DSpace at VNU: An approximate secular equation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer

DOI 10.1007/s00707-014-1090-8Chi Vinh Pham · Thi Ngoc Anh Vu An approximate secular equation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic lay

DOI 10.1007/s00707-014-1090-8

Chi Vinh Pham · Thi Ngoc Anh Vu

An approximate secular equation of Rayleigh waves

in an isotropic elastic half-space coated with a thin

isotropic elastic layer

Received: 28 October 2013 / Revised: 25 December 2013

© Springer-Verlag Wien 2014

Abstract In this paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic

half-space coated with a thin isotropic elastic layer The contact between the layer and the half-half-space is assumed

to be welded The main purpose of the paper is to establish an approximate secular equation of the wave

By using the effective boundary condition method, an approximate secular equation of fourth order in terms

of the dimensionless thickness of the layer is derived It is shown that this approximate secular equation has high accuracy From the secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established

1 Introduction

The structures of a thin film attached to solids, modeled as half-spaces coated by a thin layer, are widely applied in modern technology The determination of mechanical properties of thin films deposited on half-spaces before and during loading plays an important role in health monitoring of these structures [1,2] Among various measurement methods, the surface/guided wave method is most widely used [2], and for this method, the Rayleigh wave is a versatile and convenient tool [3,4]

For the Rayleigh wave approach, the explicit dispersion relations of Rayleigh waves supported by thin film/substrate interactions are employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data They are therefore the main purpose of the investigations of Rayleigh waves propagating in half-spaces covered with a thin layer Taking the assumption of a thin layer, explicit secular

equations can be derived by replacing the entire effect of the thin layer on the half-space by the so-called

effective boundary conditions, which relate the displacements with the stresses of the half-space at its surface.

For obtaining the effective boundary conditions, Achenbach and Keshava [5], and Tiersten [6] replaced the thin layer by a plate modeled by different theories: Mindlin’s plate theory and the plate theory of low-frequency extension and flexure, while Bovik [7] expanded the stresses at the top surface of the layer into Taylor series in its thickness The Taylor expansion technique was then developed by Rokhlin and Huang [8,9], Niklasson [10], Benveniste [11], Steigmann and Ogden [12], Ting [13], Vinh and Linh [14,15], Vinh and Anh [16] and Vinh et

al [17] Malischewsky [18] puts Tiersten’s theory into a broader context by formulating very general impedance conditions Godoy et al [19] used Malischewsky’s formalism for a similar problem with vanishing normal traction on the surface

Tiersten [6] and Bovik [7] assumed that the layer and the substrate are both isotropic and derived approxi-mate secular equations of second order In Vinh and Linh [14], the layer and the half-space were both assumed

C V Pham (B) · T N A Vu

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science,

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

E-mail: pcvinh@gmail.com; pcvinh@vnu.edu.vn

Tel.: +84-4-35532164

Fax: +84-4-38588817

to be orthotropic, and an approximate secular equation of third order was obtained In order to get more accu-rate solutions of the inverse problem (evaluating the mechanical properties of the thin films from experimental data), we need approximate secular equations of higher order

The main purpose of this paper is to establish an approximate secular equation of fourth order of Rayleigh waves propagating in an isotropic half-space coated with a thin isotropic elastic layer The contact between the layer and the half-space is assumed to be welded Starting from the basic equations in matrix form and employing the Taylor expansion technique, the effective boundary conditions are derived With these bound-ary conditions, the fourth-order approximate secular equation is obtained by considering the propagation of Rayleigh waves in the half-space without the layer It is shown that this approximate secular equation has high accuracy, and it is much more better than Bovik’s second-order approximate dispersion relation From the secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established

Note that, Achenbach and Keshava [5] derived an approximate secular of fourth order for the isotropic case However, this approximate secular equation includes the shear coefficient, originating from Mindlin’s plate theory [20], whose usage should be avoided as noted by Touratier [21], Muller and Touratier [22] and Stephen [23]

