DSpace at VNU: Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer

DSpace at VNU: Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic...

Exact secular equations of Rayleigh waves in an orthotropic elastic

half-space overlaid by an orthotropic elastic layer

Pham Chi Vinh, Vu Thi Ngoc Anh, Nguyen Thi Khanh Linh

DOI: 10.1016/j.ijsolstr.2015.12.032

To appear in: International Journal of Solids and Structures

Received date: 20 April 2015

Revised date: 28 December 2015

Accepted date: 30 December 2015

Please cite this article as: Pham Chi Vinh, Vu Thi Ngoc Anh, Nguyen Thi Khanh Linh, Exact secularequations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer,

International Journal of Solids and Structures (2016), doi:10.1016/j.ijsolstr.2015.12.032

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• For the compressible/compressible case, the exact secular equation is derived

by using the effective boundary condition method

• For three remaining cases, the exact secular equations are obtained by theincompressible limit technique

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Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an

orthotropic elastic layer

Pham Chi Vinha ∗, Vu Thi Ngoc Anhaand Nguyen Thi Khanh Linhb

aFaculty of Mathematics, Mechanics and Informatics

Hanoi University of Science

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

bDepartment of Engineering Mechanics Water Resources University of Vietnam

175 Tay Son Str., Hanoi, Vietnam

Abstract

In this paper, the propagation of Rayleigh waves in an orthotropic elastichalf-space overlaid by an orthotropic elastic layer of arbitrary uniform thick-ness is investigated The layer and the half-space may be compressible orincompressible and they are in welded contact with each other The main aim

of the paper is to derive explicit exact secular equations of the wave for fourpossible combinations of a (compressible/incompressible) half-space coated

by a (compressible/incompressible) layer For the compressible/compressiblecase, the explicit secular equation is derived by using the effective boundarycondition method For three remaining cases, the explicit secular equationsare derived from the secular equation for the compressible/compressible case

by using the incompressible limit technique along with the expressions of thereduced elastic compliances in terms of the elastic stiffnesses Based on theobtained secular equations, the effect of incompressibility on the Raleigh wavepropagation is considered numerically It is shown that the incompressibilitystrongly affects the Raleigh wave velocity and it makes the Raleigh wave ve-locity increasing

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

it is connected with reduced cost, less inspection time, and greater coverage (Hess

et al., 2013) Among surface/guided waves, the Rayleigh wave is a versatile andconvenient tool (Kuchler & Richter, 1998; Hess et al., 2013) Since the explicit dis-persion relations of Rayleigh waves are employed as theoretical bases for extractingthe mechanical properties of the layers from experimental data, they are thereforethe main purpose of any investigation of Rayleigh waves propagating in elastic half-spaces covered by an elastic layer

When the half-space and the layer are both isotropic (the isotropic case), theexplicit secular equation of Rayleigh waves was derived by Haskell (1953), Ben-Menahem (2000) (Eq (3.113), p.117) For the orthotropic case, the explicit secular

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equation of Rayleigh waves was derived by Sotiropoulos (1999) by expanding directly

a six-order determinant that comes from the traction-free conditions at the uppersurface of the layer and the continuity conditions of the displacements and stresses

at the interface of the layer and the half-space In this investigation, the layerand the half-space are assumed to be compressible Since there are four possiblecombinations: the layer and the half-space are both compressible (the compress-ible/compressible case) or incompressible (the incompressible/incompressible case),one is compressible and the other is incompressible (the compressible/incompressibleand incompressible/compressible case), the derivation of explicit secular equations

of Rayleigh waves for the remaining combinations is needed

The main purpose of this paper is to derive explicit secular equations of Rayleighwaves propagating in an orthotropic elastic half-space overlaid by an orthotropicelastic layer of arbitrary thickness for all possible combinations For the compress-ible/compressible case, the explicit secular equation is derived by employing theeffective boundary condition method that proved successful in deriving approximateexplicit secular equations of Raleigh waves in elastic half-spaces coated by a thinfilm, see Tiersten (1969), Bovik (1996), Steigmann & Ogden (2007), Vinh & Linh(2012, 2013), Vinh et al (2014a, 2014b), Vinh & Anh (2014a, 2014b) For threeremaining cases, the explicit secular equations are deduced directly from the secularequation for the compressible/compressible case by using the incompressible limitapproach, proposed by Destrade el al (2002), along with the expressions of the

the compressible/compressible case

First, the entire effect of the layer on the half-space is replaced by the exact effectiveboundary conditions at the interface The wave is then considered as a Rayleighwave propagating in the half-space, without the coating layer, that is subjected tothe effective boundary conditions

Consider an orthotropic elastic half-space x2 ≥ 0 overlaid by an orthotropic elasticlayer with uniform thickness h occupying the domain −h ≤ x2 ≤ 0 The layer isassumed to be perfectly bonded to the half-space and they are both compressible.Note that same quantities related to the half-space and the layer have the samesymbol but are systematically distinguished by a bar if pertaining to the layer Weare interested in the plane strain such that:

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

where ¯σij are stress components, commas signify differentiation with respect to xk,

a dot indicates differentiation with respect to t For an orthotropic material thestrain-stress relation is of the form (Ting, 1996):

which are necessary and sufficient conditions for the strain energy of the material

to be positive definite Substituting (3) into (2) and taking into account (1) yield:

¯

u1 = ¯U1(y)eik(x1 −ct), ¯u2 = ¯U2(y)eik(x1 −ct), y = kx2 (6)

[f ; g] := f2g1− f1g2, [f ] := f2− f1 (17)

Since the layer and half-space are welded at the interface x2 = 0, it follows ¯uk =

uk (k = 1, 2) and ¯σk2 = σk2 (k = 1, 2) at x2 = 0, or equivalently by (6) and (9):

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where Uk(0) and Σk(0) are the amplitudes of displacement and stress of the space at the interface x2 = 0 This is the desired exact effective boundary conditionthat replaces exactly the entire influence of the layer on the half-space

Now we consider the propagation of a Rayleigh wave, traveling along surface x2 = 0

of the half-space with velocity c and wave number k in the x1-direction, decaying

in the x2-direction, and satisfying the exact effective boundary condition (18) cording to Vinh & Ogden (2004), the displacements of the Rayleigh wave in thehalf-space x2 > 0 are given by:

Ac-u1 = U1(y)eik(x1 −ct), u2 = U2(y)eik(x1 −ct), y = kx2 (19)

where:

U1(y) = B1e−b1 y+ B2e−b2 y, U2(y) = i(α1B1e−b1 y+ α2B2e−b2 y) (20)

B1 and B2 are constants to be determined, and:

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It is not difficult to show that if a Rayleigh wave exists (→ the real parts of b1 and

b2 must be positive), then (see Vinh & Ogden, 2004):

A0+ B0chε1chε2+ C0shε1shε2+ D0chε1shε2+ E0chε2shε1 = 0 (32)

Equation (32) is the desired exact secular equation It is totally explicit.

When ε = 0, Eq (32) becomes A0+ B0 = 0, or equivalently:

(c66− X)c212− c22(c11− X)+ X√

c22c66

p(c11− X)(c66− X) = 0 (34)

This equation is the secular equation of Rayleigh waves propagating along thetraction-free surface of a compressible orthotropic half-space (see Vinh & Ogden2004)

It is useful to convert the secular equation (32) into dimensionless form To do

As a checking example, we use Eq (36) to compute the dimensionless wavevelocity x (of mode ”0”) for a given set of parameters, namely:

e1 = 3.5, e2 = 2.8, e3 = 1, ¯e1 = 2.5, ¯e2 = 1.2, ¯e3 = 0.5, rv = 2.8, rµ= 0.5

and then comparing the exact velocity curve (solid line) with the correspondingthird-order approximate velocity curve (dashed line) draw by solving the approxi-mate secular equation (29) in Vinh & Linh (2012) in the interval ε ∈ [0; 1.5] It is

(42)

−2(2 − ¯x)(2b1b2+ x− 2)r−1µ +

4b1b2− (2 − x)2

rµ−2o,

vx Therefore, for the isotropic case, the exactsecular equation is of the form:

of the secular equation Eq (36)

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elastic stiffnesses in the incompressible limit

Consider an arbitrary incompressible orthotropic elastic solid We are interested inthe plane strain:

u1 = u1(x1, x2, t), u2 = u2(x1, x2, t), u3 ≡ 0 (45)

The incompressibility is of the form:

u1,1+ u2,2 = 0 (46)

For incompressible orthotropic elastic materials, the stress-strain relations in terms

of elastic stiffnesses are (Ogden & Vinh, 2004):

, c66 = 1

s0 66

(50)

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where δ = c11+ c22− 2c12

Proof : According to Destrade et al (2002): 0 < s011 < +∞, 0 < s0

66 < +∞.From (47)3 and (48)3, we have:

c66s066σ12= σ12 ⇔ (c66s066− 1)σ12 = 0 (51)

Since σ12 is arbitrary it implies:

c66s066− 1 = 0, → c66 = 1

s0 66

, s011= 1

The relations (50) are proved

incompress-ible cases

In the rest of the paper, the symbol ”lim” indicates the limit that is taken as thecompressible orthotropic material goes to the incompressible limit To obtain the

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explicit secular equations of Rayleigh waves for three remaining incompressible cases

we will take the ”lim” of two sides of the equalities Eq (32)

Proposition 2: The following equalities hold:

lim ∆ = 0, lim ∆c11= lim ∆c22 = lim ∆c12= s011, lim ∆c66 = 0 (56)

Taking the limit two sides of the equalities (57) and using the relations (49) yield Eqs

lim s0 66

=0 because lim ∆ = 0and lim s0

66 > 0 (see Destrade et al., 2002)

