A study reporton

Wave Propagationin

Elastic Continua

A term project conducted

for the partial fulfillment of

the course of

Wave Propagation in Continuous Media

by

Soumyajit Roy13ME91R02

Department of Mechanical Engineering

INDIAN INSTITUTE OF TECHNOLOGY KHARAGPURKharagpur - 721302February 14, 2015

Contents

1 Introduction 1

2 Theory 12.1 Hooke’s law and Navier’s equation . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Plane waves in unbounded continua . . . . . . . . . . . . . . . . . . . . . . . 2

3 Different Scattering Problems 23.1 Elastic half-spaces having fixed surface . . . . . . . . . . . . . . . . . . . . . 3

3.1.1 s-H wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1.2 p-wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.3 s-V wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Two elastic half-spaces firmly attached to each other . . . . . . . . . . . . . 93.3 Elastic half-spaces attached to an impedance . . . . . . . . . . . . . . . . . . 11

3.3.1 s-H wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.2 p-wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 Surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Conclusions 14

References 14

List of Figures

1 A plane wave travelling in arbitrary direction [1] . . . . . . . . . . . . . . . . 22 Reflection of an s-H wave from a fixed surface . . . . . . . . . . . . . . . . . 33 Variation of σ23 with incident angle for different values of ν . . . . . . . . . . 44 Reflection of p/s-V waves from a fixed surface . . . . . . . . . . . . . . . . . 55 Ratio of amplitudes of reflected waves to incident wave . . . . . . . . . . . . 66 Variation of σ22 and σ12 with respect to incident angle for different ν values . 77 Variation of σ22 and σ12 with respect to incident angle for different ν values . 88 Variation of σ22 and σ12 with respect to incident angle for different ν values . 99 Reflection and refraction of s-H wave from an interface of two half-spaces . . 1010 Reflected and Transmitted wave ratio to incident wave with respect to incident

angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111 Polynomial function of ξB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

List of Tables

1 Values of ξB for different materials with steel . . . . . . . . . . . . . . . . . . 14

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1 Introduction

Studies of waves travelling through elastic continua have a great importance as they lead tothe very primary analysis of seismic waves. It started from the middle of 19th century andgained a rapid growth in the engineering fields in early forties along with the technologicaldevelopments in different domains. In this present work, scattering of this kind of wavesfor different boundary conditions and surface waves existing in the interface of two elastichalf-spaces are studied. Only homogeneous, isotropic and elastic materials are consideredhere for analysis.

2 Theory

Theory behind this analysis is very briefly presented here. It contains three parts - (1)Hooke’slaw and Navier’s equation for an elastic continuum, (2)Wave equations and (3)Plane wavesin unbounded continua

2.1 Hooke’s law and Navier’s equation

From elementary theory of elasticity, it is known that Hooke’s law can be written as [1]

σij = λ

(3∑

k=1

εkk

)δij + 2µεij (1)

where, λ = Eν(1−2ν)(1+ν)

and µ = E2(1−ν) . Strain displacement relations are given by

εij =1

2(ui,j + uj,i) and εk,k = uk,k (2)

The Navier’s equation is known as

ρutt − (λ+ µ)∇(∇ · u)− µ∇2u = 0 (3)

where, u is the displacement field which is a function of (x1, x2, x3, t)

2.2 Wave equations

Using a theorem due to Helmholtz, one can decompose the displacement vector field u as[1], u(x1, x2, x3, t) = uL(x1, x2, x3, t) +uS(x1, x2, x3, t) such that ∇×uL = 0 and ∇ ·uS = 0.Applying these condition in equation (3) the standard wave equation can be obtained as

uL,tt − C2L∇2uL = 0 (4)

uS,tt − C2S∇2uS = 0 (5)

where, CL =√

λ+2µρ

, Cs =√

µρ

and κ = CLCS

=√

2(1−ν)1−2ν

1

2.3 Plane waves in unbounded continua

A plane wave is propagating in two dimensional plane of x1x2 in an arbitrary direction nas shown in figure 1. The longitudinal and shear harmonic plane wave solutions [1] are,respectively, written as

