Youn-Sha Chan
Department of Computer and Mathematical
Sciences,
University of Houston-Downtown,
One Main Street,
Houston, TX 77002
Glaucio H. Paulino
Department of Civil and Environmental
Engineering,
University of Illinois,
2209 Newmark Laboratory,
205 North Mathews Avenue,
Urbana, IL 61801
Albert C. Fannjiang
Department of Mathematics,
University of California,
Davis, CA 95616
1
Gradient Elasticity Theory for
Mode III Fracture in Functionally
Graded Materials—Part II: Crack
Parallel to the Material Gradation
A Mode-III crack problem in a functionally graded material modeled by anisotropic
strain-gradient elasticity theory is solved by the integral equation method. The gradient
elasticity theory has two material characteristic lengths ᐉ and ᐉ⬘, which are responsible
for volumetric and surface strain-gradient terms, respectively. The governing differential
equation of the problem is derived assuming that the shear modulus G is a function of x,
i.e., G ⫽ G共x兲 ⫽ G0e␤x, where G0 and ␤ are material constants. A hypersingular integrodifferential equation is derived and discretized by means of the collocation method and a
Chebyshev polynomial expansion. Numerical results are given in terms of the crack
opening displacements, strains, and stresses with various combinations of the parameters
ᐉ, ᐉ⬘, and ␤. Formulas for the stress intensity factors, KIII, are derived and numerical
results are provided. 关DOI: 10.1115/1.2912933兴
Introduction
This work is a continuation of the paper on “Gradient Elasticity
Theory for Mode III Fracture in Functionally Graded Materials—
Part I: Crack Perpendicular to the Material Gradation” by Paulino
et al. 关1兴 共hereinafter referred to as Part I兲. In Part I, the authors
considered a plane elasticity problem in which the medium contains a finite crack on the y = 0 plane and the material gradation is
perpendicular to the crack. In “Part II,” the material gradation is
parallel to the crack 共see Fig. 1兲. In Part I, the shear modulus G
共that rules the material gradation兲 is a function of y only, G
⬅ G共y兲 = G0e␥y; while in Part II, it is a function of x, i.e., G
⬅ G共x兲 = G0e␤x. An immediate consequence of the difference in
geometry, which is indicated in Fig. 1, is that the location of the
crack in Part I is rather irrelevant to the problem and thus can be
shifted so that the center is at the origin point 共0, 0兲. On the other
hand, if the material gradation is parallel to the crack, then the
location of the crack is pertinent to the solution of the problem.
The method of solution is essentially the same in both Parts I
and II, i.e., the integral equation method. However, because of
differences in the geometrical configurations, some changes are
expected. For instance, in Part I, the crack opening displacement
profile is symmetric with respect to the y-axis, while in Part II, the
symmetry of the crack profiles no longer exists. Thus, some interesting questions arise.
•
•
•
•
How are the crack opening displacement profiles affected by
the gradient elasticity and the gradation of the material?
How are the stresses influenced under the gradient elasticity?
How are the stress intensity factors 共SIFs兲 calculated?
How do the results compare to the classical linear elastic
fracture mechanics 共LEFM兲?
We will address all the above questions. The remainder of the
paper is organized as follows. First, the constitutive equations of
anisotropic gradient elasticity for nonhomogeneous materials subjected to antiplane shear deformation are given. Then, the govern-
ing partial differential equations 共PDEs兲 are derived, and the Fourier transform method is introduced and applied to convert the
governing PDE into an ordinary differential equation 共ODE兲. Afterward, the crack boundary value problem is described, and a
specific complete set of boundary conditions is given. The governing hypersingular integrodifferential equation is derived and
discretized using the collocation method. Next, various relevant
aspects of the numerical discretization are described in detail.
Subsequently, numerical results are given, conclusions are inferred, and potential extensions of this work are discussed. One
appendix, providing the hierarchy of the PDEs and the corresponding integral equations, supplements the paper.
2
Constitutive Equations of Gradient Elasticity
A schematic demonstration of continuously graded microstructure in functionally graded materials 共FGMs兲 is illustrated by Fig.
2. The linkage between gradient elasticity and graded materials
within the framework of fracture mechanics and its related work
has been addressed in Part I. For the sake of completeness, the
notation and constitutive equations of gradient elasticity for an
antiplane shear crack in a FGM are briefly given in this section
and particularized to the case of an exponentially graded material
along the x-direction.
For an antiplane shear problem, the relevant displacement components are as follows:
u = v = 0,
w = w共x,y兲
共1兲
and the nontrivial strains are as follows:
⑀xz =
1 ⳵w
,
2 ⳵x
⑀yz =
1 ⳵w
2 ⳵y
共2兲
The constitutive equations of gradient elasticity for FGMs are
关1,2兴 as follows:
␴ij = ␭共x兲⑀kk␦ij + 2G共x兲共⑀ij − ᐉ2ⵜ2⑀ij兲 − ᐉ2关⳵k␭共x兲兴共⳵k⑀ll兲␦ij
Contributed by the Applied Mechanics Division of ASME for publication in the
JOURNAL OF APPLIED MECHANICS. Manuscript received February 23, 2007; final manuscript received February 27, 2007; published online August 21, 2008. Review conducted by Robert M. McMeeking.
Journal of Applied Mechanics
− 2ᐉ2关⳵kG共x兲兴共⳵k⑀ij兲
共3兲
␶ij = ␭共x兲⑀kk␦ij + 2G共x兲⑀ij + 2ᐉ⬘␯k关⑀ij⳵kG共x兲 + G共x兲⳵k⑀ij兴 共4兲
Copyright © 2008 by ASME
NOVEMBER 2008, Vol. 75 / 061015-1
y
y
βx
G = G0 e
-a
a
Material gradation perpendicular to the crack.
x
c
d
x
Fig. 3 Geometry of the crack problem and material gradation
y
␮xyz = 2G共x兲ᐉ2⳵⑀yz/⳵x
␮yxz = 2G共x兲共ᐉ2⳵⑀xz/⳵y − ᐉ⬘⑀xz兲
␮yyz = 2G共x兲共ᐉ2⳵⑀yz/⳵y − ᐉ⬘⑀yz兲
c
d
Material gradation parallel to the crack.
x
␴xx = ␴yy = ␴zz = 0,
Fig. 1 A geometric comparison of the material gradation with
respect to the crack location
␮kij = 2ᐉ⬘␯kG共x兲⑀ij + 2ᐉ2G共x兲⳵k⑀ij
␴xy = 0
␴xz = 2G共x兲共⑀xz − ᐉ2ⵜ2⑀xz兲 − 2ᐉ2关⳵xG共x兲兴共⳵x⑀xz兲 ⫽ 0
␴yz = 2G共x兲共⑀yz − ᐉ ⵜ ⑀yz兲 − 2ᐉ 关⳵xG共x兲兴共⳵x⑀yz兲 ⫽ 0
2
2
2
␴xy = 0
␴xz = 2G共⑀xz − ᐉ2ⵜ2⑀xz兲 ⫽ 0
␴yz = 2G共⑀yz − ᐉ2ⵜ2⑀yz兲 ⫽ 0
共5兲
where ᐉ is the characteristic length of the material responsible for
volumetric strain-gradient terms, ᐉ⬘ is responsible for surface
strain-gradient terms, ␴ij is the stress tensor, and ␮ijk is the
couple-stress tensor. The Lamé moduli ␭ ⬅ ␭共x兲 and G ⬅ G共x兲 are
assumed to be functions of x. Moreover, ⳵k = ⳵ / ⳵xk. The parameter
ᐉ⬘ is associated with surfaces and ␯k, ⳵k␯k = 0, is a director field
equal to the unit outer normal nk on the boundaries.
