Dispersion of Stoneley waves through the irregular common...

Scientia Iranica B (2021) 28(2), 837{846

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

Research Note

Dispersion of Stoneley waves through the irregularcommon interface of two hydrostatic stressed MTImedia

S. Chowdhurya, S. Kundua, P. Alamb;�, and Sh. Guptaa

a. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004,Jharkhand, India.

b. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, Tamil Nadu, India.

Received 9 January 2019; received in revised form 4 January 2020; accepted 1 February 2020

KEYWORDSStoneley wave;Magneto-elasticity;Hydrostatic stress;Transversely isotropic;Corrugation;Maxwell's equations.

Abstract. The present study investigates the mathematical inspection of Stoneley wavepropagation through the corrugated irregular common interface of two dissimilar Magneto-elastic Transversely Isotropic (MTI) half-space media under the impression of hydrostaticstresses. To enumerate the Lorentz's force besmeared in the structure, the generalizedOhm's law and Maxwell's equations were considered. The interior deformations wereanalytically calculated to obtain the wave frequency equation through the prescribedboundary conditions. To investigate the impacts of irregularity and various a�ectingparameters such as magnetic couplings and hydrostatic stresses on the wave propagation,frequency curves formed for the phase velocity of the wave.

© 2021 Sharif University of Technology. All rights reserved.

1. Introduction

Earth is a bizarre collection of elastic media. Tocollect enough information on the composition of theinterior of the Earth, seismic body waves were carefullychecked. In addition, for theoretical inspection, onemust take into account a number of appropriate mod-els. These models provide indirect data on the internalstructure of the Earth that can help seismologists andgeophysicists �gure out di�erent physical properties,state, and con�guration of the interior of the Earth's

*. Corresponding author.E-mail addresses: [email protected] (S.Chowdhury); kundu [email protected] (S. Kundu);[email protected] (P. Alam);shishir [email protected] (Sh. Gupta).

doi: 10.24200/sci.2020.52653.2820

medium and explore the valuable and unknown mate-rials beneath.

Stoneley [1] demonstrated the conceivable pres-ence of another type of waves (named after him asStoneley waves) that were identical to surface wavespropagating through interfaces of either solid and liquidmedia or two elastic media and found a secular relationfor the wave; he pointed to the condition of existenceof these kinds of waves. Variations of the propagationattributes of Stoneley waves in uid-�lled media werethoroughly examined by Ashour [2]. The dispersionequation for the Stoneley waves was introduced byAbo-Dahab [3] for magneto-thermoelastic materials.Tiwana et al. [4] investigated the consequences ofspherical wave di�raction due to a point source froma Perfect Electromagnetic Conducting (PEMC) halfplane.

The internal layer of the Earth comprises di�erenttypes of substances with non-identical types of prop-

838 S. Chowdhury et al./Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 837{846

erties including homogeneous, heterogeneous, trans-versely isotropic and orthotropic, etc. Transverselyisotropic substances are a particular type of orthotropicsubstances symmetrical about an axis that is normalto a plane of isotropy. The hexagonal crystals exist inthe collection of transversely isotropic solids. Investiga-tions of the wave regulations in transversely isotropicand orthotropic (�ber-reinforced [5]) materials are ofsigni�cance to the improvement of a number of princi-pal studies on mechanics. Recently, Alam et al. [6] andKundu et al. [7] studied surface waves in transverselyisotropic layers. Plane wave propagation was examinedby Singh [8] in a thermoelastic transversely isotropichalf-space.

Realistically, the Earth surface is not a continuousand regular plane. The interfaces of two adjacent layersare irregular in nature and complicated. Irregularinterfaces are found in many forms including corruga-tion (undulated), parabolic, rectangular, etc. This isthe reason why considering the impact of irregularitywhile studying the propagation of Stoneley waves is ofsigni�cance. A number of authors have investigatedseismic waves through irregular media, among whomVishwakarma and Xu [9], Singh [10], Alam et al. [11],and Saroj et al. [12] are notable.