2 Effective boundary condition of fourth order

Consider an elastic half-space x3≥ 0 coated by a thin elastic layer −h ≤ x3≤ 0 The layer and the half-space

are both homogeneous, compressible and isotropic The layer is assumed to be perfectly bonded to the half-space Note that the respective quantities related to the half-space and the layer have the same symbol but are systematically distinguished by a bar if pertaining to the layer We consider a plane motion in the(x1, x3)-plane

with displacement components(u1, u2, u3) such that

u i = u i (x1, x3, t), ¯u i = ¯u i (x1, x3, t), i = 1, 3, u2= ¯u2≡ 0, (1)

where t is the time Since the layer is made of isotropic elastic materials, the strain–stress relations take the

form

¯σ11= (¯λ + 2 ¯μ) ¯u1,1 + ¯λ ¯u3,3,

¯σ33= ¯λ ¯u1,1 + (¯λ + 2 ¯μ) ¯u3,3,

¯σ13= ¯μ( ¯u1,3 + ¯u3,1),

(2)

where ¯σ i j is the stress of the layer, commas indicate differentiation with respect to spatial variables x kand ¯λ

and ¯μ are Lamé constants In the absent of body forces, the equations of motion for the layer are:

¯σ11,1 + ¯σ13,3 = ¯ρ ¨¯u1,

¯σ13,1 + ¯σ33,3 = ¯ρ ¨¯u3, (3) where a dot signifies differentiation with respect to t From Eqs (2), (3), we have

 ¯U

¯T



=



M1 M2

M3 M4

  ¯U

¯T



where

¯U =¯u1 ¯u3

T

, ¯T =¯σ13 ¯σ33

T

, the symbol “T ” indicate the transpose of a matrix, the prime signifies differentiation with respect to x3, and

M1=

⎡

⎢

⎣

−¯λ

¯λ + 2 ¯μ ∂1 0

⎤

⎥

⎦ , M2=

⎡

⎢

⎣

1

¯μ 0

¯λ + 2 ¯μ

⎤

⎥

⎦ ,

M3=

⎡

⎢

⎣−

(¯λ + 2 ¯μ)2

− ¯λ2

¯λ + 2 ¯μ ∂12+ ¯ρ∂2

⎤

⎥

⎦ , M4= M1T.

(5)

Here, we use the notations∂1= ∂/∂x1, ∂2

1 = ∂2/∂x1 , ∂2

t = ∂2/∂t2 Equation (4) is called the matrix form

of basic equations (see also Vinh and Seriani [24,25]) From (4), it follows that

 ¯U (n)

¯T (n)



= M n ¯U

¯T



, M =

⎡

⎣M1 M2

M3 M4

⎤

⎦ , n = 1, 2, 3, , x3∈ [−h, 0]. (6)

Let h be small (i e the layer is thin), then expanding into a Taylor series ¯ T (−h) at x3= 0 up to the fourth

order of h, we have:

¯T (−h) = ¯T(0) − ¯T(0)h + 1

2! ¯T

(0)h2− 1

3! ¯T

(0)h3+ 1

4!h4¯T

Suppose that the surface x3= −h is free of traction, i.e., ¯T (−h) = 0 Introducing (6) with n = 1, 2, 3, 4 at

x3= 0 into (7) yields

I − hM4+1

2h

2M6− 1

6h

3M8+ 1

24h

4M10 ¯T (0)

=h M3−1

2h

2M5+1

6h

3M7− 1

24h

4M9 ¯U(0), (8)

where I is the identity matrix of order 2 Since the layer and the half-space are bonded perfectly to each other

at the plane x3= 0, if follows that ¯U(0) = U(0), ¯T (0) = T (0) From (8), we have

I − hM4+1

2h

2M6−1

6h

3M8+ 1

24h

4M10 T (0)

=h M3−1

2h

2M5+1

6h

3M7− 1

24h

M3, M4are defined by (5) and

M5= M3M1+ M4M3,

M6= M3M2+ M2

4,

M7= M3M12+ M4M3M1+ M3M2M3+ M2

4M3,

M8= M3M1M2+ M4M3M2+ M3M2M4+ M3

4,

M9= M3M13+ M4M3M12+ M3M2M3M1+ M2

4M3M1

+ M3M1M2M3+ M4M3M2M3+ M3M2M4M3+ M3

4M3,

M10= M3M12M2+ M4M3M1M2+ M3M2M3M2+ M2

4M3M2

+ M3M1M2M4+ M4M3M2M4+ M3M2M42+ M4

4.