For a Mode-III problem, the constitutive equations above become
␴xx = ␴yy = ␴zz = 0,
共6兲
If G is constant, i.e., the material is homogeneous, then the constitutive equations 关3,4兴 are1 as follows:
␮xxz = 2Gᐉ2⳵⑀xz/⳵x
␮xyz = 2Gᐉ2⳵⑀yz/⳵x
␮yxz = 2G共ᐉ2⳵⑀xz/⳵y − ᐉ⬘⑀xz兲
␮yyz = 2G共ᐉ2⳵⑀yz/⳵y − ᐉ⬘⑀yz兲
共7兲
It is worth to point out that each of the total stresses ␴xz and ␴yz in
Eq. 共6兲 has an extra term than the ones in 共7兲 due to the material
gradation interplay with the strain-gradient effect 关2兴.
3 Governing Partial Differential Equation and Boundary Conditions
By imposing the only nontrivial equilibrium equation
␮xxz = 2G共x兲ᐉ2⳵⑀xz/⳵x
⳵␴xz ⳵␴yz
+
=0
⳵x
⳵y
共8兲
the following PDE is obtained:
冋 冉
冋
⳵
⳵w
⳵w
G共x兲
− ᐉ 2ⵜ 2
⳵x
⳵x
⳵x
− ᐉ2
冊册 冋 冉
+
⳵
⳵w
⳵w
G共x兲
− ᐉ 2ⵜ 2
⳵y
⳵y
⳵y
册
冊册
⳵2G共x兲 ⳵2w ⳵G共x兲 ⳵3w ⳵G共x兲 ⳵3w
+
+
=0
⳵x2 ⳵x2
⳵x ⳵x3
⳵x ⳵x⳵ y 2
共9兲
If the shear modulus G is assumed as an exponential function of x
共see Fig. 3兲,
G = G共x兲 = G0e␤x
共10兲
then PDE 共9兲 can be simplified as
− ᐉ 2ⵜ 4w − 2 ␤ ᐉ 2ⵜ 2
Ceramic
phase
Ceramic
matrix
y
metallic
inclusions
with
Transition
region
Metallic
matrix
with
Metallic
phase
ceramic
inclusions
x
Fig. 2 A schematic illustration of a continuously graded microstructure in FGMs
061015-2 / Vol. 75, NOVEMBER 2008
⳵w
⳵w
⳵ 2w
=0
+ ⵜ 2w − ␤ 2ᐉ 2 2 + ␤
⳵x
⳵x
⳵x
共11兲
or
1
According to the geometry of the problem 共see Fig. 3兲, it is the upper half-plane
that is considered in the formulation. The crack is sitting on the x-axis, which is on
the boundary of the upper half-plane. Thus, the outward unit normal should be
共0 , −1 , 0兲, and not 共0, 1, 0兲. Based on Eq. 共5兲, or the last equation in Eq. 共5兲 of Ref.
关4兴, the sign in front of ᐉ⬘ in the expression for both ␮yxz and ␮yyz should be “⫺”
instead of “⫹.”
Transactions of the ASME
Table 1 Governing PDEs in antiplane shear problems
Cases
Governing PDE
ᐉ=0,␤=0
Laplace equation:
ⵜ 2w = 0
Perturbed Laplace equation:
共ⵜ2 + ␤⳵ / ⳵x兲w = 0
Helmholtz–Laplace equation:
共1 − ᐉ2ⵜ2兲ⵜ2w = 0
ᐉ=0,␤⫽0
ᐉ⫽0,␤=0
ᐉ⫽0,␤⫽0
冉
Equation 共11兲:
1 − ␤ᐉ2
⳵
− ᐉ 2ⵜ 2
⳵x
冊冉
ⵜ2 + ␤
References
tions is the same as those adopted by Vardoulakis et al. 关4兴 An
alternative treatment of boundary conditions can be found in Ref.
关9兴.
Standard textbooks
Erdogan 关7兴
4
Let the Fourier transform be defined by
Vardoulakis et al. 关4兴
Fannjiang et al. 关5兴
Zhang et al. 关8兴
Studied in this paper
冊
⳵
w=0
⳵x
Fourier Transform
F共w兲共␰兲 = W共␰兲 =
冑2␲
1 − ␤ᐉ2
⳵
− ᐉ 2ⵜ 2
⳵x
冊冉
ⵜ2 + ␤
H␤ = 1 − ␤ᐉ2
冊
⳵
w=0
⳵x
⳵
− ᐉ 2ⵜ 2,
⳵x
共12兲
w共x,y兲 =
共1 − ᐉ ⵜ 兲ⵜ w = 0
2
2
L␤ = ⵜ2 + ␤
⳵
共13兲
⳵x
− ᐉ 2ⵜ 4w =
1
or HLw = 0
ⵜ2 + ␤
冊
⳵
w=0
⳵x
共14兲
共15兲
the perturbed Laplace equation, which is the PDE that governs the
Mode-III crack problem for nonhomogeneous materials with
shear modulus G共x兲 = G0e␤x 关6,7兴. The limit of sending ᐉ → 0 will
lower the fourth order PDE 共11兲 to a second order one 共Eq. 共15兲兲,
and a singular perturbation is expected. By taking both limits ␤
→ 0 and ᐉ → 0, one obtains the harmonic equation for classical
elasticity. Various combinations of parameters ᐉ and ␤ with the
corresponding governing PDE are listed in Table 1.
One may notice that in the governing PDE 共12兲, there is no
surface term parameter ᐉ⬘ involved. However, ᐉ⬘ does influence
the solution through the boundary conditions. By the principle of
virtual work, the following boundary conditions can be derived
and are adopted in this paper:
␴yz共x,0兲 = p共x兲,
w共x,0兲 = 0,
␮yyz共x,0兲 = 0,
x 苸 共c,d兲
W共␰兲e−ix␰d␰
共18兲
−⬁
1
冑2 ␲
冕
⬁
W共␰,y兲e−ix␰d␰
共19兲
−⬁
− ᐉ2
冑2 ␲
冕冉
⬁
␰4W共␰,y兲 − 2␰2
−⬁
冊
⳵2W ⳵4W −ix␰
+ 4 e d␰
⳵y2
⳵y
− ␤ ᐉ 2ⵜ 2
⳵w − ␤ᐉ2
=
冑2 ␲
⳵x
ⵜ 2w =
− ␤ 2ᐉ 2
␤
1
冕冉
⬁
i␰3W共␰,y兲 − i␰
−⬁
冕冉
⳵2W −ix␰
e d␰
⳵y2
冊
共22兲
␰2W共␰,y兲e−ix␰d␰
共23兲
⬁
冑2 ␲
− ␰2W共␰,y兲 +
−⬁
⳵2w共x,y兲 ␤2ᐉ2
=
冑2 ␲
⳵x2
⳵w共x,y兲
␤
=
冑
⳵x
2␲
冕
冕
冊
⳵2W −ix␰
e d␰ 共21兲
⳵y2
⬁
−⬁
⬁
共− i␰兲W共␰,y兲e−ix␰d␰
共24兲
−⬁
Equations 共20兲–共24兲 are added according to Eq. 共11兲, and after
simplification, the governing ODE is obtained:
冋
ᐉ2
d2
d4
2 2
2
−
共2ᐉ
␰
+
2i
␤
ᐉ
␰
+
1兲
dy 4
dy 2
册
+ 共ᐉ2␰4 + 2i␤ᐉ2␰3 − ␤2ᐉ2␰2 + ␰2 + i␤␰兲 W = 0
5
共25兲
Solutions of the ODE
The corresponding characteristic equation to the ODE 共25兲 is
ᐉ2␭4 − 共2ᐉ2␰2 + 2i␤ᐉ2␰ + 1兲␭2
+ 共ᐉ2␰4 + 2i␤ᐉ2␰3 − ␤2ᐉ2␰2 + ␰2 + i␤␰兲 = 0
关ᐉ2␭2 − 共1 + i␤ᐉ2␰ + ᐉ2␰2兲兴共␭2 − ␰2 − i␤␰兲 = 0
共16兲
The first two boundary conditions in Eq. 共16兲 are from classical
LEFM, and the last one, involving the couple-stress ␮yyz, is
needed for the higher order theory. This set of boundary condiJournal of Applied Mechanics
⬁
共26兲
which can be further factorized as
x 苸 共c,d兲
−⬁⬍x⬍ +⬁
冑2␲
冕
共20兲
where the Helmholtz operator H = 1 − ᐉ2ⵜ2 and the Laplacian operator L = ⵜ2 are invariant under any change of variables by rotations and translations. FGM creates the perturbation and ruins the
invariance. However, the perturbing term “−␤ᐉ2共⳵ / ⳵x兲” in L␤,
which is not purely caused by the gradation of the material, involves both the gradation parameter ␤ and the characteristic
length ᐉ 共the product of ␤ and ᐉ2兲. It can be interpreted as a
consequence of the interaction of the material gradation and the
strain-gradient effect 关2兴.
