Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 4, pp. 1319-1327, Warsaw 2016
DOI: 10.15632/jtam-pl.54.4.1319

A MODE-III CRACK WITH VARIABLE SURFACE EFFECTS

Xu Wang

School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai, China

e-mail: xuwang@ecust.edu.cn

Peter Schiavone

University of Alberta, Department of Mechanical Engineering, Edmonton, Canada

e-mail: p.schiavone@ualberta.ca

We study the contribution of variable surface effects to the antiplane deformation of a
linearly elasticmaterial with amode-III crack. The surface elasticity is incorporated using a
modified version of the continuumbased surface/interfacemodel ofGurtin andMurdoch. In
our discussion, the surfacemoduli are not constant but vary along the crack surfaces. Using
Green’s function method, the problem is reduced to a single first-order Cauchy singular
integro-differential equation, which is solved numerically using Chebyshev polynomials and
a collocation method. Our results indicate that the gradient of the surface shear modulus
exerts a significant influence on the crack opening displacement and on the singular stress
field at the crack tips.

Keywords: surface elasticity, variable surface moduli, mode-III crack, Cauchy singular
integro-differential equation

1. Introduction

The analysis of deformation of an elastic solid incorporating a crack is critical for the under-
standing of failuremodes and in the general stress analysis of engineeringmaterials. Traditional
modeling via the use of linear elastic fracture mechanics (LEFM) ignores the contributions of
surface energies, surface tension and surface stresses. The high surface area to volume ratio
present at the nanoscale dictates that any continuum-basedmodel of deformation should incor-
porate the separate contribution of surface mechanics (Sharma and Ganti, 2004). Recently, the
continuum-based surface/interface model proposed by Gurtin, Murdoch and co-workers (Gur-
tin and Murdoch, 1975; Gurtin et al., 1998) has been incorporated in the analysis of several
typical crack problems (see for example, Kim et al., 2010, 2011a,b; Antipov and Schiavone,
2011; Wang, 2015; Wang and Schiavone, 2015, 2016). It was first proved by Walton (2012) and
later corroborated by Kim et al. (2013) that the contribution of surface elasticity (based on
the Gurtin-Murdoch model) to LEFM would, at best, reduce the classical strong square root
singularity to a weaker logarithmic singularity.

TheGurtin-Murdoch surface elasticity model essentially models amaterial surface as a thin
elastic membrane (of separate elasticity) perfectly bonded to the surrounding bulk material
(see, for example, Steigmann andOgden, 1997; Chen et al., 2007; Antipov and Schiavone, 2011;
Markenscoff and Dundurs, 2014). In recent studies, the incorporation of surface elasticity into
LEFM models has been confined to the simple case in which the surface moduli are constant
along the crack surfaces (Kim et al., 2010, 2011a,b; Antipov and Schiavone, 2011; Wang, 2015;
Wang and Schiavone, 2015, 2016).

Thiswork aims to study, for the first time, the effects of variable surfacemoduli in a classical
mode-III crack problem arising in the antiplane shear deformation of a linearly isotropic elastic



1320 X.Wang, P. Schiavone

solid. Specifically, the corresponding surface shear modulus is varied linearly along the upper
and lower crack surfaces. By considering a distribution of screw dislocations on the crack, the
problem is reduced to a single first-order Cauchy singular integro-differential equation for the
unknown dislocation density which is solved numerically using Chebyshev polynomials and the
collocation method. Numerical results are presented to demonstrate how the variable surface
shear modulus influences the dislocation density, crack opening displacement and the singular
stress field near the crack tips.