Figure 1: A plane wave travelling in arbitrary direction [1]

uL(x1, x2, t) = ALn exp [ikL(n · r− CLt)]= ALn exp [ikL(x1 cos θ + x2 sin θ − CLt)] (6)

uS(x1, x2, t) = ASV (a× n) exp [ikS(n · r− CSt)] + ASH a exp [ikS(n · r− CSt)]= {ASV (a× n) + ASH a} exp [ikS(x1 cos θ + x2 sin θ − CSt)] (7)

Therefore, the total wave field u(x1, x2, t) = uL(x1, x2, t) + uS(x1, x2, t), can be written incomponent wise as

u1(x1, x2, t) = AL cos θ exp [ikL(x1 cos θ + x2 sin θ − CLt)]−ASV sin θ exp [ikS(x1 cos θ + x2 sin θ − CSt)] (8)

u2(x1, x2, t) = AL sin θ exp [ikL(x1 cos θ + x2 sin θ − CLt)]+ASV cos θ exp [ikS(x1 cos θ + x2 sin θ − CSt)] (9)

u3(x1, x2, t) = AH exp [ikS(x1 cos θ + x2 sin θ − CSt)] (10)

The longitudinal waves are called primary waves or p-waves, on the other hand, shear wavesare called secondary waves or s-waves. However, shear waves have two polarizations, theyare called horizontal (s-H) and vertical (s-V) polarizations. It can be noted, also, that ans-H wave cannot excite both p and s-V waves and vice-versa.

3 Different Scattering Problems

In this term project four different situations are studied for elastic half-space. Reflected toincident, transmitted to incident and stress values are plotted for different situations and for

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different materials.

• The surface of an elastic half-space confined in x2 < 0 is fixed or clamped at x2 = 0.

• Two elastic half-spaces (one is at x2 < 0 and the other is at x2 > 0) are firmly attachedto each other at x2 = 0.

• The surface of an elastic half-space confined in x2 < 0 is attached to an impedance atx2 = 0.

• Surface waves at the interface of two elastic half-spaces (Stoneley waves).

3.1 Elastic half-spaces having fixed surface

There situations are studied here for incident s-H, p and s-V waves.

3.1.1 s-H wave

As s-H wave cannot generate s-V and p-wave, after reflection it will only generate s-H waveas shown in figure 2. According to equations (8), (9) and (10), it can be easily concludedthat the displacement field has only u3 component. The total displacement field is writtenas

Figure 2: Reflection of an s-H wave from a fixed surface

u3 = AS0 exp[ikS0(x1 sin θS0+x2 cos θS0−CSt)]+As exp[ikS(x1 sin θS−x2 cos θS−CSt)] (11)

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As the surface is fixed, the boundary condition obviously will be u3|x2=0 = 0. Applying thiscondition on equation (11), it yields

u3 = AS0 exp[ikS0(x1 sin θS0 − CSt)] + AS exp[ikS(x1 sin θS − CSt)] = 0 (12)

As the exponential terms and the two coefficients (AS0, AS) cannot be zero, equation (12)will satisfy the condition if and only if the exponential terms can be cancelled out. This isonly possible when

kS0 sin θS0 = kS sin θS and kS0CS = kSCS (13)

Equation (13) yields kS0 = kS and AS0 = −AS. Therefore, the reflected wave has a phasechange of π with respect to the incident s-H wave.

Stresses on fixed surface

Stresses on the fixed sueface can be calculated from equation (1). It can be easily seen thatall other stresses (i.e., σ22 and σ12) of surface x2 = 0 except σ23 are zero at x2 = 0 since u1and u2 are zero. Using the relations in equation (13), σ23 at x2 = 0 is obtained as

σ23|x2=0 = µu3,2|x2=0 = µAS0[ikS0 cos θS0 exp[ikS0(x1 sin θS0 − CSt)]+ikS0 cos θS0 exp[ikS0(x1 sin θS0 − CSt)]]

= 2µkS0AS0 cos θS0 exp[ikS0(x1 sin θS0 − CSt+ π/2)] (14)

The amplitude of stress is plotted with the variation of incident angle θS0 for different valuesof Poisson’s ratio. The plot is shown in figure 3.

