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Dynamic interaction of multiple arbitrarily oriented planar cracks in a piezoelectric space: A semi-analytic solution
European Journal of Mechanics A/Solids 30 (2011) 608e618
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Dynamic interaction of multiple arbitrarily oriented planar cracks in a piezoelectric space: A semi-analytic solution W.T. Ang*, L. Athanasius School of Mechanical and Aerospace Engineering, Nanyang Technological University, Republic of Singapore
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 December 2010 Accepted 11 March 2011 Available online 21 March 2011
The problem of determining the electro-elastic fields around arbitrarily oriented planar cracks in an infinite piezoelectric space is considered. The cracks which are acted upon by a transient load are either electrically impermeable or permeable. A semi-analytic method based on the theory of exponential Fourier transformation is proposed for solving the problem in the Laplace transform domain. The Laplace transforms of the jumps in the displacements and electric potential across opposite crack faces are determined by solving a system of hypersingular integral equations. Once these displacement and electric potential jumps are obtained, the displacements and electric potential and other physical quantities of interest, such as the crack tip stress and electric displacement intensity factors, can be computed with the help of a suitable algorithm for inverting Laplace transforms. The stress and electric displacement intensity factors are computed for some specific cases of the problem. 2011 Elsevier Masson SAS. All rights reserved.
Keywords: Multiple cracks Transient loads Piezoelectric materials Hypersingular integral equations
1. Introduction In recent years, the problem of determining the electro-elastic fields around cracks in piezoelectric materials has been a subject of considerable interest among many researchers. It appears that the majority of analyses on piezoelectric crack problems (see, for example, Athanasius et al. (2010), Li (2002), Shindo et al. (2000) and Wang and Mai (2002)) are concerned with electro-elastostatic deformations. According to Kuna (2010), there are comparatively fewer works on piezoelectric cracks that are acted upon by time dependent loads. The governing partial differential equations for piezoelectric materials undergoing dynamic antiplane deformations are relatively simpler in form, being reducible to a pair of equations comprising the two-dimensional Helmholtz and Laplace’s equations. Thus, most papers giving semi-analytic solutions for dynamic piezoelectric crack problems assume that the cracks undergo antiplane deformations. For example, the dynamic response of a single electrically impermeable planar crack in an infinite transversely isotropic piezoelectric material under pure electric load and undergoing an antiplane deformation is investigated by Chen (2006). Chen and Karihaloo (1999), Chen and Meguid (2000), Li and Fan (2002) and Li and Tang (2003) have solved problems involving a single planar crack in an infinitely long piezoelectric * Corresponding author. E-mail address: [email protected] (W.T. Ang). URL: http://www.ntu.edu.sg/home/mwtang/ 0997-7538/$ e see front matter 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2011.03.003
strip under antiplane deformations. Coplanar cracks undergoing antiplane deformations in piezoelectric materials are examined in Chen and Worswick (2000) and Meguid and Chen (2001). Kwon and Lee (2001) and Li and Lee (2004, 2006) have investigated the antiplane deformation of edge cracks in piezoelectric materials. There are apparently few papers in which semi-analytic solutions for cracks undergoing dynamic inplane deformations in piezoelectric materials may be found. One of them is Shindo et al. (1999). In Shindo et al. (1999), the problem of a single planar crack in a piezoelectric ceramic under normal impact is formulated in terms of a pair of dual integral equations by representing the displacement and electric potential in the Laplace transform domain by suitable Fourier sine and cosine transform representations. The dual integral equations are solved as explained in Sneddon and Lowengrub (1969), by reducing them to Fredholm integral equations of the second kind. The dynamic piezoelectric problem can also be formulated in terms of hypersingular integral equations using the approach in a recent paper by García-Sánchez et al. (2007). In García-Sánchez et al. (2007), the kernels of the hypersingular integral formulation contain second order spatial derivatives of a suitable dynamic Green’s function for piezoelectric solids. The Green’s function is derived using Radon transform. Its evaluation is a rather involved exercise, requiring the computation of a line integral over a unit circle with integrand that is expressed in terms of exponential integrals (Wang and Zhang, 2005). The present paper derives a semi-analytic solution for an electroelastic problem involving an arbitrary number of arbitrarily oriented planar cracks in an infinite piezoelectric space. The cracks are acted
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
upon by internal stresses that are time dependent and are either electrically impermeable or permeable. The displacement and electric potential in the Laplace transform domain are expressed as a linear combination of suitably constructed exponential Fourier transform representations. The integrands of the Fourier integrals contain unknown functions that are directly related to the jumps in the Laplace transforms of the displacement and electrical potential across opposite crack faces. The unknown functions are to be determined by solving numerically a system of hypersingular integral equations. Once they are determined, the displacements and electric potential and other physical quantities of interest, such as the crack tip stress and electric displacement intensity factors, can be computed with the help of a suitable algorithm for inverting Laplace transforms. The analysis presented here is general in that it covers both inplane and antiplane deformations. The crack tip stress and electric displacement intensity factors are computed for some specific cases of the problem. For the case involving a single planar crack, the values of the stress and electric displacement intensity factors computed are compared with those published in the literature. The solution presented here is applicable to cracks that are either electrically impermeable or permeable. The electrically semi-permeable cracks proposed by Hao and Shen (1994) may be of interest in engineering analysis too. The solution here may be used together with an iterative procedure (such as the one described in Ang and Athanasius (2011)) to analyze the dynamic interaction of electrically semi-permeable cracks. Numerical methods may be applied to solve dynamic piezoelectric problems involving cracks in solids of finite extent. Examples of such numerical works include: Enderlein et al. (2005) (finite element method); García-Sánchez et al. (2008), Kögl and Gaul (2000) and Wünsche et al. (2010) (boundary element method); and Sladek et al. (2006) (meshless method). The semi-analytic solution given here may be used to check the validity of those numerical techniques for solving dynamic piezoelectric crack problems.
609
Fig. 1. A geometrical sketch of the problem.
ðnÞ skj ðx1 ; x2 ; tÞmðnÞ / Pk ðx1 ; x2 ; tÞðk ¼ 1; 2; 3Þ j ðnÞ asðx1 ; x2 Þ/ðx1 ; x2 Þ˛G ðn ¼ 1; 2; /; NÞ;
(3)
and either ðnÞ
ðnÞ
dj ðx1 ; x2 ; tÞmj / P4 ðx1 ; x2 ; tÞ
(4)
asðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; /; NÞ if the cracks are electrically impermeable; or
Dfðx1 ; x2 ; tÞ/0 asðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; .; NÞ if the cracks are electrically impermeable;
2. The problem Referring to an Ox1 x2 x3 Cartesian coordinate system, consider an infinite piezoelectric space that contains N arbitrarily oriented nonintersecting planar cracks whose geometries do not change along the x3 axis. The cracks are denoted by Gð1Þ ; Gð2Þ ; .; GðN1Þ and GðNÞ. The n -th planar crack GðnÞ lies in the region ðnÞ
[ðnÞ < aj1
ðnÞ
xj cj
ðnÞ
< [ðnÞ ; aj2
ðnÞ
xj cj
where sij and di are respectively the stresses and electric displaceðnÞ ðnÞ ðnÞ ðnÞ ments, P1 ðx1 ; x2 ; tÞ;P2 ðx1 ; x2 ; tÞ;P3 ðx1 ; x2 ; tÞ and P4 ðx1 ; x2 ; tÞ are ðnÞ ðnÞ ðnÞ suitably prescribed functions for ðx1 ; x2 Þ˛G , mi ¼ ai2 are the components of a unit magnitude normal vector to the crack GðnÞ and Dfðx1 ; x2 ; tÞ denotes the jump in the electrical potential f across the crack GðnÞ as defined by
h
Dfðx1 ; x2 ; tÞ ¼ lim f x1 jejm1ðnÞ ; x2 jejmðnÞ 2 ;t
¼ 0; N < x3 < N;
i ðnÞ ðnÞ f x1 þ jejm1 ; x2 þ jejm2 ; t for ðx1 ; x2 Þ˛GðnÞ :
where ðnÞ
aij
i
0
ðnÞ sin q B ¼ @ cosqðnÞ 0
ðnÞ cos q ðnÞ sin q 0
0
1
C 0 A: 1
e/0
(1)
h
(5)
(2)
On the Ox1 x2 plane, the crack GðnÞ is a straight line cut, 2[ðnÞ is ðnÞ ðnÞ the length of the crack, ðc1 ; c2 Þ is the midpoint of the crack and qðnÞ is the angle between the crack and the vertical line passing ðnÞ ðnÞ ðnÞ through ðc1 ; c2 Þðsuch that 0 q pÞ, as shown in Fig. 1. Note that the usual convention of summing over a repeated subscript is adopted in (1) for lower case Latin subscripts that run from 1 to 3. It will be assumed that here the electro-elastic deformation of the cracked piezoelectric space does not vary along the x3 direction. The problem is to determine the displacements uk ðx1 ; x2 ; tÞ and electric potential fðx1 ; x2 ; tÞ in the piezoelectric space for time t > 0 such that suitably prescribed boundary conditions on the cracks are satisfied. More specifically, the conditions the cracks are given by
(6)
Furthermore, it is assumed here that the displacements uk and its partial derivative with respect to time (that is, vuk =vt) are both zero at time t ¼ 0 and the stresses skj ðx1 ; x2 ; tÞ and electric displacement dk ðx1 ; x2 ; tÞ generated by the cracks vanish as x21 þ x22 /N. 3. Basic equations of electroelasticity The governing equations for the displacements uk and electric potential f in a homogeneous piezoelectric material are given by
cijk[
v2 u k v2 f v2 u v2 uk v2 f þ e[ij ¼ r 2i ; ejk[ kj[ vxj vx[ vxj vx[ vxj vx[ vxj vx[ vt
(7)
where cijk[ , e[ij and ki[ are the constant elastic moduli, piezoelectric coefficients and dielectric coefficients respectively and r is the density. The constitutive equations relating ðsij ; dj Þ and ðuk ; fÞ are
610
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
vuk vf þ e[ij ; vx[ vx[ vuk vf kjp ; vx[ vxp
sij ¼ cijk[ dj ¼ ejk[
(8)
SIj ¼
CIjK[
uj for J ¼ j ¼ 1; 2; 3; f for J ¼ 4;
sij for I ¼ i ¼ 1; 2; 3; dj for I ¼ 4;
(9)
8 for I ¼ i ¼ 1; 2; 3 and K ¼ k ¼ 1; 2; 3; c > > < eijk[ for I ¼ i ¼ 1; 2; 3 and K ¼ 4; [ij ¼ e for I ¼ 4 and K ¼ k ¼ 1; 2; 3; > > : jk[ kj[ for I ¼ 4 and K ¼ 4;
(10)
In the Laplace transform domain, the problem is to solve (19) subject to
(20)
and either ðnÞ b ðnÞ ðx ; x ; sÞ b S 4j ðx1 ; x2 ; sÞmj / P 1 2 4
ð21Þ
as ðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; .; NÞ if the cracks are electrically impermeable;
vUK ðI ¼ 1; 2; 3; 4; j ¼ 1; 2; 3Þ vx[
(11)
or
b ðx ; x ; sÞ/0 as ðx ; x Þ/ðx ; x Þ˛GðnÞ DU 4 1 2 1 2 1 2
where
(19)
as ðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; .; NÞ;
and
BIK ¼
b v2 U K b ¼ 0 ðI ¼ 1; 2; 3; 4Þ: s2 BIK U K vxj vx[
ðnÞ b ðnÞ ðx ; x ; sÞðI ¼ 1; 2; 3Þ b S Ij ðx1 ; x2 ; sÞmj / P 1 2 I
v 2 UK v 2 UK ¼ BIK ðI ¼ 1; 2; 3; 4Þ vxj vx[ vt 2
SIj ¼ CIjK[
(18)
where s is the Laplace transformation parameter. Application of the Laplace transformation on both sides of (10) together with the initial conditions (13) yields
CIjK[
so that (7) and (8) may be respectively written more compactly as
CIjK[
ZN Fðx1 ; x2 ; tÞexpðstÞdt; 0
Following closely the approach of Barnett and Lothe (1975), we define
UJ ¼
b F ðx1 ; x2 ; sÞ ¼
ðn ¼ 1; 2; .; NÞ if the cracks are electrically permeable:
r if I ¼ K and Is4; 0
(12)
otherwise:
Note that uppercase Latin subscripts have values 1;2; 3 and 4: Summation is also implied for repeated uppercase Latin subscripts. It follows that the problem stated in Section 2 requires solving (10) within the piezoelectric space for time t > 0 subject to the initial-boundary conditions
UK ¼ 0 and
vUK ¼ 0 at t ¼ 0 ðK ¼ 1; 2; 3Þ; vt
ðnÞ SIj ðx1 ; x2 ; tÞmj /
ðnÞ PI ðx1 ; x2 ; tÞ ðnÞ
asðx1 ; x2 Þ/ðx1 ; x2 Þ˛G
It is also required that b S Ij ðx1 ; x2 ; sÞ/0 as x21 þ x22 /N. 5. Method of solution In this section, a method based on the theory of exponential Fourier transformation is proposed for solving (19) subject to (20)e(22).
