Multiple scattering of flexural waves from a cylindrical inclusion in a semi-infinite thin plate
JOURNAL OF
SOUND AND
VIBRATION
Journal of Sound and Vibration 320(2009)878–892
Multiple scattering of ?exural waves from a cylindrical inclusion
in a semi-in?nite thin plate
Xue-Qian Fang a,?,Xiao-Hua Wang b
a
Department of Engineering Mechanics,Shijiazhuang Railway Institute,Shijiazhuang 050043,PR China b Computer and Information Engineering Department,Shijiazhuang Railway Institute,Shijiazhuang 050043,PR China
Received 13April 2008;received in revised form 30July 2008;accepted 21August 2008
Handling Editor:https://www.sodocs.net/doc/d810387174.html,m
Available online 2October 2008Abstract
In this paper,image method and wave function expansion method are applied to investigate the multiple scattering of ?exural waves and dynamic stress concentration from a cylindrical inclusion in a semi-in?nite thin plate,and the analytical solution of this problem is obtained.The semi-in?nite plate with roller-supported boundary is considered,and the image method is used to satisfy the boundary condition.The addition theorem for Bessel functions is employed to accomplish the translation of wave ?elds between different local coordinate systems.As an example,the numerical results of dynamic stress concentration factors around the inclusion are graphically presented and analyzed.Analysis shows that the angular distribution of the dynamic stress around the inclusion shows great difference when the distance between the inclusion and the semi-in?nite edge is different.The effects of the elastic modulus,density,Poisson’s ratio,and the thickness of the inclusion on the shadow side of the inclusion are greater when the distance between inclusion and the semi-in?nite edge is small.In the region of lower frequency,the effects of the elastic modulus,density,Poisson’s ratio,and the thickness of the inclusion on the dynamic stress are https://www.sodocs.net/doc/d810387174.html,parisons with other existing models are also discussed.
r 2008Elsevier Ltd.All rights reserved.
1.Introduction
Plate structures are widely used in aviation,aerospace,shipping,and civil construction engineering.Inhomogeneities such as inclusions,cavities,or cracks in plates strongly affect the serving life of the structure.If the inhomogeneities are embedded in a plate,it is de?nitely vital to determine them and analyze their effects.The knowledge of stress concentration analysis is very important for a reliable design of the plate,so the stress concentration problems in the plate have received a considerable amount of interest over the past few decades.It is known from the literature on wave dynamics that in a certain range of wave frequency,the dynamic stress around the inhomogeneities is much greater than the static stress.So,to increase the bearing capacity of structures and the service life of structures,the investigations on elastic waves scattering and dynamic stress concentrations in plate structures are more important.
https://www.sodocs.net/doc/d810387174.html,/locate/jsvi
0022-460X/$-see front matter r 2008Elsevier Ltd.All rights reserved.
doi:10.1016/j.jsv.2008.08.023
?Corresponding author.Tel.:+8631187936542.
E-mail address:fangxueqian@https://www.sodocs.net/doc/d810387174.html, (X.-Q.Fang).
X.-Q.Fang,X.-H.Wang/Journal of Sound and Vibration320(2009)878–892879 Up to present time,stress concentration caused by cutouts or inclusions in an in?nite plate has been an interesting research topic.First,Kirsch[1]pioneered the study of the stress concentration around a circular hole in the in?nite plate under a uniform longitudinal tension.Subsequently,Ying and Truell[2]studied the scattering of sound waves by a spherical scatterer in an elastic medium.Kato[3]and White[4]considered the effect of a cylindrical obstacle on wave scattering.Afterwards,Pao and Chao[5]and Pao[6]investigated the?exural wave diffractions by a cavity in an elastic plate based on Mindlin’s https://www.sodocs.net/doc/d810387174.html,ing a combined?nite element and analytical method,Paskaramoorthy et al.studied the scattering of slow?exural waves by arbitrary-shaped cavities[7]and a?nite through-crack[8]in the in?nite elastic plate.The analogous problem of scattering of?exural waves by circular inclusions was solved by Vemula and Norris using the Mindlin plate theory[9]as well as the lower-order Kirchhoff plate theory[10].Recently,Leviatan et al.[11]presented a source-model technique to investigate the scattering of time-harmonic?exural wave in a heterogeneous thin plate.