Figure 3: Variation of σ23 with incident angle for different values of ν

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3.1.2 p-wave

As it is discussed in section 2.3, p-waves cannot excite s-H waves. But it can generate s-Vwaves. So, as shown in figure 4, an incident p-wave generates a p-wave and an s-V waveafter reflection from the fixed surface. It is very obvious, then, that u3 is zero in this case.The total displacement field is, therefore, written as

u1(x1, x2, t) = AL0 sin θL0 exp [ikL0(x1 sin θL0 + x2 cos θL0 − CLt)]+AL sin θL exp [ikL(x1 sin θL − x2 cos θL − CLt)]+ASV cos θS exp [ikS(x1 sin θS − x2 cos θS − CSt)] (15)

u2(x1, x2, t) = AL0 cos θL0 exp [ikL0(x1 sin θL0 + x2 cos θL0 − CLt)]−AL cos θL exp [ikL(x1 sin θL − x2 cos θL − CLt)]+ASV sin θS exp [ikS(x1 sin θS − x2 cos θS − CSt)] (16)

Figure 4: Reflection of p/s-V waves from a fixed surface

The boundary conditions are same as discussed in section 3.1.1, i.e., u1|x2=0 = u2|x2=0 = 0.Applying these conditions and using the same logic of section 3.1.1, the following relationscan be obtained

kL0 sin θL0 = kL sin θL = kS sin θS (17)

kL0CL = kLCL = ksCS (18)

Equations (17) and (18) yield kL0 = kL =⇒ θL0 = θL and CLCS

=√

λ+2∗µµ

= kSkL

= sin θLsin θS

= κ.

Using these relations in equations (15) and (16), one can obtain

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AL0 sin θL0 + AL sin θL + ASV cos θS = 0 (19)

AL0 cos θL0 − AL cos θL + ASV sin θS = 0 (20)

Finally, solving the above two equations the ratio of amplitudes of reflected waves to incidentwave can be found as

ALAL0

=cos(θL0 + θS)

cos(θL0 − θS)and

ASVAL0

= − sin 2θLcos(θL0 − θS)

(21)

Variations of these ratios with respect to incident angle (θL0) are shown in figure 5. It canbe noted from the expression of equation (21) that s-V wave vanishes at θL0 = 0 and π/2.The same occurs to p-wave when θL0 + θS = π/2.

(a) p-wave (b) s-V wave

Figure 5: Ratio of amplitudes of reflected waves to incident wave

Stresses on fixed surface

It is very clear that for the reflection of p-wave σ23 on the surface x2 = 0 is zero because u3is zero. σ22 and σ12 can be found from

σ22 = (λ+ 2µ)(u1,1 + u2,2)− 2µu1,1

σ12 = µ(u1,2 + u2,1) (22)

After replacing all partial derivatives and writing only the amplitude part equation (22)yields

σ22|x2=0 = µkL0[(AL0 + AL)(κ2 − 1 + cos 2θL0) + κASV sin 2θS]

σ22|x2=0 = µkL0[(AL0 − AL) sin 2θL0 − κASV cos 2θS] (23)

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Replacing AL from equation (21) in equation (23), the final expression can be obtained as

σ22|x2=0 = 2µAL0kL0cos θL0

cos(θL0 − θS)

(1− 1

κ2sin2 θL0

)1/2

[κ2 − 4 sin2 θL0] (24)

σ12|x2=0 =µAL0kL0

cos(θL0 − θS)κ

(κ2 + 2(1− κ) sin2 θL0

)(25)

