Love wave in an isotropic homogeneous elastic half-space with a functionally graded cap
layer
Zhu Hong a ,⇑,Zhang Ligang a ,Han Jiecai b ,Zhang Yumin b
a School of Mechanical Engineering,Yanshan University,Qinhuangdao 066004,China
b
Center for Composite Materials and Structures,Harbin Institute of Technology,Harbin 150001,China
a r t i c l e i n f o Keywords:
Functionally graded materials Love wave Wave velocity Dispersion
Differential equation with variable coefficient
a b s t r a c t
The dispersion characteristics of Love wave in an isotropic homogeneous half-space cov-ered with a functionally graded layer is investigated.Governing equations for the anti-plane shear wave in the graded layer are derived,and analytical solutions for the displace-ment and stress field in the layer are given.Moreover,the general dispersion relations of Love wave in both the half-space and the layer are analyzed.For the layer with shear mod-ulus and mass density varying in a parabolic form,the dispersion equations are solved in terms of iteration method.The obtained dispersion curves reveal that there exists a cut-off frequency in the lowest order vibration mode.
Ó2014Elsevier Inc.All rights reserved.
1.Introduction
In 1911,Love discovered that when an isotropic homogeneous elastic half-space is covered with a layer having a different elastic medium,there exists a special SH wave in both the covering layer and th
e half-space.This is known as Love wave.The existence of Love wave successfully explained the dispersion characteristics of the waves that arise in the seismic records.In 1973,Achenbach proved that anti-plane surface waves do not exist on a homogeneous half-space with a free surface [1].To disclose the dispersion characteristics of the waves in the seismic records,anti-plane shear waves in vertical inhomogeneous medium have also been proposed.Dutta [2,3],Bhattacharya [4],and Chattopadhyay [5]respectively discussed the propaga-tion of Love-type waves in an intermediate non-homogeneous layer lying between two semi-infinite homogeneous elastic media.Chiroiu and Ghiroiu [1]studied propagation of Love waves in an elastic homogeneous half-space covered by an elastic non-homogeneous layer.Based on the Fourier transform method,Abd-Alla and Ahmed [6]studied Love wave dispersion in an initially stressed non-homogeneous orthotropic elastic layer on a semi-infinite medium.
A recent representative paper by Achenbach and Balogun [7]addressed the propagation of anti-plane shear waves in a half-space whose shear modulus and mass density have an arbitrary dependence on the distance from the free surface.By using the WKBJ approximation method,the governing equation was solved;an equation which related the speed of sur-face waves to the wavenumber was yielded,and the propagation of waves in high-frequency range was detailedly discussed.Liu et al.[8]stu
died initial stress effect on Love wave in the inhomogeneous piezoelectric covering layer.Li et al.[9]studied the Love wave propagation in functionally graded piezoelectric materials.Collet et al.[10]studied anti-plane surface waves in a functionally graded piezoelectric half-space,and presented an exact solution for the surface wave dispersion spectra without the use of representative layering.An interesting paper on anti-plane shear waves by Shuvalov et al.[11]addressed the determination of anti-plane wave solutions,and discussed general properties of dispersion spectra in a monoclinic plate.
0096-3003/$-see front matter Ó2014Elsevier Inc.All rights reserved./10.1016/j.amc.2013.12.167
⇑Corresponding author.
E-mail address:zhuhong9@126 (H.Zhu).
Based on Biot theory,Ke et al.studied Love waves in inhomogeneous fluid saturated porous layered half-space with varying properties [12,13].
Functionally graded materials,as a kind of typtical inhomogeneous materials,is often used as a layer (or layers)attached on other bulk materials.So anti-plane shear wave in vertical inhomogeneous mediu
m is required to be concerned.Only for several special cases,analytical solution for the governing equation can be found.Even an analytical solution can be ob-tained,it generally contains extremely complicated functions,making it impractical.So,approximate approaches have to be employed.
