TÓM tắt TIẾNG ANH LUẬN án SÓNG mặt và SÓNG TRONG các cấu TRÚC MỎNG

Elastic surface waves, discovered by Rayleigh more than 120 yearsago for compressible isotropic elastic solids, have been studied exten-sively and exploited in a wide range of applicatio

INSTITUTE OF MECHANICS

———– * ———–

NGUYEN THỊ KHANH LINH

SURFACE WAVES AND WAVES

Referee 1:

Referee 2:

Referee 3:

The thesis is protected to the Council assessing

doctoral dissertation level Institute, meeting at theInstitute of Mechanics, 264 Doi Can - Ba Dinh - Ha Noi

Chapter 1 Survey

Actuality of the thesis

Problems of elastic wave propagation, especially the ones of Rayleighwave propagration, are the foundation of various practical applications

in science and technology

Elastic surface waves, discovered by Rayleigh more than 120 yearsago for compressible isotropic elastic solids, have been studied exten-sively and exploited in a wide range of applications in seismology, acous-tics, geophysics, telecommunications industry and materials science, forexample It would not be far-fetched to say that Rayleigh’s study ofsurface waves upon an elastic half-space has had fundamental and far-reaching effects upon modern life and many things that we take forgranted today, stretching from mobile phones through to the study ofearthquakes, as addressed by Adams et al A huge number of investiga-tions have been devoted to this topic As written in, one of the biggestscientific search engines Google.Scholar returns more than a millionlinks for request "Rayleigh waves" and almost 3 millions for "Surfacewaves" This data is really amazing! It shows a tremendous scale ofscientific and industrial interests in this area

The structure of a half-space coated by a thin layer is widely plied in modern technology The measurement of mechanical properties

ap-of thin films deposited on half-spaces before and during loading is ap-ofimportance and of great necessaries Note that there exists an interna-tional journal named “Thin Solid Films” which is for publishing newresearch results in the fileld of thin solid films For non-destructivelyevaluating the mechanical properties of materials, the Rayleigh wave is

a convenient tool When using the Rayleigh wave for non-destructivelyevaluating, its explicit dispersion relations are employed as theoreticalbases for extracting the mechanical properties of the thin films fromexperimental data

Nowadays, composite materials, especially the ones with fibers, arewidely used in various fields of science and industry, such as the man-ufacture of automobiles, the manufacture of aircrafts, the manufacture

of ships In order to manufacture a ship shell, for example, thinfiber/epoxy layers with different fiber-directions are attached periodi-cally to each other, up to a given thickness The ship shell can be there-

fore considered as infinite, periodically layered, elastic media Thus, theproblems on waves propagating in infinite, periodically layered, elasticmedia are needed to investigate and they attract a great attention ofresearchers.

The main purposes of the thesis

• Applying new methods to develop some problems on Rayleighwaves that were investigated previously

• Deriving approximate secular equations of Rayleigh waves agating in elastic half-spaces coated an elastic thin layer

prop-• Investigating SH wave and Lamb wave propagting in periodicaland thin structures

Research objects

Waves propagating in elastic half-spaces, waves propagating in tic half-spaces coated with a thin elastic layer, waves propagating inperiodical, thin structures

Chapter 2 Rayleigh waves

2.1 Rayleigh waves in incompressible elastic media subjected to gravity

2.1.1 Secular equation

Consider the problem of Rayleigh wave propagation in an ible isotropic elastic half-space x3≥ 0 (Figure 2.1) subjected to gravity

incompress-Figure 2.1 The problem model

The secular equation is:

(2 − x)2− 4√1 − x − δx = 0, (1)where δ = ρg/(kµ) ≥ 0, x = c2/c2, c2=pµ/ρ, k is number wave, c isvelocity, µ is Lame constant, ρ is mass density of the medium, g is theacceleration due to gravity

When δ = 0, equation (1) becomes:

Equation (2) is the secular equation of Rayleigh waves in an pressible isotropic elastic half-space without the effect of gravity.2.1.2 Formulas for the velocity of Rayleigh waves

incom-Exact formulas for the velocity of Rayleigh waves

Using the theory of cubic equation, we obtain the exact expressionsfor the velocity:

xr=2(4 + δ)

q

p16(δ + 11)(δ2+ 4)/27 + (δ3+ 12δ2+ 12δ + 136)/27

2

9q3 p16(δ + 11)(δ2+ 4)/27 + (δ3+ 12δ2+ 12δ + 136)/27

, δ ∈ [0 , 1).(3)

−13

Approximate formulas for the velocity of Rayleigh waves

By applying the method of least squares, we drive approximateformulas for the velocity of Rayleigh waves:

x2= 1 − −(2.9475724 + δ) +√δ2+ 0.1215448δ + 14.4543266

2.8868

!2

.(6)2.1.3 The existence Rayleigh waves

Theorem 2.1 Let δ ≥ 0, then:

(i) A Rayleigh wave exsists if and if 0 ≤ δ < 1

(ii) If a Rayleigh wave exists, then it is unique, and its squared mensionless velocity xr(δ) is given by Eqs (3) or (4)

di-(iii) The squared dimensionless Rayleigh wave velocity xr is a strictlymonotonously increasing function in the interval [0 , 1), from x0

to 1 (but not equal to 1), where:

x0= 1 −



26

27+

23

r113

!1/3

−89

26

27+

23

r113

!−1/3

−13





2

.(7)2.1.4 Conclusion

In this paper, the exact and highly accurate approximate formulas forthe velocity of Rayleigh waves in an incompressible isotropic elastichalf-space under gravity are derived These formulas are useful toolsfor evaluating the effect of gravity on propagation of Rayleigh wavesand for solving the inverse problem as well They are new results andhave been published in Acta Mechanica, Vol 223, 1537-1544, 2012

2.2 Rayleigh waves semi-infinite orthotropic thin plates

2.2.1 Principal Rayleigh waves

2.2.1.1 Secular equation

Consider a thin semi-infinite orthotropic medium (panel) occupyingthe half-space x2≥ 0, its principal material axes are x1, x2and x3axis(Figure 2.5) and it is in the state of plane stress

Figure 2.5 The model for pricipal Rayleigh waves

From basic equations and boundary condition, we obtain the secularequation, namely:

(B66− ρc2)[B122 − B22(B11− ρc2)]

+ρc2pB22B66

p(B11− ρc2)(B66− ρc2) = 0 (8)where: Bij are material (stiffness) coefficients which can be expressed interms of the engineering constants (Young’s and shear moduli, Poisson’sratios) as:

sec-Exact formula

Following the same procedure carried out in [Pham Chi Vinh andOgden, R W., Ach Mech., 56 (3) (2004), 247-265], formula for thevelocity of Rayleigh waves is derived:

ρc2/B66=√

b1b2b3/

(√

b1/3)(b2b3+2)+3

qR+√D+3q

R −√D

(10)

where b1= B22/B11, b2 = 1 − B122 /(B11B22), b3 = B11/B66, R and Dare given by:

Using the method of least squares, we obtain approximate formulas:

2.2.2 Non-principal Rayleigh waves

2.2.2.1 Secular equation

Consider a thin homogeneous orthotropic elastic panel occupyingthe half-space x2 ≥ 0 whose principal material axes are X, Y, Z (hình2.9) Suppose that the Z axis coincides with the x3 axis and (x1, x2)

Figure 2.9 The model of non-principal Rayleigh waves

is the rotated one from (X, Y ) by counter clockwise angle θ Supposethat the panel is subjected to the plane stress state

Using the method of first integrals, we derive the secular equation:

F (X, θ) ≡dX2[(d + d2)X − d3][d22− Q66(dX − d3)]

+ (dX −d3)[(d+d2)X −d3][Q22dX2−(d2+d21+Q22d3)X +dd3]

− 2d1X2(dX − d3)[Q26(dX − d3) − d1d2] = 0 (15)where X = ρc2 and

2.2.3 Conclusion

In this chapter we obtain the secular equation for principal Rayleighwaves that is valid for all orthotropic elastic materials and much moresimple than the ones obtained recently by Cerv Exact and approx-imate formulas for the velocity of principal Rayleigh-edge waves arealso established and they are a powerful tool for analyzing the effect ofmaterial parameters on the Rayleigh wave velocity For non-principalRayleigh waves a secular equation in explicit form is obtained by us-ing the method of first integrals They are new results and have beenpublished "Vietnam Journal of Mechanics, 34 (2) (2012), 123 – 134"