Equtions (24) and (25) are plotted for different values of ν in figure 6

(a) σ22 (b) σ12

Figure 6: Variation of σ22 and σ12 with respect to incident angle for different ν values

3.1.3 s-V wave

Reflection of s-V wave cannot generate s-H wave. So, the displacement field can be writtenfor s-V wave as

u1(x1, x2, t) = −AS0 cos θS0 exp [ikS0(x1 sin θS0 + x2 cos θS0 − CSt)]+ASV cos θS exp [ikS(x1 sin θS − x2 cos θS − CSt)]+AL sin θL exp [ikL(x1 sin θL − x2 cos θL − CLt)] (26)

u2(x1, x2, t) = AS0 sin θS0 exp [ikS0(x1 sin θS0 + x2 cos θS0 − CSt)]+ASV sin θS exp [ikS(x1 sin θS − x2 cos θS − CSt)]−AL cos θL exp [ikL(x1 sin θL − x2 cos θL − CLt)] (27)

Like the case of p-wave, here, also one can obtain the following relations

kS0 sin θS0 = kS sin θS = kL sin θL (28)

kS0CS = kSCS = kLCL (29)

The above equations yield kS0 = kS =⇒ θS0 = θS and CLCS

=√

λ+2∗µµ

= kSkL

= sin θLsin θS

=sin θLsin θS0

= κ. Using the method discussed in section 3.1.2, the final expression of ratio ofamplitudes of reflected waves to incident wave can be found as

7

ASVAS0

=cos(θS0 + θL)

cos(θS0 − θL)and

ALAS0

=sin 2θS0

cos(θS0 − θL)(30)

Variations of these ratios with respect to incident angle (θS0) are shown in figure 7. It canbe noted from the expression of equation (30) that p-wave vanishes at θS0 = 0 and π/2. Thesame occurs to s-V wave when θS0 +θL = π/2. However, more interesting phenomena can beobserved here [2]. From the relations, sin θL = κ sin θS0 is obtained. θL will have a real valuewhen κ sin θS0 6 1. So, there is a critical value of θS0 which is defined by θcr = sin−1(1/κ),beyond that, there cannot be any p-wave. At critical value, the expressions for AL and ASVare written as

AL = −2

κ

(κ2 − 1

)1/2AS0 and AS = −AS0 (31)

(a) σ22 (b) σ12

Figure 7: Variation of σ22 and σ12 with respect to incident angle for different ν values

But if incident angle is beyond critical angle, sin θL > 1 and cos θL =√

1− sin2 θL =

i√κ2 sin2 θS − 1 = iβ, where, β =

√κ2 sin2 θS − 1. According to the figure 4 the displace-

ment field for p-wave is written as uL = ALn exp[ikL(x1 sin θL−x2 cos θL−CLt)]. Replacingcos θL, one can obtain

uL = ALneβ(kSκ

)x2 exp[i(kS/κ)(x1 sin θL − CLt)] (32)

Equation 32 is an inhomogeneous wave solution. It suggests a travelling wave in x1 directionwith dying amplitude along negative x2. This type of waves are called surface waves.

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Stresses on fixed surface

Stresses on fixed surface can be obtained same as p-wave. The final expressions of amplitudeof stress are given by equations (33) and (34). Plots are also shown in figure 8

σ22|x2=0 = µκkS0AS0sin 2θS0

cos(θS0 − θL)(33)

σ12|x2=0 = 2µkS0AS0cos θS0

cos(θS0 − θL)

(1− κ2 sin2 θS0

)1/2(34)

(a) σ22 (b) σ12

Figure 8: Variation of σ22 and σ12 with respect to incident angle for different ν values

3.2 Two elastic half-spaces firmly attached to each other

Two elastic half-spaces are attached to each other firmly as shown in figure 9. Let a planes-H wave be incident on the interface. As, s-H wave only generates s-H wave, one reflectedand one transmitted s-H wave are found. For both the medium there will be only onedisplacement component along x3 direction. So, the displacement fields for medium A andB can be written as

uA3 = AS0 exp [ikS0(x1 sin θS0 + x2 cos θS0 − CAS t)] + (35)

