Bogdan Rogowski
Exact Solution of a Dielectric Crack of Mode III in Magneto-Electro-Elastic Half-Space
EXACT SOLUTION OF A DIELECTRIC CRACK OF MODE III
IN MAGNETO-ELECTRO-ELASTIC HALF-SPACE
Bogdan ROGOWSKI*
*
Department of Mechanics of Materials, Technical University of Lodz,
Al. Politechniki 6, 93-590 Lodz, Poland
[email protected]
Abstract: This paper investigated the fracture behaviour of a piezo-electro-magneto-elastic material subjected to electromagneto-mechanical loads. The PEMO-elastic medium contains a straight-line crack which is parallel to its poling direction
and loaded surface of the half-space. Fourier transform technique is used to reduce the problem to the solution of one Fredholm integral equation. This equation is solved exactly. The semi-permeable crack-face magneto-electric boundary conditions are utilized. Field intensity factors of stress, electric displacement, magnetic induction, crack displacement, electric
and magnetic potentials, and the energy release rate are determined. The electric displacement and magnetic induction
of crack interior are discussed. Strong coupling between stress and electric and magnetic field near the crack tips has been
found.
1. INTRODUCTION
2. BASIC EQUATIONS
Due to the growth in applications as smart devices,
the mechanical and fracture properties of two-phase magnetostrictive/piezoelectric composites are becoming more
and more important, see: Huang and Kuo (1997), Pan
(2001), Buchanan (2004), Chen et al. (2005), Annigeri et al.
(2006), Lee and Ma (2007), Calas et al. (2008) and Hou et
al. (2009), among other, have been published the papers
on this field. The fracture mechanics of PEMO-elastic materials also have attracted much attention and many research papers have been published; see e.g. Liu et al.
(2001), Sih and Song (2003), Gao et al. (2003), Zhou et al.
(2004), Hu and Li (2005), Wang and Mai (2006b), Li and
Kardomateas (2007), Feng et al. (2007), Tian and Rajapakse (2008), Zhan and Fan (2008); among others.
For the fracture analysis of a magneto-electro-elastic
solid of much interest are the effects of magneto-electric
boundary conditions at the crack surfaces on the crack
growth as well as the choosing of fracture criteria (Wang
and Mai, 2006b, Wang et al., 2006a). As an approximation
to a real crack, the magneto-electrically permeable and
impermeable crack face boundary conditions are prevail
in the above stated-works. However these two ideal crack
models are only the limiting cases of real dielectric crack
(Wang and Mai; 2006b; Rogowski, 2007). However, the
above-mentioned works associated with semi-permeable
crack problems are only limited to an infinite magnetoelectro-elastic solid with cracks. Additionally, the numerical procedures are used to obtain the results. Motivated by
this consideration this paper investigates a PEMO-elastic
half-space with an electrically and magnetically conducting
crack under anti-plane mechanical and in-plane electromagnetic loadings to shown exact solution in simple analytical form.
For a linearly magneto-electro-elastic medium under
anti-plane shear coupled with in-plane electric and magnetic fields there are only the non-trivial anti-plane displacement :
86
u x = 0 , u y = 0 , u z = w(x, y )
(1)
strain components ௫௭ and ௬௭ :
γ xz =
∂w
∂w
, γ yz =
∂y
∂x
(2)
stress components ௫௭ and ௬௭ , in-plane electrical and magnetic potentials and , which define electrical and magnetic field components ௫ , ௬ , ௫ and ௬ :
Ex = −
∂φ
∂ψ
∂φ
∂ψ
, Ey = −
, Hx = −
, Hy = −
∂x
∂
x
∂y
∂y
(3)
and electrical displacement components ௫ , ௬ and magnetic induction components ௫ , ௬ with all field quantities
being the functions of coordinates and .
The generalized strain-displacement relations (2)
and (3) have the form:
γ αz = w,α , Eα = −φ ,α , H α = −ψ ,α
(4)
where = , and ,ఈ = /.
For linearly magneto-electro-elastic medium the coupled constitutive relations can be written in the matrix form
[τ αz , Dα , Bα ] T
= C[γ αz ,− Eα ,− H α ] T
(5)
where the superscript denotes the transpose of a matrix
and:
acta mechanica et automatica, vol.5 no.4 (2011)
c 44
C =  e15
q15
e15
− ε 11
− d11
c~44 = c 44 + αe15 + βq15
q15 
− d11 
− µ11 
(6)
where ସସ is the shear modulus along the -direction, which
is direction of poling and is perpendicular to the isotropic
plane (
, ), ଵଵ and ଵଵ are dielectric permittivity and
magnetic permeability coefficients, ଵହ , ଵହ and ଵଵ are
piezoelectric, piezo-magnetic and magneto – electric coefficients, respectively.
The mechanical equilibrium equation (called as Euler
equation), the charge and current conservation equations
(called as Maxwell equations), in the absence of the body
force electric and magnetic charge densities, can be written
as:
] = [0,0,0]
T
T
(8)
பమ
where ∇ଶ = మ + ∂ଶ / ∂ ଶ is the two-dimensional Laplace
ப୶
operator.
Since || ≠ 0 one can decouple the equations (8):
∇ 2 w = 0 ; ∇ 2φ = 0 ; ∇ 2ψ = 0
(9)
If we introduce, for convenience of mathematics
in some boundary value problems, two unknown functions:
[χ − e15 w,η − q15 w]T
= C 0 [φ ,ψ ]T
(10)
− d11 
− µ11 
(11)
Then:
[φ ,ψ ]T
= C 0 −1 [χ − e15 w,η − q15 w] T
(12)
where:
C0
−1
=
1
2
ε 11µ11 − d11
− µ11
 d
 11
d11   e1
=
− ε 11  e2
e2 