In the present work,we study the dispersion properties of Love wave in a homogeneous elastic half-space covered with a functionally graded layer.The dispersion equation is solved by an iterative method,the general properties of Love wave in both media are obtained.The obtained results can be applied to the optimization design of functionally graded materials,nondestructive testing and inverse problem analysis.2.Governing equations
The geometry of the composite structure is depicted in Fig.1along with a Cartesian coordinate system o-xyz .Below the half-plane y P 0,an isotropic and homogeneous medium with shear elastic modulus l and mass density q is filled.This medium is covered with a layer of functionally graded materials.The shear elastic modulus and the mass density of the layer
are,respectively,l B and q B .They are further expressed in the form of l B ¼l B 0l ðy Þand q B ¼q B
0q ðy Þ,where l (y ),q (y )are func-tions of y ,and l B 0and q B 0denote the values at y =0.The layer thickness is H .The Love wave propagates along the x direction,and the anti-plane shear motion along
the z direction.2.1.Love wave in homogeneous half-space
We use w to denote the anti-plane displacement in the homogeneous half-space.It takes place along the z direction,sat-isfies the follwing governing anti-plane shear wave equation
r 2
w ¼1c T  2@2w
@t 2
;
ð0<y <þ1Þ;ð1Þ
where c T ¼ffiffiffiffiffiffiffiffiffi
l =q p is the shear wave velocity.Assuming a traveling wave solution,we have
w ¼Ae Àby exp ½ik ðx Àct Þ ;
ð2Þ
where A is a constant that need to be determined,k =x /c is the wave number,x is the angular frequency,c is the phase
velocity of Love wave and b is a constant.
Substituting Eq.(2)into Eq.(1),one obtains the value of b ,
b ¼k ½1Àð
c =c T Þ2
1=2
ð3Þ
and the two shear stress components in the homogeneous half-space:
s xz ¼l @w
@x
¼ik l Ae Àby exp ½ik ðx Àct Þ ;ð4a Þs yz ¼l
@w
@y
¼Àb l Ae Àby exp ½ik ðx Àct Þ :ð4b Þ
2.2.Love wave in functionally graded layer
The anti-plane shear displacement in the functionally graded layer is denoted as w B ,which is also along the z direction,and
satisfies
94H.Zhu et al./Applied Mathematics and Computation 231(2014)93–99
l
B
r 2
w B
þd l B dy @w B @y ¼q B @2w B
@t ðÀH <y <0Þ:ð5Þ
Assuming w B ðx ;y ;t Þtakes the form of
w B ðx ;y ;t Þ¼l ðy ÞÀ1=2U ðx ;y Þexp ðÀi x t Þ:
ð6Þ
Substituting Eq.(6)into Eq.(5),one gets
@2U @x 2þ@2U
@y
2þ½a ðy Þþb ðy Þ U ¼0;ð7Þ
where
a ðy Þ¼½4l ðy Þ
À1
l ðy Þ
À1
d l ðy Þdy
2
À2d 2l ðy Þ
dy
2 !"
#;ð8a Þ
b ðy Þ¼
x 2q B 0l B 0q ðy Þl ðy Þ¼x 2c B T
ÀÁ2
q ðy Þl ðy Þð8b Þ
and c B T ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l B 0=q B
0q is the shearing wave velocity at y =0.
Accordingly the shear stress components in the functionally graded layer can be derived:
s B xz ¼l
B
@w B @x ¼l B 0l ðy Þ1=2@U @x
exp ðÀi x t Þ;ð9a Þ
s B yz
¼l B @w B @y
¼l B 0l ðy ÞÀ1=2
l ðy Þ@U @y À12d l ðy Þdy
U
!
exp ðÀi x t Þ:
ð9b Þ
Let U ¼f ðy Þexp ðikx Þ,k =x /c is the wave number,x is the angular frequency,c is the phase velocity of Love wave,then Eq.
(7)becomes
d 2
f ðy Þdy
þk 2
c ðy Þf ðy Þ¼0;
ð10Þ
where
c ðy Þ¼
a ðy Þk
þc 2c B T
ÀÁq ðy Þl À1:
ð11Þ
The general solution of Eq.(10)can be found to be
f ðy Þ¼B 1f 1ðy ÞþB 2f 2;
ð12Þ
where B 1and B 2are two constants,and f 1(y )and f 2(y )are two linearly independent particular solutions having opposite
monotonic property with y .