The hydrostatic stress (con�ning stress) is anisotropic stress caused by the weight-induced pressureof water on a certain point of the medium. Hy-drostatic stress has considerable applications in geo-physics and geomechanics. As an isotropic stress,hydrostatic stress behaves uniformly in all directions.Addy and Chakraborty [13] examined the e�ects ofhydrostatic stress on viscoelastic Rayleigh wave prop-agation. Alam et al. [14,15] presented the outcomesof hydrostatic stress on Rayleigh and Love-type wavepropagations.

The magneto-elastic materials exhibit the mag-neto and elastic properties. The Earth's layer con-sists of di�erent kinds of sedimentary, volcanic, andmetamorphic rocks. Some ferromagnetic minerals suchas nickel, iron, and cobalt have the ability to gen-erate magnetic �elds. Rayleigh wave propagation inmagneto-elastic materials is entirely interesting dueto its various applications and properties in di�erent�elds, including geophysics, geo-tectonics, seismol-ogy, astrophysics, acoustic, defectoscopy, and optics.Said [16], Majhi et al. [17], Shaw et al. [18], and Sahuet al. [19] carried out some research works in this�eld.

In the present paper, Stoneley wave propaga-tion was investigated through the corrugated irregularcommon interface of two dissimilar Magneto-elasticTransversely Isotropic (MTI) half-space media. InSection 2, basic equations are elaborated via Lame'spotential method. To �nd a general solution, thoseequations are enlightened. In Sections 4, 5, and 6,

particular solutions are suggested. These solutionsat the corrugated joint between two dissimilar mediawith the suitable boundary condition to obtain thedispersion relation of Stoneley waves in Magneto-elasticTransversely Isotropic (MTI) media. In Section 3,We formulate the problem and in Section 7, we getthe desired dispersion relation. The outcomes arenumerically computed and graphically illuminated inSection 8. Inferred observations are given in the lastsection.

2. Basic equations

The equations of motion, introduced by Biot, for atransversely isotropic elastic half-space with magnetice�ect and hydrostatic state stress are given in thefollowing [20]:

�ij;j +�~J � ~B

�i� pir2ui = �i

@2ui@t2

: (1)

Given that the assumed half space is a perfect electricconductor, Maxwell's equations in the linear form weretaken into account for the electromagnetic �eld in theabsence of displacement current [21]:

curl ~h = ~j; curl ~E = ��e @~h@t ; div ~h = 0; (2)

where:

~h = curl�~u� ~H0

�;

and:~H = ~H0 + h(x; z; t): (3)

Here, it is assumed that MTI half-space is underconstant H0 acts on y axis.

This wave is polarized in the xz-plane. Therefore,the values of u and w are not zero, but v = 0. Moreover,u and w are independent of y, i.e.:

u = u(x; z; t); w = w(x; z; t); v = 0;

and:@@y� 0: (4)

Substituting Eqs. (1) and (4) into the two-dimensionalform, we have:

@�xx@x

+@�xz@z� pir2u+ F1 = �i

@2u@t2

; (5)

and:@�zx@x

+@�zz@z� pir2w + F3 = �i

@2w@t2

; (6)

where Fi = (J �B)i.Non-vanishing stress components of transversely

isotropic media in xz-plane are [22]:

S. Chowdhury et al./Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 837{846 839

�xx = c11@u@x

+ c13@w@z

; �zx = c44

�@u@z

+@w@x

�;

�zz = c13@u@x

+ c33@w@z

: (7)

Substituting Eqs. (2), (3), and (7) into Eqs. (5) and (6),we obtain:�

c11�pi+2�eH20�@2u@x2 +

�c13+c44+2�eH2

0� @2w@x@z

+ (c44 � pi) @2u@z2 = �i

@2u@t2

; (8)

and:

(c44 � pi)@2w@x2 +

�c13 + c44 + 2�eH2

0� @2u@x@z

+�c33 � pi + 2�eH2

0� @2w@z2 = �i

@2w@t2

: (9)

3. Formulation of the problem

In order to study Stoneley wave, a mathematicalmodel shown in Figure 1 was employed which consistedof transversely magneto elastic isotropic semi-in�nitemedia under hydrostatic stress M1 : �1 � z � �(x)and M2 : �(x) � z � 1 with di�erent elastic propertiesand corrugation at interface between two semi-in�nitemedia, as shown earlier. This Cartesian system isusually selected where the x-axis is parallel to the wavetraversal and z-axis is vertically downwards. �(x) is aperiodic and continuous function of x. To exhibit thecommon surface as represented in Figure 1, the suitableFourier series expansion for the function was suggestedby Singh [10]:

�(x) =1Xn=1

��nein�x + ��ne�in�x

�;

where �n and ��n are the coe�cients of the nth-orderFourier series expansion such that we have:

��n =

(a2 ; for n = 1An+Bn

2 ; for n = 2; 3; 4 � � �Through the aforementioned �n and ��n, �(x) wasgiven by Singh as follows [10]:

�(x) = a cos(�x) +1Xn=2

[An cos(n�x) +Bn sin(n�x)]:

4. Solution of the upper half-space

According to Helmholtz's theorem, the displacementvector ~u can be inscribed in terms of the displacementpotentials ' and , given by:

~u = ~r'+ ~r� : (10)

Eq. (10) reduces to:

u1(x; z; t) =@'1

@x� @ 1

@z;

and:

w1(x; z; t) =@'1

@z+@ 1

@x: (11)

Substituting Eq. (11) into Eqs. (8) and (9), we get:�c11 � p1 + 2�eH2

0�@2'1

@x2 +�c13 + 2c44 � p1

+ 2�eH20�@2'1

@z2 = �1@2'1

@t2; (12)

(c11 � c13 � c44 � p1)@2 1

@x2 + (c44 � p1)@2 1

@z2

= �1@2 1

@t2; (13)

Figure 1. Geometry of the problem.


(c44 � p1)@2 1

@x2 + (c33 � c13 � c44 � p1)@2 1

@z2

= �1@2 1

@t2; (14)

�c13 + 2c44 � p1 + 2�eH2

0�@2'1

@x2 +�c33 + 2�eH2

0

� p1�@2'1

@z2 = �1@2'1

@t2: (15)

While Eqs. (12) and (15) express the compressivewave associated with x and z directions, respectively,Eqs. (13) and (14) express the shear wave associatedwith x and z directions, respectively. Undoubtedly, thebody wave velocity is di�erent in x and z directions.Since the present study is primarily focused on theStoneley wave propagation only in the x direction,Eqs. (12) and (14) have been taken into account.

For the harmonic travelling wave that propagatesalong the x direction, we pursued an analytical solutionto Eqs. (12) and (14) in the following form:

['1; 1](x; z; t) = ['1(z); 1(z)] exp[ik(x� ct)]: (16)

Substituting Eq. (16) into Eqs. (12) and (14), weobtain:

(D2 � s21)'1(z) = 0; (17)

and:

(D2 � s22) 1(z) = 0; (18)

where D2 = d2

dz2 and s1 and s2 are described inAppendix.

Therefore, the solutions to Eqs. (17) and (18) aregiven as follows:

'1(z) = A1 exp(s1z) +B1 exp(�s1z); (19)

and:

1(z) = C1 exp(s2z) +D1 exp(�s2z): (20)

Thus, the potential functions '1 and 1 are given as:

'1(x; z; t) =fA1 exp(s1z) +B1 exp(�s1z)g exp

[ik(x� ct)]; (21)

and:

1(x; z; t) =fC1 exp(s2z) +D1 exp(�s2z)g exp

[ik(x� ct)]: (22)

For the upper half-space, we have:

'1(x; z; t) = A1 exp(s1z) exp[ik(x� ct)]; (23)

and:

1(x; z; t) = C1 exp(s2z) exp[ik(x� ct)]: (24)

For the upper half-space, the solutions are given by:u1(x; z; t) =[ikA1 exp(s1z)� s2C1 exp(s2z)] exp

[ik(x� ct)]; (25)

and:w1(x; z; t) =[s1A1 exp(s1z) + ikC1 exp(s2z)] exp

[ik(x� ct)]: (26)

5. Solution of the lower half-space

By solving the lower half-space similarly, we have thepotential function as shown in the following:'2(x; z; t) =fA2 exp(r1z) +B2 exp(�r1z)g exp