If we let ᐉ → 0 alone, then the perturbed Helmholtz differential
operator H␤ will become the identity operator, and one reduces
PDE 共12兲 to
冉
共17兲
where w共x , y兲 is the inverse Fourier transform of the function
W共␰ , y兲. By considering each term in Eq. 共11兲 and using Eq. 共19兲,
one obtains
where H␤ is the perturbed Helmholtz operator, L␤ is the perturbed
Laplacian operator, and the two operators commute 共H␤L␤
= L␤H␤兲. By sending ␤ → 0, we get the PDE 关4,5兴
2
w共x兲eix␰dx
−⬁
where F−1 denotes the inverse Fourier transform. Now, let us
assume that
which is the governing PDE solved in the present paper.
It may be seen, from a viewpoint of perturbation, that PDE 共12兲
can be expressed in an operator form, i.e.,
H␤L␤w = 0,
⬁
Then, by the Fourier integral formula 关10兴,
F−1共W兲共x兲 = w共x兲 =
冉
冕
1
共27兲
Clearly, the four roots ␭i 共i = 1 , 2 , 3 , 4兲 of the polynomial 共27兲
above can be written as
␭1 =
− 1冑
冑2
冑␰ 4 + ␤ 2 ␰ 2 + ␰ 2 −
i
␤␰
冑2 冑冑␰4 + ␤2␰2 + ␰2
共28兲
NOVEMBER 2008, Vol. 75 / 061015-3
Table 2 Roots ␭i together with the corresponding mechanics theory and type of material
␭2 =
1
冑2
␭3 =
Number
of roots
Roots
ᐉ=0,␤=0
2
⫾兩␰兩
ᐉ=0,␤⫽0
2
ᐉ⫽0,␤=0
4
␭1 and ␭2 in Eqs.
共28兲 and 共29兲, respectively
⫾兩␰ 兩 , ⫾ 冑␰2 + 1 / ᐉ2
ᐉ⫽0,␤⫽0
4
冑冑␰4 + ␤2␰2 + ␰2 +
Mechanics theory
and type of material
The four roots ␭1 – ␭4
in Eqs. 共28兲–共31兲
␤␰
i
冑2 冑冑␰4 + ␤2␰2 + ␰2
冑2 冑冑共␰2 + 1/ᐉ2兲2 + ␤2␰2 + ␰2 + 1/ᐉ2
1
冑2
+
共30兲
冑冑共␰2 + 1/ᐉ2兲2 + ␤2␰2 + ␰2 + 1/ᐉ2
冑2 冑冑共␰2 + 1/ᐉ2兲2 + ␤2␰2 + ␰2 + 1/ᐉ2
共31兲
If ␤ → 0, then the imaginary part of each root ␭i 共i = 1 , . . . , 4兲
disappears. Thus, we have exactly the same roots found by Vardoulakis et al. 关4兴 and Fannjiang et al. 关5兴. The root ␭1 corresponds
to the solution of the perturbed harmonic equation ⵜ2w
+ ␤⳵w / ⳵x = 0, and the root ␭3 agrees with the solution of the perturbed Helmholtz equation 共1 − ␤ᐉ2⳵ / ⳵x − ᐉ2ⵜ2兲w = 0. Various
choices of parameters ᐉ and ␤ with their corresponding mechanics
theories and materials are listed in Table 2. In contrast to the four
real roots found in Part I, the four roots here are all complex and
admit a more complicated expression.
By the symmetry of the geometry, one can only consider the
upper half-plane 共y ⬎ 0兲. By taking account of the far-field boundary condition
w共x,y兲 → 0
G共x兲
冑2 ␲
as 冑x2 + y 2 → + ⬁
共32兲
W共␰,y兲 = A共␰兲e
+ B共␰兲e
␭3 y
w共x,y兲 =
1
冑2 ␲
冕
⬁
关A共␰兲e
␭1 y
+ B共␰兲e
␭3 y
兴e
−ix␰
d␰
共34兲
冊
⬁
兵共ᐉ2␭21 − ᐉ⬘␭1兲A共␰兲e␭1y
−⬁
ᐉ⬘␭1 − ᐉ2␭21
ᐉ2␭23 − ᐉ⬘␭3
yⱖ0
共36兲
A共␰兲 = ␳共␤, ␰兲A共␰兲
共37兲
with
␳共␤, ␰兲 =
ᐉ⬘␭1 − ᐉ2␭21
ᐉ2␭23
− ᐉ ⬘␭ 3
=−
ᐉ2␰2 + i␤ᐉ2␰ + ᐉ⬘冑␰2 + i␤␰
ᐉ⬘冑␰2 + i␤␰ + 1/ᐉ2 + 共ᐉ2␰2 + i␤ᐉ2␰ + 1兲
共38兲
Denote
␾共x兲 =
1
⳵
w共x,0+兲 =
冑
⳵x
2␲
冕
⬁
共− i␰兲关A共␰兲 + B共␰兲兴e−ix␰d␰
−⬁
= F−1兵共− i␰兲关A共␰兲 + B共␰兲兴其
共39兲
The second boundary condition in Eqs. 共16兲 and 共39兲 implies that
␾共x兲 = 0,
and
冕
共33兲
where the nonpositive real part of ␭1 and ␭3 has been chosen to
satisfy the far-field condition in the upper half-plane. Accordingly,
the displacement w共x , y兲 takes the form
冕
⳵⑀yz
− ᐉ⬘⑀yz
⳵y
From the boundary condition in Eq. 共16兲 imposed on the couplestress ␮yyz 共i.e., ␮yyz共x , 0兲 = 0 for −⬁ ⬍ x ⬍ ⬁兲, one obtains the following relationship between A共␰兲 and B共␰兲:
one can express the solution for W共␰ , y兲 as
␭1 y
Vardoulakis et al. 关4兴
Fannjiang et al. 关5兴
Studied in this paper
+ 共ᐉ2␭23 − ᐉ⬘␭3兲B共␰兲e␭3y其e−ix␰d␰,
B共␰兲 =
␤␰
i
Erdogan 关7兴
冉
=
␤␰
i
Standard textbooks
␮yyz共x,y兲 = 2G共x兲 ᐉ2
共29兲
冑共␰2 + 1/ᐉ2兲2 + ␤2␰2 + ␰2 + 1/ᐉ2
冑2
References
Classical LEFM,
homogeneous materials
Classical LEFM,
nonhomogeneous materials
Gradient theories,
homogeneous materials
Gradient theories,
nonhomogeneous materials
− 1冑
−
␭4 =
Cases
x 苸 关c,d兴
共40兲
d
␾共x兲dx = 0
共41兲
c
which is the single-valuedness condition. By Eqs. 共39兲 and 共40兲,
we obtain
共− i␰兲关A共␰兲 + B共␰兲兴 =
−⬁
1
冑2 ␲
冕
⬁
1
␾共x兲eix␰dx =
冑2 ␲
−⬁
冕
d
␾共t兲ei␰tdt
c
共42兲
Both A共␰兲 and B共␰兲 are determined by the boundary conditions.