(13) 5.1. Solution in Fourier integral form
ðI ¼ 1; 2; 3Þ
For the solution of the piezoelectric crack problem, let
(14)
ðn ¼ 1; 2; .; NÞ;
b ðx ; x ; sÞ ¼ U K 1 2
and either ðnÞ
or
DU4 ðx1 ; x2 ; tÞ/0 as ðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; .; NÞ if the cracks are electrically permeable; (16) h
i ðnÞ ðnÞ UI x1 þ j3jm1 ; x2 þ j3jm2
ð17Þ
for ðx1 ; x2 Þ˛GðnÞ : In addition, it is required that SIj ðx1 ; x2 ; tÞ/0 as
a¼1 0
ðnÞ F1a ðx; sÞ
ðnÞ
3/0
where HðxÞ is the unit-step Heaviside function, F1a ðx; sÞ and ðnÞ ðnÞ F2a ðx; sÞ are functions yet to be determined, sa ðx; sÞ (n ¼ 1, 2, ., N) are roots, with positive imaginary parts, of the 8-th order polynomial equation (in s) given by
" x21
8 N
ðnÞ ðnÞ ðnÞ exp ix aj1 þ sa ðx; sÞaj2 ðnÞ ðnÞ ðnÞ þ H ar2 xr cr xj c j ðnÞ ðnÞ ðnÞ ðnÞ F2a ðx; sÞ exp ix aj1 þ sa ðx; sÞaj2 9 i = ðnÞ xj cj dx ; (23) ;
(15)
asðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; .; NÞ if the cracks are electrically impermeable;
DUI ðx1 ; x2 ; tÞ ¼ lim UI x1 j3jmðnÞ 3 ðnÞ 1 ; x2 j jm2
N X n¼1
ðnÞ
S4j ðx1 ; x2 ; tÞmj / P4 ðx1 ; x2 ; tÞ
where
(22)
þ
x22 /N:
4. Formulation in Laplace transform domain We denote the Laplace transformation of Fðx1 ; x2 ; tÞ over time t 0 by b F ðx1 ; x2 ; sÞ; that is, we define
det
s2
x
2
ðnÞ ðnÞ 2 ðnÞ ðnÞ ðnÞ ðnÞ BIK þ a11 þ sa12 CI1K1 þ a21 þ sa22 a11 þ sa12
ðCI1K2 þ CI2K1 Þ þ ðnÞ
ðnÞ a21
þ
2 saðnÞ CI2K2 22
# ¼ 0;
(24)
AK a ðx; sÞ (n ¼ 1, 2, ., N) are non-trivial solutions of the system
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
"
s2 2
x
ðnÞ ðnÞ ðnÞ 2 ðnÞ ðnÞ ðnÞ BIK þ a11 þ sa ðx;sÞa12 CI1K1 þ a21 þ sa ðx;sÞa22
Z[
ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2 a11 þ sa ðx;sÞa12 ðCI1K2 þCI2K1 Þþ a21 þ sa ðx;sÞa22 CI2K2 ðnÞ
AK a ¼ 0:
#
ð25Þ
N X
Re
n¼1
8 N 4 Z
a¼1 0
ðnÞ F1a ðx; sÞ
e2
[
gives ðnÞ
rK
ðnÞ
aj1
ðnÞ
b S Ij ðx1 ;x2 ;sÞ ¼
ðnÞ ðnÞ ðnÞ exp ix aj1 þ sa ðx; sÞaj2
F2a ðx; sÞ ¼ MaP ðx; sÞjP ðx; sÞ;
ðnÞ
mj LIja ðx; sÞMaP ðx; sÞ ¼ dIP ðn ¼ 1; 2; /; NÞ:
Z1
1
H [ðqÞ þ[ðqÞ C
¼ 0;
Use of (31) in (23) together with
(31)
ðqÞ
b ðu; sÞUðqÞ ðu; v; sÞdu DU K IK
1
ðqÞ b ðqÞ ðqÞ ðqÞ s2 GIK D U K ðu; sÞ cosh [ hjv uj ln [ hjv uj du
Z1
[ðnÞ
ðnÞ
b ðu; sÞQðnqÞ ðu; v; sÞdu DU K IK
n ¼ 1; nsq 1 ðqÞ ðqÞ ðqÞ b X1 ðvÞ; X2 ðvÞ; s ðI ¼ 1; 2; 3; 4Þ ¼ p P I for 1 < v < 1ðq ¼ 1; 2; /; NÞ;
(35)
^ ðqÞ ðu; sÞ ¼ pr ðqÞ ð[ðqÞ u; sÞ; X ðqÞ ðvÞ ¼ DU K K 1 ðqÞ ðqÞ ¼ c2 [ðqÞ v cosðq Þ, C denotes that
ðqÞ
ðqÞ
c1 þ [ðqÞ vsinðq
Þ;
the integral is to be interpreted in the Cauchy principal sense and C denotes that the ðqÞ
integral is to be interpreted in the Hadamard finite-part sense, DIK , ðqÞ
GIK and WIK ðx; sÞ are given by ðqÞ
[ðnÞ
i ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ AK a ðx;sÞMaP ðx;sÞAK a ðx;sÞM aP ðx;sÞ TPJ ðx;sÞ ¼ dKJ :
Z1
du þ [ðqÞ
ðv uÞ2
ðqÞ
(30)
pffiffiffiffiffiffiffi ðnÞ where i ¼ 1, rJ ðu; sÞ are real functions yet to be determined ðnÞ and TPJ ðx; sÞ are real functions defined by 4 h X
a ¼1 0
ðqÞ
ðqÞ
N X
(29)
ðnÞ
ðnÞ
[ðnÞ
b ðu; sÞ DIK D U K
1
Z1
þ
ðqÞ X2 ðvÞ
rJ ðu; sÞexpðixuÞdu;
8 N
ðnÞ rK ðu;sÞRe :
1
(28)
where
a ¼1
From (34), conditions (20) and (21) for electrically impermeable cracks give the hypersingular integral equations
b ðx ; x ; sÞ are continuous on the plane The functions U K 1 2 ðnÞ ðnÞ ðnÞ ar2 ðxr cr Þ ¼ 0 at points not on any of the cracks if jP ðx; sÞ are chosen to be
i
ðnÞ
xj cj
5.2. Electrically impermeable cracks
a¼1
ðnÞ jðnÞ x x P ð ; sÞ ¼ iTPJ ð ; sÞ
(27)
where are functions to be determined and the overhead bar denotes the complex conjugate of a complex number and ðnÞ MaP ðx; sÞ are defined by
Z[
ðnÞ
< [ðnÞ ; aj2
ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ LIja ðx;sÞMaP ðx;sÞexp ixsa ðx;sÞak2 xk ck ðnÞ ðnÞ ðnÞ ðnÞ þH ar2 xr cr LIja ðx;sÞM aP ðx;sÞ i ðnÞ ðnÞ ðnÞ ðnÞ TPK ðx;sÞ exp ixsa ðx;sÞak2 xk ck 9 i = h ðnÞ ðnÞ exp ix ak1 xk ck u dx du: (34) ;
(26)
jðnÞ x P ð ; sÞ
ðnÞ ðnÞ
ðnÞ
xj c j
Z[ N X n¼1
F1a ðx; sÞ ¼ MaP ðx; sÞjP ðx; sÞ;
4 X
p
ðnÞ
b ðx ; x ; sÞ in (23) is obtained by The integral representation for U K 1 2 generalizing the analyses in Ang (1987, 1991) and Clements (1981). b ðx ; x ; sÞ and b S Ij ðx1 ; x2 ; sÞ in (23) and (26) respecNote that U K 1 2 tively are represented by different integral expressions in different ðnÞ parts of the piezoelectric space. To ensure that b S Ij ðx1 ; x2 ; sÞmj are ðnÞ ðnÞ ðnÞ continuous on ar2 ðxr cr Þ ¼ 0, the functions F1a ðx; sÞ and ðnÞ F2a ðx; sÞ are chosen to be given by
ðnÞ
b ðx ; x ; sÞ are the Laplace transform of the generalized where D U K 1 2 crack opening displacements as defined in (17). The functions ^ SIj ðx1 ; x2 ; sÞ in (26) can now be written as
ðnÞ ðnÞ ðnÞ exp ix aj1 þ sa ðx; sÞaj2
h ðnÞ ðnÞ ðnÞ LIja ðx; sÞ ¼ a11 þ sna ðx; sÞa12 CIjK1 i ðnÞ ðnÞ ðnÞ þ a21 þ sna ðx; sÞa22 CIjK2 AK a :
ðnÞ
1 b ; s ¼ DU K ðx1 ; x2 ; sÞ
(33)
ðnÞ
ðnÞ
ðnÞ
xj c j
LIja ðx; sÞðn ¼ 1; 2; .; NÞ are given by
ðnÞ
(32)
h ðnÞ ðnÞ ðnÞ ixLIja ðx; sÞ H ar2 xr cr
9 i = ðnÞ xj cj dx ; ;
ðnÞ
du ¼ pjðvÞ for [ < v < [;
þ ðv uÞ2
for [ðnÞ < aj1
ðnÞ ðnÞ ðnÞ H ar2 xr cr xj c j ðnÞ F2a ðx; sÞ
e/0þ
ðnÞ
From (11) and (23), we obtain
b S Ij ðx1 ; x2 ; sÞ ¼
jðuÞ
lim e
611
DIK ¼ ðqÞ
GIK ¼ ðqÞ
lim
lim
ðqÞ
ðx=sÞ/N 2 h
ðx=sÞ/N
x s
ðqÞ
TIK ðx; sÞ; ðqÞ
ðqÞ
WIK ðx; sÞ ¼ TIK ðx; sÞ DIK ðqÞ
ðnqÞ
ðqÞ
TIK ðx; sÞ DIK ðqÞ s2 GIK x2 þ h2
i
;
ðh > 0Þ;
and UIK ðu; v; sÞ and QIK ðu; v; sÞ are respectively defined by
(36)
612
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
ZN
UðqÞ IK ðu; v; sÞ ¼
0 ðqÞ s2 GIK
where sgn ðxÞ denotes the sign of x and
ðqÞ xWIK ðx; sÞcos [ðqÞ x½v u dx
w ðnqÞ
Shi [ðqÞ hjv uj sinh [ðqÞ hjv uj
(37)
a ¼1
h
ðnqÞ xmðqÞ H Y2 ðu;vÞ j
0
ðnÞ ðnÞ LIja ðx;sÞMaP ðx;sÞexp
ðnqÞ ixsa ðx;sÞY2 ðu;vÞ
with
ðnqÞ
Yp
ðnÞ
Zu 0
ðnÞ
ZN
ZN
ðu;vÞs0;
[ðnÞ dp1 u
u
expð xÞ dx; x
ZN 0
and
Note that EiðxÞE1 ðxÞ tend to 2lnðxÞ as x/0þ . This explains the presence of the Cauchy principal integral in (35). If the cracks are electrically impermeable, the functions b ðqÞ ðu; sÞ (q ¼ 1,2, ., N) in (30) are to be determined by solving DU K the hypersingular integral equations in (35). If we make the approximation (Kaya and Erdogan, 1987) ðqÞ
J pffiffiffiffiffiffiffiffiffiffiffiffiffiffi X j¼1
The functions WIK ðx; sÞ behave as Oðs4 =x Þ for very large x: Thus, the improper integral over ½0; NÞ that appears in the definition of UðqÞ IK ðu; v; sÞ in (37) is well defined. ðnqÞ Note that the expressions for QIK ðu; v; sÞ as given in (38) are ðnqÞ ðnqÞ valid for Y2 ðu; vÞs0. If Y2 ðu; vÞ ¼ 0 then (38) has to be modified accordingly. The modification gives
8 <
w ðnqÞ
D
h
IK
: Y1ðnqÞ ðu; vÞ
ZN i2
w ðnqÞ
xW
IK
(41)
x p sinðaxÞdx ¼ sgnðaÞ½coshðjahjÞsinhðjahjÞ: 2 x2 þ h2
b ðu; sÞx 1 u2 DU K 4
ðnÞ
(42)
(39)
ðqÞ
ðnÞ
a¼1
þShiðahÞsinhðahÞðah > 0Þ;
(38)
expð xÞ dx: x
QðnqÞ IK ðu; v; sÞ ¼ Re
ðx; sÞ ¼
ðqÞ ðnÞ
mj LIja ðx; sÞMaP ðx; sÞTIK ðx; sÞ;
ðnqÞ w x2 ~ ðnqÞ ~ ðnqÞ T IK ðx; sÞ D G IK ¼ lim IK ðx=sÞ/N s 2 ~ ðnqÞ ~ ðnqÞ ðx; sÞ ¼ T~ ðnqÞ ðx; sÞ D ~ ðnqÞ s GIK ðh > 0Þ: W IK IK IK x2 þ h2 IK
0
sinhðxÞ dx; x
EiðuÞ ¼ C
u
ðqÞ
ðu; vÞ ¼ akp Xk ðvÞ ck
ShiðuÞ ¼
E1 ðuÞ ¼
ðnqÞ T~ IK ðx; sÞ;
Z1 e2 ðvuÞ2 jðuÞ Z1 jðuÞ lim du for 1 < v < 1; 2 du ¼ H 2 e/0þ 2 ðvuÞ e2 þðvuÞ 1 1 ZN x 1 cosðaxÞdx ¼ coshðahÞðEiðahÞE1 ðahÞÞ 2 x2 þ h2
ðnÞ ðnÞ ðnqÞ þH Y2 ðu;vÞ LIja ðx;sÞM aP ðx;sÞ i ðnÞ ðnqÞ exp ixsa ðx;sÞY2 ðu;vÞ 9 = ðnÞ ðnqÞ TPK ðx;sÞexp ixY1 ðu;vÞ dx ; ðnqÞ
lim
ðx=sÞ/N 4 P
The derivations of (35), (37) and (40) make use of the following results:
ðnÞ
if it is assumed that Y2
¼
ðqÞ ðnÞ If the cracks GðnÞ and GðqÞ are coplanar then mj ¼ mj , hence ðnqÞ ðnqÞ ðnqÞ ðnqÞ ðnÞ ðnÞ ~ ðnÞ ðnÞ ~ ~ T~ IK ðx;sÞ¼TIK ðx;sÞ, D IK ¼DIK ;GIK ¼GIK and W IK ðx;sÞ¼WIK ðx;sÞ. ðnqÞ ðnÞ ðqÞ Note that Y2 ðu;vÞ¼0 for all u and v if G and G are coplanar.