(10)

The relation (9) between the traction vector and displacement vector of the half-space at the plane x3= 0 is

called effective boundary condition of fourth order in the matrix form It replaces (approximately) the entire effect of the thin layer on the substrate Introducing (5) and (10) into (9), we obtain the effective boundary conditions of fourth order in component form, namely

σ13+ h(1 − 2 ¯γ)σ33,1 + 4 ¯ρ ¯c2

2(1 − ¯γ)u1,11 − ¯ρ ¨u1



+h2

2



(2 ¯γ − 3)σ13,11+ 1

¯c2 2

¨σ13+ 4 ¯ρ ¯c2

2(1 − ¯γ)u3,111 − 2 ¯ρ(1 − ¯γ) ¨u3,1



+h3

6



(4 ¯γ − 3)σ33,111+



2

¯c2 1

(1 − ¯γ) + 1

¯c2 2

(1 − 2 ¯γ)



¨σ33,1 − 8 ¯ρ ¯c2

2(1 − ¯γ)u1,1111

+ ¯ρ(5 − 4 ¯γ2) ¨u1,11− ¯ρ

¯c2 2

¨u1,tt

 + h4

24



(5 − 4 ¯γ)σ13,1111+ 2



1

¯c2 1

(1 − ¯γ) + 1

¯c2 2

(2 ¯γ2− ¯γ − 2)



¨σ13,11

+1

¯c4

2

¨σ13,tt + 4 ¯ρ(2 − ¯γ − ¯γ2) ¨u3,111 − 8 ¯ρ ¯c2

2(1 − ¯γ)u3,11111 − 2 ¯ρ(1 − ¯γ)



1

¯c2 1

+ 1

¯c2 2



¨u3,1tt



= 0

σ33+ h(σ13,1 − ¯ρ ¨u3)

+h2

2



(1 − 2 ¯γ)σ33,11+ 1

¯c2 1

¨σ33+ 4 ¯ρ ¯c2

2(1 − ¯γ)u1,111 − 2 ¯ρ(1 − ¯γ) ¨u1,1



+h3

6



(2 ¯γ − 3)σ13,111+



2

¯c2 2

(1 − ¯γ) + 1

¯c2 1



¨σ13,1 + 4 ¯ρ ¯c2

2(1 − ¯γ)u3,1111 − ¯ρ(3 − 4 ¯γ) ¨u3,11− ¯ρ

¯c2 1

¨u3,tt



+h4

24



(4 ¯γ − 3)σ33,1111+



1

¯c2 1

(4 − 6 ¯γ) + 1

¯c2 2

(2 − 6 ¯γ + 4 ¯γ2)



¨σ33,11+ 1

¯c4 1

¨σ33,tt

+ 4 ¯ρ(2 − ¯γ − ¯γ2) ¨u1,111 − 8 ¯ρ ¯c2

2(1 − ¯γ)u1,11111 − 2 ¯ρ(1 − ¯γ)



1

¯c2 1

+ 1

¯c2 2



¨u1,1tt



= 0

where

¯c1=



¯λ + 2 ¯μ

¯ρ , ¯c2=



¯μ

¯ρ , ¯γ =

¯c2 2

¯c2 1

3 An approximate secular equation of fourth order

Now, we ignore the layer and consider the propagation of a Rayleigh wave in the half-space subjected the boundary conditions (11) and (12) The Rayleigh wave travels with velocity c ( >0) and wave number k (>0)

in the x1-direction and decays in the x3-direction, i.e., its displacements and stresses vanish at x3 = +∞