To the author’s knowledge,the researches on?exural waves and dynamic stress in plates mainly focused on the models of in?nite structures.However,the models of semi-in?nite plates are more familiar in engineering application[12].Due to the complexity of multiple scattering and re?ection of elastic waves between the inhomogeneities and the boundary,an alternative,and possibly simpler,point of view is to ignore edge effects as a?rst approximation.This approach is only suitable for large plates with distant edges,and leads to much simpli?cation.To accurately describe the dynamic stress distribution,the boundary effect of the plate should be considered.In the past,only a few papers about the dynamic stress in the semi-in?nite plate are reported. The scattering of time-harmonic plane longitudinal,shear,and Rayleigh waves by a crack in two dimensions embedded in a homogeneous isotropic elastic half-space was investigated by Shah et al.[13].In a series of detailed study,Fang and his co-workers have investigated the multiple scattering and re?ection of anti-plane shear waves from cavities in the semi-in?nite plate[14,15]and a semi-in?nite slab[16],and the boundary effect were analyzed.
The multiple scattering and re?ection of?exural waves at the boundary of plates are more complicated than those of single-mode waves such as the acoustic and shear waves due to the generation of non-geometrically induced evanescent waves and the effects of higher-order boundary conditions[17].Therefore,very few papers have investigated this problem.Only recently,Fang and his co-workers have studied the multiple scattering of ?exural waves from a cutout[18]and two cutouts[19]in the semi-in?nite plate.
The main objective of this paper is to extend the work of Hu et al.[18]to the case of the multiple scattering of?exural waves from an embedded inclusion in a semi-in?nite thin plate with roller-supported boundary.The image method is applied to satisfy the boundary conditions of the semi-in?nite structure.Based on Mindlin’s theory of transverse motion in thin plates,the wave function expansion method is employed to express the wave?elds around the actual and image inclusions.The analytical solutions of the problem are obtained.As an example,the numerical results of dynamic stress concentration factor(DSCF)around the inclusion are graphically presented and discussed.
https://www.sodocs.net/doc/d810387174.html,erning equation of?exural waves in thin plates and its solution
Consider a semi-in?nite thin plate with a cylindrical inclusion,which is perfectly bonded to the exterior region along the boundary of the inclusion,as depicted in Fig.1.For simplicity,the properties of the inclusion are assumed to be uniform through their thickness.Let D,r,and h be the bending stiffness,density and thickness of the plate,and D0,r0,and h0those of the inclusion.The radius of the cylindrical inclusion is a.The distance between the center of the inclusion and the semi-in?nite edge is d.The incidence of plane?exural waves over the surface of the semi-in?nite thin plate in the x direction is considered.
For suf?ciently thin plates,approximate theories are applied.These theories take into account that the plate is thin and so the dependency of displacement?eld on the thickness coordinate may be neglected or supposed to be polynomial.Mindlin’s approximate theory for?exural waves in plates is well known,and can be found in many textbooks,for example[17,20].The theory includes shear-deformation and rotary-inertia effects,as in the Timoshenko beam theory.
Based on Mindlin’s theory,the ?exural wave equation in elastic thin plates may be described as [20]
D r 2r 2w tr h q 2w ?q ,(1)
where t is the time,q the externally applied transverse pressure,w the transverse displacement,and D the bending stiffness of the plate with
D ?Eh 3=12e1àn 2T(2)
in which E ,n ,h are the elastic modulus,Poisson’s ratio,and thickness of the plate.In the case of this paper,q is de?ned as q ?0.
Steady periodic solutions of the problem are investigated.Let w ?Re[W exp(ài o t )],then the displacement components determined by steady ?exural waves are
u x ?àz q w q x ;u y ?àz q w q y ;u z ?w ?Re ?W ex ;y Te ài o t ,(3)
where o is the incident frequency,Re(á)denotes the real part,and i ????????à1p is the imaginary unit.
According to Eqs.(1)and (3),W (x ,y )should satisfy the following equations:
r 2r 2W àk 4W ?er 2tk 2Ter 2àk 2TW ?0,
(4)er 2tk 2TW 1?0er 2àk 2TW 2?0.(5)
Here,k is the wavenumber of elastic waves in the plate,and k ?(r h o 2/D )1/4.
