Vibrations Of Poroelastic Solid Cylinder In The Presence Of Static Stresses In The Pervious Surfaces

Vibrations Of Poroelastic Solid Cylinder In The Presence Of

Static Stresses In The Pervious Surfaces

Srisailam Alety(*)

*(Department of Mathematics Kakatiya UniversityWarangal-506009, A.P., INDIA)

ABSTRACT

This paper deals with the flexural vibrations of poroelastic solid cylinders. The frequency equations for pervious

surface are obtained in the frame work of Biot’s theory of wave propagation in poroelastic solids. The gauge

invariance property is used to eliminate one arbitrary constant in solution of the problem. This would lower the number of boundary conditions actually required. For illustration purpose, three materials are considered and then discussed. In either case, phase velocity is computed against wave number

I.

INTRODUCTION

We know from daily experience that many manmade structures approximately cylindrical in shape and made of poroelastic material. Even in

man’s own body some osseous tissue, approximately

cylindrical in shape and are elastic in nature. Excess stresses and pressure in the above elements result in vibrations.

Kumar (1964) studied the propagation of axially symmetric waves in a finite elastic cylinder. Flexural vibrations of finite circular elastic cylinder is studied by Biswas et al. (1976). Mott (1972 ) investigated elastic waveguide propagation in an infinite isotropic solid cylinder that is subjected to a static axial stress

and strain. Employing Biot’s theory (Biot, 1956),

Tajuddin and Sarma (1978, 1980) studied the torsional vibrations of finite hollow poroelastic cylinders. Reddy and Tajuddin (2000) investigated plane-strain vibrations of thick-walled hollow poroelastic cylinders and discussed extreme limiting cases of plate and solid cylinder. Tajuddin and Shah (2006, 2007) studied the circumferential waves and torsional vibrations of infinite hollow poroelastic cylinders in presence of dissipation.

The analysis of the flexural vibrations in cylindrical structures has wide applications in the field of acoustics structural design and Biomechanics, where the knowledge of natural mode of the vibration is of paramount importance. There should be stress-free conditions in order to obtain natural modes

theoretically. However, to the best of author’s

knowledge, flexural vibrations of poroelastic cylinder that in presence of static stress are not investigated. Therefore, in this paper, an attempt is made to

investigate the same in the frame work of Biot’s

theory. Frequency equations are obtained for pervious boundary. Phase velocity is computed against wave number in the case of pervious surface and results are presented graphically.

This paper is organized as follows. In section 2, basic governing equations, formulation, and solution of the problem are given. In section 3, frequency equations are derived for pervious surface. Numerical results are presented in section 4. Finally, conclusions are given in section 5.

II.

Governing equations and solution of the

problem

Let

(

r

,



,

z

)

be,cylindrical,polar

coordinates.

Consider a poroelastic solid cylinder of radius

.

a

The equations of motion of a homogeneous,

isotropic poroelastic solid (Biot, 1956) in

presence of dissipation

(

b

)

are

),

(

)

(

),

(

)

(

)

(

22 12

2 2 12 11 2 2

2

 

 

  

 



















































U

u

t

b

U

u

t

R

e

Q

U

u

t

b

U

u

t

Q

e

N

A

u

N













(2.1)

Where



2 is Laplace operator,

u

(

u

,

v

,

w

)



and

)

,

,

(

U

V

W

U



are solid and liquid displacements,

e

and



are the dilatations of solid and liquid, respectively;

A

,

N

,

Q

,

R

are all poroelastic constants;



_ij are mass coefficients. The relevant

solid stresses



ij and liquid pressure

s

are





.

),

3

,

2

,

1

,

(

2









R

Qe

s

j

i

Q

Ae

Ne

_{ij ij}

ij













(2.2)

In the Eq. (2.2),



_ij is the well-known Kronecker delta function. Poroelastic constants of cylinder-I and

cylinder-II are denoted by

P

,

N

,

Q

,

R

and

P



,

N



,

Q



,

R

, respectively. We introduce the displacement potentials



’s and



’s which are functions of

r

,



and

t

as follows:

. 1 , 1 , 1 , 1 , 1 , 1 2 2 2 1 1 1                                                                           r z r z r z r z H r r H r H z W r H z H r V z H H r r U h r r h r h z w r h z h r v z h h r r u (2.3) Let

,

cos

)

(

,

cos

)

(

( ) _{2 2} ( )

1 1 t kz i t kz i

e

r

f

e

r

f













_



_



),

,

,

(

),

,

,

(

₂

1

h

r

h



h

z



H

r

H



H

z











,

sin

)

(

,

sin

)

