EFFECT OF GRAVITY ON VISCO-ELASTIC SURFACE WAVES IN SOLIDS INVOLVING TIME RATE OF STRAIN AND STRESS OF HIGHER ORDER

EFFECT OF GRAVITY ON VISCO-ELASTIC SURFACE WAVES IN SOLIDS INVOLVING TIME RATE OF STRAIN AND STRESS OF HIGHER ORDER

TAPANKUMARDASandPo R. SENGUPTA

Department

of Mathematics,Universityof Kalyani Kalyani, WestBengal,India

and LOKENATHDEBNATH

Department o

Mathematics, U’niversity of CentralFlorida Orlando,Florida 23816,

U.S.A.

(Received February 23, 1992 and in revised form March i, 1993)

ABSTRACT. A

study^ismade of the surfacewavesinahigherordervisco-elastic solid involvingtime rateofchange^{of strainand stress}under the influenceofgravity.

A

fairly general equationfor thewave velocityis derived. Thisequationis used to examine various kindsof surface waves includingRayleigh waves,

Love

waves andStoneley^waves.

It

isshown that the corresponding classical results follow from thisanalysisin the absence of gravity andviscosity.

KEY

WORDS

AND

PIIRASES: Surface waves, effects of gravityandviscosity.

1991

AMS SUBJECT CLASSIFICATION CODES:

73D20.

1.

INTRODUCTION

ConsiderableliteratureincludingBullen

[1], Flugge [2]

andStoneley

[3]

^isavailable on thetheory of surface waves inan isotropichomogeneous^elasticsolid medium. However,theeffects of gravity, viscosity andcurvature,although important, arenotincluded intheclassicalproblems. Biot

[4]

has firstinvestigatedtheeffectof gravity onRayleighwaves on thesurface of an elastic solid based on the assumptionthat gravityproducesa type of initialstressofhydrostaticin nature.

Subsequently,

Biot’s theoryhas been usedbyseveralauthorsincluding

De

and

Sengupta [5,6]

^tostudy problemsof waves and vibrations in solids under the initialstressin variousconfigurations. Further,

Sengupta

and his associates

[7-9]

have made anattempttostudy theproblemsofsurface waves in solids involving time rateof strainand viscosity.

In

spiteof these studies,relativelyless attention has beengiventosurface waveproblems inahigherorder visco-elastic solidinvolvingtime rateofstrainandstress under the influenceofgravity. Themain

purpose

of thispaperistostudysuchproblems.

A

fairly

general

equation forthe wavevelocityis derived. Thisequationis utilized to examine variouskinds ofsurface waves including Rayleigh waves,

Love

waves, and

Stoneley

waves.

It

isshownthat thecorrespondingclassical results followfrom thisanalysisin theabsence ofviscosityandgravity.

2.

FORMULATION OF THE PROBLEM AND BOUNDARY CONDITIONS

Let Mt

^and

M2

^betwohomogeneous generalvisco-elastic solid mediainvolvingtimerateofstrain andstressofhigherorder in weldedcontactunder the influence ofgravityattheircommonsurface of separation. We

suppose

^{that the} media are separatedby aplane ^horizontal boundary

72 T. K. DAS, P. R. SENGUPTA AND L. DEBNATH

infinitely greatdistancefromtheorigin,

M

being^above

M.

^Weintroduce asetoforthogonalCartesian coordinateaxes

Oxx3

^inthesemi-infiniteisotropicvisco-elasticmedia,withthe originatthecommon

bounda

surface andthe

x-axis

^isnormalto

M. ^We

^considerthe possibility ofatype of wave travelling inthedirectionof

0x

^{in sucha}manner that thedisturbanceislargelyconfinedtotheneighborhod of the

bounda

^{andatanyinstantallpaniclesonanylineparallelto}

0x2

^haveequaldisplacements.