and
8 ZN
IK
w ðnqÞ
T
1 cosh [ðqÞ hjv uj Ei [ðqÞ hjv uj 2 E1 [ðqÞ hjv uj þcosh [ðqÞ hjv uj ln [ðqÞ hjv uj ;
QðnqÞ IK ðu;v;sÞ ¼ Re:
D
ðj1Þ uðqjÞ ðuÞ; K ðsÞU
(43)
where U ðjÞ ðxÞ ¼ sinð½j þ 1arccosðxÞÞ=sinðarccosðxÞÞ is the jth order ðnjÞ Chebyshev polynomial of the second kind and uP ðsÞ are unknown coefficients, then (35) can be used to set up a system of linear ðnjÞ algebraic equations to determine uP ðsÞ for any fixed value of s (Athanasius et al., 2010).
ðnqÞ ðx; sÞexp ixY1 ðu; vÞ dx
0
w ðnqÞ
1 ðnqÞ ðnqÞ ðnqÞ Shi h Y1 ðu; vÞ sinh h Y1 ðu; vÞ cosh h Y1 ðu; vÞ 2 p ðnqÞ ðnqÞ ðnqÞ Ei h Y1 ðu; vÞ E1 h Y1 ðu; vÞ þ i sgn Y1 ðu; vÞ 2 9 = ðnqÞ ðnqÞ cosh h Y1 ðu; vÞ sinh h Y1 ðu; vÞ ; s2 G
IK
ðnqÞ
if Y2
ðu; vÞ ¼ 0;
(40)
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
613
ðnjÞ
^ ðqÞ ðu; sÞ ¼ 0 for 1 < u < 1 and q ¼ 1, 2, From (16) and (33), DU 4 ., N, if the cracks are electrically permeable. According to (14), the ^ ðqÞ ðu; sÞ;DU ^ ðqÞ ðu; sÞ and DU ^ ðqÞ ðu; sÞ that can unknown functions DU 1 2 3 be approximated as above by (43) are governed by (35) (with ðqÞ ^ ðu; sÞ ¼ 0) for I ¼ 1, 2, 3 (instead of I ¼ 1,2,3,4). DU 4
Once the coefficients uP ðsÞ in (43) are determined, the above intensity factors can be approximately calculated in the Laplace transform domain using The dynamics stress and electric displacement intensity factors at any time t may be recovered by using the numerical Laplace transform algorithm in Stehfest (1970), that is, by using the formula
6. Stress and electric displacement intensity factors
f ðtÞx
5.3. Electrically permeable cracks
The dynamic stress and electric displacement intensity factors ðnÞ ðnÞ ðnÞ ðnÞ at the tips ðX1 ð1Þ; X2 ð1ÞÞ and ðX1 ð1Þ; X2 ð1ÞÞ of the n-th ðnÞ crack G are defined as follows:
2M lnð2Þ X nlnð2Þ Vn ^f ; t n¼1 t
(46)
where ^f ðsÞ denotes the Laplace transform of f ðtÞ, M is a positive integer and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu þ 1Þ S1j X1 ðuÞ; X2 ðuÞ; t m1 þ S2j X1 ðuÞ; X2 ðuÞ; t m2 mj ; KI X1 ð1Þ; X2 ð1Þ; t ¼ lim u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu þ 1Þ S1j X1 ðuÞ; X2 ðuÞ; t m2 S2j X1 ðuÞ; X2 ðuÞ; t m1 mj ; KII X1 ð1Þ; X2 ð1Þ; t ¼ lim u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu þ 1ÞS3j X1 ðuÞ; X2 ðuÞ; t mj ; KIII X1 ð1Þ; X2 ð1Þ; t ¼ lim u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu þ 1ÞS4j X1 ðuÞ; X2 ðuÞ; t mj ; KIV X1 ð1Þ; X2 ð1Þ; t ¼ lim u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu 1Þ S1j X1 ðuÞ; X2 ðuÞ; t m1 þ S2j X1 ðuÞ; X2 ðuÞ; t m2 mj ; KI X1 ð1Þ; X2 ð1Þ; t ¼ lim þ u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu 1Þ S1j X1 ðuÞ; X2 ðuÞ; t m2 S2j X1 ðuÞ; X2 ðuÞ; t m1 mj ; KII X1 ð1Þ; X2 ð1Þ; t ¼ lim u/1þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu 1ÞS3j X1 ðuÞ; X2 ðuÞ; t mj ; KIII X1 ð1Þ; X2 ð1Þ; t ¼ lim þ u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu 1ÞS4j X1 ðuÞ; X2 ðuÞ; t mj : KIV X1 ð1Þ; X2 ð1Þ; t ¼ lim
(44)
u/1þ
J X ðnÞ ðnÞ ðnÞ ðnÞ ðj1Þ ^ X ðnÞ ð 1Þ; X ðnÞ ð 1Þ; s xp1 ffiffiffiffiffiffiffiffi DP1 m1 þ DP2 m2 uðnjÞ ð 1Þ; K P ðsÞU 1 2 I [ðnÞ j¼1 J X ðnÞ ðnÞ ðnÞ ðnÞ ðj1Þ ^ II X ðnÞ ð 1Þ; X ðnÞ ð 1Þ; s xp1 ffiffiffiffiffiffiffiffi DP1 m2 DP2 m1 uðnjÞ K ð 1Þ; P ðsÞU 1 2 ðnÞ [ j¼1 J X ðnÞ ðj1Þ ^ III X ðnÞ ð 1Þ; X ðnÞ ð 1Þ; s x p1 ffiffiffiffiffiffiffi ffi uðnjÞ K ð 1Þ; D P ðsÞU 1 2 P3 [ðnÞ j¼1 J X 1 ðnÞ ðnÞ ðnÞ ðj1Þ ^ uðnjÞ K ð 1Þ; IV X1 ð 1Þ; X2 ð 1Þ; s x pffiffiffiffiffiffiffiffi DP4 P ðsÞU [ðnÞ j¼1 J X ðnÞ ðnÞ ðnÞ ðnÞ ðj1Þ ^ I X ðnÞ ð1Þ; X ðnÞ ð1Þ; s xp1 ffiffiffiffiffiffiffiffi DP1 m1 þ DP2 m2 uðnjÞ K ð þ 1Þ; P ðsÞU 1 2 ðnÞ [ j¼1 J X ðnÞ ðnÞ ðnÞ ðnÞ ðj1Þ ^ II X ðnÞ ð1Þ; X ðnÞ ð1Þ; s xp1 ffiffiffiffiffiffiffiffi DP1 m2 DP2 m1 uðnjÞ K ð þ 1Þ; P ðsÞU 1 2 [ðnÞ j¼1 J X ðnÞ ðj1Þ ^ X ðnÞ ð1Þ; X ðnÞ ð1Þ; s x p1 ffiffiffiffiffiffiffi ffi uðnjÞ ð þ 1Þ; K D III P ðsÞU 1 2 P3 [ðnÞ j¼1 J X ðnÞ ðj1Þ ^ IV X ðnÞ ð1Þ; X ðnÞ ð1Þ; s x p1 ffiffiffiffiffiffiffiffi DP4 uðnjÞ K ð þ 1Þ: P ðsÞU 1 2 ðnÞ [ j¼1
(45)
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Vn ¼ ð1ÞnþM
minðn;MÞ X
mM ð2mÞ! ; ðM mÞ!m!ðm1Þ!ðnmÞ!ð2mnÞ! m¼½ðnþ1Þ=2 (47)
with ½r denoting the integer part of the real number r: Note that the Stehfest’s algorithm requires the problem under consideration to be solved for only real Laplace transform parameter s: It has been widely used by researchers for the numerical inversion of Laplace transforms in solving many problems in engineering (see, for example, Ang (1987); Hemker (1999) and Smith et al. (1994)). 7. Specific cases In this section, the dynamics crack tip stress and electric displacement intensity factors are computed for some specific cases of the problem. Problem 1. Consider the case of a single electrically impermeable crack in an infinite piezoelectric space. The crack lies in the region a < x1 < a; x2 ¼ 0; and the only non-zero uniform load acting on it is given by S22 ¼ HðtÞs0 ; where HðtÞ is the unit-step Heaviside function. (Note that S12 ; S32 and S42 are taken to be zero on the crack.) The electrical poling direction is taken to be along the x2 direction so that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Fig. 2. Plots of KI =ðs0 aÞ against the non-dimensionalized time t L=ðra2 Þ.
obtaining the plots in Fig. 2. Convergence is observed in the numerical values of the intensity factors when the number of terms in the formula is increased to 10: The numerical inversion of Laplace transform becomes unstable when the number of terms is increased beyond 10: To use more terms, it is necessary to refine the
C1111 ¼ C3333 ¼ A; C1133 ¼ C3311 ¼ N; C2222 ¼ C; C1122 ¼ C2211 ¼ C2233 ¼ C3322 ¼ F; C1212 ¼ C2112 ¼ C2121 ¼ C1221 ¼ C2323 ¼ C3223 ¼ C3232 ¼ C2332 ¼ L; 1 C1313 ¼ C3113 ¼ C3131 ¼ C1331 ¼ ðA NÞ; 2 C2141 ¼ C1241 ¼ C3243 ¼ C2343 ¼ C4121 ¼ C4112 ¼ C4332 ¼ C4323 ¼ e1 ; C1142 ¼ C3342 ¼ C4211 ¼ C4233 ¼ e2 ; C2242 ¼ C4222 ¼ e3 ; C4141 ¼ C4343 ¼ e1 ; C4242 ¼ e2 ;
where A, N, F, C, L, e1, e2, e3, e1 and e2 are independent constants. The piezoelectric material is PZT-BaTio3 with material constants A, N, F, C, L, e1, e2, e3, e1 and e2 and density r given by
1010 ;
1010 ; F
A ¼ 15:0 N ¼ 7:78 ¼ 6:6 C ¼ 14:6 1010 ; L ¼ 4:4 1010 ; e1 ¼ 11:4; e2 ¼ 4:35; e3 ¼ 17:5; e1 ¼ 98:7 1010 ; e2 ¼ 112 1010 ; r ¼ 5800:
1010 ; (49)
The values of A;N; F, C and L above are in N/m2, e1 ; e2 and e3 are in C/m2, e1 and e2 are in C/(Vm) and r in kg/m3. pffiffiffi The numerical stress intensity factor KI =ðs0 aÞ and electric pffiffiffi tip ða; 0Þ displacement intensity factor CKIV =ðe3 s0 aÞ at the crack pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi are plotted against the non-dimensionalized time t L=ðra2 Þ in Figs. 2 and 3 respectively. The results are obtained by using J ¼ 10 in the numerical solution of the hypersingular integral equations and M ¼ 4 (8 terms) in the Stehfest’s formula for inverting Laplace pffiffiffi transform. In Figs. 2 and 3, the numerical values of KI =ðs0 aÞ and pffiffiffi CKIV =ðe3 s0 aÞ are also compared with those extracted from Shindo et al. (1999) and García-Sánchez et al. (2007). (Numerical values of pffiffiffi CKIV =ðe3 s0 aÞ are not given in Shindo et al. (1999)). All the plots of pffiffiffi KI =ðs0 aÞ in Fig. 2 are quite close to one another, exhibiting the same general trends and reaching peak values at about the same pffiffiffi time. So are the plots of CKIV =ðe3 s0 aÞ in Fig. 3. As pointed out earlier on, 8 terms ðM ¼ 4Þ are used in the Stehfest’s formula for
(48)
calculation to obtain more a more accurate solution of the hypersingular integral equations. A higher arithmetic precision in the computing machine is needed too. Problem 2. Consider a pair of coplanar cracks, each of length 2a; as shown in Fig. 4. The distance between the inner tips of the cracks is 2d: The uniform tractions acting on the crack faces are given by S22 ¼ HðtÞs0 : Here electrically impermeable and permeable cracks will be examined separately. For the case in which the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Fig. 3. Plots of CKIV =ðe3 s0 aÞ against the non-dimensionalized time t L=ðra2 Þ.