According to Achenbach [26], the displacements and stresses of Rayleigh wave satisfying (4) (without bars) and the decay condition are given by

u1=B1e −kb1x3+ B2e −kb2x3

e i k (x1 −ct) ,

u3=α1B1e −kb1x3+ α2B2e −kb2x3

σ13= −kc2

2ρ(b1+ β1)B1e −kb1x3+ (b2+ β2)B2e −kb2x3

e i k (x1 −ct) ,

σ33= kic2

2ρ1

γ − 2 −

1

γ b1β1



B1e −kb1x3+1

γ − 2 −

1

γ b2β2



B2e −kb2x3

e i k (x1 −ct) , (15) where B1and B2are constants to be determined, and

b1=1− γ x, b2=√1− x, α k = iβ k , β1= b1, β2= 1

b2, k = 1, 2,

γ = c22

c21, c1=



λ + 2μ

μ

ρ , x =

c2

c22, 0 < x < 1.

(16)

Introducing (14), (15) into the effective boundary conditions (11), (12) leads to the following equation for

B1, B2:



f (b1)B1+ f (b2)B2= 0,

F (b1)B1+ F(b2)B2= 0, (17)

where:

f (b n ) = −(b n + β n ) + ε4r μ( ¯γ − 1) + rμ r v2x + (2 ¯γ − 1)1

γ − 2 −

1

γ b n β n



+ε2

2



(2 ¯γ − 3) + xr2

v (b n + β n ) − β n



4r μ( ¯γ − 1) + 2(1 − ¯γ)rμ r v2x

+ε3

6

1

γ − 2 −

1

γ b n β n



(4 ¯γ − 3) + (1 − 2 ¯γ2)xr2

v

− (4 ¯γ2− 5)r μ r v2x

− 8r μ(1 − ¯γ) − rμ r v4x2

 +ε4

24



(4 ¯γ − 5) + 2(2 − ¯γ2)xr2

v − x2r v4

(b n + β n )

− β n



8r μ( ¯γ − 1) + 4(2 − ¯γ − ¯γ2)xr2

v r μ − 2(1 − ¯γ2)rμ r v4x2

,

F (b n ) =1

γ − 2 −

1

γ b n β n



+ ε− (b n + β n ) + rμ r v2x β n +ε2

2



(2 ¯γ − 1) − x ¯γr2

v 1

γ − 2 −

1

γ b n β n



− 4r μ(1 − ¯γ) + 2(1 − ¯γ)xrμ r v2



+ε3

6



(2 ¯γ − 3) + (2 − ¯γ)xr2

v

(b n + β n ) − β n



4( ¯γ − 1)rμ + (3 − 4 ¯γ)xr2

v r μ

+ x2r v4r μ ¯γ+ ε4

24



(4 ¯γ − 3) + 2(1 − ¯γ − ¯γ2)xr2

v + x2r v4¯γ2 1

γ − 2 −

1

γ b n β n



+ 4(2 − ¯γ − ¯γ2)xr2

v r μ − 8(1 − ¯γ)r μ − 2(1 − ¯γ2)x2r v4r μ



,

(18)

n = 1, 2, ε = kh, r μ= μ ¯μ , rv= c2

¯c2

Due to B12+ B2

2 = 0, the determinant of coefficients of the homogeneous system (17) must vanish This provides:

From (18) and (19), after algebraically lengthy calculations whose detail omitted, we arrive at the approximate secular equation of fourth order of Rayleigh waves, namely