Finally the displacement and stress field in the functionally graded layer can be obtained:
w B ¼l ðy ÞÀ1=2f ðy Þexp ½ik ðx Àct Þ ðÀH <y <0Þ;
ð13Þs B xz ¼ik l B
0l ðy Þ
1=2f ðy Þexp ½ik ðx Àct Þ ðÀH <y <0Þ;ð14a Þs B yz
¼l B 0
l ðy Þ
À1=2
l ðy Þd f ðy Þdy À12d l ðy Þdy
f ðy Þ
!
exp ½ik ðx Àct ÞðÀH <y <0Þ:
ð14b Þ
3.Boundary conditions
On the top-most free surface and the interface between the layer and homogeneous medium,zero-stress and displace-ment/shear stress have to satisfy the continuity conditions as listed below:
s B yz ¼0;
y ¼ÀH ;ð15a Þw B ðx ;0Þ¼w ðx ;0Þ;
y ¼0;
ð15b Þs B yz ¼s yz ;
y ¼0:ð15c Þ
H.Zhu et al./Applied Mathematics and Computation 231(2014)93–99
95
4.Dispersion equations
Substituting Eqs.(2),(4),(13)and (14)into Eq.(15),we obtain the following equations:
B 1f 1ðy Þj y ¼ÀH þB 2f 2ðy Þj y ¼ÀH ¼0;
ð16a ÞB 1l ð0ÞÀ1=2f 1ð0ÞþB 2l ð0ÞÀ1=2f 2ð0ÞÀA ¼0;
ð16b ÞB 1l B 0l ð0ÞÀ1=2f 1ðy Þj y ¼0þB 2l B 0l ð0Þ
À1=2
f 2ðy Þj y ¼0þb l A ¼0;ð16c Þ
where
f 1ðy Þ¼l ðy Þd f 1ðy Þdy À12d l ðy Þ
dy f 1;ð17Þ
f 2ðy Þ¼l ðy Þ
d f 2ðy ÞÀ1d l ðy Þ
f 2
:ð18Þ
Homogeneous equations (16a)–(16c)composite a linear system for determining A ,B 1and B 2.A non-tri
vial solution re-quires the coefficient determinant of these three equations to be zero.This actually gives the general dispersion equation
of the phase velocity c of Love wave
l B 0f 2ðy Þj y ¼0þb l f 2ð0Þ
f 1ðy Þj y ¼ÀH Àl B
0f 1ðy Þj y ¼0þb l f 1ð0Þ
f 2ðy Þj y ¼ÀH ¼0:
ð19Þ
Equation (19)is a transcendental one.Analytically solving this equation often is difficult.Only when c (y )takes some spe-cial form,analytical solutions can be found,while they are commonly expressed in unfamiliar functions.For getting simple
closed-form analytical solution of Eq.(10),approximate method has to be employed.Here we employ series expansion method.First we expand c (y )in the following form,
c ðy Þ¼c j y ¼0þ
d c dy    y ¼0
y þ12!d 2c dy 2
y ¼0
y 2þÁÁÁð20Þ
Then expressing f ðy Þas:
f ðy Þ¼exp a 1y þa 2y 2þÁÁÁþa n y n þÁÁÁÀÁ
;
ð21Þ
where a 1,a 2,...are coefficients to be determined.Substituting Eqs.(20)and (21)into Eq.(10),we have
constant love什么意思ða 1Þ2þ2a 2þk 2c j y ¼0þ6a 3þ4a 1a 2þk 2
d c dy
y ¼0
!y þÁÁÁ¼0:ð22Þ
It is noted that only when c (y )>0there exists Love wave.Let the each coefficient of y in Eq.(22)to be zero,we get con-jugate complex solutions of a j ðj 2N Þ.One is a j ¼q j i ðj 2N Þ,where q j >0ðj 2N Þ;the other is a j ¼Àq j i ðj 2N Þ,the general solution of Eq.(10):
f ðy Þ¼B 1f 1ðy ÞþB 2f 2ðy Þ¼B 1exp ðq ÞþB 2exp ðÀq Þ;
ð23Þ
where q ¼ðq 1yi þq 2y 2i þÁÁÁÞ.