Hence

thewave is asurfacewaveandallpartialderivatives withrespecttothecoordinate x2 are zero. Then thecomponents ofdisplacement

u

^and

us

^atanypointmaybe expressedinthe form

u3=+-

(2.

lab)

ut Ox Ox3 Ox Ox

where

9

^{and are}the functions

ofx,x

and and

Out Ouz Ou Ou Ou

Ox

Thus the introductionof the functions

9

^{and enables us to}separate out the purely ^{dilationaland} rotational disturbances associated with the components

u

^and

ua. e

component

u,

ofcoupe, is associated withpurelydistoionalmovement.

us

9, ^and

uz

are associatedrespectivelywithP-waves,

SV-waves

and

SH-waves,

as usedbyBullen

].

e

stress-strain relationsare

Dno, DxA,i

⁺

2De, ^(2.3)

where

.o Ot ^-o Ot ^.o Ot

where0, and aretheelastic constantsand

, k

^and

(k

^1,2

n)

are the effects ofviscosity, e,iisthe straintensorand

6i

^{istheoneckersymbol.}

e

displacementequationsof motioninthehigherordergeneralvisco-elastic medium, under the influence ofavity,are

OA

Ou

(O

_x+

D,)

⁺

^D,V=u,

⁺

^pgD

ⁿ

^{pD, (2.5)}

DV=u ^pD, , u ^(2.6)

#k

du #=ux

(2.7)

where

p,q,,, ,(k

-0,1,2

,n)

denote theproperties

e

medium

M

and those with dashes the properties of themedium

M2. Substituting (2. lab)

^inequations

(2.5)-(2.7),

weobtain the wave

equations

in

M

^satisfiedby

,

^andu2,as

=

^Ot

^,v’

^+g

^{dx (2.8)}

O-sV-g (2.9)

Ot

Ox

Ou

_Ot

sVU ^{(2.1 O)}

where

v;’,=(k

+

2t,)/p. v.-t/p (2.1,,)

and

=,o ^{V2 o* O*}

^V

and similarrelationsin

M

^{withp, q,,}

^k,,

^t,replaced by

p’, q’,, ’,, U’,

^{and so on}

(where

^k

O,

1,2,...,

n).

Theboundaryconditionsare

(i) e

componentsof displacement^attheboundarysurfacebetweenthemedia

Mt

^and

M

^must

becontinuous atalltimesanddistances.

(ii)

^Thestressesoxt,

oxe, oxx

^are

+-

(a.l)

OxOx Ox Ox

au (2.14)

, 0% ] (2.15)

D,o DVZ,

+2D_u +

OxOx )

andsimilarexpressions for

Mz,

^acrosstheboundary surface between

M

^and

Mz

must be continuousal

all times anddistances.

3.

SOLUTION OF THE PROBLEM To

solveequations

(2.8)-(2.10),

^weput

(n.,q t)

(,

W,

u,_) [g(x3), gp(x,), _a=(x3)]e

formedium

Mt

and similarsolutionsfor

M:,,

the functions

,, ^a

^{beingreplacedby}

^{’, ’,} a’.

Introducing

(3.1)

ⁱⁿ

(2.8)-(2.10),

^wehavefor the medium

M,:

d

where

(3.2)

(3.3)

(3.4)

,.0 ..0 -o

Similar relationsfor

M2

can be obtainedby replacing

, ,

^fi:,,rh,

^{Vr, V,s, rl, V’r, V,}

^s,

, ,,

9 by^the same

symbols

withdashes.

Here p’, rl’,, .’,, ’, (k 0,1,

2 n are thephysical

properties

^ofthemedium

g.

We

assumethat

, ^p

and u3represent exponentially decayingsolutions in the medium

M

_{as x3} ^oo

sothat

they

canbeexpressedin theform:

t A,e -’’

⁺

^A

^{e e}

^(3.6)

74 T. K. DAS, P. R. SENGUPTA AND L. DEBNATH

P B

^{e +}

^B

^{e e}

^(3.7)

,,,v,

e,O,,-,

_(3.8)

u.