In Eq.(5),it is noted that W 1e ài o t denotes the propagating elastic waves in plates and W 2e ài o t denotes the localized evanescent wave motion.The two parts are integrated and form the motion of ?exural waves and vibration modes in thin plates.
According to Eq.(4),the general solution of the scattered waves resulting from the cylindrical inclusion can be described as
W s ?W s 1tW s 2?X
1n ?à1?A 1n H e1Tn ekr Te i n y tA 2n K n ekr Te i n y ,(6)
where A 1n and A 2n are determined by satisfying the boundary conditions are the mode coef?cients of scattered waves from inclusions,H n (1)(á)is the n th-order Hankel function of the ?rst kind,and K n (á)is the n th-order modi?ed Bessel function of the second kind.The superscript s denotes the scattered waves.It should be noted that H e1Tn eáTand K n (á)denote the outgoing waves.
Likewise,the refracted wave ?eld in the inclusion is a standing wave,which can be described as
W r ?W r 1tW r 2?X
1n ?à1?C n 1J n ek 0r Te i n y tC n 2I n ek 0r Te i n y ,(7)
y'o'
x'
r'
' o r x
y
d a W 10
e ikd e ikx + W 20e –kd e –kx Fig.1.Sketch o
f elastic waves incident upon a semi-in?nite plate with a cylindrical inclusion.
X.-Q.Fang,X.-H.Wang /Journal of Sound and Vibration 320(2009)878–892
880
where k0?(r0h0o2/D0)1/4is the wavenumber of elastic waves in the inclusion,C n1and C n2determined by satisfying the boundary conditions are the mode coef?cients of refracted waves from inclusions,J n(á)is the n th-order Bessel function of the?rst kind,and I n(á)is the n th-order modi?ed Bessel function of the?rst kind. The superscript r denotes the refracted waves.
3.Excitation of incident waves and total wave?eld in the plate
Assume that a plane?exural wave propagates along the positive x direction in the semi-in?nite plate.One is the propagating wave and the other is the localized vibration.According to the wave function expansion method,the incident waves may be described as
W i
1
?W10e i kd e i kxtW20eàkd eàkx
?W10e i kd
X1
n?à1i n J nekrTe i n ytW20eàkd
X1
n?à1
I nekrTe i n y,(8)
where W10,W20are the transverse vibration amplitudes of incident?exural waves.Note that the superscript i denotes the incident waves.
When the?exural wave propagates in the semi-in?nite structure,it is scattered by the inclusion at?rst. Then,the outgoing scattered wave from the inclusion is re?ected by the semi-in?nite edge,and the re?ected waves W1f arise.The re?ected waves are scattered by the inclusion again.This complex phenomenon is shown in Fig.1.
A semi-in?nite thin plate with roller-supported boundary is considered.To satisfy the boundary conditions at the semi-in?nite edge,the image method is applied.The re?ected waves at the edge of the semi-in?nite plate are described by the virtual image inclusion.For the image inclusion,the incident?exural waves propagate in the negative x0direction,and can be expressed as
W i
2
?W10e i kd eài kx0tW20eàkd e kx0
?W10e i kd
X1
n?à1iàn J nekr0Te i n y0tW20eàkd
X1
n?à1
eà1Tn I nekr0Te i n y0.(9)
When the exciting source is enough far from the inclusions,one may consider W20?0,and then the incident waves for the actual and image inclusions are described as
W i
1?W10e i kd e i kx?W10e i kd
X1
n?à1
i n J nekrTe i n y,(10)
W i
2?W10e i kd eài kx0?W10e i kd
X1
n?à1
iàn J nekr0Te i n y0.(11)
Considering the multiple scattering between the actual and image inclusions,the scattered?elds of?exural waves produced by the actual inclusion in the localized coordinate system(r,y)are described as
W s
1?
X1
n?à1
ˉA
n1
He1T
n
ekrTe i n yt
X1
n?à1
ˉA
n2
K nekrTe i n y,(12)
whereˉA nj?P1
l?1
A l
nj
for j?1,2are the total scattering coef?cients of the actual inclusion,and the superscript
s denotes the scattered waves.Note that l denotes the l th mode coef?cients of scattered waves.
Similarly,the scattered waves produced by the image inclusion in the localized coordinate system(r0,y0),are described as
W s
2?