By substituting Eq. 共37兲 into Eq. 共42兲 above, one gets
6 Hypersingular Integrodifferential Equation Approach
By substituting Eq. 共34兲 into Eq. 共6兲, we have
=
冑2 ␲
冕
⬁
关␭1A共␰兲e␭1y兴e−ix␰d␰,
−⬁
and
061015-4 / Vol. 75, NOVEMBER 2008
yⱖ0
冋
1
冑2␲ 共− i␰兲关1 + ␳共␤, ␰兲兴
1
册冕
d
␾共t兲ei␰tdt
共43兲
c
where
␴yz共x,y兲 = 2G共x兲共⑀yz − ᐉ2ⵜ2⑀yz兲 − 2ᐉ2关⳵xG共x兲兴⳵x⑀yz
G共x兲
A共␰兲 =
共35兲
共ᐉ2␰2 + i␤ᐉ2␰ + 1兲 + ᐉ⬘冑␰2 + i␤␰ + 1/ᐉ2
1
=
1 + ␳共␤, ␰兲
1 + ᐉ⬘冑␰2 + i␤␰ + 1/ᐉ2 − ᐉ⬘冑␰2 + i␤␰
共44兲
By replacing the A共␰兲 in Eq. 共35兲 by Eq. 共43兲, one obtains the
following integral equation in the limit y → 0+:
Transactions of the ASME
lim ␴yz共x,y兲 = lim
y→0+
y→0+
⫻
冋
G共x兲
2␲
冕
y→0+
G共x兲
2␲
冕
冕 冕冋
␾共t兲
c
d
␾共t兲ei␰tdt e␭1ye−ix␰d␰
c
⬁
−⬁
y→0+
册
−⬁
d
␭ 1共 ␤ , ␰ 兲
e ␭1 y
共− i␰兲关1 + ␳共␤, ␰兲兴
d
=
K共␰,y兲ei␰共t−x兲d␰dt,
␾共t兲
c
冋冉 冊
ᐉ⬘
2ᐉ
+
册
3ᐉ2␤2
i␰
−1
8
兩␰兩
共48兲
冕
关i␰兩␰兩e
兴e
i共t−x兲␰
−⬁
冕
y→0+
关兩␰兩e−兩␰兩y兴ei共t−x兲␰d␰ ——→
−⬁
冕
⬁
−2
共t − x兲2
共49兲
i
−⬁
册
共50兲
关1e
−兩␰兩 y
兴e
i共t−x兲␰
共51兲
冕
⬁
N共␰,0兲ei共t−x兲␰d␰
共56兲
0
Equation 共55a兲 is a Fredholm integral equation of the second kind
with the cubic and Cauchy singular kernels.
7
Numerical Solution
To numerically solve the unknown slope function ␾共t兲 in Eq.
共55a兲, we follow the general procedure outlined in the Part I paper. For the sake of clarity and completeness, each step is presented below and particularized to the problem at hand 共see Figs.
1 and 2兲.
7.1
Normalization. By the change of variables
t = 共d − c兲s/2 + 共c + d兲/2 and x = 共d − c兲r/2 + 共c + d兲/2 共57兲
the crack surface 共c , d兲 can be converted into 共−1 , 1兲, and the
main integral equation 共55a兲 can be rewritten in normalized form
冕再
冎
+
1 − 3共ᐉ/a兲2共a␤兲2/8 − 关共ᐉ⬘/a兲/共2ᐉ/a兲兴2
+ K共r,s兲 ⌽共s兲ds
s−r
+
共a␤兲共ᐉ⬘/a兲
ᐉ⬘/a
P共r兲 −␤关ar+共c+d兲/2兴
⌽共r兲 =
e
,
⌽⬘共r兲 +
2
2
G0
−⬁
with ␦共x兲 being the Dirac delta function, we obtain the limit
兩r兩 ⬍ 1
where
⌽共r兲 = ␾共ar + 共c + d兲/2兲,
P共r兲 = p共ar + 共c + d兲/2兲
K共r,s兲 = ak共ar + 共c + d兲/2, as + 共c + d兲/2兲
共52兲
共59兲
共60兲
共61兲
and k共x , t兲 is described by Eq. 共56兲.
Note that in Eq. 共58兲, G共x兲 has been written as
G共x兲 = G0e␤x = G0e␤关共共d−c兲/2兲r+关共c+d兲/2兴兴 = G0e共a␤兲r共ea␤兲共共d+c兲/共d−c兲兲
共53兲
where a␤ = 共d − c兲␤ / 2 and 共d + c兲 / 共d − c兲 are two dimensionless
quantities. Together with the terms ᐉ / a and ᐉ⬘ / a that appear in
Eq. 共58兲, the following dimensionless parameters are defined:
ᐉ̃ = ᐉ/a,
y→0+
d␰ ——→ 2␲␦共t − x兲
Journal of Applied Mechanics
共55b兲
a = 共d − c兲/2 = half of the crack length
+
y→0
兩␰兩 −兩␰兩y i共t−x兲␰
−2
e
e
d␰ ——→
␰
t−x
⬁
or x ⬎ d
共58兲
−⬁
冕冋
冕
x⬍c
1
y→0+
关i␰e−兩␰兩y兴ei共t−x兲␰d␰ ——→ 2␲␦⬘共t − x兲
⬁
冎
y→0+
4
d␰ ——→
共t − x兲3
⬁
− 2ᐉ2
3␤ᐉ2
1 − 3ᐉ2␤2/8 − 关ᐉ⬘/共2ᐉ兲兴2
3 −
2 +
共t − x兲
2共t − x兲
t−x
1 = − 2共ᐉ/a兲2 3共ᐉ/a兲2共a␤兲
−
␲ −1 共s − r兲3
2共s − r兲2
Note that the real and the imaginary parts of K⬁共␰ , 0兲 given in Eq.
共49兲 are even and odd functions of ␰, respectively.
In view of the following distributional convergence,
⬁
c
c⬍x⬍d
共55a兲
k共x,t兲 =
3␤ᐉ2
ᐉ⬘
ᐉ ⬘␤
i␰ +
兩␰兩 +
2
2
2
2
冕再
d
ᐉ⬘
␤ᐉ⬘
␾⬘共x兲 +
␾共x兲 = p共x兲,
2
2
共47兲
where K⬁共␰ , y兲 is the nonvanishing part of the asymptotic expansion of K共␰ , y兲 in the powers of ␰, as 兩␰兩 → ⬁. When y is set to be
0, we have
K⬁共␰,0兲 = − iᐉ2兩␰兩␰ −
冎
3␤ᐉ2
1 − 3␤2ᐉ2/8 − 关ᐉ⬘/共2ᐉ兲兴2
− 2ᐉ2
3 −
2 +
共t − x兲
2共t − x兲
t−x
where 兰= denotes the finite-part integral 关11兴, and the regular kernel k共x , t兲 is given by
Equation 共46兲 is an expression for the stress ␴yz共x , y兲 in the limit
form of y → 0+, which is valid for x 苸 共−⬁ , ⬁兲. Note that for the
共first兲 boundary condition in Eq. 共16兲, ␴yz共x , 0兲 = p共x兲, x is restricted to the crack surface 共c , d兲. It is this boundary condition
that leads to the governing hypersingular integrodifferential equation 共see Eq. 共55a兲 below兲. However, when SIFs are calculated, x
takes values outside of 共c , d兲, and the integral 共46兲 is not singular
共see Eq. 共55b兲 below兲.