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
Fig. 4. A pair of coplanar cracks.
coplanar cracks are electrically impermeable, the condition S42 ¼ HðtÞD0 ; where D0 is a constant, applies. The electrical poling is along the x2 direction. For the purpose of obtaining some numerical results for the dynamic stress and electric displacement intensity factors at the inner and outer tips of the coplanar cracks, the material constants of PZT-BaTio3 (as in Problem 1) are used and the load ratio s0 =D0 for electrically impermeable cracks is taken to be 1010 NC1: In Fig. 5, the non-dimensionalized stress intensity factor pffiffiffi crack tips are plotted against the KI =ðs0 aÞ at the inner and outer pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi non-dimensionalized time t L=ðra2 Þ for d=a ¼ 0:125;0:50 and pffiffiffi 0:25: The plots of KI =ðs0 aÞ for electrically permeable and impermeable cracks are almost indistinguishable. Thus, plots of pffiffiffi KI =ðs0 aÞ are given in Fig. 5 for only electrically impermeable pffiffiffi cracks. In each of the plots, KI =ðs0 aÞ increases rapidly to a peak value before settling down to the corresponding value of the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Fig. 5. Plots of KI =ðs0 aÞ against the non-dimensionalized time t L=ðra2 Þ at inner and outer crack tips of electrically impermeable cracks for selected values of d/a.
615
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Fig. 6. Plots of CKIV =ðe3 s0 aÞ against the non-dimensionalized time t L=ðra2 Þ at inner and outer crack tips of electrically impermeable cracks for selected values of d=a:
static stress intensity factor. For each of the values of d=a in Fig. 5, pffiffiffi the peak values of KI =ðs0 aÞ at the inner and the outer crack tips are significantly different and the peak value at the inner tips is larger than that at the outer tips. As may be expected, the peak pffiffiffi value of KI =ðs0 aÞ at each crack tip is larger if the cracks are closer to each other : Further calculations show that the plot of pffiffiffi KI =ðs0 aÞ at the inner tips is almost identical as that at the outer tips for d/a > 3. Plots of the non-dimensionalized electric displacement intenpffiffiffi sity factors CKIV =ðe3 s0 aÞ at the inner and the outer crack tips pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi against t L=ðra2 Þ for d/a ¼ 0.125, 0.50 and 0.25 are given in Figs. 6 and 7 for electrically impermeable and permeable cracks pffiffiffi respectively. The plots of CKIV =ðe3 s0 aÞ for electrically impermeable cracks are distinct from those for electrically permeable pffiffiffi cracks. For a fixed d=a, the peak value of CKIV =ðe3 s0 aÞ at each crack tip is apparently higher for the electrically permeable cracks than that for the impermeable cracks.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Fig. 7. Plots of CKIV =ðe3 s0 aÞ against the non-dimensionalized time t L=ðra2 Þ at inner and outer crack tips of electrically permeable cracks for selected values of d/a.
616
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pffiffiffi Fig. 10. Plots of CKII =ðF s0 aÞ against the non-dimensionalized time for selected values of d/a.
Fig. 8. Two parallel cracks.
Problem 3. Consider two equal length parallel electrically impermeable cracks as sketched in Fig. 8. The half length of each crack is given by a. The centers of the cracks lie on a vertical line and are separated by a distance denoted by d: The non-zero constant loads acting on the crack faces are given by S22 ¼ HðtÞs0 and S42 ¼ HðtÞD0 , with s0 =D0 ¼ 1010 NC1. The electrical poling is along the x2 direction. Using the material constants of PZT-BaTio3 (as in Problem 1), for selected values of d/a, we plot the non-dimensionalized crack tip stress intensity factors pffiffiffi pffiffiffi KI =ðs0 aÞ and CKII =ðF s0 aÞ and the non-dimensionalized crack tip pffiffiffi electric displacement intensity factor CKIV =ðe3 s0 aÞ against the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 non-dimensionalized time t L=ðra Þ in Figs. 9, 10 and 11 respectively. In Fig. 9, for a given d/a, the non-dimensionalized stress intenpffiffiffi sity factor KI =ðs0 aÞ rises to a peak (maximum) value and then drops to a trough (mininum) value before gradually settling down to approach its static value. Both the trough and the peak values decrease in magnitude as d/a decreases. A similar observation may be made of the non-dimensionalized electric displacement intenpffiffiffi sity factor CKIV =ðe3 s0 aÞ in Fig. 11.
pffiffiffi Fig. 9. Plots of KI =ðs0 aÞ against the non-dimensionalized time for selected values of d/a.
As d/a tends to infinity, the non-dimensionalized stress pffiffiffi intensity factor CKII =ðF s0 aÞ vanishes. Nevertheless, when the cracks come close to each other, there is an increase in the pffiffiffi magnitude CKII =ðF s0 aÞ due to larger differences in the stress distribution on opposite crack faces. This is shown in Fig. 10. For pffiffiffi d/a ¼ 10, the magnitude of CKII =ðF s0 aÞ is very small at all time. Note that the fact that KII is not zero for the parallel cracks in Fig. 8 may be explained by the well known phenomenon called Poisson effect. Due to Poisson effect, compressive stresses are generated on opposite crack faces. For the parallel cracks, they are unequal, thereby giving rise to shear stresses on each of the cracks. Problem 4. Consider now the two pairs of electrically impermeable cracks, which are of equal length 2a; in the piezoelectric space, as sketched in Fig. 12. The faces of the horizontal cracks are subject to internal uniform loads given by S12 ¼ S22 ¼ 0;S32 ¼ HðtÞs0 and S42 ¼ HðtÞD0 , such that s0 =D0 ¼ 1010 NC : The internal loads on the faces of the vertical cracks are given by SK1 ¼ 0 (for K ¼ 1; 2; 3; 4). The electrical poling is in the x3 direction, so that CIjKl are given in terms of the independent constants A, N, F, C, L, e1, e2, e3, e1 and e2 by
pffiffiffi Fig. 11. Plots of CKIV =ðe3 s0 aÞ against the non-dimensionalized time for selected values of d/a.
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
C1111 ¼ C2222 ¼ A; C1122 ¼ C2211 ¼ N; C3333 ¼ C; C1133 ¼ C3311 ¼ C2233 ¼ C3322 ¼ F; C1313 ¼ C3113 ¼ C3131 ¼ C1331 ¼ C2323 ¼ C3223 ¼ C3232 ¼ C2332 ¼ L; 1 C1212 ¼ C2112 ¼ C2121 ¼ C1221 ¼ ðA NÞ; 2 C3141 ¼ C1341 ¼ C2342 ¼ C3242 ¼ C4131 ¼ C4113 ¼ C4223 ¼ C4232 ¼ e1 ; C1143 ¼ C2243 ¼ C4311 ¼ C4322 ¼ e2 ; C3343 ¼ C4333 ¼ e3 ; C4141 ¼ C4242 ¼ e1 ; C4343 ¼ e2 :
As in the problems above, the material occupying the piezoelectric space is taken to be PZT-BaTio3 with material constants as given in (49). The deformation of the cracks is antiplane. The non-dimensionalized crack tip stress and electrical displacement intensity pffiffiffi pffiffiffi factors KIII =ðs0 aÞ and KIV =ðD0 aÞ are of interest here. For b=a ¼ 1:25 and a few selected values of d=a; these intensity factors at the lower tip of the left vertical crack are plotted against the non-
617
(50)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dimensionalized time t L=ðra2 Þ in Figs. 13 and 14. For the same values of b=a and d=a; plots of the intensity factors at the left tip of the upper horizontal crack are given in Figs. 15 and 16. As shown pffiffiffi in Figs. 14 and 16, KIV =ðD0 aÞ at both crack tips does not vary with pffiffiffi time. In Figs. 13 and 15, for a fixed b/a and d/a, KIII =ðs0 aÞ rises quickly to a peak value, drops to a trough value and gradually approaches the corresponding static value. As expected, as d=a pffiffiffi pffiffiffi increases, both KIII =ðs0 aÞ and KIV =ðD0 aÞ for the vertical cracks decrease in magnitude, becoming closer to zero.
Fig. 12. Two pairs of cracks.
pffiffiffi Fig. 14. Plots of KIV =ðD0 aÞ at the upper tip of the left vertical crack against the nondimensionalized time for b/a ¼ 1.25 and selected values of d/a.
pffiffiffi Fig. 13. Plots of KIII =ðs0 aÞ at the upper tip of the left vertical crack against the nondimensionalized time for b=a ¼ 1:25 and selected values of d=a.
pffiffiffi Fig. 15. Plots of KIII =ðs0 aÞ at the left tip of the upper horizontal crack against the nondimensionalized time for b/a ¼ 1.25 and selected values of d/a.
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pffiffiffi Fig. 16. Plots of KIV =ðD0 aÞ at the left tip of the upper horizontal crack against the non-dimensionalized time for b/a ¼ 1.25 and selected values of d/a.
8. Summary Through the use of Laplace and exponential Fourier transforms, a semi-analytic solution is derived for an electroelastodynamic problem involving an arbitrary number of arbitrarily located planar cracks in a piezoelectric space. The problem is eventually reduced to a solving a system of hypersingular integral equations. The unknown functions in the hypersingular integral equations are the Laplace transforms of the jumps in the displacements and electric potential across opposite crack faces. Once they are determined, the crack tip stress and electric displacement intensity factors can be easily computed in the Laplace transform domain. A numerical technique for inverting Laplace transforms is employed to recover the intensity factors in the physical domain. The solution is applied to study some specific cases of the problem. For the case of a single crack under impact loadings, the computed crack tip stress and electric displacement intensity factors are found to be in reasonably good agreement with those published in the literature. Numerical results are also obtained for other cases which include one which involves four interacting planar cracks under antiplane deformations. Acknowledgements The first author (WT Ang) acknowledges the sponsorship of Singapore Ministry of Education Tier 1 Research Grant RG9/09. The second author (L Athanasius) is supported by a research scholarship from Nanyang Technological University. The authors would like to thank the anonymous reviewers for their constructive comments which led to significant improvement in the paper.
References Ang, W.T., Athanasius, L., 2011. A boundary integral approach for plane analysis of electrically semi-permeable planar cracks in a piezoelectric solid. Engineering Analysis with Boundary Elements 35, 647e656. Ang, W.T., 1987. Transient response of a crack in an anisotropic strip. Acta Mechanica 70, 97e109. Ang, W.T., 1991. A pair of arbitrarily-oriented coplanar cracks in an anisotropic elastic slab. Journal of Australian Mathematical Society B 32, 284e295.