Accordingly,the displacement and stress field in the functionally graded layer can be expressed as:
w B ¼ðB 1exp ðq ÞþB 2exp ðÀq ÞÞl ðy ÞÀ1=2exp ½ik ðx Àct Þ ;
ð24Þs B xz ¼B 1exp ðq ÞþB 2exp ðÀq ÞðÞik l B
0l ðy Þ
1=2exp ½ik ðx Àct Þ ;ð25a Þs B yz ¼ðB 1g 1þB 2g 2Þexp ½ik ðx Àct Þ ;
ð25b Þ
where
g 1¼l B
l ðy Þ
À1=2
exp ðq Þl ðy Þdq À
1d l ðy Þ
g 2¼Àl
B 0
l ðy Þ
À1=2
exp ðÀq Þl ðy Þdq þ
1d l ðy Þ
:Substituting Eq.(23)into Eq.(19),the general form of dispersion equation of Love wave becomes
96H.Zhu et al./Applied Mathematics and Computation 231(2014)93–99
tan q 1H Àq 2H 2
þÁÁÁ  Àb l h 1j
y ¼ÀH þl B 0h 2j y ¼0h 3j y ¼ÀH Àl B 0h 3j y ¼0h 1j y ¼ÀH b l h 3j y ¼ÀH þl B 0h 2j y ¼0h 1j y ¼ÀH Àl B
0h 3j y ¼0h 3j y ¼ÀH
i ¼0;ð26Þ
where h 1¼l ðy Þdq dy
.h 2¼q 1l ðy Þi ;h 3¼
12d l ðy Þ
dy
:
Several remarks about this dispersion equation are given below:
(1)Because Love wave exists in both the cap layer and in an adjacent strip below the interface,c (y )depends weakly on y .
Hence just several lower terms in c (y )are enough for the accuracy for solving dispersion equation.(2)Love wave exists only when c (y )>0,hence we have
1Àa ðy Þk
2  l ðy Þq ðy Þ  1=2
c B T <c <c T :ð27Þ
The value of c is confined in the range given in Eq.(27).
(3)The dispersion equation for the case where both the half-space and cap layer are homogeneous has previously been
obtained [14]to read
kH ðc =c B T Þ2
À1  1=2¼tan À1l l B 1Àðc =c T Þ2c =c B T
ÀÁ2À1 !120@1
A þn p ;n ¼0;1;2; (28)
To compare,we rewrite the dispersion equation (26)for the homogeneous half space with a graded layer as
q 1H Àq 2H 2þÁÁÁ¼tan À1
b l h 1j y ¼ÀH þl B 0h 2j y ¼0h 3j y ¼ÀH Àl B
0h 3j y ¼0h 1j y ¼ÀH b l h 3j y ¼ÀH þl 0h 2j y ¼0h 1j y ¼ÀH Àl 0h 3j y ¼0h 3j y ¼ÀH
i
!
þn p ;
n ¼0;1;2; (29)
It is noted that the Love waves in both cases share the following features:
(1)Phase velocity c of Love wave is dispersive.
(2)There is a dispersion curve for each given n in both Eqs.(28)and (29).
(3)In both dispersion equations,for each given n there exists a critical wave number k ðn Þ
cr below which no real value solu-tion can be found.And accordingly,x ðn Þ
cr ¼k ðn Þcr c T is the cut-off frequency of the n th order vibration mode of Love wave.The solution of dispersion Eq.(29)is extremely complicated.Also the existence of solution and its properties are greatly influenced by both material parameters (such as l B ,l ,q B ,q )and the layer’s thickness H .
Since most energy is carried by the first few vibration modes of the Love wave,higher order modes may be neglected for the engineering purposes.In the following several numerical examples for the dispersion property of the lowest order vibra-tion mode will be given.5.Numerical examples
The material parameters of homogeneous half-space are:shear elastic modulus l =80GPa,mass density q =7800kg/m 3,and hence the shear wave velocity c T =3202m/s.The material of the cap layer is SiC/Al,and its shear elastic modulus l B and mass density q B are assumed to vary in a parabolic form with the depth y ,
l B ¼l B 01þe 1y þe 2y 2
ÀÁ
;ð30a Þq B ¼q B 01þe 3y þe 4y 2ÀÁ
;
ð30b Þ
where l B 0=26.3GPa,q B 0=2700kg/m 3,hence shear wave velocity is c B
T =3121m/s.At the cap layer top free surface,material
parameters are taken to be l B ÀH =29.5GPa,q B
ÀH =3200kg/m 3.The relation between the thickness of the graded layer H and e 1,e 2,e 3,e 4can be written as:
e 2¼l B ÀH =l B
0À1ÀÁ
=H 2þe 1=H ;ð31a Þe 4¼q B ÀH =q B 0À1À
Á
=H 2þe 3=H :
ð31b Þ
Plugging these conditions into Eq.(15),we have
H.Zhu et al./Applied Mathematics and Computation 231(2014)93–99
97

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