use a non-equilibrium formulation and the corresponding auxiliary (secondary) fields
tailored for FGMs. Graded finite elements have been developed to account for the spatial gradation of thermomechanical
properties. This paper presents various numerical examples in which the accuracy of the present method is verified.
2007 Elsevier Ltd. All rights reserved.
Keywords: Functionally graded material (FGM); Interaction integral; Finite element method (FEM); Thermal fracture; Stress intensity
factor; T-stress
1. Introduction
Functionally graded materials (FGMs) are a new class of advanced composites characterized by the gradual variation in composition, microstructure and material properties. These materials have emerged from the
need to enhance material performance. Hence, they are designed for specific functions and applications taking
advantage of the ideal behaviour of their constituents. For instance, a functionally graded material composed
of partially stabilized zirconia (PSZ) and CrNi alloy makes use of heat and corrosion resistance properties of
ceramics and mechanical strength and toughness of metals [1]. The FGM concept has been utilized in various
applications [2–12] including a solid oxide fuel cell, which is an electrochemical device converting the chemical
energy of hydrocarbon fuels into electrical power at elevated temperatures [13–18].
Eischen [19] extended the eigenfunction expansion technique of Williams [20] to derive the general form of
the crack-tip fields in FGMs by assuming the material gradation to be a continuous, differentiable and
*
Corresponding author. Tel.: +1 860 486 2746; fax: +1 860 486 2298.
E-mail address: jhkim@engr.uconn.edu (J.-H. Kim).
0013-7944/$ - see front matter 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engfracmech.2007.07.011
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
Nomenclature
a
B
Cijkl or
d
e
E
Etip
E0
E1
E2
Ebc
Ec
Es
f
f
g
J
Jaux
Js
J
J1
k
k1
k2
kbc
kc
ks
KI
KII
K aux
I
K aux
II
L
M
Mglobal
Mlocal
mi, ni
Ni
P
pl-e
pl-r
q
r
T
t
ui
uaux
i
ui,j
uaux
i;j
half crack length for an
internal
crack and full crack length for an edge crack
pffiffiffiffiffi
ffi
biaxiality ratio; B ¼ T pa=K I
C constitutive tensor; i, j, k, l = 1, 2, 3
the coordinate of a fixed point on the x1-axis
natural logarithm base, e = 2.71828182 . . .
Young’s modulus
Young’s modulus at the crack-tip
Young’s modulus evaluated at the origin
Young’s modulus at X1 = 0; E1 = E(0)
Young’s modulus at X1 = W; E2 = E(W)
Young’s modulus of bond coat
Young’s modulus of Zirconia–Yttria
Young’s modulus of substrate
point force applied to the crack-tip
representative functions for auxiliary displacement fields used for SIFs
representative functions for auxiliary displacement fields used for the T-stress
path-independent J-integral for the actual field
J-integral for the auxiliary field
J-integral for the superimposed fields (actual and auxiliary)
Jacobian matrix
inverse of the Jacobian matrix
thermal conductivity coefficient
thermal conductivity coefficient on the left edge
thermal conductivity coefficient on the right edge
thermal conductivity coefficient of bond coat
thermal conductivity coefficient of Zirconia–Yttria
thermal conductivity coefficient of substrate
mode I stress intensity factor
mode II stress intensity factor
auxiliary mode I stress intensity factor
auxiliary mode II stress intensity factor
length of a plate
interaction integral (M-integral)
M-integral evaluated in global coordinates
M-integral evaluated in local coordinates
unit normal vectors on the contour of the domain integral
shape function for node i of the element; Ni = Ni(n, g)
field variables
plane strain
plane stress
weight function in the domain integral
radial direction in polar coordinates
T-stress
thickness of a plate
displacements for the actual field; i = 1, 2
displacements for the auxiliary field; i = 1, 2
displacement derivatives for the actual field; i, j = 1, 2
displacement derivatives for the auxiliary field; i, j = 1, 2
2543
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A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
W
W
wgp
xi
Xi
a
a1
a2
ab
abc
as
b, c, d
C
C0
Cs
C+
C
dij
eij
eaux
ij
etij
em
ij
h
h0
h1
h2
Dh
h
j
jtip
l
ltip
m
mtip
rij
raux
ij
width of a plate
strain energy density
Gauss weights
local Cartesian coordinates; i = 1, 2
global Cartesian coordinates; i = 1, 2
thermal expansion coefficient
thermal expansion coefficient on the left edge
thermal expansion coefficient on the right edge
thermal expansion coefficient of bond coat
thermal expansion coefficient of Zirconia–Yttria
thermal expansion coefficient of substrate
material nonhomogeneity parameters
contour for J- and M-integrals
outer contour
inner contour
contour along the upper crack face
contour along the lower crack face
Kronecker delta; i, j = 1, 2
strains for the actual fields; i, j = 1, 2
strains for the auxiliary fields; i, j = 1,2
total strain; i, j = 1, 2
mechanical part of the strain; i, j = 1, 2
angular direction in polar coordinates
initial temperature
temperature on the left edge
temperature on the right edge
temperature difference
angle of crack orientation
material parameter, j = (3 m)/(1 + m) for plane stress and j = 3 4m for plane strain
j evaluated at the crack-tip
shear modulus
shear modulus evaluated at the crack-tip
Poisson’s ratio
Poisson’s ratio at the crack-tip
stresses for the actual fields; i, j = 1, 2
stresses for the auxiliary fields; i, j = 1, 2
bounded function of spatial position. Fig. 1 shows a crack in a non-homogeneous elastic body. The asymptotic stress and displacement fields around the crack-tip in FGMs are given by [19]
KI
K II
rij ðr; hÞ ¼ pffiffiffiffiffiffiffi fijI ðhÞ þ pffiffiffiffiffiffiffi fijII ðhÞ þ T di1 dj1 þ Oðr1=2 Þ;
2pr
2pr
rffiffiffiffiffiffi
rffiffiffiffiffiffi
KI
r I
K II
r II
ui ðr; hÞ ¼
gi ðhÞ þ
g ðhÞ þ OðrÞ;
ltip 2p
ltip 2p i
ð1Þ
ð2Þ
where KI and KII are the mode-I and mode-II SIFs respectively, T is the T-stress, dij is Kronecker delta, ltip is
the shear modulus at the crack tip, fij(h) and gi(h) (i, j = 1, 2) are the angular functions for stresses and displacements [21]. Stress intensity factors (SIFs) and the T-stress depend on the size, geometry and external loadings
in the case of homogeneous material. In FGMs, fracture parameters are also affected by material gradation
[19,22]. However, material gradation does not affect the order of singularity and angular functions [19,22].
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
2545
t
x
r
x1
2
θc
C (x)
α (x)
θ (x)
k (x)
Fig. 1. Cartesian (x1, x2) and polar (r, h) coordinates originating from the crack-tip in a nonhomogeneous material subjected to
temperature loading (h), traction (t) and displacement boundary conditions.