Athanasius, L., Ang, W.T., Sridhar, I., 2010. Electro-elastostatic analysis of multiple cracks in an infinitely long piezoelectric strip: a hypersingular integral approach. European Journal of Mechanics-A/Solids 29, 410e419. Barnett, D.M., Lothe, J., 1975. Dislocations and line charges in anisotropic piezoelectric insulators. Physica Status Solidi (b) 67, 105e111. Chen, Z.T., Karihaloo, B.L., 1999. Dynamic response of a cracked piezoelectric ceramic under arbitrary electro-mechanical impact. International Journal of Solids and Structures 36, 5125e5133. Chen, Z.T., Meguid, S.A., 2000. The transient response of a piezoelectric strip with a vertical crack under electromechanical impact load. International Journal of Solids and Structures 37, 6051e6062. Chen, Z.T., Worswick, M.J., 2000. Antiplane mechanical and inplane electric timedependent load applied to two coplanar cracks in piezoelectric ceramic material. Theoretical and Applied Fracture Mechanics 33, 173e184. Chen, Z.T., 2006. Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load. International Journal of Fracture 141, 395e402. Clements, D.L., 1981. Boundary Value Problems Governed by Second Order Elliptic Systems. Pitman, London. Enderlein, M., Ricoeur, A., Kuna, M., 2005. Finite element techniques for dynamic crack analysis in piezoelectrics. International Journal of Fracture 134, 191e208. García-Sánchez, F., Zhang, Ch, Sládek, J., Sládek, V., 2007. 2-D transient dynamic crack analysis in piezoelectric solids by BEM. Computational Materials Science 39, 179e186. García-Sánchez, F., Zhang, Ch, Sáez, A., 2008. 2-D transient dynamic analysis of cracked piezoelectric solids by a time domain BEM. Computer Methods in Applied Mechanics and Engineering 197, 3108e3121. Hao, T.H., Shen, Z.Y., 1994. A new electric boundary condition of electric fracture mechanics and its applications. Engineering Fracture Mechanics 47, 793e802. Hemker, C.J., 1999. Transient well flow in vertically heterogeneous aquifer. Journal of Hydrology 225, 1e18. Kaya, A.C., Erdogan, F., 1987. On the solution of integral equations with strongly singular kernels. Quarterly of Applied Mathematics 45, 105e122. Kögl, M., Gaul, L., 2000. A boundary element method for transient piezoelectric analysis. Engineering Analysis with Boundary Elements 24, 591e598. Kuna, M., 2010. Fracture mechanics of piezoelectric materials-where are we right now? Engineering Fracture Mechanics 77, 309e326. Kwon, S.M., Lee, K.Y., 2001. Edge cracked piezoelectric ceramic block under electromechanical impact loading. International Journal of Fracture 112, 139e150. Li, X.F., Fan, T.Y., 2002. Transient analysis of a piezoelectric strip with a permeable crack under anti-plane impact loads. International Journal of Engineering Science 40, 131e143. Li, X.F., Lee, K.Y., 2004. Dynamic behavior of a piezoelectric ceramic layer with two surface cracks. International Journal of Solids and Structures 41, 3193e3209. Li, X.F., Lee, K.Y., 2006. Transient response of a semi-infinite piezoelectric layer with a surface permeable crack. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 57, 636e651. Li, X.F., Tang, G.J., 2003. Transient response of a piezoelectric ceramic strip with an eccentric crack under electromechanical impacts. International Journal of Solids and Structures 40, 3571e3588. Li, X.F., 2002. Closed-form solution for a piezoelectric strip with two collinear cracks normal to the strip boundaries. European Journal of Mechanics-A/Solids 21, 981e989. Meguid, S.A., Chen, Z.T., 2001. Transient response of a finite piezoelectric strip containing coplanar insulating cracks under electromechanical impact. Mechanics of Materials 33, 85e96. Shindo, Y., Narita, F., Ozawa, E., 1999. Impact response of a finite crack in an orthotropic piezoelectric ceramic. Acta Mechanica 137, 99e107. Shindo, Y., Watanabe, K., Narita, F., 2000. Electroelastic analysis of a piezoelectric ceramic strip with a central crack. International Journal of Engineering Science 38, 1e19. Sladek, J., Sladek, V., Zhang, Ch, García-Sánchez, F., Wünsche, M., 2006. Meshless local Petrov-Galerkin method for plane piezoelectricity. Computer, Material and Continua 4, 109e118. Smith, R.S., Edwards, R.N., Beselli, G., 1994. An automatic technique for presentation of coincident-loop, impulse-response, transient electromagnetic data. Geophysics 59, 1542e1550. Sneddon, I.N., Lowengrub, M., 1969. Crack Problems in the Classical Theory of Elasticity. Wiley, New York. Stehfest, H., 1970. Numerical inversion of the Laplace transform. Communications of ACM 13, 47e49 (see also p624). Wang, B.L., Mai, Y.W., 2002. A piezoelectric material strip with a crack perpendicular to its boundary surfaces. International Journal of Solids and Structures 39, 4501e4524. Wang, C.Y., Zhang, Ch, 2005. 3-D and 2-D dynamic Green’s functions and timedomain BIEs for piezoelectric solids. Engineering Analysis with Boundary Elements 29, 454e465. Wünsche, M., García-Sánchez, F., Sáez, A., Zhang, Ch, 2010. A 2D time-domain collocation Galerkin BEM for dynamic crack analysis in piezoelectric solids. Engineering Analysis with Boundary Elements 34, 377e387.
Contents lists available at ScienceDirect
European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol
Dynamic interaction of multiple arbitrarily oriented planar cracks in a piezoelectric space: A semi-analytic solution W.T. Ang*, L. Athanasius School of Mechanical and Aerospace Engineering, Nanyang Technological University, Republic of Singapore
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 December 2010 Accepted 11 March 2011 Available online 21 March 2011
The problem of determining the electro-elastic fields around arbitrarily oriented planar cracks in an infinite piezoelectric space is considered. The cracks which are acted upon by a transient load are either electrically impermeable or permeable. A semi-analytic method based on the theory of exponential Fourier transformation is proposed for solving the problem in the Laplace transform domain. The Laplace transforms of the jumps in the displacements and electric potential across opposite crack faces are determined by solving a system of hypersingular integral equations. Once these displacement and electric potential jumps are obtained, the displacements and electric potential and other physical quantities of interest, such as the crack tip stress and electric displacement intensity factors, can be computed with the help of a suitable algorithm for inverting Laplace transforms. The stress and electric displacement intensity factors are computed for some specific cases of the problem. 2011 Elsevier Masson SAS. All rights reserved.
Keywords: Multiple cracks Transient loads Piezoelectric materials Hypersingular integral equations
1. Introduction In recent years, the problem of determining the electro-elastic fields around cracks in piezoelectric materials has been a subject of considerable interest among many researchers. It appears that the majority of analyses on piezoelectric crack problems (see, for example, Athanasius et al. (2010), Li (2002), Shindo et al. (2000) and Wang and Mai (2002)) are concerned with electro-elastostatic deformations. According to Kuna (2010), there are comparatively fewer works on piezoelectric cracks that are acted upon by time dependent loads. The governing partial differential equations for piezoelectric materials undergoing dynamic antiplane deformations are relatively simpler in form, being reducible to a pair of equations comprising the two-dimensional Helmholtz and Laplace’s equations. Thus, most papers giving semi-analytic solutions for dynamic piezoelectric crack problems assume that the cracks undergo antiplane deformations. For example, the dynamic response of a single electrically impermeable planar crack in an infinite transversely isotropic piezoelectric material under pure electric load and undergoing an antiplane deformation is investigated by Chen (2006). Chen and Karihaloo (1999), Chen and Meguid (2000), Li and Fan (2002) and Li and Tang (2003) have solved problems involving a single planar crack in an infinitely long piezoelectric * Corresponding author. E-mail address: [email protected] (W.T. Ang). URL: http://www.ntu.edu.sg/home/mwtang/ 0997-7538/$ e see front matter 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2011.03.003
strip under antiplane deformations. Coplanar cracks undergoing antiplane deformations in piezoelectric materials are examined in Chen and Worswick (2000) and Meguid and Chen (2001). Kwon and Lee (2001) and Li and Lee (2004, 2006) have investigated the antiplane deformation of edge cracks in piezoelectric materials. There are apparently few papers in which semi-analytic solutions for cracks undergoing dynamic inplane deformations in piezoelectric materials may be found. One of them is Shindo et al. (1999). In Shindo et al. (1999), the problem of a single planar crack in a piezoelectric ceramic under normal impact is formulated in terms of a pair of dual integral equations by representing the displacement and electric potential in the Laplace transform domain by suitable Fourier sine and cosine transform representations. The dual integral equations are solved as explained in Sneddon and Lowengrub (1969), by reducing them to Fredholm integral equations of the second kind. The dynamic piezoelectric problem can also be formulated in terms of hypersingular integral equations using the approach in a recent paper by García-Sánchez et al. (2007). In García-Sánchez et al. (2007), the kernels of the hypersingular integral formulation contain second order spatial derivatives of a suitable dynamic Green’s function for piezoelectric solids. The Green’s function is derived using Radon transform. Its evaluation is a rather involved exercise, requiring the computation of a line integral over a unit circle with integrand that is expressed in terms of exponential integrals (Wang and Zhang, 2005). The present paper derives a semi-analytic solution for an electroelastic problem involving an arbitrary number of arbitrarily oriented planar cracks in an infinite piezoelectric space. The cracks are acted
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
upon by internal stresses that are time dependent and are either electrically impermeable or permeable. The displacement and electric potential in the Laplace transform domain are expressed as a linear combination of suitably constructed exponential Fourier transform representations. The integrands of the Fourier integrals contain unknown functions that are directly related to the jumps in the Laplace transforms of the displacement and electrical potential across opposite crack faces. The unknown functions are to be determined by solving numerically a system of hypersingular integral equations. Once they are determined, the displacements and electric potential and other physical quantities of interest, such as the crack tip stress and electric displacement intensity factors, can be computed with the help of a suitable algorithm for inverting Laplace transforms. The analysis presented here is general in that it covers both inplane and antiplane deformations. The crack tip stress and electric displacement intensity factors are computed for some specific cases of the problem. For the case involving a single planar crack, the values of the stress and electric displacement intensity factors computed are compared with those published in the literature. The solution presented here is applicable to cracks that are either electrically impermeable or permeable. The electrically semi-permeable cracks proposed by Hao and Shen (1994) may be of interest in engineering analysis too. The solution here may be used together with an iterative procedure (such as the one described in Ang and Athanasius (2011)) to analyze the dynamic interaction of electrically semi-permeable cracks. Numerical methods may be applied to solve dynamic piezoelectric problems involving cracks in solids of finite extent. Examples of such numerical works include: Enderlein et al. (2005) (finite element method); García-Sánchez et al. (2008), Kögl and Gaul (2000) and Wünsche et al. (2010) (boundary element method); and Sladek et al. (2006) (meshless method). The semi-analytic solution given here may be used to check the validity of those numerical techniques for solving dynamic piezoelectric crack problems.
609
Fig. 1. A geometrical sketch of the problem.
ðnÞ skj ðx1 ; x2 ; tÞmðnÞ / Pk ðx1 ; x2 ; tÞðk ¼ 1; 2; 3Þ j ðnÞ asðx1 ; x2 Þ/ðx1 ; x2 Þ˛G ðn ¼ 1; 2; /; NÞ;
(3)
and either ðnÞ
ðnÞ
dj ðx1 ; x2 ; tÞmj / P4 ðx1 ; x2 ; tÞ
(4)
asðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; /; NÞ if the cracks are electrically impermeable; or
Dfðx1 ; x2 ; tÞ/0 asðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; .; NÞ if the cracks are electrically impermeable;
2. The problem Referring to an Ox1 x2 x3 Cartesian coordinate system, consider an infinite piezoelectric space that contains N arbitrarily oriented nonintersecting planar cracks whose geometries do not change along the x3 axis. The cracks are denoted by Gð1Þ ; Gð2Þ ; .; GðN1Þ and GðNÞ. The n -th planar crack GðnÞ lies in the region ðnÞ
[ðnÞ < aj1
ðnÞ
xj cj
ðnÞ
< [ðnÞ ; aj2
ðnÞ
xj cj
where sij and di are respectively the stresses and electric displaceðnÞ ðnÞ ðnÞ ðnÞ ments, P1 ðx1 ; x2 ; tÞ;P2 ðx1 ; x2 ; tÞ;P3 ðx1 ; x2 ; tÞ and P4 ðx1 ; x2 ; tÞ are ðnÞ ðnÞ ðnÞ suitably prescribed functions for ðx1 ; x2 Þ˛G , mi ¼ ai2 are the components of a unit magnitude normal vector to the crack GðnÞ and Dfðx1 ; x2 ; tÞ denotes the jump in the electrical potential f across the crack GðnÞ as defined by
h
Dfðx1 ; x2 ; tÞ ¼ lim f x1 jejm1ðnÞ ; x2 jejmðnÞ 2 ;t
¼ 0; N < x3 < N;
i ðnÞ ðnÞ f x1 þ jejm1 ; x2 þ jejm2 ; t for ðx1 ; x2 Þ˛GðnÞ :
where ðnÞ
aij
i
0
ðnÞ sin q B ¼ @ cosqðnÞ 0
ðnÞ cos q ðnÞ sin q 0
0
1
C 0 A: 1
e/0
(1)
h
(5)
(2)
On the Ox1 x2 plane, the crack GðnÞ is a straight line cut, 2[ðnÞ is ðnÞ ðnÞ the length of the crack, ðc1 ; c2 Þ is the midpoint of the crack and qðnÞ is the angle between the crack and the vertical line passing ðnÞ ðnÞ ðnÞ through ðc1 ; c2 Þðsuch that 0 q pÞ, as shown in Fig. 1. Note that the usual convention of summing over a repeated subscript is adopted in (1) for lower case Latin subscripts that run from 1 to 3. It will be assumed that here the electro-elastic deformation of the cracked piezoelectric space does not vary along the x3 direction. The problem is to determine the displacements uk ðx1 ; x2 ; tÞ and electric potential fðx1 ; x2 ; tÞ in the piezoelectric space for time t > 0 such that suitably prescribed boundary conditions on the cracks are satisfied. More specifically, the conditions the cracks are given by
(6)
Furthermore, it is assumed here that the displacements uk and its partial derivative with respect to time (that is, vuk =vt) are both zero at time t ¼ 0 and the stresses skj ðx1 ; x2 ; tÞ and electric displacement dk ðx1 ; x2 ; tÞ generated by the cracks vanish as x21 þ x22 /N. 3. Basic equations of electroelasticity The governing equations for the displacements uk and electric potential f in a homogeneous piezoelectric material are given by
cijk[
v2 u k v2 f v2 u v2 uk v2 f þ e[ij ¼ r 2i ; ejk[ kj[ vxj vx[ vxj vx[ vxj vx[ vxj vx[ vt
(7)
where cijk[ , e[ij and ki[ are the constant elastic moduli, piezoelectric coefficients and dielectric coefficients respectively and r is the density. The constitutive equations relating ðsij ; dj Þ and ðuk ; fÞ are
610
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
vuk vf þ e[ij ; vx[ vx[ vuk vf kjp ; vx[ vxp
sij ¼ cijk[ dj ¼ ejk[
(8)
SIj ¼
CIjK[
uj for J ¼ j ¼ 1; 2; 3; f for J ¼ 4;
sij for I ¼ i ¼ 1; 2; 3; dj for I ¼ 4;
(9)
8 for I ¼ i ¼ 1; 2; 3 and K ¼ k ¼ 1; 2; 3; c > > < eijk[ for I ¼ i ¼ 1; 2; 3 and K ¼ 4; [ij ¼ e for I ¼ 4 and K ¼ k ¼ 1; 2; 3; > > : jk[ kj[ for I ¼ 4 and K ¼ 4;
(10)
In the Laplace transform domain, the problem is to solve (19) subject to
(20)
and either ðnÞ b ðnÞ ðx ; x ; sÞ b S 4j ðx1 ; x2 ; sÞmj / P 1 2 4
ð21Þ
as ðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; .; NÞ if the cracks are electrically impermeable;
vUK ðI ¼ 1; 2; 3; 4; j ¼ 1; 2; 3Þ vx[
(11)
or
b ðx ; x ; sÞ/0 as ðx ; x Þ/ðx ; x Þ˛GðnÞ DU 4 1 2 1 2 1 2
where
(19)
as ðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; .; NÞ;
and
BIK ¼
b v2 U K b ¼ 0 ðI ¼ 1; 2; 3; 4Þ: s2 BIK U K vxj vx[
ðnÞ b ðnÞ ðx ; x ; sÞðI ¼ 1; 2; 3Þ b S Ij ðx1 ; x2 ; sÞmj / P 1 2 I
v 2 UK v 2 UK ¼ BIK ðI ¼ 1; 2; 3; 4Þ vxj vx[ vt 2
SIj ¼ CIjK[
(18)
where s is the Laplace transformation parameter. Application of the Laplace transformation on both sides of (10) together with the initial conditions (13) yields
CIjK[
so that (7) and (8) may be respectively written more compactly as
CIjK[
ZN Fðx1 ; x2 ; tÞexpðstÞdt; 0
Following closely the approach of Barnett and Lothe (1975), we define
UJ ¼
b F ðx1 ; x2 ; sÞ ¼
ðn ¼ 1; 2; .; NÞ if the cracks are electrically permeable:
r if I ¼ K and Is4; 0
(12)
otherwise:
Note that uppercase Latin subscripts have values 1;2; 3 and 4: Summation is also implied for repeated uppercase Latin subscripts. It follows that the problem stated in Section 2 requires solving (10) within the piezoelectric space for time t > 0 subject to the initial-boundary conditions
UK ¼ 0 and
vUK ¼ 0 at t ¼ 0 ðK ¼ 1; 2; 3Þ; vt
ðnÞ SIj ðx1 ; x2 ; tÞmj /
ðnÞ PI ðx1 ; x2 ; tÞ ðnÞ
asðx1 ; x2 Þ/ðx1 ; x2 Þ˛G
It is also required that b S Ij ðx1 ; x2 ; sÞ/0 as x21 þ x22 /N. 5. Method of solution In this section, a method based on the theory of exponential Fourier transformation is proposed for solving (19) subject to (20)e(22).