X1
n?à1
ˉB
n1
He1T
n
ekr0Te i n y0t
X1
n?à1
ˉB
n2
K nekr0Te i n y0,(13)
X.-Q.Fang,X.-H.Wang/Journal of Sound and Vibration320(2009)878–892881
whereˉB n1;ˉB n2are the total mode coef?cients of scattered waves of the image inclusion.They are determined by satisfying the boundary conditions of the inclusions.In fact,they are also related to the boundary conditions of the edge of plates.
From Eqs.(10)and(11),the total incident wave W t i,the original incident wave plus its image as the re?ected wave from the roller-supported edge,can be written as
W i
t ?W10e i kdee i kxteài kx0T?W10e i kd
X1
n?à1
ei nteà2i kd iànTJ nekrTe i n y
"#
.(14)
Then,the total?eld of elastic waves in the semi-in?nite plate should be produced by the superposition of the incident?eld,the scattered?elds and the re?ected?elds at the edge of the plate,i.e.,
W?W i
1tW s
1
tW f
1
?W i
1
tW s
1
tW s
2
.(15)
Now,for the straight roller-supported edge,the boundary conditions at the edge are considered:
q WetTq x ?
q3WetT
q x
?0at x?àb.(16)
From Eq.(14),it is clear that the total incident wave has satis?ed these conditions.
Applying the boundary conditions(16)to Eq.(15),and from the fact that r?r0,y0?pày on the plane boundary of the semi-in?nite structure,and the identity H n(1)e i n p?Hàn(1),the following relations between the total mode coef?cients of scattered waves are obtained:
ˉB
n1
?ˉAàn1;ˉB n2?eà1TànˉAàn2.(17) From Eq.(17),the relations between the total scattering coef?cients of the image inclusion and those of the actual inclusion are obtained.
Then,by using the following translational addition theorems of Bessel functions[21]
He1T
n ekr0Te i n y0?
X1
m?à1
eà1Tmàn He1T
màn
e2kdTJ mekrTe i m y,(18)
K nekr0Te i n y0?
X1
m?à1
eà1Tm K màne2kdTI mekrTe i m y,(19)
the scattered?elds W2s of the image inclusion can be represented in the local coordinate systems(r,y). After some manipulations,the total scattered?eld W s around the actual inclusion is expressed as
W s?
X1
n?à1ˉA
n1
He1T
n
ekrTe i n yt
X1
n?à1
X1
m?à1
ˉA
m1_mn J nekrTe i n y
t
X1
n?à1ˉA
n2
K nekrTe i n yt
X1
n?à1
X1
m?à1
ˉA
m2
k mn I nekrTe i n y,(20)
where
_mn?eà1Tntm He1Tntme2kdTand k mn?eà1Tntm Ke1Tntme2kdT.
4.Boundary conditions around the inclusion in the plate
There are four continuous conditions around the inclusion.It is required that W t,q W t/q r,M rr and V r are continuous across the boundary of the inclusion.The expressions of M rr and V r are given by
M rr?àD
q2W t
q r2
tv
1
r
q W t
q r
t
1
r2
q2W t
q y2
,(21)
X.-Q.Fang,X.-H.Wang/Journal of Sound and Vibration320(2009)878–892 882
V r?Q rt1
r
q M r y
q y
?àD
q
q r
er2W tTàDe1àvT
1
r
q
q y
q2W t
q r q y
à
1
r
q W t
q y
,(22)
where M rr and V r are the bending moment and the equivalent shear force around the inclusions,respectively.