SIFs play a significant role in linear elastic fracture mechanics as they characterize the crack-tip stress and
strain fields. A single parameter (KI or J) characterizes the crack-tip condition under small scale yielding condition which involves high degree of triaxiality at the crack-tip and it can be used as a material property. Single
parameter K-dominance requires that plastic zone size be small compared to the other dimensions of the
cracked structure, e.g. crack length, size of uncracked ligament and thickness. However, under excessive plasticity, the single parameter is not sufficient to represent crack-tip fields. An additional parameter, called the
elastic T-stress, is required which affects the shape and size of the plastic zone, crack-tip constraint and fracture toughness [23–25]. The T-stress represents the stress parallel to crack faces. For small amounts of crack
growth under mode-I loading, a straight crack path has shown to be stable when T < 0, whereas the path will
be unstable and will deviate from being straight when T > 0 [26]. A similar trend has been observed in threedimensional (3D) crack propagation studies by Xu et al. [27]. Hutchinson and Suo [28] also showed how the
advancing crack path is influenced by the T-stress once cracking initiates under mixed-mode loading.
For the
pffiffiffiffi
mode-I case, the biaxiality ratio can be represented as a non-dimensional parameter, i.e. B ¼ T K Ipa [29], where a
is the crack length. The biaxiality ratio does not depend on loading magnitude but it depends on the geometry
and type of loading. In the case of FGMs, material gradation also affects the biaxiality ratio.
Many researchers have considered various crack problems in FGMs under thermal loads using different
analytical approaches [30–37]. The original Rice’s J-integral [38] loses path independence for the thermal loading case [39], and a path-independent form of J-integral was derived for thermally stressed crack problems
[40]. Yildirim [41] have used the equivalent domain integral based on J-integral for fracture analysis of FGMs
and calculated the mode-I SIF under steady-state and transient thermal loading conditions. Walters et al. [42]
have used J-integral and displacement correlation techniques to evaluate surface cracks in FGMs under modeI thermomechanical loading. Yildirim et al. [43] studied the 3D surface crack problems in functionally graded
coatings subjected to mode-I mechanical and transient thermal loadings using the displacement correlation
technique. Yildirim and Erdogan [44] have used the enriched element technique to evaluate mixed-mode SIFs
under uniform thermal loading. All the aforementioned works focus on the evaluation of SIFs. Dag [45] has
recently used the Jk-integral [19,22] to evaluate the mixed-mode SIFs and the T-stress in FGMs under thermal
loads, but the formulation for the T-stress works for only mixed-mode cases, i.e. KII 5 0.
Interaction integral provides an accurate and efficient numerical framework for evaluating mixed-mode
SIFs and the T-stress in FGMs. The method is formulated on the basis of conservation laws, which lead to
the establishment of a conservation integral for two admissible states of elastic solid, actual and auxiliary.
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A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
Interaction integrals have been successfully used in evaluating SIFs [46–54] and the T-stress [50,55,56,52] in
FGMs under mechanical loading. The interaction integral has also been used in the evaluation of mixed-mode
SIFs in FGMs under thermal loadings [57]. But no work has been done for the evaluation of the T-stress in
FGMs under thermal loads using the interaction integral. Thus, this paper presents the novel formulation of
the interaction integral method to evaluate the non-singular T-stress (as well as mixed-mode stress intensity
factors) in FGMs under steady-state thermal loads.
This paper is organized as follows. Section 2 presents the auxiliary fields selected for extracting mixed-mode
SIFs and the T-stress using the interaction integral method. Section 3 provides the derivation of M-integral for
thermal fracture using the non-equilibrium formulation. Sections 4 and 5 explain the relationship of mixedmode SIFs and the T-stress to the M-integral. Section 6 addresses numerical implementations of the M-integral and the steady-state thermal diffusion. Section 7 presents numerical examples to examine the accuracy and
performance of the interaction integral in evaluating mixed-mode SIFs and the T-stress for FGMs under thermal loads. Finally, Section 8 provides some discussions and Section 9 concludes this work.
2. Auxiliary fields
The interaction integral makes use of auxiliary fields, such as displacements (uaux), strains (eaux), and stresses (raux). There are various choices for the auxiliary fields for FGMs. These auxiliary fields have to be suitably
defined in order to evaluate mixed-mode SIFs and T-stress. In this paper, we adopt displacement and strain
fields for a homogeneous material under mechanical loads, and construct new auxiliary stress fields based on
the non-equilibrium formulation using raux = Cijkl(x) eaux, where Cijkl(x) is the constitutive tensor of FGM.
The auxiliary displacement and strain fields adopted for SIFs and the T-stress are described below.
For SIFs, we select the auxiliary displacement and strain fields from the Williams’ [20] crack-tip asymptotic
fields (i.e. O(r1/2) for the displacements and O(r1/2) for the strains) with the material properties sampled at the
crack-tip location (e.g. [19]). The auxiliary displacement and strain fields are given by [20,50]:
I 1=2
uaux ¼ K aux
; h; ltip Þ þ K aux
I f ðr
II f
e
aux
¼ ðsymrÞu
aux
II
ðr1=2 ; h; ltip Þ;
;
ð3Þ
ð4Þ
where K aux
and K aux
I
II are the auxiliary mode I and mode II SIFs, respectively, and ltip denotes the shear modulus evaluated at the crack-tip. The functions f(r1/2, h, ltip) are given in many references, e.g. [21].
For the non-singular T-stress, we choose the auxiliary displacement and strain fields (i.e. O(lnr)) for the
displacements and O(r1) for the strains) from Michell’s [58] solutions for a point force applied to the
crack-tip in an infinite homogeneous body. The auxiliary displacements and strains are given by [58,50]:
uaux ¼ gðln r; h; f ; ltip ; jtip Þ;
e
aux
¼ ðsymrÞu
aux
;
ð5Þ
ð6Þ
where f is the point force applied to the crack-tip, and jtip denotes jtip = (3 mtip)/(1 + mtip) for plane stress
and jtip = 3 4mtip for plane strain evaluated at the crack-tip. The functions g(lnr, h, f, ltip, jtip) are given in
many references, e.g. [58].
3. Interaction integral for thermal fracture
The J-integral is given by [38]
Z
J ¼ lim
ðWd1j rij ui;1 Þnj dC;
Cs !0
ð7Þ
Cs
where nj is the outward normal vector to the contour Cs as shown in Fig. 2. The parameter W is the strain
energy density given by
1
1
t
W ¼ rij em
ij ¼ rij ðeij aDhdij Þ;
2
2
ð8Þ
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
2547
x2
Γs
A
C (x)
x
x1
Ctip
C (x) Ctip
Fig. 2. Motivation for development of non-equilibrium formulation. Notice that C(x) 5 Ctip for x 5 0. The area A denotes a
representative region around the crack-tip.
