Rayleigh waves with impedance boundary conditions in anisotropic solids

Rayleigh waves with impedance boundary conditions in anisotropic solids tài liệu, giáo án, bài giảng , luận văn, luận án...

To appear in: Wave Motion

Received date: 8 January 2014

Accepted date: 4 May 2014

Please cite this article as: P.C Vinh, T.T Thanh Hue, Rayleigh waves with impedance

boundary conditions in anisotropic solids, Wave Motion (2014),

http://dx.doi.org/10.1016/j.wavemoti.2014.05.002

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Rayleigh waves with impedance boundary

conditions in anisotropic solids

Pham Chi Vinha ∗ and Trinh Thi Thanh Hueb

aFaculty of Mathematics, Mechanics and Informatics

Hanoi University of Science

334, Nguyen Trai Str., Thanh Xuan, Hanoi,Vietnam

bFaculty of Civil and Industrial Construction National University of Civil Engineering

55 Giai Phong Str., Hanoi, Vietnam

Abstract The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions The half-space is assumed to be orthotropic and monoclinic with the symmetry plane x3 = 0 The main aim of the paper is to derive explicit secular equations of the wave For the orthotropic case, the secular equation is obtained by employing the traditional approach It is an irrational equation From this equation, a new version of the secular equation for isotropic materials is derived For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions.

Key words: Rayleigh waves, Impedance boundary conditions, Orthotropic, clinic, Explicit secular equation

Mono-∗ Corresponding author: Tel:+84-4-35532164; Fax:+84-4-38588817; E-mail address: inh@vnu.edu.vn (P C Vinh)

pcv-*Manuscript

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1 Introduction

Elastic surface waves, discovered by Rayleigh [1] more than 120 years ago for pressible isotropic elastic solids, have been studied extensively and exploited in awide range of applications in seismology, acoustics, geophysics, telecommunicationsindustry and materials science, for example It would not be far-fetched to say thatRayleigh’s study of surface waves upon an elastic half-space has had fundamentaland far-reaching effects upon modern life and many things that we take for grantedtoday, stretching from mobile phones through to the study of earthquakes, as ad-dressed by Adams et al [2] A huge number of investigations have been devoted

com-to this com-topic As written in [3], one of the biggest scientific search engines, GoogleScholar returns more than a million links for request ”Rayleigh waves” and almost

3 millions for ”Surface waves” This data is really amazing! It shows a tremendousscale of scientific and industrial interests in this area

For Rayleigh waves their explicit secular equation are important in practicalapplications They can be used for solving the direct (forward) problems: evaluatingthe dependence of the wave velocity on material parameters, especially for solvingthe inverse problems: to determine material parameters from measured values ofwave velocity Therefore, explicit secular equations are always the main purpose forany investigation of Rayleigh waves

In the context of Rayleigh waves, it is almost always assumed that the half-spacesare free of traction As mentioned in [4], in many fields of physics such as acous-

tics and electromagnetism, it is common to use impedance boundary conditions,that is, when a linear combination of the unknown function and their derivatives isprescribed on the boundary In the other hand, when studying the propagation ofRayleigh waves in a half-space coated by a thin layer, the researchers often replacethe effect of the thin layer on the half-space by the effective boundary conditions

on the surface of the half-space (see, for examples, Achenbach and Keshava [5],Tiersten [6], Bovik [7], Steigmann and Ogden [8], Vinh and Linh [9, 10], Vinh andAnh [11], Vinh et al [12]) These conditions lead to the impedance-like boundaryconditions on the surface As addressed in [13, 14], a thin layer on a half-space is