5.Solution of scattering mode coef?cients of elastic waves
By substituting the expressions of wave?elds into the boundary conditions of the actual inclusion,four equations determining the mode coef?cients are obtained.Multiplying by eài s y and integrating from0to2p on both sides of the equations,one can obtain a set of system of linear equations for the unknown mode coef?cients fˉA n1;ˉA n2;C n1;C n2g,
ˉA n1He1T
n
ekaTtJ nekaT
X1
m?à1
_mnˉA m1tK nekaTˉA n2tI nekaT
X1
m?à1
k mnˉA m2
àJ nek0aTC n1àI nek0aTC n2?àW10e i kdei nteà2i kd iànTJ nekaT,(23)
ˉA
n1_1Ht_1J
X1
m?à1
_mnˉA m1t_1KˉA n2t_1I
X1
m?à1
k mnˉA m2
à_2
J C n1à_2
I
C n2?àW10e i kdei nteà2i kd iànT_1J,(24)
<1 H ˉA
n1
t<1
J
X1
m?à1
_mnˉA m1t<1KˉA n2t<1I
X1
m?à1
k mnˉA m2à
D0
D
?<2
J
C n1t<2
I
C n2
?àW10e i kdei nteà2i kd iànT<1
J
,(25)
I1HˉA n1tI1J
X1
m?à1_mnˉA m1tI1KˉA n2tI1I
X1
m?à1
k mnˉA m2à
D0
D
?I2
J
C n1tI2
I
C n2
?àW10e i kdei nteà2i kd iànTI1
J
.(26) Here,the following notations are used:
_1X??nX nekaT?kaX nt1ekaT ,(27)
_2X??nX nek0aT?k0aX nt1ek0aT ,(28)
<1 X ??n2e1àvT?ekaT2 X nekaTàe1àvTkaX0
n
ekaT,(29)
<2 X ??n2e1àv0T?ek0aT2 X nek0aTàe1àv0Tk0aX0
n
ek0aT,(30)
I1X?n2e1àvTX nekaTà?n2kae1àvT?ekaT3 X0nekaT,(31)
I2X?n2e1àv0TX nek0aTà?n2k0ae1àv0T?ek0aT3 X0nek0aT,(32) where the upper and lower signs refer to X?H(1),J and X?K,I,respectively.
After arrangement,Eqs.(23)–(26)can be simpli?ed as
EA?f,(33) where E is the coef?cient matrix of(8n+4)?(8n+4),f the vector of(8n+4)ranks,and A the mode coef?cients.
According to the de?nition of DSCF,the dynamical bending moment concentration factor is the ratio of amplitude hoop bending moment around the inclusion and the maximum bending moment in the incident direction of elastic waves[22].Thus,the expression of the DSCF around the cylindrical inclusion X.-Q.Fang,X.-H.Wang/Journal of Sound and Vibration320(2009)878–892883
described as
DSCF ?M n
yy ?M yy =M 0?à1
W 10k 2?r 2W àe1àv Tq 2W =q r 2 ,(34)
where M 0is the maximum amplitude of bending moment of incident waves and M 0?Dk 2W 10.Then,the DSCF around the inclusions a and b are,respectively,written as DSCF ?1k a 2e i kd X 1n ?à1ei n te à2i kd i àn TT J e i n y tX 1n ?à1ˉA n 1T H e i n y tX 1n ?à1X 1m ?à1ˉA m 1_e1Tmn T J e i n y "tX 1n ?à1ˉA n 2T K e i n y tX 1n ?à1X 1m ?à1ˉA m 2k e1Tmn
T I e i n y #.(35)
Here,the following notations are used:
T X ?e1àv TkaX 0n eka Tà?e1àv Tn 2?v eka T2 X n eka T,
(36)
where the upper and lower signs refer to X ?H (1),J and X ?K ,I ,respectively.
6.Numerical examples and discussion
Fatigue failures often occur in the regions with high stress concentration,so an understanding of the distribution of the dynamic stress around the inclusion is quite useful in structural design.According to the expression of DSCF,the DSCFs around the cylindrical inclusion are simulated by using MATLAB.
In the following analysis,it is convenient to make the variables dimensionless.To accomplish this step,a representative length scale a ,where a is the radius of inclusion,is introduced.The following dimensionless variables and quantities have been chosen for computation:the incident wavenumber is k *?ka ?0.01–2.0,the distance between the center of the inclusion and the semi-in?nite boundary is d *?d /a ?1.1–10.0,the ratio of elastic modulus E *?E 0/E ?0.1–5.0,the ratio of mass density r *?r 0/r ?0.1–2.0,the ratio of thickness h *?h 0/h ?1.0–2.0,v ?0.3,and v 0?0.2,0.3,0.4.
To validate the present dynamical model,Figs.2and 3are given.Fig.2shows the angular distribution of the DSCFs around a hole in the in?nite thin plate.The DSCFs obtained from Ref.[7]are represented by line
120
90
2.56030
330
300
270240
21018015021.5
1
https://www.sodocs.net/doc/d810387174.html,parison of the angular distribution of dynamic stress concentration factors (k *?0.5,d *?10.0).