t
where em
ij denotes the mechanical part of the strain, eij the total strain, a = a(x) the thermal expansion coefficient that varies with spatial coordinates, Dh = h h0 with h0 as the initial temperature (see Fig. 1), and dij the
Kronecker delta. The equivalent domain integral (EDI) form of the J-integral is obtained as
Z
Z
J ¼ ðrij ui;1 Wd1j Þq;j dA þ ðrij ui;1 Wd1j Þ;j q dA;
ð9Þ
A
A
where q is a weight function. In this paper we used the plateau function [59,50]. The J-integral of the superimposed fields (actual and auxiliary) is obtained as
Z 1
aux
aux
m
aux
ðr
Js ¼
ðrij þ raux
Þðu
þ
u
Þ
þ
r
Þðe
þ
e
Þd
i;1
ik
1j q;j dA
ij
i;1
ik
ik
ik
2
A
Z 1
aux
aux
aux
m
aux
þ
ðrij þ rij Þðui;1 þ ui;1 Þ ðrik þ rik Þðeik þ eik Þd1j Þ q dA:
ð10Þ
2
A
;j
Eq. (10) is decomposed into
J s ¼ J þ J aux þ M;
ð11Þ
1
where the interaction integral (M) is given by
Z 1
aux
aux
aux m
ðr
M¼
rij uaux
þ
r
u
e
þ
r
e
Þd
i;1
ik ik
1j q;j dA
i;1
ij
ik ik
2
A
Z 1
aux
aux
aux
aux m
þ
rij ui;1 þ rij ui;1 ðrik eik þ rik eik Þd1j q dA:
2
A
;j
ð12Þ
This general form of M-integral becomes a specific form for the non-equilibrium formulation as follows. The
auxiliary stress field used is
aux
raux
ij ¼ C ijkl ðxÞekl ;
ð13Þ
which does not satisfy equilibrium because it differs from
aux
raux
ij ¼ ðC ijkl Þtip ekl ;
ð14Þ
where (Cijkl)tip is the constitutive tensor at the crack-tip (see Fig. 2).
1
Here, the so-called M-integral should not be confused with the M-integral (conservation integral) of Knowles and Sternberg [60],
Budiansky and Rice [61], and Chang and Chien [62]. Also, see the book by Kanninen and Popelar [63] for a review of conservation
integrals in fracture mechanics.
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A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
Based on the non-equilibrium formulation, one obtains that
m aux
aux m
aux m
rij eaux
ij ¼ C ijkl ðxÞekl eij ¼ rkl ekl ¼ rij eij ;
ð15Þ
and rewrites Eq. (12) as
Z n
Z n
o
o
aux
aux
aux
aux
aux
rij ui;1 þ rij ui;1 rik eik d1j q;j dA þ
rij uaux
þ
r
u
r
e
d
q dA:
M ¼ M1 þ M2 ¼
i;1
ik
1j
i;1
ij
ik
A
A
;j
ð16Þ
The last term of the integral M2 in Eq. (16) is expressed as
aux
aux
m aux
m aux
m aux
m aux
ðrik eaux
ik d1j Þ;j ¼ ðrik eik Þ;1 ¼ ðrij eij Þ;1 ¼ ðC ijkl ekl eij Þ;1 ¼ C ijkl;1 ekl eij þ C ijkl ekl;1 eij þ C ijkl ekl eij;1
aux
aux m
aux
¼ C ijkl;1 em
kl eij þ rij eij;1 þ rij eij;1 :
ð17Þ
Substitution of Eq. (17) into M2 of Eq. (16) leads to
Z Z aux
aux
aux
m aux
aux m
aux
rij;j uaux
þ
r
u
þ
r
u
þ
r
u
C
e
e
þ
r
e
þ
r
e
q
dA
M2 ¼
ij
i;1
i;1j
ijkl;1
ij
i;1
i;1j
ij;j
ij
kl ij
ij
ij;1
ij;1 q dA:
A
ð18Þ
A
Using compatibility (actual and auxiliary) and equilibrium (actual) (i.e. rij,j = 0 with no body force), one simplifies Eq. (18) as
Z n
o
m aux
aux
m
M2 ¼
raux
ð19Þ
ij;j ui;1 C ijkl;1 ekl eij þ rij ðui;1j eij;1 Þ q dA:
ZA n
o
m aux
aux
¼
raux
ð20Þ
ij;j ui;1 C ijkl;1 ekl eij þ rij ða;1 ðDhÞ þ aðDhÞ;1 Þdij q dA:
A
Therefore, the resulting interaction integral (M) becomes
M ¼ M local
Z n
o
aux
aux
¼
rij uaux
þ
r
u
r
e
d
q;j dA
i;1
ik
1j
i;1
ij
ik
A
Z n
o
m aux
aux
þ
raux
ij;j ui;1 C ijkl;1 ekl eij þ rij ða;1 ðDhÞ þ aðDhÞ;1 Þdij q dA;
ð21Þ
A
where the underlined term is a non-equilibrium term that appears due to non-equilibrium of the auxiliary
stress fields.
4. Evaluation of stress intensity factors
The relationship between J-integral and the mode I and mode II SIFs is given by
J local ¼
K 2I þ K 2II
;
Etip
ð22Þ
where Etip ¼ Etip for plane stress and Etip =ð1 m2tip Þ for plane strain. One obtains Mlocal as [50]
M local ¼
2
ðK I K aux
þ K II K aux
I
II Þ:
Etip
ð23Þ
The mode I and mode II SIFs are evaluated as follows:
Etip ð1Þ
M local ; ðK aux
¼ 1:0; K aux
ð24Þ
I
II ¼ 0:0Þ;
2
Etip ð2Þ
M local ; ðK aux
K II ¼
¼ 0:0; K aux
ð25Þ
I
II ¼ 1:0Þ:
2
The relationships of Eqs. (24) and (25) are the same as those for homogeneous materials [64] except that, for
FGMs, the material properties are evaluated at the crack-tip location [46–48].
KI ¼
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
2549
x2
r
crack
θ
f
x1
uaux
εaux
Fig. 3. A point force applied at the crack-tip in the direction parallel to the crack surface.
5. Evaluation of the T-stress
The T-stress can be also evaluated from the interaction integral with no contributions of both singular (i.e.
O(r1/2)) and higher-order (i.e. O(r1/2) and higher) terms in the crack-tip asymptotic fields. The derivation is
given by Kim and Paulino [65] and Paulino and Kim [56]. From the above Eq. (12), the M-integral in the form
of a line integral is obtained as
Z n
o
aux
aux
M local ¼ lim
rik eaux
ð26Þ
ik d1j rij ui;1 rij ui;1 nj dC:
Cs !0
Cs
Here we can consider only the stress parallel to the crack direction, i.e.
rij ¼ T d1i d1j ;
where T denotes the T-stress. One obtains that
!
r11
t
þ C tip atip Dhtip di1 ;
ui;1 ¼ e11 di1 ¼
Etip
ð27Þ
ð28Þ
where Ctip = 1 for plane stress and Ctip = 1 + mtip for plane strain. Substituting Eqs. (27) and (28) into Eq. (26),
one obtains
!
Z
Z
T
aux
rij nj ui;1 dC ¼ þ C tip atip Dhtip lim
raux
ð29Þ
M local ¼ lim
ij nj dC:
Cs !0 C
Cs !0 C
E
tip
s
s
Because the force f is in equilibrium (see Fig. 3)
Z
raux
f ¼ lim
ij nj dC;
Cs !0
ð30Þ
Cs
and thus the following relationship is obtained:
M local Etip
C tip atip Dhtip Etip :
T ¼
f
ð31Þ
Note that, for FGMs, the material properties are sampled at the crack-tip location.
6. Numerical implementations
6.1. M-integral
For numerical computation by means of the FEM, the M-integral is evaluated first in global coordinates
((Mm)global) (m = 1,2) and then transformed to local coordinates (Mlocal). The M-integral in Eq. (21) is numerically evaluated using the following form:
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A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
a
θ0
b
dθ = 0
d X2
X2
θ2
L =8
θ1
a
X1
E2
α2
E1
α1
W =1
dθ = 0
d X2
c
d
Fig. 4. Example 1: (a) An exponentially graded strip with an edge crack under thermal loads; (b) complete finite element mesh; (c) mesh
detail showing 12 sectors (S12) and 4 rings (R4) around the crack-tip employed in the 2D analysis; (d) mesh detail showing 10 sectors (S10)
and 14 rings (R14) around the crack-tip employed in the 3D analysis.