e3 
(13)
The governing field variables are:
τ zk = ~
c 44 w,k − α D k − β B k
φ = αw + e1 χ + e2η
ψ = βw + e2 χ + e3η
(14)
Consider a PEMO-elastic half-space containing straightline crack of length 2 , parallel to the surface of a halfspace which is subjected to electric, magnetic and mechanical loads. The crack is located along the -axis from −
to at a depth ℎ from the loaded surface with a rectangular
coordinate system, as shown in Fig. 1. The PEMO-elastic
half-space is poled in the -direction.
To solve the crack problem in linear elastic solids,
the superposition technique is usually used. Thus we first
solve the magneto-electro-elastic field problem without the
cracks in the medium under electric, magnetic and mechanical loads. This elementary solution is:
 D0 , case I



e2 
e q 
e
D y = D =  15 τ 0 +  ε 11 + 15  E 0 +  d 11 + 15 15  H 0 ,


c 44 
c 44 

 c 44


case II

 B0 , case I



e q 
q2 
q
B y = B =  15 τ 0 +  d 11 + 15 15  E 0 +  µ11 + 15  H 0 ,

c 44 
c 44 

 c 44


case II

τ zy ( x, h ± ) = −τ 0 , D y (x, h ± ) = − D + d 0 ,
zy
(15)
(18)
Then, we use equal and opposite values as the crack surface traction and utilize the unknowns ଴ and ଴ in the
crack region. Thus, in this study, −଴ ; −(
− ଴ ), −( −
଴ ) are, respectively, mechanical, electrical and magnetic
loadings applied on the crack surfaces (the so called perturbation problem).
The boundary conditions can be written as:
x‹ a
[τ ] = 0 , [D ] = 0 , [B ] = 0 ,
B k = η , k ; k = x, y
where:
(17)
3. FORMULATION OF THE CRACK PROBLEM
B y (x, h ± ) = − B + b0 ;
D k = χ ,k
∇ 2 w = 0 ; ∇ 2 χ = 0 ; ∇ 2η = 0
α
β
1