2. Bulk and surface elasticity

2.1. The bulk elasticity

In a fixed rectangular coordinate system xi (i = 1,2,3), the equilibrium and stress-strain
relations for an isotropic elastic bulk solid are well-known to be

σij,j =0 σij =2µεij +λεkkδij εij =
1

2
(ui,j +uj,i) (2.1)

Here, i,j,k = 1,2,3 and we sum over repeated indices; λ and µ are the Lame constants of the
bulkmaterial; σij and εij are respectively the components of the stress and strain tensors in the
bulk; ui is the i-th component of the displacement vector u and δij is the Kronecker delta.
For the antiplane shear deformation of an isotropic elastic material, the two shear stress

components σ31 and σ32 and the out-of-plane displacement w = u3(x1,x2) can be expressed in
terms of a single analytic function f(z) of the complex variable z = x1+ix2 as

σ32+iσ31 = µf
′(z) w =Im{f(z)} (2.2)

2.2. The surface elasticity

The equilibrium conditions on the surface incorporating interface/surface elasticity can be
expressed as (Gurtin andMurdoch, 1975; Gurtin et al., 1998; Ru, 2010)

σαjnjeα+σ
s
αβ,βeα =0 tangential direction

[σijninj] = σ
s
αβκαβ normal direction

(2.3)

where α,β = 1,3; eα are the bases for the surface; ni are the components of the unit normal
vector to the surface; [·] denotes the jump of the corresponding quantities across the surface;
σsαβ are the components of the surface stress tensor and καβ are those of the curvature tensor
of the surface. In addition, the constitutive equations on the isotropic surface are given by

σsαβ = σ0δαβ +2(µ
s−σ0)εsαβ +(λs+σ0)εsγγδαβ +σ0∇su (2.4)

where εsαβ is the surface strain tensor, σ0 is the surface tension, λ
s and µs are the two surface

Lame parameters,∇s is the surface gradient and γ =1,3. In contrast to previous studies in this
area, here it is assumed that λs and µs can vary along the surface.

3. A mode-III crack with variable surface effects

Consider the antiplane shear deformation of a linearly elastic and homogeneous isotropic solid
weakened by a finite crack {−a ¬ x1 ¬ a, x2 =0}. The crack surfaces are traction-free and the
solid is subjected to a uniform remote shear stress σ∞32. Let the upper and lower half-planes be
designated the “+” and “−” sides of the crack, respectively.



A mode-III crack with variable surface effects 1321

From Eq. (2.3), the boundary conditions on the crack surfaces can be written as

σs13,1+(σ23)
+− (σ23)− =0 on the upper crack face

σs13,1+(σ23)
+− (σ23)− =0 on the lower crack face

(3.1)

where (σ23)
− in Eq. (3.1)1 and (σ23)

+ in Eq. (3.1)2 are zero.

In the current setting, we have from the surface constitutive equations in Eq. (2.4) that

σs13 =2[µ
s(x1)−σ0]εs13 (3.2)

which indicates that the surface shear modulus is not constant but indeed variable along the
upper and lower crack surfaces.

By making use of Eq. (3.2) and assuming a coherent interface (εsαβ = εαβ), Eqs. (3.1) are
written as

(σ23)
+ =−[µs(x1)−σ0]u+3,11−

d[µs(x1)−σ0]
dx1

u+3,1 on the upper crack face

(σ23)
− =+[µs(x1)−σ0]u−3,11+

d[µs(x1)−σ0]
dx1

u−3,1 on the lower crack face

(3.3)

or equivalently

(σ23)
++(σ23)

− =−[µs(x1)−σ0](u+3,11−u
−
3,11)−

d[µs(x1)−σ0]
dx1

(u+3,1−u
−
3,1)

(σ23)
+− (σ23)− =−[µs(x1)−σ0](u+3,11+u

−
3,11)−

d[µs(x1)−σ0]
dx1

(u+3,1+u
−
3,1)

(3.4)

Theproblemcanbe formulated by considering adistribution of line dislocationswithdensity
b(x1) on the crack. Consequently, the analytic function f(z) can bewritten in the following form

f(z)=
1

2π

a
∫

−a

b(ξ)ln(z− ξ) dξ+
σ∞32
µ
z (3.5)

From the above expression, it follows that

f ′+(x1)=−
ib(x1)

2
+
1

2π

a
∫

−a

b(ξ)

x1− ξ
dξ+

σ∞32
µ

f ′−(x1)=
ib(x1)

2
+
1

2π

a
∫

−a

b(ξ)

x1− ξ
dξ+

σ∞32
µ

(3.6)

where−a < x1 < a, The subscripts “+” and “−” here indicate limiting values as we approach
the crack from the upper and lower half-planes, respectively.