We split K共␰ , y兲 into the singular part K⬁共␰ , y兲 and the nonsingular part N共␰ , y兲:
K共␰,y兲 = K⬁共␰,y兲 + N共␰,y兲
d
+ k共x,t兲 ␾共t兲dt,
共46兲
with
−兩␰兩 y
1
␲
−⬁
␭ 1共 ␤ , ␰ 兲
e ␭1 y
共− i␰兲关1 + ␳共␤, ␰兲兴
G=
␲ c
+ k共x,t兲 ␾共t兲dt +
⬁
−⬁⬍x⬍⬁
+
=
册
−⬁
c
共45兲
冕 冕
K共␰,y兲 =
K⬁共␰,y兲ei␰共t−x兲d␰␾共t兲dt
lim
⫻ei共t−x兲␰d␰dt
G
= lim
+
2
␲
y→0
⬁
d
␭ 1共 ␤ , ␰ 兲
1
共− i␰兲关1 + ␳共␤, ␰兲兴
= lim
冕冕
冕再
⬁
共54兲
ᐉ˜⬘ = ᐉ⬘/a and ˜␤ = a␤
共62兲
They will be used in the numerical implementation and results.
7.2 Representation of the Density Function. To proceed
with the numerical approximation, a representation of ⌽共s兲 is choNOVEMBER 2008, Vol. 75 / 061015-5
sen so that one can evaluate the hypersingular and the Cauchy
singular integrals by finite part and Cauchy principal value, respectively 共see Refs. 关11,12兴兲. For the cubic hypersingular integral
equation 共58兲, the solution ⌽共s兲 can be represented as
⌽共s兲 = g共s兲冑1 − s
⬁
− 2ᐉ̃2
冋
n=1
where g共⫾1兲 is finite and g共⫾1兲 ⫽ 0 关5兴. By finding numerical
solution for g共s兲, one can find the approximate solution for ⌽共s兲.
The representation 共63兲 of ⌽共s兲 is suggested by the following
asymptotic behavior of the solution around the crack tips:
strains ⬃ 冑r,
Displacements ⬃ r3/2,
⬁
+
−
reported in Refs. 关1,5兴.
⬁
g共s兲 =
兺 a T 共s兲
n n
or g共s兲 =
兺 A U 共s兲
n
n
⌽共s兲 = 冑1 − s2
兺 A U 共s兲
n
n
共66兲
sin关共n + 1兲cos−1共s兲兴
,
sin关cos−1共s兲兴
n = 0,1,2, . . .
共67兲
Note that the single-valuedness condition 共41兲 or, equivalently,
1
⌽共s兲ds = 0 implies
兰−1
共68兲
A0 = 0
Thus, the running index n in Eq. 共66兲 can start from 1 instead of
0.
7.4 Evaluation of the Derivative of the Density Function.
The term ⌽⬘共r兲 in Eq. 共58兲 is evaluated using the expansion 共66兲
and the fact that
n+1
d
关Un共r兲冑1 − r2兴 = −
冑1 − r2 Tn+1共r兲,
dr
冋
⬁
冑1 − r2 兺 AnUn共r兲
n=0
冉 冊 册兺 冕
冕冑
n=1
Un共s兲冑1 − s2
An
–
ds
␲ −1
s−r
1
1 − s2Un共s兲K共r,s兲ds
−1
⬁
兺
2冑1 − r2 n=1
G共r兲
⬁
2
1
ᐉ̃⬘
⬁
共n + 1兲AnTn+1共r兲 +
ᐉ̃⬘˜␤
冑1 − r2 AnUn共r兲
2
n=1
兺
兩r兩 ⬍ 1
,
共71兲
We have used the running index n that starts from 1 共see Eq. 共68兲兲.
7.6 Evaluation of Singular and Hypersingular Integrals.
Formulas for evaluating singular integral terms in Eq. 共71兲 are
listed below:
冕
Un共s兲冑1 − s2
1
–
ds = − Tn+1共r兲,
␲ −1
s−r
1
冕
1 = Un共s兲冑1 − s2
ds = − 共n + 1兲Un共r兲,
␲ −1 共s − r兲2
兩r兩 ⬍ 1,
n 苸 Z+ 共72兲
1
兩r兩 ⬍ 1,
n 苸 Z+
共73兲
n艌0
共69兲
冕
1 = Un共s兲冑1 − s2
ds
␲ −1 共s − r兲3
1
冦
册
=
兩r兩 ⬍ 1
共74兲
⬁
− ᐉ̃2
An关共n2 + n兲Un+1共r兲 − 共n2 + 3n + 2兲Un−1共r兲兴
2共1 − r2兲 n=1
兺
冋
⬁
冉 冊册
3ᐉ̃2˜␤2
3˜␤ᐉ̃2
ᐉ̃⬘
−
共n + 1兲AnUn共r兲 − 1 −
2 n=1
8
2ᐉ̃
兺
⬁
⫻
⬁
兺
AnTn+1共r兲 +
n=1
共70兲
061015-6 / Vol. 75, NOVEMBER 2008
冧
7.7 Evaluation of Nonsingular Integral. By combining all
the results obtained so far in the numerical approximation, one
may rewrite Eq. 共71兲 in the following form:
共n + 1兲AnTn共r兲
7.5 Formation of the Linear System of Equations. The An
coefficients are determined by transforming the integral equation
共55a兲 into a system of linear algebraic equations in terms of the
An’s. By replacing ⌽共s兲 in Eq. 共58兲 by the representation 共66兲, and
using Eq. 共70兲, one obtains the governing integral equation in
discretized form
n=0
The details of the calculation can be found in Ref. 关13兴.
⬁
冑1 − r2 兺
n=0
−1
− 1,
= 共n2 + n兲Un+1共r兲 − 共n2 + 3n + 2兲Un−1共r兲
, n艌1
4共1 − r2兲
+
d
dr
兺
1
⬁
where Un共s兲 is defined, as usual, by
⌽⬘共r兲 =
An
P共r兲
共65兲
n=0
Thus
冕
n=0
The coefficients an’s or An’s are numerically determined by the
collocation method. As shown by Chan et al. 关6兴, the two expansions should lead to consistent numerical results. In this paper, the
expansion in terms of Un共s兲 is adopted, i.e.,
Un共s兲 =
=
⬁
n=0
兺␲
n=1
stresses ⬃ r−3/2 共64兲
7.3 Chebyshev Polynomial Expansion. In view of Eq. 共63兲
and the fact that 兵Un共s兲其 are orthogonal on 共−1 , 1兲 with respect to
the weight function 冑1 − s2, g共s兲 can be most naturally expressed
in terms of the Chebyshev polynomials of the second kind Un共s兲.