ASR exp [ikSR(x1 sin θS − x2 cos θS − CAS t)]

uB3 = AST exp [ikST (x1 sin θST + x2 cos θST − CBS t)] (36)

9

Figure 9: Reflection and refraction of s-H wave from an interface of two half-spaces

In this case, there are two boundary conditions. The first boundary condition is dis-placements are continuous, i.e., uA3|x2=0 = uB3|x2=0. This yields kS0 = kSR, θS0 = θS,sin θSTsin θS

= kSRkST

=CBSCAS

= 1γ

and

AS0 + ASR = AST (37)

The second boundary condition is stresses are continuous, i.e, σA23|x2=0 = σB23|x2=0 whichimplies µAuA3,2|x2=0 = µBuB3,2|x2=0 and it gives

µAkS0(AS0 − ASR) cos θS0 = µBkSTAST cos θST (38)

Equations (37) and (38) are solved to get

ASR =µA cos θS0 − µBγ cos θSTµA cos θS0 + µBγ cos θST

AS0 (39)

AST =2µA cos θS0

µA cos θS0 + µBγ cos θSTAS0 (40)

These amplitude ratios are plotted considering A as steel and B as Aluminium with respectto incident angle and shown in figure 10. Like the case of s-V wave, discussed in section3.1.3, critical angle of incident can also be defined here as θcr = sin−1 γ. At critical angleof incident, ASR = AS0 and AST = 2AS0. For steel-aluminium interface, critical angle ofincident is 50.56o.

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(a) Reflected wave (b) Transmitted wave

Figure 10: Reflected and Transmitted wave ratio to incident wave with respect to incidentangle

3.3 Elastic half-spaces attached to an impedance

An elastic half-space confined in x2 6 0 is firmly attached to another material having specificlongitudinal and specific shear impedance of ZL and ZS. Two different cases of s-H and p-wave will be studied here.

3.3.1 s-H wave

The displacement field is same as discussed in section 3.1.1. The boundary condition isdefined by σ23|x2=0 = −ZSu3,t|x2=0, which gives the final expression as

AS =µ cos θS0 − ZSCSµ cos θS0 + ZSCS

AS0 (41)

3.3.2 p-wave

Here, also the displacement fields are already defined in section 3.1.2. The boundary condi-tions are given by

σ22|x2=0 = −ZLu2,t|x2=0 and σ12|x2=0 = −ZSu1,t|x2=0 (42)

The final expressions are formed in CAS (Computer Algebraic System) Mathematica andthey are given below.

ALAL0

= −∆1

∆and

ASVAL0

= −∆2

∆(43)

11

where,

∆1 = κµ2 − CS(ZL + ZSκ)µ cos(θL0 − 2θS) + µ2 cos 2(θL0 − θS)− CSZLµ cos(2θL0 − θS)

+CSZSκ2µ cos θS + CSZSκ

2µ cos 3θS + κµ2 cos 4θS − 2C2SZLZSκ cos(θL0 + θS)−

µ2 cos 2(θL0 + θS) + CSZLµ cos(2θL0 + θS)− CS(ZL − ZSκ)µ cos(θL0 + 2θS) (44)

∆2 = 2κ(C2SZLzS − 2µ2 cos 2θS) sin 2θL0 (45)

∆ = κµ2 + CS(ZL − ZSκ)µ cos(θL0 − 2θS) + 2C2SZLZSκ cos(θL0 − θS)− µ2 cos 2(θL0 − θS) +

CSZLµ cos(2θL0 − θS) + CSZSκ2µ cos θS + CSZSκ

2µ cos 3θS + κµ2 cos 4θS +

µ2 cos 2(θL0 + θS)− CSZLµ cos(2θL0 + θS) + CS(ZL + ZSκ)µ cos(θL0 + 2θS) (46)

3.4 Surface waves

It has been discussed earlier in section 3.1.3 that surface waves do exist in the interface oftwo elastic bodies. However it also exists in free surface also. The first is called Stoneleywaves and the later Rayleigh waves. In both the cases, waves travel in the x1 directionand amplitudes die down with the increasing distance from the interface or free surface,respectively. Theoretically the amplitude will be zero at infinity. Here, using the solution ofRayleigh waves, Stoneley waves are discussed.