It is not difficult to verify that the boundary condition inEq. (3.4)2 is automatically satisfied
with f(z) given by Eq. (3.5). On the other hand, the boundary condition in Eq. (3.4)1 leads to
the following first-order Cauchy singular integro-differential equation for the unknown density
function b(x1)

−
µ

π

a
∫

−a

b(ξ)

ξ−x1
dξ+2σ∞32 = [µ

s(x1)−σ0]b′(x1)+
d[µs(x1)−σ0]

dx1
b(x1) −a< x1< a (3.7)



1322 X.Wang, P. Schiavone

From Eqs. (3.6), we deduce that

∆w = w+−w− =−
x1
∫

−a

b(ξ) dξ −a < x1 < a (3.8)

Consequently, for a single-valued displacement in the case of a contour surrounding the crack
surface we require that

a
∫

−a

b(ξ) dξ =0 (3.9)

In what follows, we assume that µs(x1)−σ0 is a linear function of the coordinate x1 and is
given by

µs(x1)−σ0 = µ0
(

1+
k

a
x1
)

−a < x1 < a (3.10)

where µ0(> 0) and k(−1 < k < 1) are two constants. The constant k can be considered as a
parameter characterizing the gradient of the surface shear modulus µs(x1) along the surfaces.
Using Eq. (3.10), Eq. (3.7) simplifies to

−
µ

π

a
∫

−a

b(ξ)

ξ−x1
dξ+2σ∞32 = µ0

(

1+
k

a
x1
)

b′(x1)+
µ0k

a
b(x1) −a < x1 < a (3.11)

Comparing Eq. (3.11) with Eq. (23) in Kim et al. (2010) reveals that a nonzero gradient
parameter k will result in an additional term b(x1) on the right-hand side of the equation. In
the next Section, we present an approach based on Chebyshev polynomials and an adapted
collocation method to solve Eq. (3.11) numerically together with the auxiliary condition in Eq.
(3.9).

4. Solution to the singular integro-differential equation

We begin by setting x = x1/a in Eq. (3.11). For convenience, we write b(x)= b(ax)= b(x1). As
a result, Eqs. (3.9) and (3.11) can be written in the following normalized form

1
∫

−1

b̂(t)

t−x
dt =−πSe(1+kx)b̂′(x)−πSekb̂(x)+2π −1 < x < 1

1
∫

−1

b̂(t) dt=0

(4.1)

where

b̂(x)=
µb(x)

σ∞32
Se =

µ0
aµ

(4.2)

Define the inverse operator T−1 by

T−1ψ(x) =
1

π
√
1−x2

1
∫

−1

ψ(t) dt− 1
π2
√
1−x2

1
∫

−1

√
1− t2ψ(t)
t−x

dt −1 < x < 1 (4.3)



A mode-III crack with variable surface effects 1323

and apply to Eq. (4.1)1 to obtain

b̂(x)=
1

π
√
1−x2

1
∫

−1

b̂(t) dt−
1

π
√
1−x2

1
∫

−1

√
1− t2[−Se(1+kt)b̂′(t)−Sekb̂(t)+2]

t−x
dt (4.4)

Multiply both sides of Eq. (4.4) by
√
1−x2 and using the condition in Eq. (4.1)2, we obtain

b̂(x)
√

1−x2 =−
1

π

1
∫

−1

√
1− t2[−Se(1+kt)b̂′(t)−Sekb̂(t)+2]

t−x
dt (4.5)

We assume that the unknown function b̂(x) can be approximated by the following expansion

b̂(x)=
N
∑

m=0

cmTm(x) (4.6)

where Tm(x) represents the mth Chebyshev polynomial of the first kind.