1
C −1 = ~ α α 2 + c~44 e1 αβ + c~44 e2 
c44
 β αβ + c~44 e2 β 2 + c~44 e3 
τ yz = τ 0
where:
 − ε 11
C0 = 
− d11
Note that ̃ସସ is the piezo-electro-magnetically stiffened
elastic constant.
Note also that:
(7)
Subsequently, the Euler and Maxwell equations take the
form:
[
(16)
These material parameters will appear in our solutions.
τ zα ,α = 0 ; Dα ,α = 0 ; Bα ,α = 0 ; α = x, y
C ∇ 2 w, ∇ 2φ , ∇ 2ψ
µ11e15 − d11q15
= −(e1e15 + e2 q15 )
2
ε 11µ11 − d11
ε q −d e
β = 11 15 11 215 = −(e3q15 + e2e15 )
ε 11µ11 − d11
α=
y
y
[ w ] = 0 , [φ ] = 0 , [ψ ] = 0 ,
x ‹∞ , y = h
x ≥a, y = h
τ zy (x,0 ) = 0 , D y ( x,0 ) = 0 , B y (x,0 ) = 0 ; x ‹ ∞
(19)
(20)
(21)
(22)
87
Bogdan Rogowski
Exact Solution of a Dielectric Crack of Mode III in Magneto-Electro-Elastic Half-Space
where the notation || = ା − ି and ା denotes the
values for ℎ + while ି for ℎ −.
χ̂ (s, y ) = B1 (s )e − sy
>ℎ
η̂ (s, y ) = C1 (s )e − sy
wˆ (s, y ) = A2 (s )e − sy + A3 (s )e sy
χˆ (s, y ) = B2 (s )e − sy + B3 (s )e sy
0≤<ℎ
(26b)
ηˆ (s, y ) = C2 (s )e − sy + C3 (s )e sy
Fig. 1. The PEMO-elastic half-space with a crack parallel
to its surface under an anti-plane mechanical
and in-plane electric and magnetic loads.
Inside the crack the unknown electromagnetic
field appears (݀଴ and ܾ଴ are unknown to be determined)
Of course, in perturbation problem the surface of the
half-space is free. The electric displacement ଴ and magnetic induction ଴ inside the crack are obtained from semipermeable crack face boundary conditions (Rogowski
(2007)). For two different magneto-electric media: PEMOmaterial and notch space we have continuity condition
of electric and magnetic potential in both materials at interface. The semi-permeable crack-face magneto-electric
boundary conditions are expressed as follows:
d 0 = −ε c
[φ ]
2δ (x )
, b0 = − µ c
[ψ ]
(23)
2δ (x )
where () describes the shape of the notch and ௖ , ௖
are the dielectric permittivity and magnetic permeability
of crack interior. If we assume the elliptic notch profile
such that:
δ (x ) = (δ 0 a ) a − x
2
2
(24)
where ଴ is the half-thickness of the notch at = 0,
we obtain:
[ ]
2d 0 (δ 0 ε c ) a 2 − x 2 = − φ
In the domain > ℎ the solution has the form (26a)
to ensure the regularity conditions at infinity.
The transforms of Eqs (14) yield:
ϕˆ (s, y ) = e1 χˆ (s, y ) + e2ηˆ (s, y ) + αwˆ (s, y )
ψˆ (s, y ) = e2 χˆ (s, y ) + e3ηˆ (s, y ) + βwˆ (s, y )
Dˆ y = χˆ , y , Bˆ y = ηˆ, y
The unknown functions ௜ , ௜ and ௜ ,
= 1, 2, 3, are obtained from the boundary conditions (20)
and (22), which in transform domain are:
[τˆ ] = 0 ; [Dˆ ] = 0 ; [Bˆ ] = 0
y
zy
y
τˆzy = 0 ; Dˆ y = 0 ; Bˆ y = 0 ;
[ ]
where [|] = , ℎ + − (, ℎ−).
The result is:
(
B (s ) = gˆ (s )(e
C (s ) = hˆ(s )(e
)
)
)
A1 (s ) = fˆ (s ) e − sh − e sh
− sh
− e sh
− sh
− e sh
1
1
(25)
B2 (s ) = B3 (s ) = gˆ (s ) e − sh
Eqs (25) form two coupling linear equations with respect to ଴ and ଴ since || and || depends linearly
on these quantities as show boundary conditions (19) and
(21).