(13) 5.1. Solution in Fourier integral form
ðI ¼ 1; 2; 3Þ
For the solution of the piezoelectric crack problem, let
(14)
ðn ¼ 1; 2; .; NÞ;
b ðx ; x ; sÞ ¼ U K 1 2
and either ðnÞ
or
DU4 ðx1 ; x2 ; tÞ/0 as ðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; .; NÞ if the cracks are electrically permeable; (16) h
i ðnÞ ðnÞ UI x1 þ j3jm1 ; x2 þ j3jm2
ð17Þ
for ðx1 ; x2 Þ˛GðnÞ : In addition, it is required that SIj ðx1 ; x2 ; tÞ/0 as
a¼1 0
ðnÞ F1a ðx; sÞ
ðnÞ
3/0
where HðxÞ is the unit-step Heaviside function, F1a ðx; sÞ and ðnÞ ðnÞ F2a ðx; sÞ are functions yet to be determined, sa ðx; sÞ (n ¼ 1, 2, ., N) are roots, with positive imaginary parts, of the 8-th order polynomial equation (in s) given by
" x21
8 N
ðnÞ ðnÞ ðnÞ exp ix aj1 þ sa ðx; sÞaj2 ðnÞ ðnÞ ðnÞ þ H ar2 xr cr xj c j ðnÞ ðnÞ ðnÞ ðnÞ F2a ðx; sÞ exp ix aj1 þ sa ðx; sÞaj2 9 i = ðnÞ xj cj dx ; (23) ;
(15)
asðx1 ; x2 Þ/ðx1 ; x2 Þ˛GðnÞ ðn ¼ 1; 2; .; NÞ if the cracks are electrically impermeable;
DUI ðx1 ; x2 ; tÞ ¼ lim UI x1 j3jmðnÞ 3 ðnÞ 1 ; x2 j jm2
N X n¼1
ðnÞ
S4j ðx1 ; x2 ; tÞmj / P4 ðx1 ; x2 ; tÞ
where
(22)
þ
x22 /N:
4. Formulation in Laplace transform domain We denote the Laplace transformation of Fðx1 ; x2 ; tÞ over time t 0 by b F ðx1 ; x2 ; sÞ; that is, we define
det
s2
x
2
ðnÞ ðnÞ 2 ðnÞ ðnÞ ðnÞ ðnÞ BIK þ a11 þ sa12 CI1K1 þ a21 þ sa22 a11 þ sa12
ðCI1K2 þ CI2K1 Þ þ ðnÞ
ðnÞ a21
þ
2 saðnÞ CI2K2 22
# ¼ 0;
(24)
AK a ðx; sÞ (n ¼ 1, 2, ., N) are non-trivial solutions of the system
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
"
s2 2
x
ðnÞ ðnÞ ðnÞ 2 ðnÞ ðnÞ ðnÞ BIK þ a11 þ sa ðx;sÞa12 CI1K1 þ a21 þ sa ðx;sÞa22
Z[
ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2 a11 þ sa ðx;sÞa12 ðCI1K2 þCI2K1 Þþ a21 þ sa ðx;sÞa22 CI2K2 ðnÞ
AK a ¼ 0:
#
ð25Þ
N X
Re
n¼1
8 N 4 Z
a¼1 0
ðnÞ F1a ðx; sÞ
e2
[
gives ðnÞ
rK
ðnÞ
aj1
ðnÞ
b S Ij ðx1 ;x2 ;sÞ ¼
ðnÞ ðnÞ ðnÞ exp ix aj1 þ sa ðx; sÞaj2
F2a ðx; sÞ ¼ MaP ðx; sÞjP ðx; sÞ;
ðnÞ
mj LIja ðx; sÞMaP ðx; sÞ ¼ dIP ðn ¼ 1; 2; /; NÞ:
Z1
1
H [ðqÞ þ[ðqÞ C
¼ 0;
Use of (31) in (23) together with
(31)
ðqÞ
b ðu; sÞUðqÞ ðu; v; sÞdu DU K IK
1
ðqÞ b ðqÞ ðqÞ ðqÞ s2 GIK D U K ðu; sÞ cosh [ hjv uj ln [ hjv uj du
Z1
[ðnÞ
ðnÞ
b ðu; sÞQðnqÞ ðu; v; sÞdu DU K IK
n ¼ 1; nsq 1 ðqÞ ðqÞ ðqÞ b X1 ðvÞ; X2 ðvÞ; s ðI ¼ 1; 2; 3; 4Þ ¼ p P I for 1 < v < 1ðq ¼ 1; 2; /; NÞ;
(35)
^ ðqÞ ðu; sÞ ¼ pr ðqÞ ð[ðqÞ u; sÞ; X ðqÞ ðvÞ ¼ DU K K 1 ðqÞ ðqÞ ¼ c2 [ðqÞ v cosðq Þ, C denotes that
ðqÞ
ðqÞ
c1 þ [ðqÞ vsinðq
Þ;
the integral is to be interpreted in the Cauchy principal sense and C denotes that the ðqÞ
integral is to be interpreted in the Hadamard finite-part sense, DIK , ðqÞ
GIK and WIK ðx; sÞ are given by ðqÞ
[ðnÞ
i ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ AK a ðx;sÞMaP ðx;sÞAK a ðx;sÞM aP ðx;sÞ TPJ ðx;sÞ ¼ dKJ :
Z1
du þ [ðqÞ
ðv uÞ2
ðqÞ
(30)
pffiffiffiffiffiffiffi ðnÞ where i ¼ 1, rJ ðu; sÞ are real functions yet to be determined ðnÞ and TPJ ðx; sÞ are real functions defined by 4 h X
a ¼1 0
ðqÞ
ðqÞ
N X
(29)
ðnÞ
ðnÞ
[ðnÞ
b ðu; sÞ DIK D U K
1
Z1
þ
ðqÞ X2 ðvÞ
rJ ðu; sÞexpðixuÞdu;
8 N
ðnÞ rK ðu;sÞRe :
1
(28)
where
a ¼1
From (34), conditions (20) and (21) for electrically impermeable cracks give the hypersingular integral equations
b ðx ; x ; sÞ are continuous on the plane The functions U K 1 2 ðnÞ ðnÞ ðnÞ ar2 ðxr cr Þ ¼ 0 at points not on any of the cracks if jP ðx; sÞ are chosen to be
i
ðnÞ
xj cj
5.2. Electrically impermeable cracks
a¼1
ðnÞ jðnÞ x x P ð ; sÞ ¼ iTPJ ð ; sÞ
(27)
where are functions to be determined and the overhead bar denotes the complex conjugate of a complex number and ðnÞ MaP ðx; sÞ are defined by
Z[
ðnÞ
< [ðnÞ ; aj2
ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ LIja ðx;sÞMaP ðx;sÞexp ixsa ðx;sÞak2 xk ck ðnÞ ðnÞ ðnÞ ðnÞ þH ar2 xr cr LIja ðx;sÞM aP ðx;sÞ i ðnÞ ðnÞ ðnÞ ðnÞ TPK ðx;sÞ exp ixsa ðx;sÞak2 xk ck 9 i = h ðnÞ ðnÞ exp ix ak1 xk ck u dx du: (34) ;
(26)
jðnÞ x P ð ; sÞ
ðnÞ ðnÞ
ðnÞ
xj c j
Z[ N X n¼1
F1a ðx; sÞ ¼ MaP ðx; sÞjP ðx; sÞ;
4 X
p
ðnÞ
b ðx ; x ; sÞ in (23) is obtained by The integral representation for U K 1 2 generalizing the analyses in Ang (1987, 1991) and Clements (1981). b ðx ; x ; sÞ and b S Ij ðx1 ; x2 ; sÞ in (23) and (26) respecNote that U K 1 2 tively are represented by different integral expressions in different ðnÞ parts of the piezoelectric space. To ensure that b S Ij ðx1 ; x2 ; sÞmj are ðnÞ ðnÞ ðnÞ continuous on ar2 ðxr cr Þ ¼ 0, the functions F1a ðx; sÞ and ðnÞ F2a ðx; sÞ are chosen to be given by
ðnÞ
b ðx ; x ; sÞ are the Laplace transform of the generalized where D U K 1 2 crack opening displacements as defined in (17). The functions ^ SIj ðx1 ; x2 ; sÞ in (26) can now be written as
ðnÞ ðnÞ ðnÞ exp ix aj1 þ sa ðx; sÞaj2
h ðnÞ ðnÞ ðnÞ LIja ðx; sÞ ¼ a11 þ sna ðx; sÞa12 CIjK1 i ðnÞ ðnÞ ðnÞ þ a21 þ sna ðx; sÞa22 CIjK2 AK a :
ðnÞ
1 b ; s ¼ DU K ðx1 ; x2 ; sÞ
(33)
ðnÞ
ðnÞ
ðnÞ
xj c j
LIja ðx; sÞðn ¼ 1; 2; .; NÞ are given by
ðnÞ
(32)
h ðnÞ ðnÞ ðnÞ ixLIja ðx; sÞ H ar2 xr cr
9 i = ðnÞ xj cj dx ; ;
ðnÞ
du ¼ pjðvÞ for [ < v < [;
þ ðv uÞ2
for [ðnÞ < aj1
ðnÞ ðnÞ ðnÞ H ar2 xr cr xj c j ðnÞ F2a ðx; sÞ
e/0þ
ðnÞ
From (11) and (23), we obtain
b S Ij ðx1 ; x2 ; sÞ ¼
jðuÞ
lim e
611
DIK ¼ ðqÞ
GIK ¼ ðqÞ
lim
lim
ðqÞ
ðx=sÞ/N 2 h
ðx=sÞ/N
x s
ðqÞ
TIK ðx; sÞ; ðqÞ
ðqÞ
WIK ðx; sÞ ¼ TIK ðx; sÞ DIK ðqÞ
ðnqÞ
ðqÞ
TIK ðx; sÞ DIK ðqÞ s2 GIK x2 þ h2
i
;
ðh > 0Þ;
and UIK ðu; v; sÞ and QIK ðu; v; sÞ are respectively defined by
(36)
612
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
ZN
UðqÞ IK ðu; v; sÞ ¼
0 ðqÞ s2 GIK
where sgn ðxÞ denotes the sign of x and
ðqÞ xWIK ðx; sÞcos [ðqÞ x½v u dx
w ðnqÞ
Shi [ðqÞ hjv uj sinh [ðqÞ hjv uj
(37)
a ¼1
h
ðnqÞ xmðqÞ H Y2 ðu;vÞ j
0
ðnÞ ðnÞ LIja ðx;sÞMaP ðx;sÞexp
ðnqÞ ixsa ðx;sÞY2 ðu;vÞ
with
ðnqÞ
Yp
ðnÞ
Zu 0
ðnÞ
ZN
ZN
ðu;vÞs0;
[ðnÞ dp1 u
u
expð xÞ dx; x
ZN 0
and
Note that EiðxÞE1 ðxÞ tend to 2lnðxÞ as x/0þ . This explains the presence of the Cauchy principal integral in (35). If the cracks are electrically impermeable, the functions b ðqÞ ðu; sÞ (q ¼ 1,2, ., N) in (30) are to be determined by solving DU K the hypersingular integral equations in (35). If we make the approximation (Kaya and Erdogan, 1987) ðqÞ
J pffiffiffiffiffiffiffiffiffiffiffiffiffiffi X j¼1
The functions WIK ðx; sÞ behave as Oðs4 =x Þ for very large x: Thus, the improper integral over ½0; NÞ that appears in the definition of UðqÞ IK ðu; v; sÞ in (37) is well defined. ðnqÞ Note that the expressions for QIK ðu; v; sÞ as given in (38) are ðnqÞ ðnqÞ valid for Y2 ðu; vÞs0. If Y2 ðu; vÞ ¼ 0 then (38) has to be modified accordingly. The modification gives