[ik(x� ct)]; (27)

and: 2(x; z; t) =fC2 exp(r2z) +D2 exp(�r2z)g exp

[ik(x� ct)]; (28)

where r1, r2 are given in Appendix.For the lower half-space, we have:

'2(x; z; t) = B2 exp(�r1z) exp[ik(x� ct)]; (29)

and: 2(x; z; t) = D2 exp(�r2z) exp[ik(x� ct)]: (30)

The solutions for the lower half-space are given by:u2(x; z; t) =[ikB2 exp(�r1z) + r2D2 exp(�r2z)]

exp[ik(x� ct)]; (31)

and:w2(x; z; t) =[�r1B2 exp(�r1z) + ikD2 exp(�r2z)]

exp[ik(x� ct)]: (32)

6. Boundary conditions

The suitable boundary conditions must be satis�ed atthe common corrugated interface (i.e., at z = �(x) =a cos(�x)):a. Displacement components are continuous:

u1 = u2; and w1 = w2:

b. Stresses are continuous:i.

��1zz + �M1

zz�� 0(x)

��1zx + �M1

zx�

=��2zz + �M2

zz�� 0(x)

��2zx + �M2

zx�;

ii.��1xz + �M1

xz�� 0(x)

��1xx + �M1

xx�

=��2xz + �M2

xz�� 0(x)

��2xx + �M2

xx�:


7. Dispersion relation

By substituting the values of u1, w1, u2, and w2 intothe aforementioned Boundary Conditions (BCs), thefollowing homogeneous algebraic system of equationsfor unknowns A1, B2, C1, and D2 would be achieved.This algebraic system of equations can be expressed bythe matrix-vector term as follows:

Z4�4Y4�1 = 0; (33)

where Z4�4 is a four-by-four matrix given byZ4�4 = [�ij ], (i; j = 1; 2; � � � ; 6) and Y4�1 is afour-by-one column matrix described by Y4�1 =�A1 B2 C1 D2

�T . In addition, the values for �ijare given in Appendix.

To achieve a non-trivial solution to the aforemen-tioned homogeneous system of equations, the determi-nant of the given matrix should be zero, i.e., j�ij j = 0;this gives the required dispersion relation of Stoneleywaves in a presumed structure.

8. Numerical conclusion and discussion

An interpretive inspection was made to emphasizethe e�ects of various elastic parameters against thevelocity of Stoneley wave propagation. The dispersionof Eq. (33) was used for numerical interpretation andgraphical depiction.

i. For the upper half-space M1, take the example ofMagnesium material [23] into account:

c11 = 0:597� 1011 N/m2;

c13 = 0:217� 1011 N/m2;

c33 = 0:617� 1011 N/m2;

c44 = 0:164� 1011 N/m2;

and:

�1 = 1740 kg/m3:

ii. For the lower half-space M2, take the example ofBeryllium material [24] into account:

~c11 = 2:293� 1011 N/m2;

~c13 = 0:14� 1011 N/m2;

~c33 = 3:364� 1011 N/m2;

~c44 = 1:625� 1011 N/m2;

and:

�2 = 1040 kg/m3:

The present study aimed to examine the consequencesof magneto-elastic coupling parameter for both upperand lower half-spaces (mH = �eH2

0c11

; lH = ~�e ~H20

~c11) of

hydrostatic stress for the upper and lower half-spaces(P1 = p1

c11; P2 = p2

~c11) and position parameter (t =

xh ) for the proposed model, respectively. To thisend, dimensionless phase velocity (c=c1) against thedimensionless common surface corrugation parameter(a�) is plotted for each of Figures 2{6 for their di�erentvalues. In every �gure, the vertical axis shows the

Figure 2. In uence of Stoneley wave phase velocity(c=c1) at increasing values of mH against corrugationparameter (a�).

Figure 3. In uence of Stoneley wave phase velocity(c=c1) at di�erent values of lH against corrugationparameter (a�).


Figure 4. Stoneley wave phase velocity (c=c1) at di�erentvalues of P1 against corrugation parameter (a�).

Figure 5. Stoneley wave phase velocity (c=c1) atincreasing values of P2 against corrugation parameter(a�).

dimensionless phase velocity (c=c1) and the horizontalaxis presents the dimensionless common surface corru-gation parameter (a�).