ðM m Þglobal ¼
X
X n
o
aux
aux
rij uaux
þ
r
u
r
e
d
i;m
ik ik
mj q;j detðJÞwgp
i;m
ij
elems Gauss pts:
n
o
m aux
aux
þ raux
ij;j ui;m C ijkl;m ekl eij þ rij ða;m DhÞ þ aDðhÞ;m Þdij q detðJÞwgp ;
ð32Þ
where the outer summation includes all the elements within the domain and the inner summation includes all
Gauss points with corresponding weights wgp, and det(J) is the determinant of the standard Jacobian matrix
relating (X1, X2) with (n, g) [66].
For the sake of generality, we determine derivatives of thermomechanical properties and temperature by
using shape function derivatives of finite elements [22,67]. These include
aux
aux
raux
ij;j ¼ C ijkl;j ekl þ C ijkl ekl;j ;
C ijkl;1 ;
a;1 ðDhÞ þ aðDhÞ;1 :
ð33Þ
Thus, the derivatives of a field variable P (e.g. Cijkl, a, or Dh) are obtained as
n
X
oP
oN i
¼
P i ; ðm ¼ 1; 2Þ;
oX m
oX m
i¼1
ð34Þ
where n is the number of element nodes and Ni = Ni(n, g) are the element shape functions which can be found
in many references, e.g. [66]. The derivatives oNi/oXm are obtained as
oN i =oX 1
oN i =oX 2
¼J
1
oN i =on
oN i =og
;
where J1 is the inverse of the standard Jacobian matrix.
ð35Þ
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
2551
6.2. Steady-state thermal diffusion
The present work addresses one-way coupling of thermomechanical analyses by which the field quantities
such as displacements, strains and stresses are affected by temperature loading, and not vice versa. We assume
that the crack faces are insulated, and heat flux is directed along the horizontal axis. Hence, the problems considered involve one-dimensional diffusion. The temperature distribution is obtained by solving the one-dimensional steady-state diffusion equation:
o
oh
k
ð36Þ
¼ 0; with k ¼ kðX 1 Þ and h ¼ hðX 1 Þ:
oX 1
oX 1
Temperature fields in Examples 4 and 5 are calculated using the Runge–Kutta method which solves the onedimensional diffusion equation (i.e. second-order ordinary differential equation).
7. Numerical examples
The performance of the interaction integral in evaluating mixed-mode SIFs and the T-stress for FGMs
under thermal loads is examined by means of numerical examples. The following examples are presented:
(1)
(2)
(3)
(4)
(5)
An edge crack in a plate: exponential gradation.
An edge crack in a plate: linear gradation.
An inclined center crack in a plate: exponential gradation.
An edge crack in a plate: hyperbolic-tangent gradation.
A crack in a functionally graded thermal barrier coating (TBC).
All the examples are analyzed using the FEM code FGM-FRANC2D.2 Examples 1 and 2 are also analyzed
using an in-house 3D FEM code for further numerical verification. Both codes incorporate the gradation of
thermomechanical material properties at the size-scale of the element. The specific graded elements used here
are based on the Direct Gaussian Formulation [67].
All the geometry is discretized with isoparametric graded elements [22]. The specific elements used in the 2D
analysis consist of singular quarter-point six-node triangles (T6qp) for crack-tip discretization, eight-node serendipity elements (Q8) for a circular region around crack-tip elements and for a far-field region, and regular
six-node triangles (T6) in a transition zone to Q8 elements. For the 3D analysis, we used 15-node quarter-point
wedge element for crack-tip discretization and 20-node brick elements for other regions.
All the examples consist of SIFs and T-stress results for FGMs, and those results are obtained by the interaction integral in conjunction with the FEM. In the first example, the FEM results for the mode-I SIF are
compared with available semi-analytical [30]pand
ffiffiffiffiffiffi numerical [42,43,41] solutions. In the second example, the
FEM results for the biaxiality ratio ðB ¼ T pa=K I Þ in a homogeneous plate are compared with numerical
solutions by Sladek and Sladek [70]. Also the new FEM results are provided for the FGMs. In the third example, the FEM results for SIFs and T-stress for thermal loads are compared with mechanically equivalent loads,
which enables us to verify the present method used in evaluating the T-stress and mixed-mode SIFs for FGMs.
The fourth example deals with hyperbolic-tangent material gradation which can realistically model the interface diffusion in bi-material systems. In the last example, the FEM results for SIFs and T-stress are provided
for a functionally graded thermal barrier coating.
7.1. An edge crack in a plate: exponential gradation
Fig. 4a shows an edge crack of length ‘‘a’’ in an exponentially graded plate subjected to steady-state thermal
loads. Fig. 4b shows the complete mesh configuration. Fig. 4c shows the mesh detail showing 12 sectors (S12)
2
The FEM code FGM-FRANC2D is based on I-FRANC2D [22] at the University of Illinois at Urbana-Champaign, and also
FRANC2D [68,69] developed at Cornell University.
2552
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
and 4 rings (R4) of elements around the crack-tip employed in the 2D analysis. Fig. 4d illustrates the mesh
detail with S10 and R14 crack-tip template used in the 3D analysis. The 2D mesh discretization consists of
907 Q8, 47 T6, and 12 T6qp elements, with a total of 966 elements and 2937 nodes and the 3-D representative
mesh discretization consists of 10 15-node quarter-point wedge elements and 528 20-node brick elements, with
total of 538 elements and 4054 nodes.
Young’s modulus and thermal expansion coefficient (a) are exponential functions of X1, while Poisson’s
ratio is constant. In this example, we considered a constant Poisson’s ratio because it has negligible effect
on fracture behavior of FGMs under pure mode-I conditions and some mixed-mode conditions (see the paper
[71] for more information).
The following data were used in the FEM analyses:
plane strain and plane stress;
a ¼ 0:5;
W ¼ 1;
bX 1
EðX 1 Þ ¼ E1 e
1
E2
b ¼ ln
W
E1
L ¼ 8;
and
and
t ¼ 0:1ð3D FEAÞ;
aðX 1 Þ ¼ a1 ecX 1 ;
1
a2
c ¼ ln
;
W
a1
E1 ¼ EðX 1 ¼ 0Þ ¼ 1:0
and
E2 ¼ EðX 1 ¼ W Þ ¼ 5 or 10;
a1 ¼ aðX 1 ¼ 0Þ ¼ 0:01ð C1 Þ and
h1 ¼ hðX 1 ¼ 0Þ
and
mðX 1 Þ ¼ m ¼ 0:3;
a2 ¼ aðX 1 ¼ W Þ ¼ 0:02ð C1 Þ;
h2 ¼ hðX 1 ¼ W Þ;
h0 ¼ 10 C:
Table 1 compares the present FEM results for normalized mode-I SIF in FGMs under various thermal
loads with the solutions provided by Erdogan and Wu [30], Walters et al. [42], Yildirim et al. [43] and Yildirim
[41]. The FEM results show good agreement with the reference results. For Case 1, we considered a constant
thermal conductivity coefficient (k), and for Case 2, we considered
1
k2
kðX 1 Þ ¼ k 1 edX 1 ; where d ¼
ln
; k 1 ¼ 1 and k 2 ¼ 10:
W
k1
The temperature distribution for Case 2 is obtained in the close-form solution as
hðX 1 Þ ¼ AedX 1 þ B;
where the unknowns A and B are obtained from temperature boundary conditions.
