118

On the Wave Propagation in Inhomogeneous Non-Isotropic Media By SASADHAR DE 1) Summary - Following EASON, we have discussed here the propagation of elastic waves in nonhomogeneous spheres and cylinders when the curved surface is given a uniform normal loading. The material is assumed to be transversely isotropic with respect to a direction of symmetry, the stress and displacement components within the body may be assumed to depend on one space co-ordinate and time None. The particular case in which the elastic parameters are proportional to (radius)'~ has been considered.

. The wave propagation in inhomogeneous media has recently been discussed by EASON [1] 2), the results of which may be applied in the field of geophysics. The same problem in an inhomogeneous rod and uniform shear loading of the curved surface of a spherically or cylindrically symmetrical inhomogeneous solid have also been considered by EASON [2, 3]. When the whole curved surface of the body receives a uniform time dependent loading, the stress and displacement components within the body will depend on space - coordinate and time alone. The present problem is an extension of this problem of normal loading in the case of non-isotropic material. The spheres and cylinders under investigation are assumed to be transversely isotropic with respect to a radius vector through the center of the sphere or cylinder. For such an inhomogeneous body, the elastic parameters are assumed to be functions of the radial coordinate alone. The fundamental equations governing the motion of an anisotropic sphere of non-homogeneous material are established in 2 and a particular simple solution of these equations is obtained in 3. In 4 we have considered the case when the elastic parameters are proportional to (radius)". In 5 the basic equations for the anisotropic cylinder under conditions of plane strain are established. Solution of these equations is obtained in 6 and a special case in which the elastic parameters are proportional to (radius)" is considered in 7. 2. F u n d a m e n t a l e q u a t i o n s

In spherical coordinates r, 0, ~b, we consider a sphere which is composed of a material exhibiting transverse isotropy with respect to the radius r. For complete 1) Department of Physics, Visva-Bharati, Old Engineering Office, Santiniketan, West Bengal, India. 2) Numbers in brackets refer to References, page 125.

On the Wave Propagation in InhomogeneousNon-Isotropic Media

119

spherical symmetry, the stress-displacement relations are (LovE) 0u a, = cll ~rr + 2

C12

u/r, (2.1)

#u ~o = '*, = c~2 ~ + (c= + c~) u/r,

where o-,, o-0, ao are the stress components, u is the radial displacement, and c~i are the elastic constants. The only non-trivial equation of motion for the completely symmetrical problem is da, 2 O2u 8~- + -r (a~ - ao) = 0 gt2 , (2.2) where t is the time and 0 is the density of the material of the solid. From (2.1) and (2.2), we can write the equation of motion as Cll

\~I~ 2 -Jr---r Or - 2(cz2 + c23 -

c,2) u/r'-

+ &

02U

d (c1~ - q2) u/r

- 2d~

Assuming

C22 "1- C23 ~,~CI1 "1- r

we

~r + 2 ulr

= 0 &2

9

have ~2 u

+ c 2 3 - 2 c 1 2 ) u / r = o - -~t2

"

(2.3) Let Cll ~ air r -2(n+l) ~

u ~---~ /~n,

= R r -z("+~).

(2.4)

The equation (2.3) becomes

a2~ da,~ Oe~ all ~~r + -dr t?r

O2q~ q5 F dall R ~ + rz Lr(n + 2) - -dr - all(n + 2) (n + 3)

(2.5)

_ 2 d z r2,+3~ = O,

dr

d

where ~=C22"1-r

--

2 c12.

Equation (2.5) has the form

~2 c~ a11~ -+

da i l O~b d~- dr

a2 q~ (2.6)

R~=0,

provided a 11 and Z satisfy the equation

r(. + 2) d 11 _ ar

+ 2)(. + 3)= 2

aZ r2n+ 3 . dr

(2.7)

120

S. De

(Pageoph,

Equation (2.6) is the equation governing the propagation of longitudinal waves in a rod and the propagation of shear waves in spherical or cylindrical coordinates. It was shown in [2] that (2.6) may be transformed into the inhomogeneous wave equation provided (2.7) is satisfied. From (2.7) we have after integration, (n + 2) cli = 0~r 1-" + 2 r l - " [ d~xdrr'-I dr,

(2.8)

FI/

where :~ is a constant. Two special cases are, n=l,

3ca~=~+2

x (2.9)

= cll + 2 c~2 and n = - 2,

cli

--

C12

-~-

(2.10)

constant.