However, the orthogonality is not required in the implementation
of the numerical procedures, and either Chebyshev polynomials of
the first kind Tn共s兲 or of the second kind Un共s兲 may be employed
in the approximation, i.e.,
⬁
1
3ᐉ̃2˜␤2
ᐉ̃⬘
−
+ 1−
8
2ᐉ̃
共63兲
2
兺
冕
3ᐉ̃2˜␤
An = Un共s兲冑1 − s2
An = Un共s兲冑1 − s2
ds
−
ds
␲ −1 共s − r兲3
2 n=1 ␲ −1 共s − r兲2
兺
n=1
⬁
An
␲
冕冑
2
1
1 − s2Un共s兲K共r,s兲ds
−1
⬁
ᐉ̃⬘˜␤
冑1 − r2 AnUn共r兲
共n
+
1兲A
T
共r兲
+
−
n
n+1
2
2冑1 − r2 n=1
n=1
ᐉ̃⬘
=
P共r兲
G共r兲
,
兺
兩r兩 ⬍ 1
兺
共75兲
The regular kernel in Eq. 共75兲 is actually a double integral, i.e.,
Transactions of the ASME
冕冑
1
冕冑
冕冑
冕
ᐉKIII共d兲 = lim 2冑2␲共x − d兲共x − d兲␴yz共x,0兲
1
1 − s2Un共s兲K共r,s兲ds =
−1
1 − s2Un共s兲ak共ar,as兲ds
x→d+
−1
1
=
= lim 2
1 − s2Un共s兲
r→1+
−1
⬁
⫻
⫻␴yz
aN共␰,0兲sin关a␰共s − r兲兴d␰ds
共76兲
+ 共c + d兲/2,0兲
冑2␲共x − d兲␴yz共x,0兲
冉 冊冕
− 2ᐉ2
␲a2
共x ⬎ d兲
KIII共d兲 = 2冑2␲a
共77兲
N
x→c−
⫻
共x ⬍ c兲
共78兲
冕
1
−1
1
␲
冕
冉
兩r兩
Un共s兲冑1 − s2
ds = − r − 冑r2 − 1
s−r
r
1
−1
冉
冉
冕
1
−1
=
冉
兩r兩
−1
共n + 1兲 r − 冑r2 − 1
2
r
冤冉
⫻ n 1−
兩r兩
冑r 2 − 1
冊
冊
r−
2
+
冊
共84兲
冉 冊
− 2ᐉ
G0e␤d lim 共r − 1兲3/2
a
r→1+
冉
兩r兩
冑r 2 − 1
冊
r−
2
+
⬁
冊
n−1
兩r兩 2
冑r − 1
r
冑r 2 − 1 3
冥
= 冑␲aG0e␤d共ᐉ/a兲 兺 共n + 1兲An
An
共85兲
兩r兩 2
冑r − 1
r
共冑r2 − 1兲3
冥
⬁
冊
n艌0
,
Similarly,
共86兲
n=0
共80兲
n−1
n 艌 0 共81兲
共x ⬎ d兲
共82兲
ᐉKIII共c兲 = lim 2冑2␲共c − x兲共c − x兲␴yz共x,0兲
共x ⬍ c兲
共83兲
x→d+
Journal of Applied Mechanics
共r ⬎ 1兲
KIII共c兲 = 冑␲aG0e␤c共ᐉ/a兲 兺 共− 1兲n共n + 1兲An
ᐉKIII共d兲 = lim 2冑2␲共x − d兲共x − d兲␴yz共x,0兲
Thus,
⌽共s兲
ds
共s − r兲3
兩r兩
− 共n + 1兲
r − 冑r2 − 1
2
r
冤冉
n 艌 0 共79兲
,
The highest singularity in Eqs. 共79兲–共81兲 appears in the last
term in Eq. 共81兲 and it behaves like 共r2 − 1兲−3/2 as r → 1+ or r →
−1−. Motivated by such asymptotic behavior, we define the SIFs
for strain-gradient elasticity as
x→c−
−1
⫻ n 1−
n+1
n
兩r兩 2
冑r − 1 ,
r
Un共s兲冑1 − s2
ds
共s − r兲3
1
n=0
兩r兩
Un共s兲冑1 − s
ds = − 共n + 1兲 1 − 2
冑r − 1
共s − r兲2
2
⫻ r−
1
␲
冊
兺
n=0
After normalization, the crack surfaces are located in the interval
共−1 , 1兲. The density function ⌽共t兲 is expanded in terms of Chebyshev polynomials of the second kind Un, which, when substituted
into Eq. 共55b兲, give rise to the following formulas for 兩r兩 ⬎ 1 共see
Ref. 关13兴兲:
1
␲
共r ⬎ 1兲
r→1+
and
C
共c兲 = lim 冑2␲共c − x兲␴yz共x,0兲
KIII
共r ⬎ 1兲
By using Eq. 共66兲 in conjunction with Eq. 共81兲, we obtain from
Eq. 共84兲
In classical LEFM, the SIFs are defined by
x→d+
冊
= 2a冑2␲a lim 冑共r − 1兲共r − 1兲G0ea␤re␤共d+c兲/2
⫻
C
KIII
共d兲 = lim
d−c
c+d
r+
,0
2
2
r→1+
The Fourier sine transform in Eq. 共76兲 can be efficiently evaluated
by applying fast Fourier transform 共FFT兲 关14兴. The integral along
关−1 , 1兴 can be readily obtained by the Gaussian quadrature
method 关15兴.
Stress Intensity Factors
冉
册
c+d
d−c
r+
− d 共ar − a兲
2
2
2␲
= 2a冑2␲a lim 冑共r − 1兲共r − 1兲␴yz共共d − c兲r/2
0
8
冑 冋冉 冊
共x ⬎ d兲
9
Results and Discussion
The numerical results include crack surface displacements,
strains, stresses, and SIFs.
9.1 Crack Surface Displacements. The 共normalized兲 crack
surface displacements shown in Figs. 4–8 are obtained by integrating the slope function 共see Eq. 共87兲 below兲. Figure 4 shows a
full normalized crack sliding displacement profile for a homogeneous medium 共˜␤ = 0兲 with the strain-gradient effect. The crack
profile in Fig. 4 is symmetric because the material is homogeneous. Figures 5 and 6 are for classical LEFM. Figures 7 and 8 are
for the strain-gradient theory. As ␤ ⬍ 0, the material has larger
shear modulus at the left side of the crack than at the right side,
and thus the material is stiffer on the left and more compliant on
the right, as shown in Figs. 5 and 7. Similarly, Figs. 6 and 8
illustrate the case of ␤ ⬎ 0 and confirm that the material is stiffer
on the right and more compliant on the left. The variation of the
shear modulus destroys the symmetry of the displacement profiles. The most prominent feature is the cusping phenomena
around the crack tips, as shown in Figs. 4, 7, and 8. The difference
between Figs. 5 and 6 and Figs. 7 and 8 is the cusp at the crack
tips. In Figs. 5 and 6, one may observe that the profiles have a
tangent line with infinite slope at the crack tips, which is a common crack behavior exhibited in the classical LEFM. However,
such is not the case in gradient theory as evidenced by the numerical results shown.
NOVEMBER 2008, Vol. 75 / 061015-7
0.6
1.4
0.4
1.2
0.2
1
y
G(x) = G eβ x
0
x
w(x,0)/(ap0 / G0 )
Normalized displacement, w(x,0)
1.6
0
0.8
−0.2
0.6
−0.4
0.5
0.4
−0.6
β = 0.1
0.25
1.0
1.5
0.2
2.0
−1
0
0
0
1
Normalized crack length, x
Fig. 4 Full crack displacement profile for homogeneous mate˜ = 0… under uniform crack surface shear loading ␴ „x , 0…
rial „␤
yz
= −p0 with choice of „normalized… 艎̃ = 0.2 and 艎̃⬘ = 0
9.2 Strains. We have used the strain-like field, ␾共x兲 共the slope
function兲, as the unknown density function in our integral equation formulation. The normalized version, ⌽共x兲, with various ᐉ̃ is
plotted in Fig. 9. Note that ⌽共⫾1兲 = 0 while in classical LEFM,
⌽共⫾1兲 = ⫾ ⬁. The vanishing slope is equivalent to the cusping at
the crack tips. The 共normalized兲 crack displacement profile w共r , 0兲
can be obtained by
w共r,0兲 =
冕
r
⌽共s兲ds =
−1
冕
r
−1
N
冑1 − s2 兺 AnUn共s兲ds
共87兲
n=0
As ᐉ decreases, ⌽共x兲 seems to converge to the slope function of
the classical LEFM case in the region away from the crack tips,
where ⌽共x兲 is very different from its classical counterpart near the
crack tips.
9.3 Stresses. Similar to classical LEFM, the stress ␴yz共x , 0兲
diverges as x approaches the crack tips along the ligament 共Fig.
0.5
1
1.5
x/a
2
Fig. 6 Classical LEFM, i.e., 艎̃ = 艎̃⬘ \ 0. Crack surface displacement in an infinite nonhomogeneous plane under uniform
crack surface shear loading ␴yz„x , 0… = −p0 and shear modulus
G„x… = G0e␤x. Here, a = „d − c… / 2 denotes the half crack length.