Table 1 also compares the mode-I SIF obtained by using two types of crack-tip elements: T6qp and regular
T6 elements. The results are very similar for the given mesh discretization involving S12 and R4 crack-tip temTable 1
pffiffiffiffiffiffi
Example 1: Normalized mode-I SIF in FGMs under thermal loads. The normalizing factor K 0 ¼ ½ðE1 a1 h0 Þ=ð1 m1 Þ pa. Case 1:
E2/E1 = 5, a2/a1 = 2; Case 2: E2/E1 = 10, a2/a1 = 2, k2/k1 = 10 (see Fig. 4)
Case
Load
Analysis
type
KI/K0
Present
Erdogan and Wu
[30]
Walters et al.
[42]
Yildirim et al.
[43]
Yildirim
[41]
T6qp(2D)
T6(2D)
3D
1
h1 = 0.5h0
h2 = 0.5h0
h1 = 0.05h0
h2 = 0.05h0
pl-e
pl-r
pl-e
0.0128
0.0090
0.0244
0.0129
0.0091
0.0246
0.0129
–
0.0245
0.0125
–
0.0245
0.0127
–
0.0241
0.0124
–
0.0238
0.0128
0.0090
–
2
h1 = 0.2h0
h2 = 0.5h0
h1 = 0.05h0
h2 = 0.5h0
pl-e
pl-r
pl-e
0.0334
0.0235
0.0406
0.0335
0.0236
0.0406
0.0338
–
0.0411
0.0335
–
0.0410
0.0335
–
0.0409
0.0331
–
0.0404
0.034
0.024
–
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
2553
Table 2
Example 1: 2D FEA results for the mode-I SIF, the T-stress and the biaxiality ratio in FGMs under thermal loads (see Fig. 4c for mesh
detail)
pffiffiffiffiffiffi
Case
Material variation
Load
Analysis type
KI
T
B ¼ T pa=K I
1
E2/E1 = 5
a2/a1 = 2
2
E2/E1 = 10
a2/a1 = 2
h1 = h2 = 0.5h0
h1 = h2 = 0.05h0
h1 = 0.2h0
h2 = 0.5h0
h1 = 0.05h0
h2 = 0.5h0
pl-e
pl-r
pl-e
0.00229
0.00161
0.00437
0.0067
0.0046
0.0126
3.66
3.58
3.61
pl-e
pl-r
pl-e
0.00599
0.00421
0.00728
0.0183
0.0128
0.0228
3.82
3.81
3.92
Table 3
Example 1: 3D FEA results for the mode-I SIF, the T-stress and the biaxiality ratio in FGMs under thermal loads (see Fig. 4d for mesh
detail)
pffiffiffiffiffiffi
Case
Material variation
Load
Analysis type
KI
T
B ¼ T pa=K I
1
E2/E1 = 5
a2/a1 = 2
h1 = h2 = 0.5h0
h1 = h2 = 0.05 h0
pl-e
pl-e
0.00231
0.00439
0.0060
0.0115
3.25
3.28
2
E2/E1 = 10
a2/a1 = 2
h1 = 0.2h0
h2 = 0.5h0
h1 = 0.05h0
h2 = 0.5h0
pl-e
0.00606
0.0174
3.60
pl-e
0.00736
0.0218
3.71
plates as shown in Fig. 4c. However, when the S12 and R2 template around the crack-tip is used, we observe
that T6qp elements (KI = 0.0128) showed better performance than T6 elements (KI = 0.0132) for the problem
in the first row of Case 1 in Table 1. The SIF result is also shown to be more sensitive in the case of using T6
element than T6qp element. Although we use a relatively fine mesh around crack tips in all the examples, we
employ the quarter-point T6 crack-tip elements for the sake of generality.
Table 1 also provides FEM results for the mode-I SIF obtained from the 3D fracture analysis using the Mintegral implemented in an in-house MATLAB code and show good agreement with other reference results.
This 3D analysis employs a group of ten 15-node quarter-point wedge element surrounding the crack-front
region, 13 concentric semi-circular domains consisting of 20-node brick elements around crack-tip elements
and 20-node brick elements for a far-field region (see Fig. 4d). For the plain-strain case, out-of-plane displacements are constrained.
Table 2 presents FEM results for the mode-I SIF, the T-stress and the biaxiality ratio. We observe that, for
both cases, the T-stress and SIF values are lower for the plane-stress condition than those for the plane-strain
condition. However, there is no significant difference in the biaxiality ratio for plain-stress and plain-strain
cases. Table 3 provides the results obtained from the 3D analysis which are in a good agreement with the
2D results. It is observed that the SIFs obtained from the 3D analysis show better agreement with the 2D
results than those for T-stress.
7.2. An edge crack in a plate: linear gradation
Fig. 5a show an edge crack of length ‘‘a’’ in a linearly graded plate subjected to steady-state thermal loads.
Fig. 5b shows the complete mesh configuration. Fig. 5c shows the mesh detail showing 12 sectors (S12) and 4
rings (R4) of elements around the crack-tip employed in the 2D analysis. Fig. 5d illustrates the mesh detail
with S10 and R14 crack-tip template employed in the 3D analysis. The 2D mesh discretization consists of
508 Q8, 48 T6 and 12 T6qp elements, with a total of 568 elements and 1163 nodes and the 3D representative
mesh discretization consists of 10 15-node quarter-point wedge elements and 318 20-node brick elements, with
a total of 328 elements and 2479 nodes.
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A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
a
b
dθ = 0
d X2
X2
θ2
θ0
L= 4
θ1
a
X1
E2
α2
E1
α1
W =1
dθ = 0
d X2
c
d
Fig. 5. Example 2: (a) A linearly graded strip with an edge crack under thermal loads; (b) complete finite element mesh; (c) mesh detail
showing 12 sectors (S12) and 4 rings (R4) around the crack-tip employed in the 2D analysis; (d) mesh detail showing 10 sectors (S10) and
14 rings (R14) around the crack-tip employed in the 3D analysis.