When we assume X=/3 cll, where/3 is a constant, (2.7) becomes

r(n+2) d C i l r Z ( ' + l ) + 2 ( n + l ) r ( n + 2 ) c l l r 2"+1 dr

r2("+1) -

r

x (n + 2) (n + 3) = 2/3 at11 r2.+3

dr

or

(n + 2 - 2 13) d~-ri - Cilr (1 - n) (n + 2)

i.e., Cll = c~ r N,

(2.11)

where N =

(1 - n) (n + 2) n+2-2/3

(2.12)

so that when all the elastic parameters are of the form (radius) s , the basic equation reduces to (2.6) and the results given by EASON [2, 3] may be applied.

3. Particular solution Assuming n = - 2 , equation (2.5) reduces to 02r dall 0r a l l 0~r2 + dr Or

02d? ROt 2

2 (~ dz r 3 dr - O.

(3..1)

We now introduce Laplace transform of r defined by o0

~(r, s) = f qS(r, t) e -st dt. 0

(3.2)

Vol. 89, 1971/VI) On the Wave Propagation in Inhomogeneous Non-Isotropic Media

121

Multiplying (3.1) by e -st and integrating over all t > O, we find that d2~ d a ~ d~ all d r 2 + dr dr

R

S2 ~9

2 ~ dz r 3 dr - O,

(3.3)

when we assume that 4)=O0/Ot=O at t = 0 . We seek a solution of (3.3) of the form = 37f e +gs,

(3.4)

w h e r e f a n d g are functions of r alone and 37 is a function of s alone. Using (3.4), we have from (3.3)

al,

+al1

-Tz'

+_s(2al

+allf

+f

11)

(3.5)

+ s 2 f(g,Z al I _ R) = O,

where prime indicates differentiation with respect to r. Equation (3.5) will be satisfied if a l l f " --[- a l,l f ' = 2 ~ -f Z' ,

(3.6)

2f' g" ' f - + g' + al~ = 0,

(3.7)

all

R --- all

g,2,

(3.8)

are satisfied. From (3.7), we have after integration

f2 g, al 1 =

(3.9)

f2

where 6 is a constant. If two of the five unknown quantities f, g, R, al 1 and X are known, the equations may be solved for the remaining three quantities. Suppose the values of cx~ and 0 i.e. a 1~ and R are known, then the quantities g, f and X are determined by g

Z=

~" f 4 ~ / C l l dr

(3.10)

fir f -- @11 0) 1/4

(3.11)

~

dr

all d r

(3.12)

dr.

When c~a and 0 are given, the corresponding values of c22-~c23 - 2 c 1 2 = c l l - c ~ 2 and So, C22-t-1723is determined.

i.e.

C12

122

S. De

(Pageoph,

4. Particular case We consider the solid to be inhomogeneous for which ~,~ = c ~ rN,

z =/~ c1~ =/~ 7 1 rN,

= 0~0 r Iv+2M,

(4.1)

where fl, N, c~ Co and M are constants. We have from equations (3.10) and (3.11)

g=

~/O~l1 FM+I

(4..2)

M+I

6 f = ( o I ~Oo)~/~ r~/~+~/~-,

(4.3)

and equation (3.12) can be written as

fic~ rN= f r3(c2~~

dr d ( c'l rN-zdf) dr(''' n = -

(1 - N / 2 - M/2) c~

=

rN/2+~/z+2 __ ( r N / 2 _ M / 2 _ 2 ) d r ,

2

dr

i.e., p rN= =

1 - N / 2 - M / 2 [ rN - (N/2 + M/2 + 2) rN] 2 1 - N/2 - M/2 N/2 - M/2 - 2 2 N

So, (3.12) will be satisfied provided 8 fl N = (M Again, from (4.1), we have

Z=flql

(/~ -

+ 1) 2 -

(N -

3) 2 .

(4.4)

or, 1) c l l + c12 = 0 .

Since c11 > cl 2, so it is expected that 0 < p < 1, which is the condition for the value of ft. If (4.4) is satisfied, equation (3.4) with (4.2) and (4.3) gives ~ and (2.4) then gives an expression for ~. So the problems of impact loading of a spherical surface will become to carry out the Laplace inversion of (3.4) and introduce boundary conditions to determine 37.