a model finding a broad range of applications, including: the Earth’s crust in mology, the foundation/soil interaction in geotechnical engineering, thermal barriercoatings, tissue structures in biomechanics, coated solids in material science, andmicro-electro-mechanical systems Rayleigh waves with impedance boundary condi-tions are therefore significant in many fields of science and technology However, veryfew investigations on Rayleigh waves with impedance boundary conditions have beendone In [15] Malischewsky considered the propagation of Rayleigh waves with Tier-sten’s impedance boundary conditions and provided a secular equation Recently,Godoy et al [4] investigated the existence and uniqueness of Rayleigh waves withimpedance boundary conditions which are a special case of Tiersten’s impedanceboundary conditions In works [4] and [15] the half-space is assumed to be isotropic.Nowadays, anisotropic materials are widely used in various fields of modern tech-

seis-nology The investigations of Rayleigh waves with impedance boundary conditions

in anisotropic solids are therefore significant in practical applications

The main purpose of this paper is to study the propagation of Rayleigh waveswith Tiersten’s impedance boundary conditions [15] in anisotropic elastic half-spaces.Two cases of anisotropy are considered: orthotropic materials and monoclinic oneswith the symmetry plane x3 = 0 (see [16]) For the orthotropic case the secu-lar equation is obtained by employing the traditional approach It is an irrationalequation and it provides a new version of the secular equation for isotropic materi-als For the monoclinic case, for obtaining the secular equation we use the method

of polarization vector and the secular equation obtained is an algebraic equation ofeighth-order When the impedance parameters vanish, this equation coincides withthe secular equation of Rayleigh waves with traction-free boundary conditions

Consider an elastic half-space which occupies the domain x2 ≥ 0 We are interested

in the plane strain such that:

ui = ui(x1, x2, t), i = 1, 2, u3 ≡ 0 (1)

where t is the time Suppose that the half-space is made of compressible orthotropic

elastic material, then the strain-stress relations are [16]:

where σij and cij are respectively the stresses and the material constants, commasindicate differentiation with respect to spatial variables xk The elastic constants

c11, c22, c12, c66 satisfy the inequalities:

where ρ is the mass density, a superposed dot signifies differentiation with respect

to t Introducing (2) into (4) leads to the equations governing infinitesimal motion,expressed in terms of the displacement components, namely:

c11u1,11+ c66u1,22+ (c12+ c66)u2,12 = ρ¨u1

c66u2,11+ c22u2,22+ (c12+ c66)u1,12 = ρ¨u2

(5)Now we consider the propagation of a Rayleigh wave, travelling with velocity c (> 0)and wave number k (> 0) in the x1-direction and decaying in the x2-direction, i e.:

According to Vinh and Ogden [17], the displacement components of this Rayleighwave which satisfy Eqs.(5) and the decay condition (6) are given by:

u1 = (B1e−kb1 x 2 + B2e−kb2 x 2)eik(x1 −ct)

u2 = (α1B1e−kb1 x 2 + α2B2e−kb2 x 2)eik(x1 −ct)

(8)where B1, B2 are constants to be determined from the impedance boundary condi-tions (7), b1, b2 are roots of the equation:

Substituting Eqs (8) and (14) into the impedance boundary conditions (7) gives:

2 = c66/ρ, is squared dimensionless velocity of Rayleigh waves,

δn = Zn/√ρc66(∈ R), n = 1, 2, are dimensionless impedance parameters Since

in which S and P are given by:

P = (e1− x)(1 − x)

e2(e1− x) + 1 − x − (1 + e3)2

Equation (19) is the (dimensionless) secular equation of Rayleigh waves propagating

in an orthotropic elastic half-space whose surface is subjected to the impedanceboundary conditions (7)

Taking δ1 = δ2 = 0 in Eq (19) we obtain the (dimensionless) secular equation

of Rayleigh waves propagating along a traction-free surface of an orthotropic elastichalf-space, namely:

which was derived by Chadwick [18] (see also [17, 19])

When the elastic half-space is isotropic, i e c11=c22=λ + 2µ, c66 = µ, c12 = λ,

λ and µ are Lame constants, one can see that two roots of Eq (9) having positivereal parts are:

Introducing (23) and (24) into Eq (16) and taking into account e1 = e2=1/γ,

e3 = 1/γ− 2 we arrive at the equation:

(x− 2)2− 4√1− x√1− γx + x√x(δ1√

1− x + δ2√1

− γx)+δ1δ2x(√

When δ2 = 0 Eq (25) is simplified to:

3 Monoclinic materials with the symmetry plane

x3 = 0

3.1 Basic equations in matrix form

Consider a linearly elastic half-space x2 ≥ 0, made of a monoclinic material withthe symmetry plane x3 = 0 (see [16]) For such materials, in-plane motions aredecoupled from anti-plane motions, therefore we can consider the plane strain suchthat:

Using the first of (30) and taking into account (31) yield:

3.2 Rayleigh waves Stroh’s formulation

Now we consider the propagation of a Rayleigh wave, travelling with velocity c (> 0)and wave number k (> 0) in the x1-direction and decaying in the x2-direction Then,the displacements and stresses of the Rayleigh wave are sought in the form:

un = Un(y)eik(x1 −ct), σn2= iktn(y)eik(x1 −ct), n = 1, 2, y = kx2 (39)

Substituting (39) into Eq (36) leads to:

where the prime signifies differentiation with respect to y and:

ξ =

ut

, u =



U1

U2

, t =



t1

t2

, N =



n66 n26

n26 n22

, N3=



X− η 0

, N4 = N1T (42)

that can be written in matrix form as:

, Q =

Proof: From PP−1= I it follows:

(

P1P(1−1)+ P2P(3−1)= I, P1P(2−1)+ P2P(4−1)= 0

P3P(1−1)+ P4P(3−1)= 0, P3P(2−1)+ P4P(4−1) = I (58)Taking transpose and complex conjugate two sides of the equalities (58) and using(55) yield:

The proof is completed

Lemma 2: Suppose the matrix P expressed by (54) is invertible and the ities (55) hold for the matrices Pk For all n∈ Z the matrix Pn is expressed as:

Then, the equalities (55) also hold for the matrices P(n)k

Proof: + Clearly, the equalities (55) hold for matrices P(0)k and P(1)k

+ Assume the equalities (55) hold for the matrices P(n)k , n > 1 Then, it is notdifficult to show that these equalities are satisfied for the matrices P(n+1)k Thatmeans the equalities (55) hold for P(n)k for all n∈ Z, n ≥ 0.

+ By the lemma 1, the equalities (55) hold for P(n)k for all n ∈ Z, n ≤ 0 Theproof of the lemma 2 is finished

Lemma 3: Suppose the matrix P expressed by (54) is invertible and the matrices

Pk satisfy the equalities (55) Then we have:

Proof of the proposition:

Pre-multiplying two sides of the equation (53)1 by ¯YTˆIPn we have:

¯

YTˆI PnY′ = i ¯YTˆI Pn+1Y (65)

Taking transpose and complex conjugate two sides of Eq (65) and using (64) yield:

( ¯Y′)TˆI PnY =−i ¯YTˆI Pn+1Y (66)

From (65) and (66) it follows:

ddy

h

¯

YTˆIPnYi

= 0 → ¯YTˆI PnY = C ∀ y ∈ [0 + ∞]

where C is a constant Due to the second of (53) the constant C must be zero.Therefore we have:

3.4 Explicit secular equations

Now in Eqs (56) we take P = Q that is given by (49), (50), and Y = w According

to the second of (48): σ(0) = 0, Eqs (56) are therefore simplified to:

is a complex number, α = a + ib, a, b are real Introducing the expression of u(0)into (68) and taking into account the fact that Q(n)3 is hermitian (see (52)) we have:

As the matrix Q(n)3 is hermitian, Q(n)11 , Q(n)22 (n = 1, 2), ˆQ(11−1), ˆQ(22−1) are real and