8 <
w ðnqÞ
D
h
IK
: Y1ðnqÞ ðu; vÞ
ZN i2
w ðnqÞ
xW
IK
(41)
x p sinðaxÞdx ¼ sgnðaÞ½coshðjahjÞsinhðjahjÞ: 2 x2 þ h2
b ðu; sÞx 1 u2 DU K 4
ðnÞ
(42)
(39)
ðqÞ
ðnÞ
a¼1
þShiðahÞsinhðahÞðah > 0Þ;
(38)
expð xÞ dx: x
QðnqÞ IK ðu; v; sÞ ¼ Re
ðx; sÞ ¼
ðqÞ ðnÞ
mj LIja ðx; sÞMaP ðx; sÞTIK ðx; sÞ;
ðnqÞ w x2 ~ ðnqÞ ~ ðnqÞ T IK ðx; sÞ D G IK ¼ lim IK ðx=sÞ/N s 2 ~ ðnqÞ ~ ðnqÞ ðx; sÞ ¼ T~ ðnqÞ ðx; sÞ D ~ ðnqÞ s GIK ðh > 0Þ: W IK IK IK x2 þ h2 IK
0
sinhðxÞ dx; x
EiðuÞ ¼ C
u
ðqÞ
ðu; vÞ ¼ akp Xk ðvÞ ck
ShiðuÞ ¼
E1 ðuÞ ¼
ðnqÞ T~ IK ðx; sÞ;
Z1 e2 ðvuÞ2 jðuÞ Z1 jðuÞ lim du for 1 < v < 1; 2 du ¼ H 2 e/0þ 2 ðvuÞ e2 þðvuÞ 1 1 ZN x 1 cosðaxÞdx ¼ coshðahÞðEiðahÞE1 ðahÞÞ 2 x2 þ h2
ðnÞ ðnÞ ðnqÞ þH Y2 ðu;vÞ LIja ðx;sÞM aP ðx;sÞ i ðnÞ ðnqÞ exp ixsa ðx;sÞY2 ðu;vÞ 9 = ðnÞ ðnqÞ TPK ðx;sÞexp ixY1 ðu;vÞ dx ; ðnqÞ
lim
ðx=sÞ/N 4 P
The derivations of (35), (37) and (40) make use of the following results:
ðnÞ
if it is assumed that Y2
¼
ðqÞ ðnÞ If the cracks GðnÞ and GðqÞ are coplanar then mj ¼ mj , hence ðnqÞ ðnqÞ ðnqÞ ðnqÞ ðnÞ ðnÞ ~ ðnÞ ðnÞ ~ ~ T~ IK ðx;sÞ¼TIK ðx;sÞ, D IK ¼DIK ;GIK ¼GIK and W IK ðx;sÞ¼WIK ðx;sÞ. ðnqÞ ðnÞ ðqÞ Note that Y2 ðu;vÞ¼0 for all u and v if G and G are coplanar.
and
8 ZN
IK
w ðnqÞ
T
1 cosh [ðqÞ hjv uj Ei [ðqÞ hjv uj 2 E1 [ðqÞ hjv uj þcosh [ðqÞ hjv uj ln [ðqÞ hjv uj ;
QðnqÞ IK ðu;v;sÞ ¼ Re:
D
ðj1Þ uðqjÞ ðuÞ; K ðsÞU
(43)
where U ðjÞ ðxÞ ¼ sinð½j þ 1arccosðxÞÞ=sinðarccosðxÞÞ is the jth order ðnjÞ Chebyshev polynomial of the second kind and uP ðsÞ are unknown coefficients, then (35) can be used to set up a system of linear ðnjÞ algebraic equations to determine uP ðsÞ for any fixed value of s (Athanasius et al., 2010).
ðnqÞ ðx; sÞexp ixY1 ðu; vÞ dx
0
w ðnqÞ
1 ðnqÞ ðnqÞ ðnqÞ Shi h Y1 ðu; vÞ sinh h Y1 ðu; vÞ cosh h Y1 ðu; vÞ 2 p ðnqÞ ðnqÞ ðnqÞ Ei h Y1 ðu; vÞ E1 h Y1 ðu; vÞ þ i sgn Y1 ðu; vÞ 2 9 = ðnqÞ ðnqÞ cosh h Y1 ðu; vÞ sinh h Y1 ðu; vÞ ; s2 G
IK
ðnqÞ
if Y2
ðu; vÞ ¼ 0;
(40)
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
613
ðnjÞ
^ ðqÞ ðu; sÞ ¼ 0 for 1 < u < 1 and q ¼ 1, 2, From (16) and (33), DU 4 ., N, if the cracks are electrically permeable. According to (14), the ^ ðqÞ ðu; sÞ;DU ^ ðqÞ ðu; sÞ and DU ^ ðqÞ ðu; sÞ that can unknown functions DU 1 2 3 be approximated as above by (43) are governed by (35) (with ðqÞ ^ ðu; sÞ ¼ 0) for I ¼ 1, 2, 3 (instead of I ¼ 1,2,3,4). DU 4
Once the coefficients uP ðsÞ in (43) are determined, the above intensity factors can be approximately calculated in the Laplace transform domain using The dynamics stress and electric displacement intensity factors at any time t may be recovered by using the numerical Laplace transform algorithm in Stehfest (1970), that is, by using the formula
6. Stress and electric displacement intensity factors
f ðtÞx
5.3. Electrically permeable cracks
The dynamic stress and electric displacement intensity factors ðnÞ ðnÞ ðnÞ ðnÞ at the tips ðX1 ð1Þ; X2 ð1ÞÞ and ðX1 ð1Þ; X2 ð1ÞÞ of the n-th ðnÞ crack G are defined as follows:
2M lnð2Þ X nlnð2Þ Vn ^f ; t n¼1 t
(46)
where ^f ðsÞ denotes the Laplace transform of f ðtÞ, M is a positive integer and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu þ 1Þ S1j X1 ðuÞ; X2 ðuÞ; t m1 þ S2j X1 ðuÞ; X2 ðuÞ; t m2 mj ; KI X1 ð1Þ; X2 ð1Þ; t ¼ lim u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu þ 1Þ S1j X1 ðuÞ; X2 ðuÞ; t m2 S2j X1 ðuÞ; X2 ðuÞ; t m1 mj ; KII X1 ð1Þ; X2 ð1Þ; t ¼ lim u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu þ 1ÞS3j X1 ðuÞ; X2 ðuÞ; t mj ; KIII X1 ð1Þ; X2 ð1Þ; t ¼ lim u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu þ 1ÞS4j X1 ðuÞ; X2 ðuÞ; t mj ; KIV X1 ð1Þ; X2 ð1Þ; t ¼ lim u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu 1Þ S1j X1 ðuÞ; X2 ðuÞ; t m1 þ S2j X1 ðuÞ; X2 ðuÞ; t m2 mj ; KI X1 ð1Þ; X2 ð1Þ; t ¼ lim þ u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu 1Þ S1j X1 ðuÞ; X2 ðuÞ; t m2 S2j X1 ðuÞ; X2 ðuÞ; t m1 mj ; KII X1 ð1Þ; X2 ð1Þ; t ¼ lim u/1þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu 1ÞS3j X1 ðuÞ; X2 ðuÞ; t mj ; KIII X1 ð1Þ; X2 ð1Þ; t ¼ lim þ u/1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 2[ðnÞ ðu 1ÞS4j X1 ðuÞ; X2 ðuÞ; t mj : KIV X1 ð1Þ; X2 ð1Þ; t ¼ lim
(44)
u/1þ
J X ðnÞ ðnÞ ðnÞ ðnÞ ðj1Þ ^ X ðnÞ ð 1Þ; X ðnÞ ð 1Þ; s xp1 ffiffiffiffiffiffiffiffi DP1 m1 þ DP2 m2 uðnjÞ ð 1Þ; K P ðsÞU 1 2 I [ðnÞ j¼1 J X ðnÞ ðnÞ ðnÞ ðnÞ ðj1Þ ^ II X ðnÞ ð 1Þ; X ðnÞ ð 1Þ; s xp1 ffiffiffiffiffiffiffiffi DP1 m2 DP2 m1 uðnjÞ K ð 1Þ; P ðsÞU 1 2 ðnÞ [ j¼1 J X ðnÞ ðj1Þ ^ III X ðnÞ ð 1Þ; X ðnÞ ð 1Þ; s x p1 ffiffiffiffiffiffiffi ffi uðnjÞ K ð 1Þ; D P ðsÞU 1 2 P3 [ðnÞ j¼1 J X 1 ðnÞ ðnÞ ðnÞ ðj1Þ ^ uðnjÞ K ð 1Þ; IV X1 ð 1Þ; X2 ð 1Þ; s x pffiffiffiffiffiffiffiffi DP4 P ðsÞU [ðnÞ j¼1 J X ðnÞ ðnÞ ðnÞ ðnÞ ðj1Þ ^ I X ðnÞ ð1Þ; X ðnÞ ð1Þ; s xp1 ffiffiffiffiffiffiffiffi DP1 m1 þ DP2 m2 uðnjÞ K ð þ 1Þ; P ðsÞU 1 2 ðnÞ [ j¼1 J X ðnÞ ðnÞ ðnÞ ðnÞ ðj1Þ ^ II X ðnÞ ð1Þ; X ðnÞ ð1Þ; s xp1 ffiffiffiffiffiffiffiffi DP1 m2 DP2 m1 uðnjÞ K ð þ 1Þ; P ðsÞU 1 2 [ðnÞ j¼1 J X ðnÞ ðj1Þ ^ X ðnÞ ð1Þ; X ðnÞ ð1Þ; s x p1 ffiffiffiffiffiffiffi ffi uðnjÞ ð þ 1Þ; K D III P ðsÞU 1 2 P3 [ðnÞ j¼1 J X ðnÞ ðj1Þ ^ IV X ðnÞ ð1Þ; X ðnÞ ð1Þ; s x p1 ffiffiffiffiffiffiffiffi DP4 uðnjÞ K ð þ 1Þ: P ðsÞU 1 2 ðnÞ [ j¼1
(45)
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W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
Vn ¼ ð1ÞnþM
minðn;MÞ X
mM ð2mÞ! ; ðM mÞ!m!ðm1Þ!ðnmÞ!ð2mnÞ! m¼½ðnþ1Þ=2 (47)
with ½r denoting the integer part of the real number r: Note that the Stehfest’s algorithm requires the problem under consideration to be solved for only real Laplace transform parameter s: It has been widely used by researchers for the numerical inversion of Laplace transforms in solving many problems in engineering (see, for example, Ang (1987); Hemker (1999) and Smith et al. (1994)). 7. Specific cases In this section, the dynamics crack tip stress and electric displacement intensity factors are computed for some specific cases of the problem. Problem 1. Consider the case of a single electrically impermeable crack in an infinite piezoelectric space. The crack lies in the region a < x1 < a; x2 ¼ 0; and the only non-zero uniform load acting on it is given by S22 ¼ HðtÞs0 ; where HðtÞ is the unit-step Heaviside function. (Note that S12 ; S32 and S42 are taken to be zero on the crack.) The electrical poling direction is taken to be along the x2 direction so that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Fig. 2. Plots of KI =ðs0 aÞ against the non-dimensionalized time t L=ðra2 Þ.