Young’s modulus, Poisson’s ratio and thermal expansion coefficient are linear functions of X1. The following data were used in the FEM analyses:
plane strain;
a ¼ 0:1 0:8;
E1 ¼ 1:0 10
5
W ¼ 1;
and
5 1
h1 ¼ 0:0 C
and
t ¼ 0:1ð3D FEAÞ;
E2 ¼ 0:5 105 ;
a1 ¼ 1:67 10 ð C Þ
L ¼ 4;
and
m1 ¼ 0:3
5 and
m2 ¼ 0:35;
1
a2 ¼ 1:0 10 ð C Þ;
h2 ¼ 1:0 C; h0 ¼ 0:0 C:
Tables 4 and 5 present the mode-I SIF, the T-stress and the biaxiality ratio for various crack lengths for
both homogeneous and graded materials using 2D and 3D FEMs, respectively. For the homogeneous case,
the biaxiality ratio obtained is in good agreement with that reported in Sladek and Sladek [70] within the
graphical accuracy. For all a/W ratios, the absolute magnitudes of the mode-I SIF and the T-stress decrease,
Table 4
Example 2: 2D FEA results for the mode-I SIF, the T-stress and the biaxiality ratio in FGMs under thermal loads (see Fig. 5c for mesh
detail)
a/W
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Homogeneous (cf. [70])
FGMs
KI
T
pffiffiffiffiffiffi
B ¼ T pa=K I
0.6454
0.7760
0.7951
0.7527
0.6705
0.5601
0.4288
0.2825
0.4317
0.2179
0.0314
0.1463
0.3258
0.5075
0.6980
0.8960
0.3749
0.2225
0.0383
0.2178
0.6089
1.2440
2.4139
5.0281
KI
T
pffiffiffiffiffiffi
B ¼ T pa=K I
0.4229
0.4691
0.4385
0.3742
0.2972
0.2186
0.1444
0.0795
0.2645
0.0992
0.0269
0.1312
0.2154
0.2854
0.3424
0.3886
0.3505
0.1676
0.0595
0.3930
0.9083
1.7924
3.5163
7.7491
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
2555
Table 5
Example 2: 3D FEA results for the mode-I SIF, the T-stress and the biaxiality ratio in FGMs under thermal loads (see Fig. 5d for mesh
detail)
a/W
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Homogeneous (cf. [70])
FGMs
KI
T
pffiffiffiffiffiffi
B ¼ T pa=K I
0.6447
0.7762
0.7954
0.7529
0.6706
0.5602
0.4289
0.2826
0.4114
0.2127
0.0262
0.1543
0.3339
0.5173
0.7048
0.9103
0.3576
0.2172
0.0319
0.2297
0.6240
1.2678
2.4369
5.1066
KI
T
pffiffiffiffiffiffi
B ¼ T pa=K I
0.4216
0.4624
0.4375
0.3738
0.2973
0.2190
0.1455
0.0805
0.2440
0.0906
0.0410
0.1440
0.2282
0.2961
0.3513
0.3961
0.3243
0.1553
0.0909
0.4318
0.9620
1.8562
3.5804
7.8006
and the absolute magnitudes of the biaxiality ratio increase due to the material gradation. Moreover, the
material gradation also affects the sign of the T-stress and the biaxiality ratio.
7.3. An inclined center crack in a plate
Fig. 6a and b show an inclined center crack of length ‘‘2a’’ located with a geometric angle h (counter-clockwise) in a plate subjected to a steady-state thermal load and an equivalent-mechanical fixed-grip loading,
a
b
Δ
2W=20
2W=20
2L=20
2a
θ
=2
θ
x1
=
2a
ε = Δ 2L
σ22 = E (x) ε
ε = α(x) Δθ(x)
σ22 = E (x) ε
Thermal
c
x1
2
2L=20
x2
x2
Mechanical
d
Contour 4
Contour 3
Contour 2
Contour 1
Fig. 6. Example 3: (a) FGM plate with an inclined crack with geometric angle h (a) subjected to thermal loads; (b) mechanically equivalent
fixed-grip loading; (c) typical finite element mesh; (d) mesh detail using 12 sectors (S12) and 4 rings (R4) around crack-tips and the domain
used for interaction integrals (h ¼ 30 counter-clockwise).
2556
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
respectively. Fig. 6c shows the complete mesh configuration. Fig. 6d shows the mesh detail using 12 sectors
(S12) and 4 rings (R4) of elements around crack-tips and various domains considered for the interaction integral. The mesh discretization consists of 1641 Q8, 94 T6, and 24 T6qp elements, with a total of 1759 elements
and 5336 nodes. The steady-state thermal loads and mechanical fixed-grip loading results in an uniform
mechanical strain em
e in a corresponding uncracked structure, which corresponds to
22 ðX 1 ; X 2 Þ ¼ r22 ðX 1 ; 10Þ ¼ eE0 ebX 1 for FGMs. Young’s modulus is an exponential function of X1, while the Poisson’s ratio
is constant. The following data were used in the FEM analyses:
plane stress;
nonhomogeneity parameter : ba ¼ 0:0 and 0:5;
a ¼ 1; L ¼ W ¼ 10; h ¼ 0 to 90 ; e ¼ aðX 1 ÞDhðX 1 Þ ¼ D=ð2LÞ ¼ 1:0;
EðX 1 Þ ¼ E0 ebX 1 ;
0
E ¼ 1:0;
mðX 1 Þ ¼ m;
m ¼ 0:3:
Table 6
pffiffiffiffiffiffi
Example 3: Normalized mixed-mode SIFs in FGMs for ba = 0.5 (K 0 ¼ eE0 pa) (see Fig. 6). The results for mixed-mode SIFs considering
the mechanical loads equivalent to thermal loads are identical to those for thermal loads and so are not provided in this table
Method
h
K þ =K 0
K þ =K 0
K =K 0
K =K 0
I
II
I
II
Konda and Erdogan [72]
0
18
36
54
72
90
1.424
1.285
0.925
0.490
0.146
0.000
0.000
0.344
0.548
0.532
0.314
0.000
0.674
0.617
0.460
0.247
0.059
0.000
0.000
0.213
0.365
0.397
0.269
0.000
M-integral (Thermal)
0
18
36
54
72
90
1.423
1.283
0.923
0.488
0.145
0.000
0.000
0.344
0.549
0.532
0.314
0.000
0.665
0.610
0.455
0.245
0.058
0.000
0.000
0.211
0.362
0.394
0.266
0.000
Table 7
Example 3: Normalized T-stress in FGMs under thermal loads in comparison with available reference solutions considering equivalentmechanical loads for ba = 0.0 and 0.5 (r0 ¼ eE0 ). The domains surrounded by Contours 2-4 have been used (see Fig. 6d) and the pathindependent FEM results are obtained (see Figs. 7 and 8)
ba
h
M-integral (Thermal)
Paulino and Dong [73]
M-integral (Mechanical) [56]
T(+a)/r0
T(a)/r0
T(+a)/r0
T(a)/r0
T(+a)/r0
T(a)/r0
ba = 0.0
0
15
30
45
60
75
90
0.974
0.844
0.488
0.002
0.500
0.867
1.003
0.974
0.844
0.488
0.002
0.500
0.867
1.003
0.999
0.866
0.500
0.000
0.499
0.866
1.000
0.999
0.866
0.500
0.000
0.500
0.866
1.000
0.983
0.853
0.497
0.005
0.491
0.859
0.995
0.983
0.853
0.497
0.005
0.491
0.859
0.995
ba = 0.5
0
15
30
45
60
75
90
0.879
0.757
0.418
0.049
0.525
0.878
1.003
0.854
0.743
0.431
0.016
0.490
0.857
1.003
0.867
0.748
0.420
0.039
0.513
0.870
1.000
0.876
0.763
0.444
0.010
0.490
0.858
1.000
0.896
0.773
0.434
0.036
0.513
0.868
0.994
0.858
0.747
0.436
0.011
0.484
0.850
0.994
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
2557
Table 6 compares the present FEM results for normalized mixed-mode SIFs obtained by the present Mintegral with semi-analytical solutions provided by Konda and Erdogan [72] for various geometric angles
of a crack in FGMs. The FEM results are in good agreement with those by Konda and Erdogan [72] (maximum difference 1.3%, average difference 0.6%). The FEM results for SIFs considering thermal loads are the
same as those for equivalent-mechanical loads and so not provided in the table.