(

( ) r r i(kz t)

t kz i r

r

g

r

e

H

G

r

e

h













,

cos

)

(

,

cos

)

(

i(kz t) i(kz t)

e

r

G

H

e

r

g

h

_



_



 _



_





,

cos

)

(

,

cos

)

(

3 ( )

) ( 3 t kz i z t kz i

z

g

r

e

H

G

r

e

h













(2.4)

In the Eq. (2.4),

k

is wave number,



is frequency,

i

is complex unity, and

t

is time. Then the equations of motion (Biot, 1956) in terms of displacement potential functions are

)

(

)

(

_{11 1 12 2 1 2}

2 2 1

2



_

_



_









_









_





_







Q

b

P

)

(

)

(

12 1 22 2 1 2

2 2 1

2











































R

b

Q

),

(

)

(

)

2

(

2 ₂r _{2 11 r 12 r r r}

r

h

H

b

h

H

h

r

r

h

h

N







































),

(

)

(

)

2

(

2 _ _{2 2}



_{11 }



_{12   }



h

H

b

h

H

h

r

r

h

h

N

r































),

(

)

(

11 12

2

z z z

z

z

h

H

b

h

H

h

N



























),

(

)

(

0





12

h





r





22

H





r



b

h



r



H



r

),

(

)

(

0





₁₂

h











₂₂

H









b

h







H





),

(

)

(

0





₁₂

h





_z





₂₂

H





_z



b

h



_z



H



_z (2.5)Where dot over a quantity represents differentiation

with respect to time

t

and

P



A



2

N

.

Eq. (2.5) with the help of (2.4) are reduced to

),

(

_{11 1 12 2}

2 2

1

Q

K

f

K

f

P



















),

(

_{12 1 22 2}

2 2

1

R

K

f

K

f

Q



















),

(

)

2

(

₂r ₂ 2 _{11 r 12 r}

r

K

g

K

G

r

g

r

g

g

N

















),

(

)

2

(

11 12

2 2

2  







K

g

K

G

r

g

r

g

g

N







r







),

(

)

(

11 3 12 3

2

3

K

g

K

G

g

N











(2.6) Where 2 2 2 2 2

1

k

r

n

dr

d

r

dr

d











,

.

,

,

22 22 12 12 11 11













ib

K

ib

K

ib

K













(2.7)

The general solutions of the equations (2.6) can be obtained in terms of the Bessel function of first kind

J

n. The Eq. (2.6) after a long calculation

yield:

,

cos

)

)

(

)

(

(

,

cos

)

)

(

)

(

(

) ( 2 1 2 2 2 1 1 2 1 1 2 ) ( 2 1 2 1 1 1 1 t kz i t kz i

e

r

J

C

r

J

C

e

r

J

C

r

J

C

 





















 











),

(

)

(

_{3 1 3}

3

r

C

J

r

g





),

(

2

)

(

2

g

₁

r



g

_r



g

_



C

₄

J

₂



₃

r

).

(

2

)

(

2

g

₂

r



g

_r



g

_



C

₅

J

₀



₃

r

(2.8)

The gauge invariance property (Gazis, 1959) is used to eliminate one integration constant from the Eq. (2.8). Accordingly, any one of the potential functions

3 , 2

1

,

g

g

g

can be set equal to zero without loss of

generality of the solution. Setting

g

₂



0

, we can obtai

.

1

g

g

Substituting (2.4) in (2.3), the displacements of solid

are 3 ( )

'

1

)

cos

1

(

g

ikg

e

i kz t

r

f

u







_



 ,

) ( '

3

1

)

sin

1

(

f

ikg

r

g

e

ikz t

r

v













.

cos

)

1

(

₁ '

g

_r

e

i(kz t)

r

g

g

ikf

w





_



_





 (2.10)

Substituting (2.9) in (2.10), the displacement components become ) ( 1 3 '

1

)

cos

1

(

g

ikg

e

ikz t

r

f

u









 ,

) ( '

3 1

1

)

sin

1

(

f

ikg

g

e

ikz t

r

v













) ( 1

' 1

1

)

cos

2

(

g

e

ikz t

r

g

g

ikf

w







_







(2.11) For flexural vibrations, solid displacement components of cylinder can readily be evaluated from the Eq. (2.11) are given

by . cos ) ( cos ) ( cos ) ( cos ) ( sin ) ( sin ) ( sin ) ( sin ) ( cos ) ( cos ) ( cos ) ( cos ) ( ) ( 4 3 2 1 34 33 32 31 24 23 22 21 14 13 12 11 t kz i e C C C C r A r A r A r A r A r A r A r A r A r A r A r A w v u                                                (2.12)