Ce

-*A/C-

and similar solutionsinM can beobtainedreplacing

, ap,

u2,A,A2,

B,B2,

C,

rl, V;s, , .,

^thesame

symbols^withdashes in solutions

(3.6)-(3.8).

^ltere and

t’

^(j

^1,2)

^arerespectivelytherootsof the equations

and

where

(3.9)

% igrll(to:’ ;V,s,) " 4,

ie,

n/(C _i ’ V,s’/’q,’)

(j

1,2)

In

evaluating quantitieslike

qq= -

^and

^{V’I Zto2,tl/,}

the root withpositiverealpartmustbe taken in each ease.

Using boundaryconditions

(i)

^and(ii),^weobtain

[1-ictlQt]A

+

[1 ictQ2]A:, [1 +ia’iQi]A +[1 +icQ’]A ^{(3.1 la)}

C

-C’ (3.11b)

[

^/

Qda /[c

^/

iQda-[a;-Qi]al /[-Q’da’ (3. c)

* (1 +Q)ai}At

p(Vsh]k)[{2iQt

+ +

{2iQ

+

(1

+

Q)}A]

p’(v%’)[{-2iQ’ +(1

+

Q’x)’

⁺

{-2iQ’z +(1

+

Q2)}A’] ^{(3.1 ld)}

% ^c (( "’ -

(p)[{VQ- 1)

+

2V(1- iQ)}A +{ VgQ2

^*

1)

+

2V(1- iQ)}A]

-(p’’)[{V(Q- l)+ EVj(1 +i’Q)}A

+

{V(Q2-1)+ EVj(1 +iQ)}A]. (3.11 It

follows fromequations

(3.11b)

and

(3.11e)

^thatC

=C’-0.

Thus thereisnopropagation ^ofdis-

placement

_u2.

Hence

there are no

SH-waves

in this case.

From equations (3.11a), (3.11c), (3.11d)and (3.1139,

^weeliminate the constants

A

t,

A2,Ai, A,

^to

get equationfor the wavevelocityin determinant form

M,,I- ^{o, (i,j 1,2,3,4), (3.12)}

where

andwhere m 1,2.

I1 ^[1 t(t,,Q,,,], M, ,,.. -[

+

t,t.,,,Q,,]

M-,

[,t,,,

+

Q,,], M. ,,,... -[,t’,,,

^-t

Q,’. ]"

M (pV;.hl)[2,O,,

+(I +

Q,)’:,,],

M (p/rl) V,’}(Q,-. 1)+ 2v;.(l i,t,,Q,,,)], M,, (-p’/q’)[ V;’(Q I)

+

2v,’.."(

⁺

i,t’,.Q’,.)]

Equation

(3.12)

gives thewavevelocity forthesurfacewaves in thecommonboundary,andthe strain rate andthestress rateof higherorder in thepresenceof gravity and viscosity areincluded in

(3.12).

4. PARTICULAR CASES (i) Rayleigh

Waves

In

ordertoinvestigate thepossibility ofRayleighwaves, wetake theplane boundary ^{as a}free surface with

M2

replaced by ^avacuum. Obviously,there are no

SH-waves

inthis case.

In

viewof

(3.1 ld)

and

(3.11]),

^{we obtain}

{2iQ +(1

+

Q)cq}a

+

{2iQ2 +(1 +Q_)ct2}a2-o, (4.1)

and

{V(Q- 1)+ 2Vs(l -icqQ)}A

⁺

{v(O- 1)+ 2v’.(l -icra2)}A2-O ^(4.2)

Eliminating theconstants

A

^and

A:

^{fromequations}

(4.1)-(4.2),

^weget

IM;,I

^{0 (i,j}

^{1,2) (4.3)}

where

Mi,..[2iQ, +(l +Q,.)ct,]. M’:z,.=[V(O, -1)+ 2V(1-ict, Q,)] (r-l,2). (4.44b)

Equation

(4.3)

^isthe required wave velocityequation^forvisco-elasticRayleighwavesincluding^the strainrateandstress rateofhigherorder undergravitationalfield. Whentheeffects of viscosity and gravityareneglected,^thisequationreducestothe classicalresultasdiscussedbyBullen

].