C2 (s ) = C3 (s ) = hˆ(s ) e − sh
4. THE SOLUTION FOR HALF-SPACE
Define the Fourier transform pair by equations:
0
2
π
∞
∫0
fˆ (s ) cos(sx )ds
2
∞
π ∫0
χ ( x, y ) = −
[
f̂ (s) sgn( y − h)e
−s ( y−h)
ĝ(s )[sgn( y − h)e
π∫
2
∞
]
− e−s( y+h) cos(sx)ds
]
−s ( y−h )
− e−s( y+h) cos(sx)ds
−s ( y−h )
− e−s( y+h) cos(sx)ds
0
(26)
Considering the symmetry about -axis the Fourier cosine transform is only applied in Eqs (15) resulting in ordinary differential equations and their solutions:
wˆ (s, y ) = A1 (s )e − sy
(26a)
88
(29)
Finally, the solution for the half-space with dislocation
density functions (), () and ℎ() in the domain ≥ 0
|| < ∞ is given by:
w(x, y) = −
WITH DISCONTINUITY AT y = h
fˆ (s ) = ∫ f (x ) cos(sx)dx , f ( x ) =
(28)
y=0
A2 (s ) = A3 (s ) = fˆ (s ) e − sh
2b0 (δ 0 µ c ) a 2 − x 2 = − ψ
∞
(27)
τˆzy (s, y ) = c~44 wˆ , y − αDˆ y − βBˆ y
η(x, y ) = −
2
∞
π ∫0
[
ĥ(s ) sgn( y − h)e
]
(30)
acta mechanica et automatica, vol.5 no.4 (2011)
[
∞
]
where ଴ () is the Bessel function of the first kind
and zero order and (), (), ℎ() are new auxiliary
functions. This representation satisfies equations (31b)
automatically and converts equations (32) to the Abel one
integral equation, which can be solved explicitly. The result
is the Fredholm integral equation of the second kind:
2
−s ( y−h)
τ zy (x, y ) = ~
c44 ∫ sf̂ (s) e
− e−s( y+h) cos(sx)ds +
π 0
−
s[αĝ (s) + βĥ(s)][e
π∫
2
∞
−s ( y−h)
]
− e−s( y+h) cos(sx)ds
0
D y (x , y ) =
sĝ (s )[e
π∫
2
∞
−s (y −h)
]
− e − s ( y + h ) cos(sx )ds
a
f (u ) − ∫ f (v )K (u, v ) dv = 1
0
B y (x , y ) =
sĥ(s )[e
π∫
2
∞
−s (y −h)
]
with the kernel:
− e − s ( y + h ) cos(sx )ds
∞
0
where ‫݊݃ݏ‬ሺ‫ ݕ‬− ℎሻ = +1 or ‫ > ݕ‬ℎ or ‫ < ݕ‬ℎ, respectively.
The potentials ߮(‫ݔ‬, ‫ )ݕ‬and ߰(‫ݔ‬, ‫ )ݕ‬are obtained from
Eqs (14).
5. FREDHOLM INTEGRAL EQUATION
OF THE SECOND KIND
The unknown functions ݂(‫)ݏ‬, ݃(‫ )ݏ‬and ℎ(‫ )ݏ‬can be
obtained from the mixed boundary conditions (19)
and (21)which yield:
K (u, v ) = v ∫ se −2 sh J 0 (su )J 0 (sv )ds
0
−2
0
0
f (u ) = g (u ) = h(u )
2
π
2
π
∫ sgˆ (s )[1 − e ]cos(sx )ds = −(D − d )
∞
of course (), () and ℎ () are dissimilar since
are proportional to ଴ + − ଴ + ( − ଴ ) ; − ଴
and − ଴ , respectively, and ଴ , ଴ are dissimilar functions, defined by Eqs. (23).
6. THE SOLUTION OF FREDHOLM INTEGRAL
EQUATION OF THE SECOND KIND
−2 sh
(31a)
0
0
∞
[
]
−2 sh
∫ shˆ(s )1 − e cos(sx )ds = −(B − b0 ) ;
(36)
0
44
0
(35)
and:
τ + (D − d )α + (B − b )β
sfˆ (s)[1 − e sh ]cos(sx)ds = −
π∫
c~
2
∞
(34)
0
The kernel function (, ) may be presented in more
useful form. Using the Neumann’s theorem (Watson,
1966):
J 0 (su )J 0 (sr ) =
x ‹a
0
π
∫ J 0 (sR )dα ;
1
π
(37)
0
R 2 = u 2 + r 2 − 2ur cos α
∞
∫ f̂ (s )cos(sx )ds = 0
and the integral:
∞
0
∞
∫ sJ (Rs )e
∫ ĝ (s )cos(sx )ds = 0
− 2 sh
0
(31b)
0
∞
ds =
0
[R
2h
2
+ (2h )
]
2 3 2
(38)
the kernel function becomes:
∫ ĥ(s ) cos(sx )ds = 0
;
x ≥a
K (u, v ) =
0
4hv
πl 3 2
The integral equations (31a) may be rewritten as:
τ0 + (D − d0 )α + (B − b0 )β 
x
 fˆ(s)1− e−2sh