whereε = kh is called the dimensionless thickness of the layer, and

A0= (2 − x)2− 4b1b2,

A1 = xr μ



[r2

v x − 4(1 − ¯γ)]b2+ r2

v xb1



,

A2= 1

2



− A0



xr v2(1 + ¯γ) − 4(1 − ¯γ)+ 2r2

μ r v2x

4(1 − ¯γ) − r2

v x

+ 2r μ(2b1b2− 2 + x)4(1 − ¯γ) − 2r2

v x ¯γ,

A3= −1

6xr μ



[( ¯γ + 3)r4

v x2+ 4(2 ¯γ − 3)r2

v x + 8(1 − ¯γ)]b1+ [(3 ¯γ + 1)r4

v x2

+ 4( ¯γ2− 2)r2

v x + 8(1 − ¯γ)]b2



,

0 0.5 1 1.5 0.5

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

ε

x = c/c2in the intervalε ∈ [0 1.5] that are calculated by the exact

secular equation (6) in Ref [ 7] (solid line), by Bovik’s second-order approximate secular equation (Eq 38, Ref [7]) (dash-dot line)

and by the approximate secular equation of fourth-order ( 24) (dashed line) Here we take r μ = 1.9266, r v = 1.6070, γ = 0.3556

and ¯γ = 0.2258

A4= 1

24A0

8(1 − ¯γ2) − 8xr2

ν − 12xr2

ν ¯γ + 12xr2

ν ¯γ2+ x2r ν4+ 6x2r ν4¯γ +x2r ν4¯γ2 +1

6(1 − b1b2)− 8r2

μ ( ¯γ − 1)2+ 8r2

μ r ν2x (2 − 3 ¯γ + ¯γ2)

+ 2r2

μ r ν4x2( ¯γ2+ 2 ¯γ − 4) + r2

μ r ν6x3(1 + ¯γ) + 16rμ ¯γ( ¯γ − 1)

+1

3b1b2r μ r ν2

r ν2x2( ¯γ2+ 4 ¯γ − 1) + 2x(1 − 8 ¯γ + 5 ¯γ2)

+1

6r μ x

r ν4( ¯γ2+ 4 ¯γ − 1)(x2− 2x) + 2r2

ν (1 − 8 ¯γ + 5 ¯γ2)(x − 2) − 8 ¯γ( ¯γ − 1) (22)

Since A0[x(ε)] = O(ε), from (22), it follows that A4ε4= A∗

4ε4+ O(ε5), where

A∗

4= 1

6(1 − b1b2)− 8r2

μ ( ¯γ − 1)2+ 8r2

μ r v2x (2 − 3 ¯γ + ¯γ2)

+2r2

μ r v4x2( ¯γ2+ 2 ¯γ − 4) + r2

μ r v6x3(1 + ¯γ) + 16rμ ¯γ( ¯γ − 1)

+1

3b1b2r μ r v2

r v2x2( ¯γ2+ 4 ¯γ − 1) + 2x(1 − 8 ¯γ + 5 ¯γ2)

+1

6r μ x

r v4( ¯γ2+ 4 ¯γ − 1)(x2− 2x) + 2r2

v (1 − 8 ¯γ + 5 ¯γ2)(x − 2) − 8 ¯γ( ¯γ − 1) (23) Therefore, Eq (20) can be written as

A0+ A1ε + A2ε2+ A3ε3+ A∗4ε4+ O(ε5) = 0. (24) Equations (20) and (24) are the desired approximate secular equations Up to the third order ofε, Eqs (20) and (24) coincide with Eq (38) in Ref [14] Taking into account the fact that A0[x(ε)] = O(ε), one can see

that up to the second order, Eqs (20), (24) and Bovik’s second-order approximation (Eq 38 in Ref [7]) are identical to each other and they coincide with Eq (43) in Ref [14]

Figure1presents the dependence onε ∈ [0 1.5] of the dimensionless Rayleigh wave velocity√x = c/c2

that is calculated by the exact secular equation (6) in Ref [7] by Bovik’s second-order approximate secular equation (Eq 38, Ref [7]) and by the approximate secular equation of fourth order (24) Here, we take

r μ = 1.9266, r v = 1.6070, γ = 0.3556 and ¯γ = 0.2258.