Chapter 3 Rayleigh waves in elastic spaces underlying a water layer

half-3.1 The exact secular equation

Consider an incompressible isotropic elastic half-space x3 ≥ 0 that isoverlaid with a layer of incompressible non-viscous water occupying thedomain 0 ≤ x3 ≤ h (see Figure 3.1) Both the elastic half-space andthe water layer are assumed to be under the gravity

Figure 3.1 The problem modelFrom the basic equations, the boundary conditions at x3 = h and thecontinuity conditions at x3= 0, we obtain the secular equation:(2 − x)2− 4√1 − x − δ x + r δ x − r f (x, δ, ε)x2= 0, 0 < x < 1 (17)where: x = c2/c2, c2=pµ/ρ , δ = g/kc2, ε = kh, r = ρρ0, f (x, δ, ε) =(δ − xthε)/(x − δthε), c is velocity, µ and ρ are Lame contants and themass density of the medium, g is the acceleration due to gravity, k isthe wave number, ρ0 is the mass density of the water

Remark: To the best knowledge of the authors the exact secular tion (17) did not appear in the literature It is shown the approximatesecular equation obtained by Bromwich is a special case of (17).3.1.2 On existence of Rayleigh waves

equa-Theorem 3.1

i) A Rayleigh wave is impossible if either {δ ≥ 1, 0 < δthε ≤ 1,

r ≥ 1 + 2/δ} or {δ ≥ 1, δthε > 1, r ∈ (0, m] ∪ [1 + 2/δ, +∞)}.ii) There exists a unique Rayleigh wave, namely CRW, if {0 < δ < 1,

3.1.3 Approximate secular equation

Suppose that h is small (the layer is thin) From the equation (17),

we obtain the fourth-order approximate secular equation, namely:



ε4= 0, δthε < x < 1

(18)

3.2 Approximate formulas for the velocity

Both δ and ε being small

By using the perturbation method, we obtain the second-order proximation of the squared dimensionless velocity of Rayleigh waves:

ap-r x

x0 = 1+0.1089δ−0.0994rε−0.0782δ2+0.1211r δε−0.0453r2ε2 (19)Only ε being small

The second-order approximation of x(ε) is:

Suppose that the water layer is thin: ε << 1, by using the method

of least squares, we derive the global approximation:

Chapter 4 Rayleigh waves in a half-space coated by a thin layer

4.1 Rayleigh waves in a compressible orthotropic elastic half-space coated by a thin orthotropic elastic layer

4.1.1 Effective boundary condition of third-order

Consider an elastic half-space x2≥ 0 coated by a thin elastic layer −h ≤

x2≤ 0 Both the layer and the half-space are homogeneous, orthotropicand elastic The thin layer is assumed to be perfectly bonded to thehalf-space

Figure 4.1 The problem moldel

By using the effective boundary condition method, the entire effect ofthe layer on the half-space is replaced by effective boundary conditions,namely:

σ12+ h(r1σ22,1− r3u1,11− ¯ρ¨u1)+h

(23)

where σij, ui are respectively the stresses and the displacements of thesubstrate, ¯cij, ¯ρ are respectively the material constants and the massdensity of the layer, a dot signifies differentiation with respect to t,commas indicate differention with respect to spatial variables xk and

and cij, ρ are respectively the material constants and the mass density

of the half-space, c (> 0) is wave velocity, k (> 0) is the wave number ,

ε = kh It is clear that the squared dimensionless Rayleigh wave velocity

x = c2/c2 depend on 9 dimensionless parameters: rµ, rv, ek, ¯ek, k =

where x0= x(0) is the squared dimensionless velocity of Rayleigh wavespropagating in an orthotropic elastic half-space that is given by:

x0=√

s1s2s3/

(√

s1/3)(s2s3+ 2) + 3

q

R +√

D + 3q

R −√D

(26)