Table 7 compares the FEM results for normalized T-stress obtained by the present M-integral with those
reported by Paulino and Dong [73] who used the singular integral equation method and with those for the
equivalent mechanical loading. The present FEM results are in good agreement with those by Paulino and
Dong [73]. Comparing the two equivalent systems, we observe that, for the homogeneous case with
ba = 0.0, the average difference was 1.2%; and for the FGM case with ba = 0.5, the average difference was
1.4%. These calculations considered all the given geometric angles except for 45 which involves reference
solutions of very small (or zero) magnitude. Note that the FEM results for the T-stress considering thermal
loads, however, are not identical but very similar to those for equivalent-mechanical loads. This may be
due to finite discretization of two equivalent continuum mechanics problems. For the same discretization,
we observe that the T-stress is more sensitive to the present M-integral for two equivalent loads than SIFs.
Figs. 7 and 8 show the path-independence of the M-integral in evaluating mixed-mode SIFs and the T-stress,
respectively, for the crack inclined by h ¼ 30 . Four integration domains as shown in Fig. 6d are used. The SIFs
4
K : all terms
I
K : excluding noneq–term
I
K : all terms
II
KII : excluding noneq–term
3.5
3
SIFs
2.5
2
1.5
1
0.5
0
1
2
3
4
Domain
Fig. 7. Example 3: Path-independence of the SIFs obtained by the interaction integral (h ¼ 300 ). The SIFs are evaluated either including
all terms or all terms except for the non-equilibrium term in Eq. (21) (see Fig. 6(d) for domains surrounded by each contour). The solid
lines indicate the path-independence and convergence of the M-integral.
0
T–stress
Including all terms
–1
Including all terms
except for the non–eq term
–2
1
2
3
4
No. of Domain
Fig. 8. Example 3: Path-independence of the T-stress obtained by the interaction integral (h ¼ 30 ). The T-stress is evaluated either
including all terms or all terms except for the non-equilibrium term in Eq. (21) (see Fig. 6(d) for domains surrounded by each contour).
The solid line indicates the path-independence and convergence of the M-integral.
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A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
and the T-stress are evaluated either including all terms or all terms except for the non-equilibrium term in Eq.
(21). One must consider all terms in Eq. (21) to get converged solutions for FGMs under thermal loads.
7.4. An edge crack in a plate: Hyperbolic-tangent function
Fig. 9a shows an edge crack of length ‘‘a’’ in a graded plate, and Fig. 9b shows the complete mesh discretization
using 12 sectors (S12) and 4 rings (R4) of elements around the crack-tip. The displacement boundary condition is
prescribed such that u2 = 0 along the lower and upper edges and u1 = 0 for the node at the left-bottom corner.
Young’s modulus, Poisson’s ratio, thermal expansion coefficient (a) and thermal conductivity coefficient (k)
are hyperbolic-tangent functions as follows (see Fig. 10):
E þ Eþ E Eþ
þ
tanh ½bðX 1 þ dÞ;
2þ
2þ
m þm
m m
mðX 1 Þ ¼
þ
tanh½dðX 1 þ dÞ;
2 þ
2 þ
a þa
a a
aðX 1 Þ ¼
þ
tanh½dðX 1 þ dÞ;
2
2 þ
þ
k þk
k k
þ
tanh½dðX 1 þ dÞ;
kðX 1 Þ ¼
2
2
EðX 1 Þ ¼
a
b
X2
d θ =0
d X2
o
Δθ= 0 C
o
Δθ= −10 C
L=4
ð37Þ
a
X1
E ( X 1)
α( X 1)
k ( X 1)
d θ =0
d X2
W= 2
Fig. 9. Example 4: An edge crack in hyperbolic-tangent materials: (a) geometry and BCs; (b) complete finite element mesh with 12 sectors
(S12) and 4 rings (R4) around the crack-tip.
3.5
E+
Young’s Modulus
3
2.5
2
FGM
1.5
E–
1
–1
–0.5
0
0.5
1
X1
Fig. 10. Example 4: Variation of Young’s modulus with ba = 15.
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
2559
where d is considered to be zero. The mesh discretization consists of 208 Q8, 37 T6, and 12 T6qp elements,
with a total of 257 elements and 1001 nodes. The following data were used for the FEM analysis:
plane stress and plane strain;
a=W ¼ 0:1 0:8;
L=W ¼ 2:0;
b ¼ 15:0; d ¼ 5:0
ðE ; Eþ Þ ¼ ð1; 3Þ; ðm ; mþ Þ ¼ ð0:3; 0:1Þ;
ða ; aþ Þ ¼ ð0:01; 0:03Þ;
ðk ; k þ Þ ¼ ð1; 3Þ:
Considering the thermal conductivity coefficient, the temperature distribution is obtained by solving the
one-dimensional steady-state diffusion equation as
Z
dX 1
hðX 1 Þ ¼ A
þ B;
kðX 1 Þ
where A and B are constants obtained from temperature boundary conditions. Fig. 11 illustrates the variation
of thermal conductivity and the resulting temperature field. We also used the Runge-Kutta method that also
provides the same temperature distribution.
Table 8 shows the present 2D FEM results for the mode I SIF (KI), the T-stress and the biaxiality ratio for
both plane stress and plain strain conditions. It is noted that the biaxiality ratio (a qualitative index of cracktip constraint) decreases rapidly as the ratio a/W approaches to 0.5 for which the crack-tip is located near the
pseudo-interface with steep gradation.
0
–1
–2
–3
1
k (x1 )
–5
k (x )
1
θ (x )
–4
3
–6
2
–7
1
θ(x1 )
–8
–9
–10
–1
–0.5
0
0.5
1
x1
Fig. 11. Example 4: Variations of the thermal conductivity coefficient (k(X1)) and the resulting temperature field (h(X1)).
Table 8
Example 4: The mode-I SIF, T-stress and the biaxiality ratio for for various a/W ratios in a hyperbolic-tangent material (see Fig. 9)
a/W
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Plane stress
Plane strain
KI
T
pffiffiffiffiffiffi
B ¼ T pa=K I
KI
T
pffiffiffiffiffiffi
B ¼ T pa=K I
0.7841
1.0520
1.1510
1.1240
0.8092
0.3002
0.2006
0.1338
0.4307
0.3751
0.3476
0.4206
0.8695
0.0427
0.0278
0.0339
0.4354
0.3997
0.4146
0.5932
1.9045
0.2761
0.2906
0.5680
0.8713
1.1700
1.2840
1.2620
0.9417
0.3737
0.2588
0.1774
0.4823
0.4188
0.3972
0.5112
1.1640
0.1228
0.0130
0.0302
0.4387
0.4012
0.4247
0.6421
2.1908
0.6380
0.1053
0.3816
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A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
7.5. A crack in a functionally graded thermal barrier coating (TBC)
Fig. 12a shows a functionally graded thermal barrier coating deposited on the bond coat and the metallic
substrate [41]. The metallic substrate is made of a nickel-based superalloy. The FGM coating is 100% Zirconia–Yttria at X1 = 0 and 100% nickel–chromium–aluminum–zirconium (NiCrAlY) bond coat at X1 = W1.