By inserting Eq. (4.6) into Eq. (4.5), andmaking use of the following identities

dTm(x)

dx
= mUm−1(x) 2xUm(x)= Um+1(x)+Um−1(x)

1
∫

−1

Tm(t) dt =







1+(−1)m
1−m2

m 6=1
0 m =1

1
∫

−1

Um(t)
√
1− t2

t−x
dt =−πTm+1(x)

1
∫

−1

Tm(t)
√
1− t2

t−x
dt =

1
∫

−1

[Um(t)− tUm−1(t)]
√
1− t2

t−x
dt

=

1
∫

−1

Um(t)
√
1− t2

t−x
dt−x

1
∫

−1

Um−1(t)
√
1− t2

t−x
dt−

1
∫

−1

Um−1(t)
√

1− t2 dt

=−πTm+1(x)+πxTm(x)−
π

2
δm1−πxδm0

(4.7)

with Um(x) being the m-th Chebyshev polynomial of the second kind, we finally arrive at

c0
(
√

1−x2+Sekx
)

+
N
∑

m=1

cm
[

Sek
(

1+
m

2

)

Tm+1(x)

+
(

√

1−x2+Sem−Sekx
)

Tm(x)+
Sekm

2
Tm−1(x)

]

=2x

(4.8)

If we select the collocation points given by x = −cos(iπ/N) for i = 1,2, . . . ,N, Eqs. (4.8)
and (4.1)2 further reduce to the following algebraic equations



1324 X.Wang, P. Schiavone

c0

(

√

1−
(

cos
iπ

N

)2
−Sekcos

iπ

N

)

+
N
∑

m=1

cm

[

(−1)m+1Sek
(

1+
m

2

)

cos
(m+1)iπ

N

+(−1)m
(

√

1−
(

cos
iπ

N

)2
+Sem+Sekcos

iπ

N

)

cos
miπ

N

+(−1)m−1
Sekm

2
cos
(m−1)iπ

N

]

=−2cos
iπ

N
i =1,2, . . . ,N

N
∑

m=0,m6=1

1+(−1)m
1−m2

cm =0

(4.9)

The (N +1) unknowns cm, m = 0,1,2, . . . ,N can be uniquely determined by solving the
(N +1) independent equations in Eqs. (4.9).

5. Numerical results and discussion

In Figs. 1a and 1b, we illustrate the distributions of the dislocation density b(x) and the crack
opening displacement ∆w for four values of the gradient parameter k with Se =1. It is observed
from the two figures that: (i) b(x) is no longer an odd function of x and ∆w is no longer an
even function of x for k 6=0; (ii) as k increases from zero, themagnitude of b(−1) < 0 increases
considerably whereas that of b(1) > 0 decreases only marginally; (iii) as k increases from zero,
∆w increases significantly for the majority of the left portion of the crack and shrinks only
marginally for a small part of the right portion of the crack. It is observed fromEq. (3.10) that
the surface shear modulus for x < 0 always decreases and that for x > 0 always increases as
k increases from zero. This means that the left section of the crack becomes softer as opposed
to the right portionwhich becomes stiffer as k increases from zero. For example, when k =0.99,
µs(−1)−σ0 =0.01µ0 and µs(+1)−σ0 =1.99µ0. In this case, the crack surface in the immediate
neighbourhood of the left crack tip exhibits a minimal surface effect. Thus −b̂(−1) should be
considerably large since it becomes infinite in the absence of any surface effect. In fact, the
numerical result shows that b̂(−1)≈−30.

Fig. 1. The distribution of b(x) (a) and ∆w (b) for four values of the gradient parameter
k =0,0.4,0.8,0.99with S

e
=1

In Figs. 2a and 2b, we illustrate the variations of b(x) and ∆w for three sets of surface
parameters: Se =1, k =0.8; Se =1.8, k =0; Se =0.2, k =0. The surface shearmodulus at the
left crack tip for Se = 1, k = 0.8 is simply equal to the constant surface shear modulus in the
case Se = 0.2, k = 0; the surface shear modulus at the right crack tip when Se = 1, k = 0.8 is