10兲. Moreover, the sign of the stress changes, and as ᐉ decreases,
the interior part 共i.e., the region apart from the two crack tips兲 of
␴yz共x , 0兲 seems to converge to the solution of classical LEFM.
The finding of the negative near-tip stress is consistent with the
results by Zhang et al. 关8兴 who also investigated a Mode-III crack
in elastic materials with strain-gradient effects; this negative stress
may be considered as a necessity for the crack surface to reattach
near the tips. The point worth noting here is that not all straingradient theories possess the negative-stress feature near the crack
tips. For instance, the strain-gradient elasticity theory for cellular
materials 关16兴 and elastic-plastic materials with strain-gradient effects 关17兴, which fall within the classical couple-stress theory
framework, shows a positive-stress singularity near the crack tip.
On the other hand, the strain-gradient theory proposed by Fleck
and Hutchinson 关18兴, which does not fall into the above framework, predicts a compressive stress near the tip of a tensile
Mode-I crack 关19,20兴.
14
y
β = −2
10
−1.8
12
10
βx
w(x,0)/(ap0 / G0 )
0
w(x,0)/(ap / G )
12
G(x)=G0e
8
−1.6
0
x
6
−1.4
−1.2
4
0
0
0
1
x/a
1.5
2
Fig. 5 Classical LEFM, i.e., 艎̃ = 艎̃⬘ \ 0. Crack surface displacement in an infinite nonhomogeneous plane under uniform
crack surface shear loading ␴yz„x , 0… = −p0 and shear modulus
G„x… = G0e␤x. Here, a = „d − c… / 2 denotes the half crack length.
061015-8 / Vol. 75, NOVEMBER 2008
6
β = −1.5
4
β = −1.0
β = −0.5
β = −0.1
0
−0.4
0.5
8
2
−0.8
2
β = −2.0
G(x)=G0eβx
0
0.4
0.8
x/a
1.2
1.6
2.0
Fig. 7 Crack surface displacement in an infinite nonhomogeneous plane under uniform crack surface shear loading
␴yz„x , 0… = −p0 and shear modulus G„x… = G0e␤x with choice of
„normalized… 艎̃ = 0.10 and 艎̃⬘ = 0.01. Here, a = „d − c… / 2 denotes the
half crack length.
Transactions of the ASME
G(x)=G0eβx
1.0
0.8
β = 0.1
0.6
β = 0.5
0.4
0.001
0.005
20
10
0.01
0.05
0.1
0
−20
β = 2.0
0
0
0.0005
−10
β = 1.0
0.2
0.0001
30
σyz(x/a,0)/G0
w(x,0)/(ap0 / G0 )
1.2
0.4
0.8
1.2
x/a
1.6
−30
2.00
2.0
Fig. 8 Crack surface displacement in an infinite nonhomogeneous plane under uniform crack surface shear loading
␴yz„x , 0… = −p0 and shear modulus G„x… = G0e␤x with choice of
„normalized… 艎̃ = 0.10 and 艎̃⬘ = 0.01. Here, a = „d − c… / 2 denotes the
half crack length.
2.02
2.04
2.06
x/a
2.10
2.08
˜ = 0.5, 艎̃⬘
Fig. 10 Stress ␴yz„x / a , 0… / G0 along the ligament for ␤
= 0, and various 艎̃. Crack surface „c , d… = „0 , 2… located in an infinite nonhomogeneous plane is assumed to be under uniform
crack surface shear loading ␴yz„x , 0… = −p0 and shear modulus
G„x… = G0e␤x. Here, a = „d − c… / 2 denotes the half crack length.
N
KIII共c兲
9.4 Stress Intensity Factors. Besides using ␾共x兲 as the unknown density function, one may also use displacement w共x兲 to be
the unknown in the formulation of the integral equation 共see ApC
pendix兲. By rewriting KIII
in terms of the coefficients in the expansion for w, one obtains 共see Ref. 关6兴兲 the following.
•
classical LEFM
40
1.4
G0冑␲共d − c兲/2
=e
KIII共d兲
G0冑␲共d − c兲/2
␤c
兺 共− 1兲 共n + 1兲A
n
共89兲
n
0
N
= e ␤d
兺 共n + 1兲A
n
0
With Tn expansion,
N
C
共c兲
KIII
G0冑␲共d − c兲/2
= e ␤c
共88兲
0
N
C
KIII
共d兲
G0冑␲共d − c兲/2
•
兺
共− 1兲nan
= e ␤d
兺a
n
0
With Un expansion,
Table 3 contains the 共normalized兲 SIFs for the case of classical
LEFM by using both Tn and Un expansions 关see Eqs. 共63兲 and
共65兲兴. The SIFs in Table 3 have been obtained by using Eqs. 共88兲
and 共89兲, and they are close to the results reported by Erdogan 关7兴.
Table 4 contains the SIFs for strain-gradient elasticity at ᐉ̃
= 0.1 and ᐉ̃⬘ = 0.01. One observes that the dependence of KIII共c兲
and KIII共d兲 is similar to the classical case reported in Table 3.
10
8
Concluding Remarks
This paper has shown that the integral equation method is an
effective means of formulating crack problems for a FGM considering strain-gradient effects. The theoretical framework and numerical analysis has been utilized to solve antiplane shear crack
problems in FGMs by using Casal’s continuum. The behavior of
classical LEFM
0.01
6
0.05
4
Table 3 Normalized SIFs for Mode-III crack problem in a FGM
„艎 = 艎⬘ \ 0…
φ(x/a)
0.1
2
0.2
Un representation
0.1
0
␤
0.2
−2
0.05
0.01
−4
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
x/a
Fig. 9 Strain ␾„x / a… along the crack surface „c , d… = „0 , 2… for
␤˜ = 0.5, 艎̃⬘ = 0, and various 艎̃ in an infinite nonhomogeneous
plane under uniform crack surface shear loading ␴yz„x , 0… =
−p0 and shear modulus G„x… = G0e␤x. Here, a = „d − c… / 2 denotes
the half crack length.
Journal of Applied Mechanics
共 兲
d−c
2
−2.00
−1.50
−1.00
−0.50
−0.10
0.00
0.10
0.50
1.00
1.50
2.00
KIII共c兲
KIII共d兲
Tn representation
KIII共c兲
KIII共d兲
p0冑␲共d − c兲 / 2
p0冑␲共d − c兲 / 2
p0冑␲共d − c兲 / 2
p0冑␲共d − c兲 / 2
1.21779
1.17801
1.14307
1.09036
1.02289
1.00000
0.97312
0.85676
0.72845
0.63007
0.55672
0.55672
0.63007
0.72845
0.85676
0.97312
1.00000
1.02289
1.09036
1.14307
1.17801
1.21779
1.21779
1.17801
1.14307
1.09036
1.02289
1.00000
0.97312
0.85676
0.72845
0.63007
0.55672
0.55672
0.63007
0.72845
0.85676
0.97312
1.00000
1.02289
1.09036
1.14307
1.17801
1.21779
NOVEMBER 2008, Vol. 75 / 061015-9
Table 4 Normalized generalized SIFs for a Mode-III crack at 艎̃
˜
= 0.1, 艎̃ = 0.01, and various values of ␤
KIII共c兲
˜␤
p0冑␲共d − c兲 / 2
1.23969
1.12585
1.04849
0.96814
0.94385
0.91677
0.80277
0.67637
0.49938
0.49938
0.67600
0.80248
0.91658
0.94385
0.96828
1.04854
1.12584
1.23969
−2.00
−1.00
−0.50
−0.10
0.00
0.10
0.50
1.00
2.00
•
KIII共d兲
p0冑␲共d − c兲 / 2
2. Classical LEFM, Nonhomogeneous Materials 共G ⬅ G共y兲
= G 0e ␥ y 兲
•
冕冋
G.H.P. would like to thank the support from the National Science Foundation under Grant No. CMS 0115954 共Mechanics and
Materials Program兲 and from the NASA-Ames “Engineering for
Complex Systems Program,” and the NASA-Ames Chief Engineer 共Dr. Tina Panontin兲 through Grant No. NAG 2-1424.