Figure 2 presents the e�ect of mH on the Stoneleywaves phase velocity. It also shows the uctuationin the phase velocity of Stoneley wave against thedimensionless common surface corrugation parameterat di�erent values of mH . According to di�erent valuesof mH and the corresponding curves, upon increasingthe value of mH , the phase velocity of the Stoneleywave in Magnesium material would increase.

Figure 3 demonstrates some curves that are plot-ted for dimensionless phase velocity against the di-mensionless common surface corrugation parameter at

Figure 6. Stoneley wave phase velocity (c=c1) at di�erentvalues of position parameter (t) against corrugationparameter (a�).

di�erent values of magneto-elastic coupling parameter(lH) in the lower half-space. As observed in this setof curves (1, 2, 3, 4), the magneto-elastic parameter(lH) has proportional impacts on the phase velocity inBeryllium material.

Figure 4 shows the e�ect of hydrostatic stressparameter (P1). According to this �gure, for aparticular dimensionless corrugation parameter, thehydrostatic stress parameter has unfavorable e�ectson the dimensionless phase velocity in Magnesiummaterial.

Figure 5 elaborates the e�ect of hydrostatic stressparameter (P2) associated with the lower half-space.As observed, while the phase velocity increased uni-formly for the frequency region a� < 1:3, it decreasedin the frequency region a� > 1:3 upon increasing themagnitude of the hydrostatic stress parameter.

Figure 6 depicts the dispersion curves of Stoneleywave in case the position parameter (t) is taken intoaccount for the variation. A meticulous observationof this �gure reveals that the phase velocity follows adecreasing trend.

Figure 7 shows the correlative study of dispersioncurves in di�erent cases. Curve 1 shows a case when theupper half-space is free from magnetic �eld (mH = 0).Curve 2 implicates a case when the lower half-space isfree from magnetic �eld (lH = 0). Curve 3 refers to acase when the upper half-space is free from hydrostaticstate of stress (P1 = 0). Curve 4 is a case when thelower half-space is free from hydrostatic state of stress(P2 = 0). As observed, Curve 3 (mH = 0) and Curve 1(P1 = 0) support higher and lower phase velocities,respectively, than all other cases.

Given that the phase velocity (c=c1) is dependent


Figure 7. Stoneley wave phase velocity (c=c1) in theabsence of di�erent parameters against corrugationparameter (a�).

Figure 8. Variation of Stoneley wave velocity (c=c1) withrespect to P1 and P2.

on the magneto-elasticity and hydrostatic stresses, thesurface plot of phase velocity against varying hydro-static stress parameters (P1 and P2) and magneto-elasticity parameters (mH and lH) are shown in Fig-ures 8 and 9, respectively.

9. Conclusions

In this study, an analytical approach was employedto examine the Stoneley wave propagation throughthe corrugated interface of two Magneto-elastic Trans-versely Isotropic (MTI) half-spaces. A dispersionequation that could measure the velocity of the Stone-ley waves was obtained. All the wave fronts were

Figure 9. Variation in Stoneley wave velocity (c=c1) withrespect to mH and lH .

signi�cantly exerted by di�erent parameters and weregraphically depicted using 2D and 3D plots. Basedon the theoretical and numerical studies, it can beconcluded that:

1. The magneto-elastic coupling parameters for bothhalf-spaces followed an increasing trend in termsof the phase velocity. The phase velocity wasa�ected the most by the absence of magneto-elasticparameter in the upper half-space and the leastby the absence of the magneto-elastic couplingparameter;

2. Although the hydrostatic stress parameter for theupper half-space had inverse e�ect on the phasevelocity, for the lower half-space, it had a mixedvariation in the phase velocity.