In the Eq. (2.12),

C

₁

,

C

₂

,

C

₃

,

C

₄are all arbitrary constants and

),

(

)

(

1

)

(

_{1 1 1 2 1}

11

J

r

J

r

r

r

A











),

(

)

(

1

)

(

_{1 2 2 2 2}

21

J

r

J

r

r

r

A











),

(

1

)

(

1 3

13

J

r

r

r

A





A

14

(

r

)



ikJ

2

(



3

r

),

),

(

1

)

(

_{1 1}

21

J

r

r

r

A







₂₂

(

)

1

J

₁

(

₂

r

),

r

r

A







),

(

)

(

1

)

(

_{1 3 3 2 3}

23

J

r

J

r

r

r

A













),

(

)

(

2 3

24

r

ikJ

r

A





),

(

)

(

_{1 1}

31

r

ikJ

r

A





A

₃₂

(

r

)



ikJ

₁

(



₂

r

),

,

0

)

(

33

r



A

A

₃₄

(

r

)







₃

J

₁

(



₃

r

),

(2.13)

Where

.

3

,

2

,

1

,

2 2 2

2

_

_

_

i

k

V

_i i





(2.14)

In the Eq. (2.14),

V

₁

,

V

₂ and

V

3 are dilatational

wave velocities of first and second kind, and shear wave velocity, respectively (Biot 1956). It is shown in the paper (Mott, 1972) that effective shear stress component



_zr' in view of static axial stress is

given by

z

u

T

zr zr









330 '





(2.15)

Where

T

330 is the applied static axial stress. By

substituting the displacements in the stress – displacement relations given by the Eq (2.2), the relevant stresses and liquid pressure are obtained they are, . cos ) ( cos ) ( cos ) ( cos ) ( cos ) ( cos ) ( cos ) ( cos ) ( sin ) ( sin ) ( sin ) ( sin ) ( cos ) ( cos ) ( cos ) ( cos ) ( ) ( 4 3 2 1 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 ' t kz i zr r rr e C C C C r M r M r M r M r M r M r M r M r M r M r M r M r M r M r M r M s s                                                             (2.16)

In the equations (2.16)

),

(

)

))

(

)

((

))

(

)

(((

)]

(

3

)

(

[

2

)

(

1 1 2 2 1 2 1 2 1 1 2 1 1 3 2 1 11

r

J

k

Q

A

R

Q

Q

A

R

Q

r

J

r

r

J

N

r

M





































)

(

12

r

M

Is similar expression as M11(r) with



1,

1



replaced by



₂,



₂

)),

(

(

2

)

(

3 _{2 3}

13

J

r

r

N

r

M









)

(

2

)

(

2

)

(

2 3 3 3 3

14

J

r

Nik

J

r

r

Nik

r

M











)

(

2

)

(

1 _{2 1}

21

J

r

r

r

)

(

22

r

M

Is similar as

M

₂₁

(

r

)

with



1 replaced,by

2



(

)

2

3 ₂

(

₃

)

₃2 ₃

(

₃

),

23

J

r

J

r

r

r

M













)

(

)

(

3 3 3

24

r

ik

J

r

M









))

(

)

(

1

(

)

(

2

)

(

2

)

(

1 1 1 2 1 330 1 1 1 2 1

31

J

r

J

r

r

ik

T

r

J

Nik

r

J

r

Nik

r

M





















)

(

32

r

M

Is similar as

M

31

(

r

)

with



1 replaced by



2

)

(

1

)

(

2

)

(

_{1 3 330 1 3}

33

J

r

r

ik

T

r

J

r

Nik

r

M









)

(

)

(

)

(

)

(

)

(

3 _{1 3 330} 2 _{2 3}

3 2 2 2 3

34

J

r

T

k

J

r

r

N

r

J

k

N

r

M



















),

(

)

)(

(

)

(

2 2 _{1 1}

1 2

1

41

r

R

Q

k

J

r

M













M

₄₂

(

r

)

is similar as

M

₄₁

(

r

)

with



₁,



₁ replaced by



₂,



₂

,

0

)

(

43

r



M

M

₄₄

(

r

)



0

,

(2.17)

In the Eq. (2.17),

.