It is seen from Fig.1that the exact velocity curve and the approximate velocity curve of fourth order almost totally coincide with each other That means the fourth-order approximate secular equation (24) has high accuracy Figure 1also shows that it is much better than Bovik’s second-order approximate secular equation

4 An approximate formula of third order for the velocity

In this section, we establish an approximate formula of third order for the squared dimensionless Rayleigh

wave velocity x (ε) that is of the form

x (ε) = x(0) + x(0) ε + x(0)

2 ε2+ x(0)

6 ε3+ O(ε4), (25)

where x (0) is the squared dimensionless velocity of Rayleigh waves propagating in an isotropic elastic

half-space that is given by [27]

x (0) = 4(1 − γ )



2−4

3γ +3

R+√D+3

R−√D

−1

in which

R = 2(27 − 90γ + 99γ2− 32γ3)/27,

D = 4(1 − γ )2(11 − 62γ + 107γ2− 64γ3)/27, (27)

the roots in the formula (26) taking their principal values

From (20), it follows that

x(0) = − A1

A 0x

x =x(0) , x(0) = − 2 A2A20x − 2A 0x A1A 1x + A 0x x A21

A30x

x =x(0) ,

x(0) =−6 A3+ 6A 2x x(0) + 3A 1x x x2(0) + 3A 1x x(0)

+ 3A 0x x x(0)x(0) + A 0x x x x3(0)/A 0x 

x =x(0) ,

(28)

where A1, A2and A3are given by (21) and

A0x = 2x− 2 + √1+ γ − 2γ x

1− x√1− γ x



, A 0x x = 2 + (γ − 1)2

(1 − x)3

(1 − γ x)3,

A 0x x x = 3(γ − 1)2 1+ γ − 2γ x

2

(1 − x)5

(1 − γ x)5, A1x = r μ

r2

v x (4 − 5x) + 12x(1 − ¯γ) − 8(1 − ¯γ)

2√

r v2x (4 − 5γ x)

2√

1− γ x ,

A 1x x = r μ

r2

v (8 − 24x + 15x2) + 4(1 − ¯γ)(4 − 3x)

4

(1 − x)3 +r v2(8 − 24γ x + 15 ¯γ2x2)

4

A 2x = −1

2r

2

v (1 + ¯γ)A0−1

2



(1 + ¯γ)r2

v x − 4(1 − ¯γ)A 0x

+ 2r μ r v2

1−√1− x1− γ xr μ(2 − 2 ¯γ − r2

v x ) + 2 ¯γ− ¯γ x

+1

2r

2

μ r v2x

4(1 − ¯γ) − r2

v x  1 + γ − 2γ x

√

1− x√1− γ x + 2r μ



1−√1+ γ − 2γ x

1− x√1− γ x



2(1 − ¯γ) − r2

v x ¯γ.

(29)

Note that, x (0) can be calculated by another exact formula derived by Malischewsky [28] or by the approximate expressions with high accuracy obtained recently by Vinh and Malischewsky [29–31]

Figure2presents the dependence onε ∈ [0 0.8] of the dimensionless Rayleigh wave velocity√x = c/c2

that is calculated by the exact secular equation (6) in Ref [7] and by the approximate formula (25) Here, we

take r μ = 0.1, r v = 3.85, γ = 0.39 and ¯γ = 0.13 It is seen that the approximate velocity curve is very close

to the exact one

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.5

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

x (ε) in the interval [0 0.8] that are calculated by the exact secular

equation (6) in Ref [ 7] (solid line) and by the formula (25) (dashed line) Here we take r μ = 0.1, r v = 3.85, γ = 0.39 and

¯γ = 0.13

5 Conclusions

In this paper, the propagation of Rayleigh waves in an isotropic elastic half-space covered with a thin isotropic elastic layer is investigated The contact between the layer and the half-space is welded An approximate secular equation of fourth order in terms of the dimensionless thickness of the layer is derived using the effective boundary condition method It is shown that the approximate secular equation obtained has high accuracy From the approximate secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established Since the secular equation and the formula for the Rayleigh wave velocity obtained are totally explicit, they are good tools for evaluating the mechanical properties of thin films deposited to half-spaces

Acknowledgments The work was supported by the Vietnam National Foundation for Science and Technology Development

(NAFOSTED).

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