Nomenclature

�ij Component of stress vectorpi Hydrostatic stress�i Mass densityc Common wave velocityk Wave number~J Electric current densityj Electric intensity vector~B Magnetic induction vectorFi Components of Lorentz's force in the

ith directionui Displacement component~E Electric intensity vector~H Magnetic �eld vector


H0; ~H0 Primary magnetic �eld of upper andlower half-spaces, respectively

~h Perturbed magnetic �eld vector�e; ~�e Magnetic permeability of upper and

lower half-space respectivelycij ; ~cij Elastic constants of upper and lower

half-space respectivelyci Wave velocity of respective layersAn Cosine coe�cients of Fourier series

expansion of order n respectivelyBn Sine coe�cients of nth order Fourier

series expansiona Amplitude of the corrugated surface� Wave number of corrugations'; Lame's potentialsmH ; lH Dimensionless magnetic coupling

parameter of upper and lowerhalf-spaces, respectively

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Appendix

Mathematical relation between all the parameters aregiven below:

c1 =rc11

�1; s2

1 = k2 c11 � p1 + 2�eH20 � �1c2

c13 + 2c44 � p1 + 2�eH20;

r21 = k2 ~c11 � p2 + 2~�e ~H2

0 � �2c2

~c13 + 2~c44 � p2 + 2~�e ~H20; c2 =

s~c11

�2;

s22 = k2 c44 � p1 � �1c2

c33 � c13 � c44 � p1;

r22 = k2 ~c44 � p2 � �2c2

~c33 � ~c13 � ~c44 � p2;

�11 = ik exp(s1a cos(�x));

�12 = �ik exp(�r1a cos(�x));

�13 = �s2 exp(s2a cos(�x));

�14 = �r2 exp(�r2a cos(�x));

�21 = s1 exp(s1a cos(�x));

�22 = r1 exp(�r1a cos(�x));

�23 = ik exp(s2a cos(�x));

�24 = �ik exp(�r2a cos(�x));

�31 =�s2

1c33 � k2c13 + �eH20 (s2

1 � k2)

+ 2iks1a�c44 sin(�x)�

exp (s1a cos(�x)) ;

�32 =h~c13k2 � ~c33r2

1 � ~�e ~H20 (r2

1 � k2)

+ 2ika�r1~c44 sin(�x)i

exp (�r1a cos(�x)) ;

�33 =�iks2(c33 � c13)� a�c44(s2

2 + k2) sin(�x)�

exp(s2a cos(�x));

�34 =�ikr2(~c33 � ~c13) + a�~c44(r2

2 + k2) sin(�x)�

exp(�r2a cos(�x));

�41 =�2iks1c44 + a� sin(�x)

�s2

1c13 � k2c11

+ �eH20 (k2 � s2

1)�

exp(s1a cos(�x));

�42 =�2ikr1~c44 � a� sin(�x)

�r21~c13 � ~c11k2

+ ~�e ~H20 (k2 � r2

1)oi

exp(�r1a cos(�x));

�43 =�ika�s2 sin(�x)(c13 � c11)� c44(s2

2 + k2)�

exp(s2a cos(�x));

�44 =�~c44(r2 + k2)� ika�r2 sin(�x) (~c11 � ~c13)

�exp (�r2a cos(�x)) :

Biographies

Soumitra Chowdhury is currently pursuing thePhD degree at Indian Institute of Technology (ISM)Dhanbad at the Department of Mathematics and Com-puting. He received his BSc degree from the PresidencyCollege Kolkata, under the University of Calcuttain 2014 and MSc degree from IIT(ISM) Dhanbad in2016.

Santimoy Kundu currently works as an AssociateProfessor at the Department of Mathematics and Com-puting, Indian Institute of Technology (ISM) Dhanbad,India. His research areas of interest are theoreticalseismology and wave propagation aspects.

Parvez Alam currently works at the Departmentof Mathematics, School of Advanced Sciences, Vel-lore Institute of Technology, Vellore as an AssistantProfessor. He completed his PhD in Applied Mathe-matics form IIT(ISM) Dhanbad in 2018 and MSc inMathematics from AMU, Aligarh in 2013. His main


research interests include theoretical seismology andsolid mechanics.

Shishir Gupta is a Professor at the Departmentof Mathematics and Computing, Indian Institute of

Technology (ISM) Dhanbad, India. He served as theHead of the Department of the Applied Mathematics,IIT(ISM) Dhanbad from 2015 to 2018. His researchareas are theoretical seismology, numerical methodsand special functions.

Dispersion of Stoneley waves through the irregular common...

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