c~44
∞
2   −2sh

 (32)
(D − d0 )x
gˆ(s)1− e sin(sx)ds = −

∫
π 0  ˆ  −2sh


(
h(s) 1− e 
B − b0 )x
 




We introduce the integral representation of the unknown
functions:
τ 0 + (D − d0 )α + (B − b0 )β 
 fˆ (s) 
~
a  f (u)
c44

 
 

D − d0
gˆ (s) = −
∫ g(u)uJ0 (su)du
 hˆ(s)  
0  h(u) 
B − b0


 
 


π 2
∫ (1 − k
dα
2
0
cos 2 α
)
32
(39)
4uv
l = (u + v ) + 4h , k = 2
l
2
2
2
2
The kernel function is presented by means of elliptic integral. The integral equation (34) can be solved by iterative
method.
The recurrence formula is:
a
f i (u ) = 1 + ∫ f i −1 (v )K (u, v )dv , f (v ) = 1 , i = 1,2,...n
0
(40)
0
(33)
The n-th approximation gives:
89
Bogdan Rogowski
Exact Solution of a Dielectric Crack of Mode III in Magneto-Electro-Elastic Half-Space
f (u ) = 1 +
a
a+u
 4h K (k 0 ) 
1 −
+
 π l0 
τ zy (x, h ±) = − τ 0 ∫ f (u )udu∫ sJ0 (su)(1 − e−2sh )cos(sx)ds
0
π 0
 a   4h K (k 0 ) 
+
 1 −
 + .... +
 a + u   π l0 
2
2
 a 
+

a+u
 4h K (k 0 ) 
1 −

 π l0 
n
∞
∫0 e
n
π /2
∫
0
(42)
4au
l = (a + u ) + 4h , k = 2
l0
2
2
0
2
The sum of infinite geometric series converges to the
solution as ݊ → ∞, giving:

a 
2 K (k 0 ) 
1 −

f (u ) = 1 −
 a + u  π l0 2h 
−1
u ≤a
(43)
The range of convergence is given by inequality:
u ≤a
(44)
and is satisfied for all of ‫ ݑ‬and ܽ/ℎ.
For ℎ → ∞, ሺ2/ߨሻ‫݇(ܭ‬଴ ) → 1 and ݈଴ /2ℎ → 1, while for
ℎ → 0, we have the logarithmic singularity of ‫݇(ܭ‬଴ ) for
‫ܽ = ݑ‬:
K (k0 ) ~
(45)
2 au
1−
a+u
But ℎ‫݇(ܭ‬଴ )/݈଴ tends to zero as ܽ/ℎ → ∞.
Thus we have the values:
 2
 δ
1
a
f   = 2 1 +
K

2
2
h
 1+ δ
 π 1 + δ
f (0 ) = 1 +
(
)(
u = x 1 + ξ 1 −η
2
2
2
(
η
x ξ 2 +η2
)
)
(48)
2h = xξη , xξ › 0, η › 0
equations (47) may be written as:
 x
Dy (x, h ±) 2 D− d0  d a
η

=− 
 ∫ f (u)udu 2 2 − 2 2
(
)
B
x
h
,
±
 x x −u x ξ +η
 y
 π B −b0  dx 0
)
a

x
d
η
− 2 2
τ zy (x, h ±) = − τ 0 ∫ f (u)udu
2
2
x ξ +η
π dx 0
 x x − u



(



(49)
(
)
The singular terms of these quantities (|‫ܽ → |ݔ‬ା ) are included in the first term in Eq. (49). Since the singular field
near the crack tip exhibits the inverse square-root singularity we define the stress, electric displacement and magnetic
induction intensity factors as follows:
τ zy 
 Kτ 
 