(ii)

Love Waves

In

^thiscaseweconsidera

layered

semi-infinite medium in which

M

is boundedby^twohorizontal planesurfaces at a finite distance

H-apart,

while

M

remains infinite as it was.

In

thiscase,we consider thedisplacementcomponent u only.

Forthe medium

M:,

we writedown thefull solution, since thedisplacement in

M:,

maynolonger diminish with the increasing distance from common boundary

x3--0

^{and for the medium}

M

^the

solutions are the same asitwas in thegeneralcase.

Therefore, forthe medium

Mz

^wewrite

-...V’, -,z.;’’;; " e,(,-,)

"N’n-’n’’/v;;

C

^e

^(4.5)

76 T.K. DAS, P. R. SENGUPTAANDL. DEBNATH

where the restrictionthat thereal part of

vfrl

^2-

^tOZrl’/V;,

^{ispositiveis notrcqutredfor}

^M.

In

this case theboundaryconditionsare

(i)

u:, ando3:,arecontinuousatx3 0 and

(ii) o’3,_

0 at x3 ^-//.

Applyingtheseboundary^conditionsand using

(3.8)

and

(4.5),

^wefind

-"V’,, -o,:,,;

"q;

-

Cte -C2e

^=0.

Eliminating

C, C’

^and

C’2

^{fromtheequations}

(4.6)-(4.8),

^weobtain

(4.7)

(pV’/’q)V/1-cZrl’dV’2s +(p’V’:z/q’)V’(cZrl,’/I/’) tan,lHVCcZq’/vS

^-0 (4.9) wherec

coal.

^{Thisis therequiredwave}velocity equationforhigherorder visco-elastic

Love

waves involvingthe strain rate and stress rate undertheinfluenceof gravity. Itisimportantto note that^l,ove waves are notaffected bygravity butby viscosity. Whenq0 and

q, rl’, Z., X’-la, la’,

⁰

(k

1, 2,...,

n)then

equation

(4.9)

^isin agreementwiththe corresponding^classicalresult

[1]

^{in a}perfectly elastic medium.

(iii)

Stoneley Waves

In

theclassicaltheory,the

Stoneley

waves area generalized from ofRayleighwavespropagating alongthe commonboundary

ofMt

^andM:,.

^In

the influence of gravity,Stoneleywavesalongthe common boundaryof thegeneralvisco-elastic solid media

M

^and

^M:,

^involvingthe strain rate andstress rateof higher order,aretherefore determinedbytherootsofthefrequency equation

(3.12). In

the absence of theseeffects,thisequationalsoagreeswiththe corresponding classical result.

[11 [21 [3]

[41 [61 [7]

[8]

REFERENCES

BULLEN, K. E. An In.troduction

^totheTheory of Seismology, CambridgeUniversity

Press,

London

(1965), p.

253.

FLUGGE, W.

Visco-elasticity,BlaisdellPublishing

Co. (1967).

STONELEY, R.

Proc.

R. Soc.

London,

A.

106

(1924),

^416-428.

BIOT, M.A.

Mechanics oflncremental Deformation

(1965),

44-45,273-281, Wiley,

New

York.

DE, S. N.

and

SENGUPTA, P. R. Ger.

Beitr.

Geophys.,

84

(1975),

^509-514.

DE, S. N.

and

SENGUPTA,

P.

R. Ge.r. Beitr.. Geophys.,

⁸⁵

(1976),

^311-318.

SENGUPTA, P. R.

and

ROY, S.

K.

Get.

Beitr.

Geophys.,

92

(1983),

^570-576.

SENGUPTA, P. R.

and

PAL, K. C..Proc.

^Ind.

Nat.n.

^Acad.,53A,

^No

^.1

(1987),

^113-123.

[9] DAS, T. K.

and

SENGUPTA, P. R.

Ind.

J. Pure App!.

Math., 21

(7) (1990),

^661-675.

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