Let’s assume the negative half-space is occupied by material A and the positive half-spaceby material B. The surface wave displacement fields for the negative half space can be foundin the literature as [1]

uA1 = (UAL e

kALx2 + UAS e

kAS x2) exp[i(kx1 − ωt)]

uA2 = −i(kALkUAL e

kALx2 +k

kASUAS e

kAS x2

)exp[i(kx1 − ωt)] (47)

where, (kAL )2 = k2 −(

ωCAL

)2and (kAS )2 = k2 −

(ωCAS

)2. Similarly, for the positive half-space

it can be written as

uB1 = (UBL e

−kBL x2 + UBS e

−kBS x2) exp[i(kx1 − ωt)]

uB2 = i

(kBLkUBL e

−kBL x2 +k

kBSUBS e

−kBS x2)

exp[i(kx1 − ωt)] (48)

where, (kBL )2 = k2 −(

ωCBL

)2and (kBS )2 = k2 −

(ωCBS

)2. It is clearly seen from the equations

(47) and (48) that they vanish at x2 = −∞ and x2 = +∞. Both the displacements andstresses are continuous, and these will be used as boundary conditions. So, uA1 |x2=0 = uB1 |x2=0

yields

UAL + UA

S − UBL − UB

S = 0 (49)

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The second condition on displacement , i.e., uA2 |x2=0 = uB2 |x2=0 provides

kALkUAL +

k

kASUAS +

kBLkUBL +

k

kBSUBS = 0 (50)

From first stress condition, i.e., σA22|x2=0 = σB22|x2=0, one can get

[k2 + (kAS )2]UAL + 2k2UA

S − β[k2 + (kBS )2]UBL − 2k2βUB

S = 0 (51)

where, β = µB

µA.The second stress condition, i.e., σA12|x2=0 = σB12|x2=0, yields

2kAS kALU

AL + [k2 + (kAS )2]UA

S + 2βkAS kALU

BL + β

(kASkBS

)[k2 + (kBS )2]UB

S = 0 (52)

Equations (49) to (52) are homogeneous equations which can be solved iff1 1 −1 −1kALk

kkAS

kBLk

kkBS

[k2 + (kAS )2] 2k2 −β[k2 + (kBS )2] −2k2β

2kAS kAL [k2 + (kAS )2] 2βkAS k

AL β

(kASkBS

)[k2 + (kBS )2]

= 0 (53)

Equation (53) yields a polynomial of ξB which is defined as CCBS

, where C is the speed of

surface wave and CBS is the shear wave speed in medium B. Equation (53) is plotted for

different materials with respect to ξB, considering A as steel. Point of inflection from theplot in the range of 0 6 ξB 6 1 gives the possible root of the equation. This polynomialfunction is plotted for steel-aluminium interface in figure 11. Values of ξB for other materialsare also tabulated.

Figure 11: Polynomial function of ξB

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Material (A) Material (B) ξBSteel Aluminium 0.9920Steel Magnesium 0.9940Steel Molybdenum 0.9815Steel Nickel 0.9995Steel Steel 1.0000

Table 1: Values of ξB for different materials with steel

4 Conclusions

In this report, a few cases have been studied. It can be extended further more to investigatepropagation of wave in layered media and wave guides.

References

[1] Hagedorn, P and DasGupta, A. (2007), ”Vibrations and Waves in Continuous MechanicalSystems”, John Wiley & Sons, Ltd.

[2] Achenbach, J. D, (1973) ”Wave Propagation in Elastic Solids”, North-Holland PublishingCompany.

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