A.C.F. acknowledges the support from the USA National Science Foundation 共NSF兲 through Grant No. DMS-9971322 and UC
Davis Chancellor’s Fellowship.
Y.-S.C. thanks the support from the Applied Mathematical Sciences Research Program of the Office of Mathematical, Information, and Computational Sciences, U.S. Department of Energy,
under Contract No. DE-AC05-00OR22725 with UT-Battelle,
LLC; he also thanks the grant 共No. W911NF-05-1-0029兲 from the
U.S. Department of Defense, Army Research Office. Organized
Research Award 共2007兲 from University of Houston-Downtown is
also acknowledged by Y.-S.C.
Appendix: Hierarchy of Governing Integral Equations
In this appendix, we list the type of the physical problem under
antiplane shear loading, its governing PDE, and integral equation
associated with the choice of the density function. The corresponding references in the literature are also provided.
Erdogan and Ozturk 关22兴 have investigated this problem as
bonded nonhomogeneous materials with an interface cut.
3. Classical LEFM, Nonhomogeneous Materials 共G ⬅ G共x兲
= G 0e ␤x兲
•
•
•
冕冋
•
d
G0 = ␾共t兲
dt = p共x兲,
␲ c 共t − x兲2
c⬍x⬍d
共A2兲
Many standard textbooks have covered the Laplace equation 共see,
for example, Ref. 关21兴兲.
061015-10 / Vol. 75, NOVEMBER 2008
册
c⬍x⬍d
共A4兲
冕冋
册
d
1
␤
G共x兲 =
+ N共x,t兲 ␾共t兲dt
2 +
␲
共t
−
x兲
2共t
− x兲
c
= p共x兲,
c⬍x⬍d
共A5兲
The regular kernels Ñ共x , t兲 in Eq. 共A4兲 and N共x , t兲 in Eq. 共A5兲 can
be found in Ref. 关6兴. Erdogan 关7兴 has studied this problem for
bonded nonhomogeneous materials.
4. Gradient Elasticity, Homogeneous Materials 共G ⬅ G0兲
•
•
PDE: Helmholtz–Laplace equation 共1 − ᐉ2ⵜ2兲ⵜ2w共x , y兲
= 0.
Integral equation with the density function ␾共x兲
= ⳵w共x , 0兲 / ⳵x:
冕再
d
冎
1 = − 2ᐉ2 1 − 共ᐉ⬘/ᐉ兲2/4
+
+ K0共t − x兲 ␾共t兲dt
␲ c 共t − x兲3
t−x
共A1兲
Integral equation with the density function ␾共x兲
= w共x , 0兲:
c
1
␤
+ log兩t − x兩 + Ñ共x,t兲 ␾共t兲dt
t−x 2
Integral equation with the density function ␾共x兲
= w共x , 0兲:
d
c⬍x⬍d
d
= p共x兲,
PDE: Laplace equation ⵜ2w共x , y兲 = 0.
Integral equation with the density function ␾共x兲
= ⳵w共x , 0兲 / ⳵x:
冕
PDE: Perturbed Laplace equation 共ⵜ2 + ␤共⳵ / ⳵x兲兲w共x , y兲
= 0.
Integral equation with the density function ␾共x兲
= ⳵w共x , 0兲 / ⳵x:
G共x兲
–
␲
1. Classical LEFM, Homogeneous Materials 共G ⬅ G0兲
冕
−a⬍x⬍a
共A3兲
Acknowledgment
␾共t兲
G0
dt = p共x兲,
–
␲ ct − x
册
a
1
G0
–
+ K̃␥共x,t兲 ␾共t兲dt = p共x兲,
␲ −a t − x
the solution around the crack tips is affected by the strain-gradient
theory, and not by the gradation of the materials. Also, the integral
equation formulation has been found to be an adequate tool for
implementing the numerical procedures and to assess physical
quantities such as crack surface displacements, strains, stresses,
and SIFs. Further experiments are needed for justifying the physical aspects of the method. Future work includes extension of the
theory to Mode-I crack problems.
•
•
PDE: Perturbed Laplace equation 共ⵜ2 + ␥共⳵ / ⳵y兲兲w共x , y兲
= 0.
Integral equation with the density function ␾共x兲
= ⳵w共x , 0兲 / ⳵x:
+
ᐉ⬘
p共x兲
␾⬘共x兲 =
2
G0
共A6兲
Fannjiang et al. 关5兴 have studied Eq. 共A6兲 in detail.
5. Gradient Elasticity, Nonhomogeneous Materials 共G ⬅ G共y兲
= G 0e ␥ y 兲
•
•
PDE: 共1 − ␥ᐉ2共⳵ / ⳵y兲 − ᐉ2ⵜ2兲共ⵜ2 + ␥共⳵ / ⳵y兲兲w共x , y兲 = 0.
Integral equation with the density function ␾共x兲
= ⳵w共x , 0兲 / ⳵x:
Transactions of the ASME
冕再
a
− 2ᐉ2 5ᐉ2␥2/8 + ᐉ⬘␥/4 + 1 − 共ᐉ⬘/ᐉ兲2/4
G0 =
+
␲ −a 共t − x兲3
t−x
冎
关3兴
+ k␥共x,t兲 ␾共t兲dt + G共ᐉ⬘/2 + ᐉ2␥兲␾⬘共x兲
= p共x兲,
兩x兩 ⬍ a
共A7兲
This is the Part I paper by Paulino et al. 关1兴.
6. Gradient Elasticity, Nonhomogeneous Materials 共G ⬅ G共x兲
= G 0e ␤x兲
• PDE: 共1 − ␤ᐉ2共⳵ / ⳵x兲 − ᐉ2ⵜ2兲共ⵜ2 + ␤共⳵ / ⳵x兲兲w共x , y兲 = 0.
• Integral equation with the density function ␾共x兲
= ⳵w共x , 0兲 / ⳵x:
冕再
关2兴
关4兴
关5兴
关6兴
关7兴
关8兴
d
3␤ᐉ2
1 − 3ᐉ2␤2/8 − 关ᐉ⬘/共2ᐉ兲兴2
1 = − 2ᐉ2
−
+
␲ c 共t − x兲3 2共t − x兲2
t−x
冎
+ k共x,t兲 ␾共t兲dt +
= p共x兲/G,
•
␤ᐉ⬘
ᐉ⬘
␾⬘共x兲 +
␾共x兲
2
2
关10兴
关11兴
c⬍x⬍d
This is the main governing integral equation 共55a兲.
Integral equation with the density function ␾共x兲
= w共x , 0兲:
冕冦
d
3ᐉ2␤
1 = − 6ᐉ2
+
4 −
␲ c 共t − x兲
共t − x兲3
冋 冉 冊册
1−
冉 冊
ᐉ⬘ 2 3ᐉ2␤2
−
8
2ᐉ
2
共t − x兲
冧
␤
ᐉ␤
ᐉ⬘
1−
+
16
2
2ᐉ
+
+ k̃共x,t兲 ␾共t兲dt
t−x
2
+
2 3
冋冉 冊 册
1 ᐉ⬘
ᐉ⬘
ᐉ ⬘␤
␾⬘共x兲 −
␾⬙共x兲 −
2
2
ᐉ 2ᐉ
p共x兲
,
=
G共x兲
3
+
关9兴
ᐉ⬘
␾共x兲
8ᐉ2
c⬍x⬍d
关12兴
关13兴
关14兴
关15兴
关16兴
关17兴
关18兴
关19兴
关20兴
关21兴
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