 
 K D  = lim+ 2( x − a ) D y 
 K  x →a
B 
 B
 y
(50)
The intensity factors are obtained as:
1
K w = ~ (Kτ + αK D + βK B )
c44
K φ = α K w + e1 K D + e2 K B
1
ln
sin (sx )J 0 (su )ds =
2
2
0
u l

2
K (k0 ) ‹  2 +  0
a  2h
π

− 2 sh
2
dα
2
( 1 − k 0 cos 2 α )1 2
∞
Using the integral:
(41)
where ‫݇(ܭ‬଴ ) is the elliptic integral of the first kind defined
by:
K (k 0 ) =
a
2




−1
(46)
a
δ
, δ=
h
4
2
The values of ݂(ܽ/ℎ) changes from 1 to 2 for all
of ܽ/ℎ and ݂(‫ )ݑ‬is given explicitly by Eq. (43).
Kψ = β K w + e2 K D + e3 K B
Kτ =
2
τ f (a ) a
π 0
KD =
2
(D − d 0 ) f (a ) a
π
KB =
2
(B − b0 ) f (a ) a
π
(51)
The jumps of displacement, electric potential and magnetic potential of the crack surfaces can be expressed as:
[ w ] = π4 τ + (D − d c~)α + (B − b )β ∫ f (u )udu
a
0
0
0
44
7. FIELD INTENSITY FACTORS
x
u2 − x2
[φ] =π4 α[τ +(D−dc~)α+(B−b )β] +e (D−d ) +e (B−b )∫ f (u)udu
a
0
The electric displacement, magnetic induction and shear
stress outside of the crack surface can be expressed by:
Dy (x, h ±) 2 D− d0 a
∞
−2sh

= − 
∫ f (u)udu∫0 sJ0 (su) 1−e cos(sx)ds (47)
(
)
±
B
x
,
h
−
B
b
π
0 0

 y

(
90
)
0
0
1

0
2
0
44
(52)
x u −x
2
2
[ψ] =π4 β[τ +(D−dc~)α+(B−b )β] +e (D−d ) +e (B−b )∫ f (u)udu
a
0
0
0
2
0
3
0
44

x u2 −x2
Substituting Eqs (52) into Eqs (25) and differentiating
both obtained equations with respect to and using
the following rule of differentiation under integral sign:
acta mechanica et automatica, vol.5 no.4 (2011)
d
f (u )du
xf (a )
d  f (u )  du
=−
+ x∫ 

∫
2
2
2
2
dx x u − x
du  u  u 2 − x 2
a a −x
x
a
a
(53)
equations (25) may be converted to two equations in which
singular terms at → − 0, appear. For the singularity
to vanish at → − 0, it must be true that:
α

d0 = −ε0 f (a)~ [τ0 +(D−d0 )α +(B−b0 )β] +e1(D−d0 ) +e2(B−b0 )
c44

β

b0 = −µ0 f (a)~ [τ0 +(D−d0 )α +(B−b0)β] +e2(D−d0) +e3(B−b0 )
c
 44

2 a
2 a
ε c , µ0 =
µ
π δ0
π δ0 c
w
, Kφ , Kψ
Thus:
[
(
)

 α2

 αβ
 
B − b0 = Dµ0 f  ~ + e2  + B1− ε0 f  ~ + e1  +

 c44
 
 c44

[
1
(Kτ K w + K D Kφ + K B Kψ )
2
(61)
2
π2
f
2
(a )a ~
)[(
)
1
2 2
ε11µ11 − d11
τ0 +
2
c44 ε11µ11 − d11
)
(
(
)
(62)
+ 2(c44 d11 + e15 q15 )(D − d 0 )(B − b0 )]
8. SOLUTIONS BASED ON IDEAL CRACK-FACE
BOUNDARY CONDITIONS
−1
)
where = (/ℎ).
The electric and magnetic intensity factors are obtained
by substitution of Eqs (56) into Eqs (51).
Furthermore we consider the behaviour of the jumps
of the displacement, electric and magnetic potentials
and define the following intensity factors:
[ ]
[ ]
[ ]
K w 
w
1
 