X.-Q.Fang,X.-H.Wang /Journal of Sound and Vibration 320(2009)878–892
884
with dots.The bold line obtained from the method in this paper represents the DSCFs in the thin plate without any inhomogeneities.Excellent agreement with Ref.[7]can be observed in Fig.2.A thin plate without any inhomogeneities is identical to the plate with an inclusion which has the same mechanical properties and thickness as the plate itself.As expected,the DSCFs in this case are uniformly 1.0,and this also can validate the dynamical model.
Fig.3illustrates the DSCF at the position of y ?p /2as a function of the dimensionless wavenumber with E 0?h 0?0.When the distance ratio is d *?8.0,the wave ?eld close to the edge of the plate is almost the same as that of the semi-in?nite plate with no inclusion.E 0?h 0?0means that the inclusion in the plate reduces to a cavity.From Fig.3,one can see that when the incident wavenumber is k *-0,the DSCF is the maximum,and its value is about M y *?1.90,which is consistent with the numerical results of in?nite plates in Refs.[18,22].It can be seen that peaks and troughs occur in Fig.3.This is due to the variation in wavenumber changing the distance between the inclusion and the nodal line in the total standing wave ?eld created by the incident,re?ected,and scattered waves.At some frequencies,the inclusion is at nodal line,while at others it is at an anti-nodal line.
Figs.4–9illustrate the angular distribution of DSCFs around the inclusion when the values of d *,E *,r *,and v 0are different.
In Fig.4,the inclusion in the plate reduces to a cavity.From Fig.4,it can be seen that when the distance between the inclusion and the semi-in?nite edge is great,the dynamic stresses at the positions around the cavity show little variation.The dynamic stress at the positions of y ?0is the maximum.If the distance between the inclusion and the semi-in?nite edge becomes small,the DSCF at the position of y ?p is the maximum.The DSCF at the position of y ?0is the minimum.The maximum value of DSCF is much greater than that at other positions.
In Fig.5,the elastic modulus and density of the inclusion are greater than those of the plate.It can be observed that when the distance between the inclusion and the semi-in?nite edge is small,the DSCF at the position of y ?p is still the https://www.sodocs.net/doc/d810387174.html,paring the results with those in Fig.4,it is clear that if the distance between the inclusion and the semi-in?nite edge is great,the effects of the elastic modulus and density of the inclusion on the DSCFs are little.However,when the distance between the inclusion and the semi-in?nite edge is small,the effects of the elastic modulus and density of the inclusion on the DSCFs become great,especially on the shadow side of the inclusion.Due to the effect of the elastic modulus and density of the inclusion,the dynamic stresses on the shadow side of the inclusion become great.
In Fig.6,the elastic modulus and density of the inclusion are less than those of the plate.It can be observed that when the distance between the inclusion and the semi-in?nite edge is small,the DSCF at the position of y ?p is still the https://www.sodocs.net/doc/d810387174.html,paring the results with those in Fig.4,it can be seen that if the distance
00.10.20.30.40.50.60.70.80.9 1.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
D S C F Dimensionless wave number k ?
Fig.3.Dynamic stress concentration factors versus dimensionless wavenumber with different values of d *(E 0?r 0?0,y ?p /2).X.-Q.Fang,X.-H.Wang /Journal of Sound and Vibration 320(2009)878–892885
between the inclusion and the semi-in?nite edge is great,the effects of the elastic modulus and density of the inclusion on the DSCFs are little.However,when the distance between the inclusion and the semi-in?nite edge is small,the effects of the elastic modulus and density of the inclusion on the DSCFs become great,especially on the shadow side of the inclusion.Due to the effect of the elastic modulus and density of the inclusion,the dynamic stresses on the shadow side of the inclusion become little.
Comparing the results in Fig.6with those in Fig.5,it is interesting to note that no matter whether the elastic modulus and density of the inclusion is greater than those of the plate,the dynamic stresses on the shadow side of the inclusion become great.When the distance between the inclusion and the semi-in?nite edge is small,the dynamic stress on the position of y ?p is greater in the case of E *41.0,r *41.0than that in the case of E *o 1.0,r *o 1.0.When the distance between the inclusion and the semi-in?nite edge is great,the
90
2.5
6030
330
300
270240
2101801501202
1.5
1
0.5Fig.5.Angular distribution of dynamic stress concentration factors around the inclusion with different values of d *;k *?0.5,E *?5.0,r *?2.0,h *?1.0,v ?v 0?0.3.