obtaining the plots in Fig. 2. Convergence is observed in the numerical values of the intensity factors when the number of terms in the formula is increased to 10: The numerical inversion of Laplace transform becomes unstable when the number of terms is increased beyond 10: To use more terms, it is necessary to refine the
C1111 ¼ C3333 ¼ A; C1133 ¼ C3311 ¼ N; C2222 ¼ C; C1122 ¼ C2211 ¼ C2233 ¼ C3322 ¼ F; C1212 ¼ C2112 ¼ C2121 ¼ C1221 ¼ C2323 ¼ C3223 ¼ C3232 ¼ C2332 ¼ L; 1 C1313 ¼ C3113 ¼ C3131 ¼ C1331 ¼ ðA NÞ; 2 C2141 ¼ C1241 ¼ C3243 ¼ C2343 ¼ C4121 ¼ C4112 ¼ C4332 ¼ C4323 ¼ e1 ; C1142 ¼ C3342 ¼ C4211 ¼ C4233 ¼ e2 ; C2242 ¼ C4222 ¼ e3 ; C4141 ¼ C4343 ¼ e1 ; C4242 ¼ e2 ;
where A, N, F, C, L, e1, e2, e3, e1 and e2 are independent constants. The piezoelectric material is PZT-BaTio3 with material constants A, N, F, C, L, e1, e2, e3, e1 and e2 and density r given by
1010 ;
1010 ; F
A ¼ 15:0 N ¼ 7:78 ¼ 6:6 C ¼ 14:6 1010 ; L ¼ 4:4 1010 ; e1 ¼ 11:4; e2 ¼ 4:35; e3 ¼ 17:5; e1 ¼ 98:7 1010 ; e2 ¼ 112 1010 ; r ¼ 5800:
1010 ; (49)
The values of A;N; F, C and L above are in N/m2, e1 ; e2 and e3 are in C/m2, e1 and e2 are in C/(Vm) and r in kg/m3. pffiffiffi The numerical stress intensity factor KI =ðs0 aÞ and electric pffiffiffi tip ða; 0Þ displacement intensity factor CKIV =ðe3 s0 aÞ at the crack pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi are plotted against the non-dimensionalized time t L=ðra2 Þ in Figs. 2 and 3 respectively. The results are obtained by using J ¼ 10 in the numerical solution of the hypersingular integral equations and M ¼ 4 (8 terms) in the Stehfest’s formula for inverting Laplace pffiffiffi transform. In Figs. 2 and 3, the numerical values of KI =ðs0 aÞ and pffiffiffi CKIV =ðe3 s0 aÞ are also compared with those extracted from Shindo et al. (1999) and García-Sánchez et al. (2007). (Numerical values of pffiffiffi CKIV =ðe3 s0 aÞ are not given in Shindo et al. (1999)). All the plots of pffiffiffi KI =ðs0 aÞ in Fig. 2 are quite close to one another, exhibiting the same general trends and reaching peak values at about the same pffiffiffi time. So are the plots of CKIV =ðe3 s0 aÞ in Fig. 3. As pointed out earlier on, 8 terms ðM ¼ 4Þ are used in the Stehfest’s formula for
(48)
calculation to obtain more a more accurate solution of the hypersingular integral equations. A higher arithmetic precision in the computing machine is needed too. Problem 2. Consider a pair of coplanar cracks, each of length 2a; as shown in Fig. 4. The distance between the inner tips of the cracks is 2d: The uniform tractions acting on the crack faces are given by S22 ¼ HðtÞs0 : Here electrically impermeable and permeable cracks will be examined separately. For the case in which the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Fig. 3. Plots of CKIV =ðe3 s0 aÞ against the non-dimensionalized time t L=ðra2 Þ.
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
Fig. 4. A pair of coplanar cracks.
coplanar cracks are electrically impermeable, the condition S42 ¼ HðtÞD0 ; where D0 is a constant, applies. The electrical poling is along the x2 direction. For the purpose of obtaining some numerical results for the dynamic stress and electric displacement intensity factors at the inner and outer tips of the coplanar cracks, the material constants of PZT-BaTio3 (as in Problem 1) are used and the load ratio s0 =D0 for electrically impermeable cracks is taken to be 1010 NC1: In Fig. 5, the non-dimensionalized stress intensity factor pffiffiffi crack tips are plotted against the KI =ðs0 aÞ at the inner and outer pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi non-dimensionalized time t L=ðra2 Þ for d=a ¼ 0:125;0:50 and pffiffiffi 0:25: The plots of KI =ðs0 aÞ for electrically permeable and impermeable cracks are almost indistinguishable. Thus, plots of pffiffiffi KI =ðs0 aÞ are given in Fig. 5 for only electrically impermeable pffiffiffi cracks. In each of the plots, KI =ðs0 aÞ increases rapidly to a peak value before settling down to the corresponding value of the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Fig. 5. Plots of KI =ðs0 aÞ against the non-dimensionalized time t L=ðra2 Þ at inner and outer crack tips of electrically impermeable cracks for selected values of d/a.
615
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Fig. 6. Plots of CKIV =ðe3 s0 aÞ against the non-dimensionalized time t L=ðra2 Þ at inner and outer crack tips of electrically impermeable cracks for selected values of d=a:
static stress intensity factor. For each of the values of d=a in Fig. 5, pffiffiffi the peak values of KI =ðs0 aÞ at the inner and the outer crack tips are significantly different and the peak value at the inner tips is larger than that at the outer tips. As may be expected, the peak pffiffiffi value of KI =ðs0 aÞ at each crack tip is larger if the cracks are closer to each other : Further calculations show that the plot of pffiffiffi KI =ðs0 aÞ at the inner tips is almost identical as that at the outer tips for d/a > 3. Plots of the non-dimensionalized electric displacement intenpffiffiffi sity factors CKIV =ðe3 s0 aÞ at the inner and the outer crack tips pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi against t L=ðra2 Þ for d/a ¼ 0.125, 0.50 and 0.25 are given in Figs. 6 and 7 for electrically impermeable and permeable cracks pffiffiffi respectively. The plots of CKIV =ðe3 s0 aÞ for electrically impermeable cracks are distinct from those for electrically permeable pffiffiffi cracks. For a fixed d=a, the peak value of CKIV =ðe3 s0 aÞ at each crack tip is apparently higher for the electrically permeable cracks than that for the impermeable cracks.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Fig. 7. Plots of CKIV =ðe3 s0 aÞ against the non-dimensionalized time t L=ðra2 Þ at inner and outer crack tips of electrically permeable cracks for selected values of d/a.
616
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
pffiffiffi Fig. 10. Plots of CKII =ðF s0 aÞ against the non-dimensionalized time for selected values of d/a.
Fig. 8. Two parallel cracks.
Problem 3. Consider two equal length parallel electrically impermeable cracks as sketched in Fig. 8. The half length of each crack is given by a. The centers of the cracks lie on a vertical line and are separated by a distance denoted by d: The non-zero constant loads acting on the crack faces are given by S22 ¼ HðtÞs0 and S42 ¼ HðtÞD0 , with s0 =D0 ¼ 1010 NC1. The electrical poling is along the x2 direction. Using the material constants of PZT-BaTio3 (as in Problem 1), for selected values of d/a, we plot the non-dimensionalized crack tip stress intensity factors pffiffiffi pffiffiffi KI =ðs0 aÞ and CKII =ðF s0 aÞ and the non-dimensionalized crack tip pffiffiffi electric displacement intensity factor CKIV =ðe3 s0 aÞ against the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 non-dimensionalized time t L=ðra Þ in Figs. 9, 10 and 11 respectively. In Fig. 9, for a given d/a, the non-dimensionalized stress intenpffiffiffi sity factor KI =ðs0 aÞ rises to a peak (maximum) value and then drops to a trough (mininum) value before gradually settling down to approach its static value. Both the trough and the peak values decrease in magnitude as d/a decreases. A similar observation may be made of the non-dimensionalized electric displacement intenpffiffiffi sity factor CKIV =ðe3 s0 aÞ in Fig. 11.
pffiffiffi Fig. 9. Plots of KI =ðs0 aÞ against the non-dimensionalized time for selected values of d/a.
As d/a tends to infinity, the non-dimensionalized stress pffiffiffi intensity factor CKII =ðF s0 aÞ vanishes. Nevertheless, when the cracks come close to each other, there is an increase in the pffiffiffi magnitude CKII =ðF s0 aÞ due to larger differences in the stress distribution on opposite crack faces. This is shown in Fig. 10. For pffiffiffi d/a ¼ 10, the magnitude of CKII =ðF s0 aÞ is very small at all time. Note that the fact that KII is not zero for the parallel cracks in Fig. 8 may be explained by the well known phenomenon called Poisson effect. Due to Poisson effect, compressive stresses are generated on opposite crack faces. For the parallel cracks, they are unequal, thereby giving rise to shear stresses on each of the cracks. Problem 4. Consider now the two pairs of electrically impermeable cracks, which are of equal length 2a; in the piezoelectric space, as sketched in Fig. 12. The faces of the horizontal cracks are subject to internal uniform loads given by S12 ¼ S22 ¼ 0;S32 ¼ HðtÞs0 and S42 ¼ HðtÞD0 , such that s0 =D0 ¼ 1010 NC : The internal loads on the faces of the vertical cracks are given by SK1 ¼ 0 (for K ¼ 1; 2; 3; 4). The electrical poling is in the x3 direction, so that CIjKl are given in terms of the independent constants A, N, F, C, L, e1, e2, e3, e1 and e2 by
pffiffiffi Fig. 11. Plots of CKIV =ðe3 s0 aÞ against the non-dimensionalized time for selected values of d/a.
W.T. Ang, L. Athanasius / European Journal of Mechanics A/Solids 30 (2011) 608e618
C1111 ¼ C2222 ¼ A; C1122 ¼ C2211 ¼ N; C3333 ¼ C; C1133 ¼ C3311 ¼ C2233 ¼ C3322 ¼ F; C1313 ¼ C3113 ¼ C3131 ¼ C1331 ¼ C2323 ¼ C3223 ¼ C3232 ¼ C2332 ¼ L; 1 C1212 ¼ C2112 ¼ C2121 ¼ C1221 ¼ ðA NÞ; 2 C3141 ¼ C1341 ¼ C2342 ¼ C3242 ¼ C4131 ¼ C4113 ¼ C4223 ¼ C4232 ¼ e1 ; C1143 ¼ C2243 ¼ C4311 ¼ C4322 ¼ e2 ; C3343 ¼ C4333 ¼ e3 ; C4141 ¼ C4242 ¼ e1 ; C4343 ¼ e2 :
As in the problems above, the material occupying the piezoelectric space is taken to be PZT-BaTio3 with material constants as given in (49). The deformation of the cracks is antiplane. The non-dimensionalized crack tip stress and electrical displacement intensity pffiffiffi pffiffiffi factors KIII =ðs0 aÞ and KIV =ðD0 aÞ are of interest here. For b=a ¼ 1:25 and a few selected values of d=a; these intensity factors at the lower tip of the left vertical crack are plotted against the non-
617
(50)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dimensionalized time t L=ðra2 Þ in Figs. 13 and 14. For the same values of b=a and d=a; plots of the intensity factors at the left tip of the upper horizontal crack are given in Figs. 15 and 16. As shown pffiffiffi in Figs. 14 and 16, KIV =ðD0 aÞ at both crack tips does not vary with pffiffiffi time. In Figs. 13 and 15, for a fixed b/a and d/a, KIII =ðs0 aÞ rises quickly to a peak value, drops to a trough value and gradually approaches the corresponding static value. As expected, as d=a pffiffiffi pffiffiffi increases, both KIII =ðs0 aÞ and KIV =ðD0 aÞ for the vertical cracks decrease in magnitude, becoming closer to zero.
Fig. 12. Two pairs of cracks.
pffiffiffi Fig. 14. Plots of KIV =ðD0 aÞ at the upper tip of the left vertical crack against the nondimensionalized time for b/a ¼ 1.25 and selected values of d/a.
pffiffiffi Fig. 13. Plots of KIII =ðs0 aÞ at the upper tip of the left vertical crack against the nondimensionalized time for b=a ¼ 1:25 and selected values of d=a.
pffiffiffi Fig. 15. Plots of KIII =ðs0 aÞ at the left tip of the upper horizontal crack against the nondimensionalized time for b/a ¼ 1.25 and selected values of d/a.
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pffiffiffi Fig. 16. Plots of KIV =ðD0 aÞ at the left tip of the upper horizontal crack against the non-dimensionalized time for b/a ¼ 1.25 and selected values of d/a.
8. Summary Through the use of Laplace and exponential Fourier transforms, a semi-analytic solution is derived for an electroelastodynamic problem involving an arbitrary number of arbitrarily located planar cracks in a piezoelectric space. The problem is eventually reduced to a solving a system of hypersingular integral equations. The unknown functions in the hypersingular integral equations are the Laplace transforms of the jumps in the displacements and electric potential across opposite crack faces. Once they are determined, the crack tip stress and electric displacement intensity factors can be easily computed in the Laplace transform domain. A numerical technique for inverting Laplace transforms is employed to recover the intensity factors in the physical domain. The solution is applied to study some specific cases of the problem. For the case of a single crack under impact loadings, the computed crack tip stress and electric displacement intensity factors are found to be in reasonably good agreement with those published in the literature. Numerical results are also obtained for other cases which include one which involves four interacting planar cracks under antiplane deformations. Acknowledgements The first author (WT Ang) acknowledges the sponsorship of Singapore Ministry of Education Tier 1 Research Grant RG9/09. The second author (L Athanasius) is supported by a research scholarship from Nanyang Technological University. The authors would like to thank the anonymous reviewers for their constructive comments which led to significant improvement in the paper.
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