A mode-III crack with variable surface effects 1325

just the constant surface shearmodulus for the case Se =1.8, k =0. It is observed fromFig. 2a
that b̂(−1)=−4.4524 for Se =1, k =0.8 and b̂(−1)=−4.8444 for Se =0.2, k =0. These two
values of b̂ are clearly close to each other. In addition, b̂(1) = 1.1282 for Se = 1, k = 0.8 and
b̂(1) = 0.9538 for Se = 1.8, k = 0. Again, these two values of b̂ are close. From Fig. 2b we see
that ∆w for Se =1, k =0.8 is greater than that for Se =1.8, k =0 and is smaller than that for
Se = 0.2, k = 0. This observation is in agreement with the conclusion reached in Antipov and
Schiavone (2011) that surface effects decrease the crack opening displacement. Intuitively, our
observations are consistent with the physics of the problem.

Fig. 2. The distribution of b(x) (a) and ∆w (b) for three sets of the surface parameters: S
e
=1, k =0.8;

S
e
=1.8, k =0; S

e
=0.2, k =0

Once b̂(x) is known, the stress field can be obtained from

σ32
σ∞32
+i

σ31
σ∞32
=
1

2π

1
∫

−1

b̂(t)

ẑ− t
dt+1 (5.1)

where ẑ = z/a. Since b̂(x) is finite at x = ±1, the stresses exhibit a logarithmic singularity at
the crack tips as follows

σ32
σ∞32
+i

σ31
σ∞32
=−

b̂(1)

2π
ln(z −a)+O(1) as z → a

σ32
σ∞32
+i

σ31
σ∞32
=
b̂(−1)
2π
ln(z +a)+O(1) as z →−a

(5.2)

We illustrate in Fig. 3 the stress component σ32 along the negative real axis for four values
of the gradient parameter k with Se =1.As k increases from zero, thematerial in the proximity
of the left crack tip becomes softer. Consequently, as illustrated in Fig. 3, the stress increases.
For any value of k, the stress is consistently lower than that found from the corresponding
classical solution σ32/σ

∞
32 = |x|/

√
x2−1 in the absence of surface effects. In order to verify

the logarithmic singularity at the crack tips, the near tip distribution of σ32 along the negative
real axis outside the crack is shown in Fig. 4. Seemingly, σ32 is a linear function of ln(−x−1)
for a fixed value of k. Thus the logarithmic singularity at the crack tip is verified numerically.
From Fig. 4 we can also calculate the pre-factors of the logarithmic term as:−0.2738 for k =0;
−0.3798 for k =0.4;−0.7758 for k =0.8;−4.8376 for k =0.99. The theoretical values fromEq.
(5.2)2 give: −0.2481 for k =0; −0.3448 for k =0.4; −0.7086 for k =0.8; −4.7396 for k =0.99.
Clearly, the calculated pre-factors well approximate the theoretical values.



1326 X.Wang, P. Schiavone

Fig. 3. σ32 along the negative real axis for four values of the gradient parameter k =0,0.4,0.8,0.99
with S

e
=1

Fig. 4. The near tip distribution of σ32 along the negative real axis outside the crack

6. Conclusions

In this paper, we utilize amodified version of theGurtin-Murdochmodel to examine the effects
of variable surface shear modulus in a mode-III fracture problem arising in the antiplane shear
deformation of a linearly elastic solid.ThemethodofGreen’s functions is used to obtain an exact
complete solution valid throughout the entire domain of interest (including at the crack tips) by
reducing theproblemto aCauchy singular integro-differential equation of thefirst-orderwhich is
solved numerically using an adapted collocation method. Numerical results demonstrate clearly
that the gradient of the surface shearmodulus exerts a significant influence on the distributions
of dislocation density on the crack, crack opening displacement and stress distribution near the
crack tips. The numerical results also verify that the resulting analysis is correct and that the
proposed collocationmethod is an effective tool in the analysis of crack problems in the presence
of variable surface effects.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No.: 11272121)

and through a DiscoveryGrant from the Natural Sciences and Engineering Research Council of Canada

(Grant#RGPIN155112).



A mode-III crack with variable surface effects 1327

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Manuscript received November 27, 2015; accepted for print March 7, 2016