90
2.5
6030
330
300
270240
2101801501202
1.5
1
0.5Fig.4.Angular distribution of dynamic stress concentration factors around the cavity with different values of d *;k *?0.5,E 0?r 0?0.
X.-Q.Fang,X.-H.Wang /Journal of Sound and Vibration 320(2009)878–892
886
dynamic stress on the position of y ?0becomes greater in the case of E *o 1.0,r *o 1.0than that in the case of E *41.0,r *41.0.
The effect of increasing the thickness of the inclusion on the angular distribution of DSCF is shown in Fig.7.It can be seen that when the distance between the inclusion and the semi-in?nite edge is small,due to the effect of the thickness of the inclusion,the dynamic stress on the shadow side of the inclusion become much greater,especially at the position of y ?p .However,if the distance between the inclusion and the semi-in?nite edge is great,the dynamic stresses on the illuminated side of the inclusion become much greater.The dynamic stresses at the positions of y ?p /2and àp /2show little variation with the thickness of the inclusion.The effect of Poisson’s ratio of the inclusion on the angular distribution of DSCF is shown in Figs.8and 9.In Fig.8,Poisson’s ratio of the inclusion is greater than that of the plate.In Fig.9,Poisson’s ratio of the
90
2.5
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1.5
1
0.5Fig.6.Angular distribution of dynamic stress concentration factors around the inclusion with different values of d *;k *?0.5,E *?0.1,r *?0.2,h *?1.0,v ?v 0?0.3.
90
2.5
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1.5
1
0.5Fig.7.Angular distribution of dynamic stress concentration factors around the inclusion with different values of d *;k *?0.5,E *?5.0,r *?2.0,h *?1.5,v ?v 0?0.3.
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inclusion is less than that of the https://www.sodocs.net/doc/d810387174.html,paring the results with those in Fig.5,it can be seen that only if the distance between the inclusion and the semi-in?nite edge is small,does a change of the inclusion’s Poisson’s ratio have any observable effect of on the DSCFs around the inclusion.The effect at the position of y ?p is the maximum.When Poisson’s ratio of the inclusion is greater than that of the plate,the dynamic stress becomes small.When Poisson’s ratio of the inclusion is less than that of the plate,the dynamic stress becomes great.
Fig.10illustrates the DSCFs at the position of y ?p /2of the inclusion as a function of the incident wavenumber when the values of E *and r *are different.It can be seen that the dynamic stress at the position of y ?p /2decreases with the increase of dimensionless wavenumber.In the region of low frequency,the
90
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6030
330
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270240
2101801501202
1.5
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0.5Fig.8.Angular distribution of dynamic stress concentration factors around the inclusion with different values of d *;k *?0.5,E *?5.0,r *?2.0,h *?1.0,v ?0.3,v 0?0.2.
90
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1.5
1
0.5Fig.9.Angular distribution of dynamic stress concentration factors around the inclusion with different values of d *;k *?0.5,E *?5.0,r *?2.0,h *?1.0,v ?0.3,v 0?0.4.
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dynamic stress decreases with the increase of the values of E *and r *.In the region of higher frequency,the dynamic stress increases with the increase of the values of E *and r *.
Fig.11illustrates the DSCFs at the position of y ?p of the inclusion as a function of the incident wavenumber when the values of E *and r *are different.It can be seen that the dynamic stress at the position of y ?p decreases with the increase of dimensionless wavenumber.In the region of low frequency,the dynamic stress increases with the increase of the values of E *and r *.In the region of higher frequency,the dynamic stress decreases with the increase of the values of E *and r *.Comparing with the results in Fig.10,it is observed that the effect of the values of E *and r *on the dynamic stress at the position of y ?p is greater.Figs.12and 13illustrate the DSCFs as a function of the incident wavenumber with different values of v 0at the positions of y ?p /2and p of the inclusion,respectively.It can be seen that in the region of low frequency,
00.10.20.30.40.50.60.70.80.9 1.000.2
0.4
0.6
0.8
1.0
1.2
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1.6
1.8
2.0
D S C F Dimensionless wave number k ?