Q(n)12 , Q(n)12 (n = 1, 2), ˆQ(12−1), ˆQ(12−1) are complex numbers whose real and imaginary

parts are denoted, respectively, by Q(n,r)12 and Q(n,i)12 (n = 1, 2), ˆQ(12−1,r) and ˆQ(12−1,i).Substituting α = a + ib, Q(n)12 = Q(n,r)12 + iQ(n,i)12 (n = 1, 2), ˆQ(−1)12 = ˆQ(−1,r)12 + i ˆQ(−1,i)12into Eqs (70) we arrive at a system of three linear equations, namely:

where D is the determinant of the 3×3 matrix in Eq (74), Dk are the determinants

of matrices obtained by replacing this matrix’s kth column with the vector on theright-hand side of Eq (74) From Eq (75) it follows:

that is the desired explicit secular equation of the wave The expressions of thedeterminants D, Dk are lengthy and are not displayed here, but they are easilycomputed by using the expressions in Eqs (71)-(73) Also from (71)-(73) one cansee that the secular equation (76) is an algebraic equation of eighth-order in X.Remark 2: To obtain the secular equation of the wave we can apply the funda-mental equations (56) with the impedance boundary condition (44) in which P = N.However, the derivation is more complicated, especially when the size of the squarematrix N is higher than 4

When the impedance parameters vanish, i e δ1 = δ2 = 0, from (71)-(73) we

Q(1,i)12 = Q(2,i)12 = ˆQ(12−1,i) = Q(1,r)12 = Q(2)22 = 0 (77)

With (77) Eq (74) is simplified to:



whose determinant of this system’s matrix must be zero, i e.:

equation obtained by Destrade [22] using the method of first integrals (Eq (28))and by Ting [23] using the cofactor-matrix approach (Eq (4.12b)).

Note that, from the first three equations of (77) it implies D ≡ D1 ≡ D3 ≡ 0.Therefore Eq (76) becomes D2 = 0, i e it simplifies to Eq (79) that is equivalent

to Eq (82) Thus, Eq (82) is a special case of Eq (76)

When the material is orthotropic, c16 = c26 = 0 (see [16]) From (32) we have

(85)

in which the dimensionless material parameters ek (k = 1, 2, 3) are defined by (17),

δ1 and δ2 are the dimensionless impedance parameters, x = c2/c2

2, c2

2 = c66/ρ From(85) one can see that Eq (84) is an algebraic equation of sixth-order in x

Remark 3: i) By squaring (two times) two sides of Eq (19) and then dividingthe resulting equation by (e1− x)2/e2

2 we arrive at Eq (84) Equation Eq (19) istherefore considered as the original version of Eq (84)

ii) Equation (19) is much more simple than Eq (84) This fact says that it isreasonable to consider separately the case of orthotropic materials

iii) Equation (19) is will be useful in investigating the uniqueness of Rayleighwaves in orthotropic half-spaces with impedance boundary conditions by the com-plex function method [24, 25]

In this paper, the propagation of Rayleigh waves in anisotropic half-spaces subjectedimpedance boundary conditions is investigated Two cases of anisotropy are con-sidered: orthotropic materials and monoclinic materials with the symmetry plane

x3= 0 For orthotropic case, the secular equation is derived by using the traditionaltechnique and it is an irrational equation From this equation we obtain a newsecular equation of the wave for the isotropic case For the monoclinic half-spaces,first the impedance boundary condition is replaced by a traction-free-like boundarycondition Then the secular equation is derived by using the method of polarizationvector This equation is an algebraic equation of eighth-order

In this paper, the... of Rayleigh waves in anisotropic half-spaces subjectedimpedance boundary conditions is investigated Two cases of anisotropy are con-sidered: orthotropic materials and monoclinic materials with. .. class="text_page_counter">Trang 22

equation obtained by Destrade [22] using the method of first integrals (Eq (28))and by Ting [23] using the cofactor-matrix approach (Eq