 
 Kφ  = lim−
φ 
 K  x →a 2 2(a − x )  ψ 
 
 ψ
(57)
When the crack is one of four ideal crack models: magneto-electrically permeable, magneto-electrically impermeable, magnetically permeable and electrically impermeable,
magnetically impermeable and electrically permeable are
the limiting cases of the magneto-electrically dielectric
crack model.
− fully impermeable case: ଴ → 0 and ଴ → 0, − ଴ →
, − ଴ → and the intensity factors are given by:
Kτimp.imp.
K Bimp.imp.
(58)
Kψ = βK w + e2 K D + e3 K B
These field intensity factors satisfy the constitutive
equations:
2 f (a ) a
(τ 0 + Dα + Bβ )
π c~44
2
= f (a ) aτ 0
π
2
= f (a ) a D
π
2
= f (a ) a B
π
K wimp.imp. =
K Dimp.imp.
In view of the results in Eq. (52), we have:
Kφ = αK w + e1K D + e2 K B
(60)
+ 2(q15ε11 − e15 d11 )τ 0 (B − b0 ) +
(56)
2



  β2
c
  α

×1− f ε0 ~ +e1  + µ0 ~ +e3  −ε0µ0 ~44 e1e3 −e22 
c
 c

c44


   44   44 

1
K w = ~ (Kτ + αK D + β K B )
c44
[ ]
+ 2(e15 µ11 − q15 d11 )τ 0 (D − d 0 ) +

τµ
+ ~0 0 f β + ε0q15 e1e3 − e22 f ×
c44

(
{
where [||], [||] and [||] are the jumps of displacement,
electric potential and magnetic potential field intensity
factors given by Eqs. (52).
The energy release rate is defined as:
(
)]
(
δ
2
− c44 µ11 + q15
(D − d 0 )2 − c44ε11 + e152 (B − b0 )2 +
−1

 α

β


c


× 1− f ε0  ~ + e1  + µ0  ~ + e3  − ε0µ0 ~44 e1e3 − e22 
c

c

c
44


 44

  44



2
(59)
[ ]
B y (r + a ,0)[ψ ](r + a − δ )}dr
G=

τε
+ ~0 0 f α + µ0e15 e1e3 − e22 f ×
c44

2
T
or :
)]
(
= C −1 [Kτ , K D , K B ]
D y (r + a ,0) φ (r + a − δ ) +
G=

 β2

 αβ

 
D − d0 = D1− µ0 f  ~ + e3  + Bε 0 f  ~ + e2  +

 c44

 c44

 
T
1
1
lim ∫ τ yz (r + a ,0) w (r + a − δ ) +
2 δ →0 δ 0
(54)
(55)
]
The energy release rate is derived in the following
in similar manner to proposed by Pak (1990).
The energy release rate of the crack-tip is obtained from
the following integral:
G=
where:
ε0 =
[K
(63)
K φimp.imp. = αK wimp.imp. + e1 K Dimp.imp. + e2 K Bimp.imp.
Kψimp.imp. = βK wimp.imp . + e2 K Dimp.imp. + e3 K Bimp.imp.
Equations (63) indicate that ఛ , ஽ and ஻ are independent on the material constants, while௪ , థ and ట ,
depend. Since () depend on the parameter of location
91
Bogdan Rogowski
Exact Solution of a Dielectric Crack of Mode III in Magneto-Electro-Elastic Half-Space
of the crack(the thickness ℎ), strictly speaking on ℎ/,
these quantities depend on this location.
− full permeable case: ଴ → 0 and ଴ → 0
Then:
e15τ 0
q τ
B − b0 = 15 0
c44 ,
c44
D − d0 =
(64)
K
per. per.
Kτ
f1 (µ ) =
2
= f (a ) aτ 0
π
=
e15 K wper . per .
(65)
K Bper . per . = q15 K wper . per .
per . per .
Kφ
=0
(66)
− electrically impermeable and magnetically permeable:
଴ → 0 and ଴ → 0, − ଴ → , ଴ → 0
K Dimp. per = K Dimp.imp.
  αβ
 

τ β   β 2
B − b0 =  D ~ + e2  f (a ) + ~0  1 +  ~ + e3  f 
c44    c44
  c44

 
−1
(67)
K
=
K wimp. per. =
π
− Let ଴ tends to infinity and ଴ is finite
Then:
K Dε c perm. = K Dimp.imp. (1 − f 2 (ε )) + K Dper. per. f 2 (ε )
f 2 (ε ) =
2 τ 02 af 2 (a )
c44
π2
2
(70)
(71)
where:
The energy release rate is:
imp . per .
B
q152 
1
π µ11 δ 0 1 