s1= e2/e1, s2= 1 − e2/(e1e2), s3= e1

x0(0) = −D1

D0x

x=x 0

, x00(0) = −D0xxD

2− 2D1xD1D0x+ D2D2

0x

D3 0x

x=x 0

and R, D are given by (11) in which b1, b2, b3 are substitutied by

s1, s2, s3

4.2 Rayleigh waves in an incompressible orthotropic elastic half-space coated by a thin orthotropic elastic layer

4.2.1 Effective boundary condition of third-order

Consider an elastic half-space x2 ≥ 0 coated by a thin elastic layer

−h ≤ x2 ≤ 0 The layer and the half-space are both homogeneous,incompressible, orthotropic The thin layer is assumed to be perfectlybonded to the half-space (see Figure 4.1)

By using the effective boundary condition method, the entire fect of the layer on the half-space is replaced by effective boundaryconditions, namely:

ef-σ12+ h(σ22,1+ ¯δu1,11− ¯ρ¨u1) +h

2

2 (r1σ12,11+

¯ρ

¯

c66

¨

σ12+ ¯δu2,111− 2¯ρ¨u2,1)+h

3

6 (r1σ22,111+

¯ρ

4.2.2 An approximate secular equation of third-orderSubtituting expressions of the stresses and the displacements of thehalf-space into the effective boundary conditions (27), (28) leads to thedispersion equation of the wave, namely:

It is clear that the squared dimensionless Rayleigh wave velocity

x = c2/c2depends on 5 dimensionless parameters: eδ, ¯eδ, rµ, rvand ε.4.3 Rayleigh waves in pre-stressed elastic half-space coated by a thin elastic layer

4.3.1 Effective boundary condition of third-order

We consider a homogeneous surface layer of uniform thickness h lying a homogeneous half-space, both being pre-stressed compressible

over-isotropic elastic materials with the underlying deformations ing to pure homogeneous strains (see Figure 4.5) The principal direc-tions of strain in the two solids are aligned, ones direction being normal

correspond-to the planar interface defined by x2= 0 A rectangular Cartesian dinate system (x1, x2, x3) is employed with its axes coinciding with theprincipal directions of the pure strain The layer occupies the domain

coor-−h < x2< 0 and the half-space corresponds to the region x2> 0

Figure 4.5 The problem model

By using the effective boundary condition method, the entire effect ofthe layer on the half-space is replaced by effective boundary conditions,namely:

4.3.2 An approximate secular equation of third-order

By introducing expressions of the incremental stresses and the mental displacements of the half-space into the effective boundary con-ditions (30), (31), we arrive at the approximate secular equation ofthird-order of Rayleigh waves, namely

It is clear that the squared dimensionless Rayleigh wave velocity x =

c2/c2 depend on 13 dimensionless parameters: ek, ¯ek (k = 1, 2, 3, 4, 5),

elas-Obtained results are:

• The effective boundary conditions

• By using these conditions, the approximate secular equations ofthird order are derived in explicit for the wave velocity

This new results have been published in "Wave Motion, Vol 49,681-689, 2012", "International Journal of Non-Linear Mechanics, 50(2013) 91–96", "Procceding of the 9thNational Congress on Mechanics"(Hanoi, December 8-9, 2012)

Chapter 5 Waves in a periodical and thin structures

The main aim of this chapter is to establish recurrent formulas lating Ω2n+1(n ≥ 2) in the asymptic expression of the wave velocity for

calcu-SH waves in infinite, periocally, compressible isotropic media and Lambwaves in infinite, periodically layered, incompressible, pre-strained me-dia

5.1 SH waves in infinite, periodically layered elastic media containing thin layers

5.1.1 Setting problem

Consider a SH propagating in infinite periodically layered of whicheach periodicity cell consists of N different isotropic elastic layers (N ≥2) with the thickness h1, h2, , hN The thickness of the periodicity cell

is h = h1+ +hN Assume that the angle between the wave propagationdirection and the x1-axis is θ (Figure 5.1)

o

Figure 5.1 The problem model5.1.2 Formulas for calculating Ωk

By using the asymptotic method, we derive:

• Formula for determining Ω1

Ω1= h1/µi−1

hρi cos

2θ + 1hρisin



f221 −sin

2θ2

,(38)