5. Fundamental equations, Non-isotropic cylinder We consider a cylinder, composed of a material exhibiting transverse isotropy about the radial direction. Then for conditions of plane strain, the stress-displacement rela-

Vol. 89, 1971/VI) On the Wave Propagation in Inhomogeneous Non-Isotropic Media

123

tions for radial symmetry are ,r~ = ell ~ + cl~ u/r Ou

,

Cro = el~ ~ + c2~ u/r

(5.1)

I I

where u is the component of displacement in the radial direction and r, 0 are polar coordinates. The only non-trivial equation of motion for the cylinder which is subjected to uniform normal loading over the cylindrical surface is

Oa~ a~- ao ~r +

r

02u - 0 ~t 2

(5.2)

which, with equation (5.1), gives

dcll Ou dea2 OZu (O~u 1 ou) c,l \ ~ r 2 + -r & - c ~ u/r 2 + - -dr Or + Tar u / r = O - . Ot2 Assuming cl~ ~,~ e22, w e have

) c11( u

{02u 10u cll \ & 2 + - r - O r - ~'/r2 + ~ r

)

o r + u/r

d - u/r dr (cll -- el~) ---- 0 0t~ 9

(5.3)

Let u

=

4~ r" ~

Cll

=

all

r -(l+2n)

~

Q~R

r -(l+2n).

(5.4)

The equation (5.3) becomes

gzc~ da l l &[9 020 d? aal or2 + dr Or R o t f + r ~ x[

(5.5)

daxl r (n+ l ) - a l l ( n + l ) ( n +

dr = 0 ,

where ~- C l l

-- el2

(5.5) takes the form (2.6) if

r(n + 1) dall dr -

-

-

all(n + 1) (n + 2) = r 2"+2 d z. dr

(5.6)

Equation (5.6) has the general solution

(. + 1)ell = ~ ; - - + where ~ is a constant.

rl n IdX r"-i ~r,

3 dr

(5.7)

124

S. De

(Pageoph,

T w o special cases are

n=l,

2cl~=c~+Z

i.e.,

(5.8)

(Z = C l l "4- C12

n = - 1,

cx,

(5.9)

constant.

- - C12 =

I f we n o n assume X,= fl cl 1, where fl is a constant,

(5.1o) (5.6) becomes

~(, + 1) d C l l

r (2n+i)

q- (2 n + 1) r(n + 1) cl,

F2n

- ell

/.(2n + 1)

dr

dcll dr

x (n+l)(n+Z)=fl

r2n+2

or,

(n + 1 - ~)

dcll dr

e l i {(n + 1) (n + 2 ) -

(2 n + 1) (n + 1)}

r

i.e.,

dc lj

1 --

dr

n2

l + n - fl r

cli

Hence, ell

= Co ll

rN

(5.tl)

where 1 --

N-

tl 2

1 + n - fl'

(5.12)

so that when cl 1 - c i 2 and c 11 are related by (5.10) and (5.11), equation (2.6) holds and the results given in [2, 3] m a y be applied.

6. Particular solution Setting n = - 1, we have f r o m (5.5)

~2r

dalt 0~ a 1 1 0 ~ - + dr ~r

~2~ R &2

r dx r2 d r - O "

(6.1)

Solving like before, we shall get the set of equations g =

ar

a,/r f - (cll ~)1/4

(6.2) (6.3)

Vol. 89, 1971/vi) On the Wave Propagation in Inhomogeneous Non-Isotropic Media

x=

f-dr

a l i d~r d r ,

125

(6.4)

where 6 is a constant. Like before, if c u and e are known, g, f a n d cl2 can be determined by (6.2), (6.3) and (6.4). 7. Particular case

We consider the solid to be inhomogeneous with elastic parameters defined by (4.1). Then we have from (6.2) and (6.3), g =

M-4- i

(7.1)

6 f = (coi ~o)1/4 r(N+M- I)/2

(7.2)

and (6.4) becomes fl cO1 rN =

x

f

r @11 Qo) 1/4 r(N+M-1)/2 d t5 dr

Ic~ rN- 1 x

(cOl eo)l/4 x

I - N - M _(I+N+M)]21 2 r dr

i.e., =

Hence, 4 fl N = (M + 1) 2 - ( N -

2) 2 ,

(7.3)

and the condition on the value of fl remains the same as before (4.5). REFERENCES [I] G. EASON(1970), Appl. Sci. Research. 21 (Jan. 1970). [2] G. EASON,Bull. Seism. Soc. Amer. 57, (1967) 1267. [31 G. EASON,Acta Mech. 7, (1969) 137. [4] A. E. H. Love (1944), A Treatise on the Mathematical Theory of Elasticity (Dover Publications, 1944). (Received 30th October 1970)