The hyperbolic-tangent function is used to simulate potential interfacial diffusion using the steep gradation
between the bond coat and the substrate. The FGM coating is considered to contain a periodic crack of length
‘‘a’’ with an interval ‘‘b’’. Due to periodicity, only one crack is modelled. Fig. 12b shows complete mesh configuration. Fig. 12c shows the mesh detail using 12 sectors (S12) and 4 rings (R4) of elements around the cracktip. The representative mesh discretization consists of 374 Q8, 226 T6 and 12 T6qp elements, with a total of
612 elements and 1693 nodes. The TBC system is assumed to be initially at an uniform temperature (h0) and is
subjected to a change in temperature due to steady-state diffusion involving temperature boundary conditions.
Table 9 shows thermomechanical material properties considered in this example. The following material gradation and data were used:
a
X2
FGM coating
b =2
a =0.1–1.0
Ec
αc
kc
W1 =1
X1
Bond
coat
Metal Substrate
(Homogeneous)
E
α
bc
bc
Es
αs
k
bc
ks
W2 =0.5
dθ
=0
d X2
dθ
=0
d X2
W3 =5
b
c
Fig. 12. Example 5: (a) A crack in a functionally graded thermal barrier coating; (b) complete finite element mesh; (c) mesh detail using 12
sectors (S12) and 4 rings (R4) around the crack-tip.
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
2561
Table 9
Thermomechanical properties of the monolithic components used in the FGM TBC [41,76]
Material
E (GPa)
m
a (C1)
k (W/m K)
Zirconia–Yttria
Bond coat(NiCrAlY)
Substrate(Ni)
27.6
137.9
175.8
0.25
0.27
0.25
10.01 · 106
15.16 · 106
13.91 · 106
1
25
7
plane strain;
a ¼ 0:1 1:0;
b ¼ 2;
W 1 ¼ 1:0;
h1 ¼ hðX 1 ¼ 0Þ ¼ 0:2h0 ; and
For the FGM coating region;
EðX 1 Þ ¼ Ec þ ðEbc Ec ÞX 2 ;
W 2 ¼ 0:5;
W 3 ¼ 5:0
with h0 ¼ 1000 C:
h3 ¼ hðX 1 ¼ W 1 þ W 2 þ W 3 Þ ¼ 0:5h0
ð38Þ
mðX 1 Þ ¼ mc þ ðmbc mc ÞX
kðX 1 Þ ¼ k c þ ðk bc k c ÞX 2
aðX 1 Þ ¼ ac þ ðabc ac ÞX ;
For the region with the bond coat and the substrate
P s þ P bc P s P bc
P ðX 1 Þ ¼
þ
tanh bðX 1 Þ with b ¼ 100
2
2
P ¼ E; m; a and k
and
Fig. 13 illustrates the variation of thermal conductivity of graded TBCs and the resulting (normalized) temperature field. We used Runge–Kutta method to obtain the temperature field. Table 10 presents the mode-I
3
30
Bond
Coat
Substrate
25
2
20
1.5
15
θ(x1 )/θ0
1
2.5
k (x )
θ (x1 ) /θ0
FGM
coating
k (x )
1
10
1
5
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0
6.5
x1 /W1
Fig. 13. Example 5: Variations of the thermal conductivity coefficient (k(X1)) and the resulting normalized temperature field (h(X1)/h0).
Table 10
Example 5: The mode-I SIF, the T-stress and the biaxiality ratio for various a/W1 ratios in a functionally graded thermal barrier coating
(see Fig. 12)
pffiffiffiffiffiffi
a/W1
KI
T
B ¼ T pa=K I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
180.6
258.2
338.4
433.6
550.3
703.6
889.9
1118
1400
1751
141.7
131.0
119.2
137.2
177.5
237.3
321.7
442.0
618.5
1026
0.4398
0.4022
0.3420
0.3547
0.4043
0.4630
0.5301
0.6268
0.7429
1.0386
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A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
SIF, the T-stress and the biaxiality ratio for various a/W1 ratios. The absolute magnitudes of the mode-I SIF
and T-stress increase with the increase in the a/W1 ratios, but the biaxiality ratio does not.
8. Discussion
This paper presents a novel formulation of the interaction integral to evaluate the T-stress and SIFs in isotropic FGMs under steady-state thermal loads. The asymptotic stress field around the crack-tip in FGMs has
the same form as the Irwin–Williams [74,20] solution for homogeneous materials. The correspondence of the
crack-tip behavior between homogeneous and graded materials, that is so-called local homogenization near the
crack-tip [75], provides a basis for the present formulation. The present method relates the asymptotically
defined interaction integral (M-integral) to the T-stress and SIFs, converts the M-integral to an equivalent
domain integral (EDI) using mechanical auxiliary fields, and calculates such fracture parameters using a finite
domain.
A single parameter (SIFs or J) characterizes crack-tip fields under small scale yielding conditions which
involve high degree of triaxiality at the crack-tip. The single parameter fracture mechanics is valid if the plastic
zone size be small compared with other dimensions of the cracked structure. However, under large scale yielding conditions, the single parameter is not sufficient to represent crack-tip fields, and so the additional parameter, i.e. the elastic T-stress, is needed which is evaluated in this paper. For mode I problems in Examples 1, 2,
4 and 5, we also evaluate the biaxiality ratio (B) which plays a role as a qualitative index of the relative cracktip constraint of various geometries [59].
Numerical examples presented in this paper demonstrate the accuracy and performance of the T-stress and
mixed-mode SIFs obtained by the M-integral. The present formulation is capable of dealing with any kinds of
smooth material gradations including micromechanics models and continuum functions by using shape function derivatives. The present study has the following characteristics:
• The FEM results for mixed-mode SIFs and the T-stress agree well with the available reference results. For
the verification of the T-stress and mixed-mode SIFs for thermal loads, two equivalent mechanical and
thermal systems are considered and well compared.
• In general, for the same mesh discretization, the accuracy of SIFs is higher than that of the T-stress.
• The path-independence of the M-integral has been observed for both SIFs and the T-stress. However, the
T-stress is more dependent on the size of domain than SIFs. This may be due to the nature of the non-singular T-stress and the auxiliary fields used for the T-stress.
• Material gradation affects the magnitudes and signs of the T-stress and SIFs; however, it does not affect the
crack-tip singularity (i.e. r1/2).
9. Conclusions
In this paper, mixed-mode stress intensity factors and the non-singular T-stress in FGMs under steady-state
thermal loads are evaluated by means of a novel interaction integral in conjunction with the 2D and 3D finite
element analyses. We used the non-equilibrium formulation and the corresponding auxiliary fields tailored for
FGMs. Various numerical examples are presented to verify the accuracy and performance of the present
method. The FEM results showed very good agreement with the reference results. The potential extension
of this work includes the evaluation of the T-stress and mixed-mode stress intensity factors in 3D functionally
graded solid oxide fuel cells (SOFCs) under transient thermal loading.
Acknowledgements
We gratefully acknowledge the support from the National Science Foundation (NSF) under the Faculty
Early Career Development (CAREER) Grant CMMI-0546225 (Material Design & Surface Engineering Program). We also acknowledge start-up support from the University of Connecticut. Any opinions expressed
herein are those of the writers and do not necessarily reflect the views of the sponsors.
A. KC, J.-H. Kim / Engineering Fracture Mechanics 75 (2008) 2542–2565
2563
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