µ
=
1
+
1+ µ ,
2 µ c a f (a )  µ11c44 
K Bε c perm. = K Bimp. per. (1 − f 2 (ε )) + K Bper. per. f 2 (ε )
Kψper . per . = 0
G=
(69)
where:
2 f (a ) a
=
τ0
π c44
K Dper . per .
K Dperm.µc = K Dper.imp. (1 − f1 (µ )) + K Dper. per. f1 (µ )
K Bperm.µc = K Bimp.imp. (1 − f1 (µ )) + K Bper. per. f1 (µ )
And:
per . per .
w
electric displacement.
In practical applications the following cases appear:
− Let ଴ tends to infinity and ଴ is finite
Then:
e2 
π ε 11 δ 0 1 
1
1 + 15 
, ε =
2 ε c a f (a )  ε 11c44 
1+ ε
(72)
In above equations the notation ௣௘௥௜௠௣ denotes the intensity factors (51) and (58) for electrically permeable
and magnetically impermeable crack boundary conditions
i.e. for the values (68). Similarly ௜௠௣.௣௘௥. are defined
by Eqs (51) or (58) and (67).
The functions of permittivity ௖ and permeability ௖
approaches zero as ௖ and ௖ tends to zero and are unity
as ௖ and ௖ tends to infinity.
The solution perfectly matches the exact solution
in both limiting cases, namely permeable and/or impermeable electric and/or magnetic boundary conditions.
f (a ) a (B − b0 )
9. RESULT AND DISCUSSION
τ + Dα
2
f (a ) a 0 ~
+ βK Bimp. per.
π
c44
The solutions for the electrically impermeable and magnetically permeable crack are independent of the applied
magnetic field.
− electrically permeable and magnetically impermeable:
ε଴ → 0 and μ଴ → 0, B − b଴ → B, b଴ → 0
K Bper.imp. = K Bimp.imp.

τ α
 

αβ 
α2 
D − d 0 =  ~0 + B e2 + ~  f (a)1 +  e1 + ~  f (a)
c44 
c44 

 c44
 

K Dper.imp. =
−1
(68)
2
f (a ) a (D − d 0 )
π
τ + Bβ
2
f (a ) a 0 ~
+ αK Dper.imp.
π
c44
The solution for the electrically permeable and magnetically impermeable crack are independent of the applied
K wper.imp =
92
Fig. 2 Variation of (/ℎ) versus ratio of /ℎ; stress, electric
displacement and magnetic induction intensity factors
are proportional to (/ℎ) since: /ℎ = ఛ /଴ √(2/
) = ஻ /√(2/) in fully impermeable case
The electric and magnetic response, in fully impermeable case, is proportional to the applied electric and magnetic load, respectively, and is independent on the mechanical loads, as Eq. (63) implies. Similarly is for stress inten-
acta mechanica et automatica, vol.5 no.4 (2011)
sity factor. The intensity factors of stress, electric displacement and magnetic induction, therefore, are just a function
of the geometry of the cracked PEMO-elastic half-space
as shown in Fig. 2.
From the Fig. 2 we can see that the SIF, EDIF and MIIF
increase with /ℎ. For small values of /ℎ these quantities
grow at an approximately constant rate with increasing
/ℎ. For very large /ℎ (the crack near the boundary
of a half-space) (/ℎ) increases slowly tending to 2.
10. CONCLUSIONS
From analytical and numerical results, several
conclusions can be formulated:
− The electric displacement intensity factor is independent
of the applied magnetic field in the special case of electrically impermeable and magnetically permeable crack;
− The magnetic induction intensity factor is independent
of the applied electric displacement in the special case
of electrically permeable and magnetically impermeable
crack;
− Applications of electric and magnetic fields do not alter
the stress intensity factors;
− The analytical solution (43) is new to the author’ best
knowledge. Accordingly, the behaviour of a crack
which lies near of the boundary of the medium may be
investigated exactly;
− Note that the plane = 0 is a plane of symmetry
(௫௭ = 0, ௫ = 0 and ௫ = 0 on this plane). In consequence the solutions are valid for quarter-plane
with edge crack of length .
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93