Fig.10.Dynamic stress concentration factors versus dimensionless wavenumber with different values of E *and r *at y ?p /2(h*?1.0,d *?1.5,v ?v 0?0.3).
00.10.20.30.40.50.60.70.80.9 1.0
00.4
0.8
1.2
1.6
2.0
2.4
2.8
D S C F Dimensionless wave number k
?Fig.11.Dynamic stress concentration factors versus dimensionless wavenumber with different values of E *and r *at y ?p (h *?1.0,d ?1.5,v ?v 0?0.3).
X.-Q.Fang,X.-H.Wang /Journal of Sound and Vibration 320(2009)878–892889
the dynamic stress shows little variation with the value of v 0.However,in the region of higher frequency,the dynamic stress decreases with the increase of the value of v https://www.sodocs.net/doc/d810387174.html,paring the results in Figs.12and 13,it is observed that the effect of the value of v 0on the dynamic stress at the position of y ?p is greater.
Figs.14and 15illustrate the DSCFs as a function of the incident wavenumber with different values of h *at the positions of y ?p /2and p of the inclusion,respectively.At the position of y ?p /2,the dynamic stress increases with the increase of the value of h *in the region of higher frequency;the dynamic stress shows little variation in the region of low frequency.However,at the position of y ?p ,the dynamic stress increases greatly with the increase of the value of h *in the region of low frequency;the dynamic stress shows little variation in the region of higher https://www.sodocs.net/doc/d810387174.html,paring the results in Figs.14and 15,it is observed that the effect of the value of h *on the dynamic stress at the position of y ?p is greater.
00.10.20.30.40.50.60.70.80.9 1.000.2
0.4
0.6
0.8
1.0
1.2
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1.6
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2.0
D S C F Dimensionless wave number k ?
Fig.12.Dynamic stress concentration factors versus dimensionless wavenumber with different values of v 0at y ?p /2(E *?5.0,r *?2.0,h *?1.0,d *?1.5).
00.10.20.30.40.50.60.70.80.9 1.0
00.4
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1.2
1.6
2.0
2.4
2.8
D S C F Dimensionless wave number k ?
Fig.13.Dynamic stress concentration factors versus dimensionless wavenumber with different values of v 0at y ?p (E *?5.0,r *?2.0,h *?1.0,d *?1.5.
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7.Conclusion
In this study,based on the dynamical equation of ?exural waves in elastic thin plates,and applying the image method and the wave function expansion method,the multiple scattering of ?exural waves and dynamic stress from a cylindrical inclusion in a semi-in?nite thin plate are investigated.The semi-in?nite edge with roller-supported boundary conditions is considered.Analytical solutions and numerical results of the problem are presented and https://www.sodocs.net/doc/d810387174.html,parisons with previous literature demonstrate the validity of the analytical method.
It can be seen that the analytical results of dynamic stress in semi-in?nite plates are different from those in in?nite https://www.sodocs.net/doc/d810387174.html,paring with the results in Hu et al.[18],the existence of the inclusion has great effect on the distribution of the dynamic stress in the semi-in?nite plate,especially in the region of higher frequency.If the distance between the inclusion and the semi-in?nite edge is small,due to the multiple scattering of elastic
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D S C F Dimensionless wave number k
?
Fig.14.Dynamic stress concentration factors versus dimensionless wavenumber with different values of h *at y ?p /2(E *?5.0,r *?2.0,v ?v 0?0.3,d *?1.5).
00.10.20.30.40.50.60.70.80.9 1.0
00.5
1.0
1.5
2.0
2.5
3.0
D S C F Dimensionless wave number k ?
Fig.15.Dynamic stress concentration factors versus dimensionless wavenumber with different values of h *at y ?p (E *?5.0,r *?2.0,v ?v 0?0.3,d *?1.5).
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waves,the DSCF at the position of y ?p around the inclusion is the maximum.The effects of the elastic modulus,density,thickness,and Poisson’s ratio of the inclusion on the DSCFs are greater when the distance between the inclusion and the semi-in?nite edge is small,especially on the shadow sides of the inclusion.In contrast to the elastic modulus,density,thickness of the inclusion,the effect of Poisson’s ratio of the inclusion on the angular distribution of DSCF is less.
The analysis of this paper can provide a theoretical basis and reference data for strength designs and non-destructive evaluation of plate structures with holes near their edges.
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