2

,

1

)),

(

)

((

1

2 2

12 11 22 11

2















i

Q

PR

V

Qk

Rk

Qk

Rk

i

i



(2.18)

III. Boundary conditions and frequency equation

The boundary conditions for the stress-free surface at

a

r



in the case of pervious surface are

0

)

(



_rr



s



,

0







z ,

0

'



zr



0



s

, (3.1)

The Eq. (3.1) results in a system of four equations in four arbitrary constants

C

1

,

C

2

,

C

3

,

C

4 . A

nontrivial solution can be obtained if the determinant of the coefficients vanishes. Accordingly, we obtain the frequency equation for a pervious surface and is,given,under:

4

,

3

,

2

,

1

,

,

0

)

(

a



i

j



M

_ij ,

(3.2)

Where

M

ijs are defined in the Esq. (2.16)

IV. Numerical results

Due to dissipative nature of the medium, waves are attenuated. Attenuation presents some difficulty in the definition of phase velocity; therefore, the case

b



0

is considered for the numerical results. Even if we make

b

zero, problem would be poroelastic in nature as the coefficients

R

Q

N

A

,

,

,

would not vanish, only thing is mass

coefficients

K

ij

would be real and will be

reduced to



ij

. The following non-dimensional

parameters are introduced to investigate the

frequency equation (3.2):

,

~

,

~

,

~

,

,

,

,

,

,

,

2

3 0 2

2 0 2

1 0

22 3 12 2 11 1

4 3

2 1

























































V

V

z

V

V

y

V

V

x

d

d

d

H

N

a

H

R

a

H

Q

a

H

P

a













,

0

c

d

m



(4.1)

In the Eq. (4.1),

d

is the phase velocity ,

m

is the non-dimensional phase velocity,

R

0 is the aspect

ratio,

R

₁is the ratio of lengths of the cylinders,

R

Q

P

H





2



and







₁₁



2



₁₂





₂₂

.

Also,

c

0 and

V

0 are reference velocities and are

given by ₀2

,



N

c





H

V

₀2



. Employing these

non-dimensional quantities we obtain the implicit relation between non dimensional phase velocity

m

and non

dimensional wave number

ka

for fixed 330

.

H

T

Non

coupling parameter is present. From the figure it is clear that phase velocity is higher for material-I than that of material-II and material III . This is due to presence of mass coupling parameter present in material-II and material-III.

V.

F

IGURES AND

T

ABLES

TABLE-I Para

mete

r

a

1

a

2

a

3

a

4

d

1

d

2

d

3

x

~

~

_y

~

_z

Ma

terial -I

0.61 0.04 25

0.30 5

0.0 341 93

0. 5

0 0.

5 1. 67 1

0. 81 2

14.6 23

Ma

terial -II

0.61 0.04 25

0.30 5

0.0 341 93

0. 65

-0.15 0. 65

2. 38 8

0. 90 9

18.0 02

Ma

terial -III

0.84 3

0.06 5

0.02 8

0.0 341 93

0. 90 1

-0.00 1

0. 10 1

0. 99 9

4. 76 3

3.85 1

Fig.1.Variation of non dimensional phase velocity

with non dimensional wave number when

.

5

.

0

.

330



H

T

VI. CONCLUSION

Flexural vibrations in isotropic poroelastic solid cylinder are investigated in the framework of Biot’s

theory in the case of pervious surface. Phase velocity against wave number is investigated for three different poroelastic materials. This kind of analysis can be made for any poroelastic solid cylinder if the values of all constants of pertinent materials are available.

References

[1]. Biot M.A., 1956, The theory of propagation of elastic waves in fluid-saturated porous solid, J.Acous.Soc.Am, 28, 168-178.

[2]. Biswas R.N., 1976, Nangia A.K. and Ram Kumar, Flexural vibrations of a finite circular cylinder, Acustica, 35, 26-31.

[3]. Mott G., Elastic waveguide propagation in an infinite isotropic solid cylinder that is subjected to a static axial stress and strain,

Journal of Acoustical Society of America,

1972, pp. 1129-1133.

[4]. Malla Reddy P. and Tajuddin M, 2000, Exact analysis of the plane strain vibrations of thick walled hollow poroelastic cylinders, International Journal of Solids & Structures, 37, 3439-3456.

[5]. Ram Kumar, 1964, Axially symmetric vibrations of a finite isotropic cylinder, J.Acoust. Soc. Am., 38, 851-854.

[6]. Tajuddin M. and Sarma K.S., 1978, Torsional vibrations of finite hollow poroelastic circular cylinders, Def. Sci. Jn., 28, 97-102.

[7]. Tajuddin M. and Sarma K.S., 1980, Torsional vibrations of poroelastic circular cylinders, Trans. ASME, J. Appl. Mech., 47, 214-216. [8]. Tajuddin M. and Ahmed Shah, S., 2006,

Circumferential waves of infinite hollow poroelastic cylinders, Trans. ASME, J. Appl. Mech., 73, 705-708.

[9]. Tajuddin M. and Shah S.A., 2007, On torsional vibrations of infinite hollow poroelastic cylinders, J. Mech. Materials and Structures, 2(1), 189-200.