Remark 2: From (57) it follows that for the compressible orthotropic elasticmaterials we have:

D = 1

∆ where D = c11c22− c2

12 > 0 (58)Note that the equalities (50), (56) and (58) are valid for both the layer and thehalf-space

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Proposition 3: Let the orthotropic elastic material of the layer approaches theincompressible limit Then we have:

¯bk

, lim ¯γk = c66

¯0 k

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Taking the limit of the equality (63) and using (56) give the third of (59) From(11)1 and (59)3 we derive the fourth of Eqs (59) Using (8)2 and (11)2 and takinginto account (58) provide:

Taking the limit of the equality (64) and using (56) give the last of (59)

Proposition 4; Let the orthotropic elastic material of the half-space approachesthe incompressible limit Then the following equalities hold:

lim P = 1− x, lim S = eδ− x − 2 (65)

where eδ = δ/c66, δ = c11+ c22− 2c12

Proof: The proof of (65) are similar to the one of (59)1,2

In order to derive the explicit secular equation for the incompressible/incompressiblecase, first we multiply two sides of Eq (32) by ∆ (> 0), then take the limit twoside of the obtained equation as the materials of both the layer and the half-spaceapproach to the incompressible limit and take into account (59) and (65) Aftermultiplying two sides of the resulting equation by the factor ¯b2

ex-When ε = 0 (the layer is absent), from (66) and (67) we have:

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Figure 2: Rayleigh wave velocity curves draw by solving the exact secular equation(66) (solid line) and the third-order approximate secular equation (3.14) in Vinh et

al (2014a) (dashed line) Here we take eδ = 3, ¯eδ = 2.8, rµ= 1, rv = 2.8

and then comparing the exact velocity curve with the corresponding third-orderapproximate velocity curve (dashed line) draw by solving the approximate secularequation (3.14) in Vinh et al (2014a) in the interval ε∈ [0; 1.5] It is seen from Fig

2 that in this interval the exact and approximate curves are almost coincide witheach other This asserts that the secular equation Eq (66) as well as the relations(59) and (65) are correct

Analogously, to derive the explicit secular equation for the compressible/incompressiblecase, we multiply two sides of Eq (32) by ∆ (> 0), then take the limit two side ofthe resulting equation as the materials of the half-space approach to the incompress-ible limit and take into account (65) After multiplying two sides of the resulting

where S and P are given by (68) Equation (70) with the coefficients A1, B1, C1, D1, E1

calculated by (71) is the explicit exact secular equation of Rayleigh waves in sionless form for the compressible/incompressible case

dimen-When ε = 0 (the layer is absent), from (70) and (71) we have:

(eδ− x)√P − x = 0 (72)

This is the dimensionless secular equation of Rayleigh waves propagating in anincompressible orthotropic half-space (see Ogden & Vinh, 2004)

For obtaining the explicit secular equation for the incompressible/compressible case,

we take the limit two side of Eq (32) as the materials of the layer approach to the

where S and P are given by (38) Equation (73) with the coefficients A1, B1, C1, D1, E1

calculated by (74) is the explicit exact secular equation of Rayleigh waves in sionless form for the incompressible/compressible case

dimen-When ε = 0 (the layer is absent), from (73) and (74) it follows:



e23− e2(e1− x)√P + x(e1− x) = 0 (75)

that is the dimensionless secular equation of Rayleigh waves in a compressible thotropic half-space (see Vinh & Ogden, 2004)

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Remark 3: The explicit secular equations of Rayleigh waves for the ible cases can be derived by employing the effective boundary condition method andthey are the same as Eqs (66), (70) and (73) This proves the validity of theobtained secular equations

In this Section, as an example of application of the obtained approximate secularequations, we consider numerically the effect of the incompressibility on the Rayleighwave velocity For this aim we consider four examples In the first example, a com-pressible half-space is coated either by a compressible layer or by an incompressiblelayer Two these layers have the same elastic constants In the second example,the compressible half-space is replaced by an incompressible In the third (fourth)example, two different (compressible and incompressible) half-spaces with the sameelastic constants are covered with the same compressible (incompressible) layer

In particular, in the first example, we take e1 = 2.8, e2 = 2.5, e3 = 0.6 for thehalf-space and ¯e1 = 3.5, ¯e2 = 0.5, ¯e3 = 1 (¯eδ = 3.5) for the layers and rµ = 1,

rv = 2.2

In the second example, we choose eδ= 3.2 for the half-space and ¯e1 = 2.2, ¯e2 = 1,

¯

e3 = 0.6 for the layers and rµ= 1, rv = 2.8

In the third example, the dimensionless parameters are taken as ¯e1 = 2.5, ¯e2 =0.6, ¯e3 = 1 for the layer and e1 = 3.0, e2 = 1.8, e3 = 0.5 for the half-spaces and