VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MECHANICS ———– * ———– NGUYEN THỊ KHANH LINH SURFACE WAVES AND WAVES IN THIN STRUCTURES Major: Engineering mechanics Code: 62 52 01 01 SUMMARY OF DOCTORAL THESIS HANOI – 2013 Supervisor: Assoc. Prof. Dr. Pham Chi Vinh Referee 1: Referee 2: Referee 3: The thesis is protected to the Council assessing doctoral dissertation level Institute, meeting at the Institute of Mechanics, 264 Doi Can - Ba Dinh - Ha Noi. At hours and minutes, , 20 1 Chapter 1. Survey Actuality of the thesis Problems of elastic wave propagation, especially the ones of Rayleigh wave propagration, are the foundation of various practical applications in science and technology. Elastic surface waves, discovered by Rayleigh more than 120 years ago 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, for example. It would not be far-fetched to say that Rayleigh’s study of surface waves upon an elastic half-space has had fundamental and far- reaching effects upon modern life and many things that we take for granted today, stretching from mobile phones through to the study of earthquakes, as addressed by Adams et al A huge number of investiga- tions have been devoted to this topic. As written in, one of the biggest scientific search engines Google.Scholar 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 tremendous scale of scientific and industrial interests in this area. The structure of a half-space coated by a thin layer is widely ap- plied in modern technology. The measurement of mechanical properties of thin films deposited on half-spaces before and during loading is of importance and of great necessaries. Note that there exists an interna- tional journal named “Thin Solid Films” which is for publishing new research results in the fileld of thin solid films. For non-destructively evaluating the mechanical properties of materials, the Rayleigh wave is a convenient tool. When using the Rayleigh wave for non-destructively evaluating, its explicit dispersion relations are employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data. Nowadays, composite materials, especially the ones with fibers, are widely 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, thin fiber/epoxy layers with different fiber-directions are attached periodi- cally to each other, up to a given thickness. The ship shell can be there- 2 fore considered as infinite, periodically layered, elastic media. Thus, the problems on waves propagating in infinite, periodically layered, elastic media are needed to investigate and they attract a great attention of researchers. The main purposes of the thesis • Applying new methods to develop some problems on Rayleigh waves that were investigated previously. • Deriving approximate secular equations of Rayleigh waves prop- agating in elastic half-spaces coated an elastic thin layer. • Investigating SH wave and Lamb wave propagting in periodical and thin structures. Research objects Waves propagating in elastic half-spaces, waves propagating in elas- tic half-spaces coated with a thin elastic layer, waves propagating in periodical, thin structures. Research range Exact and approximate secular equations of Rayleigh waves and formulas for the Rayleigh wave velocity. Research methods The method of cubic equation, the method of least squares, the perturbation method, the method of effective boundary condition, the method of polarization vector and the asymptotic method. 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 incompress- ible isotropic elastic half-space x 3 ≥ 0 (Figure 2.1) subjected to gravity. 3 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 = c 2 /c 2 2 , c 2 =  µ/ρ, k is number wave, c is velocity, µ is Lame constant, ρ is mass density of the medium, g is the acceleration due to gravity. When δ = 0, equation (1) becomes: (2 − x) 2 − 4 √ 1 − x = 0. (2) Equation (2) is the secular equation of Rayleigh waves in an incom- pressible isotropic elastic half-space without the effect of gravity. 2.1.2 Formulas for the velocity of Rayleigh waves Exact formulas for the velocity of Rayleigh waves Using the theory of cubic equation, we obtain the exact expressions for the velocity: x r = 2(4 + δ) 3 − 3   16(δ + 11)(δ 2 + 4)/27 + (δ 3 + 12δ 2 + 12δ + 136)/27 + 8 − 8δ − δ 2 9 3   16(δ + 11)(δ 2 + 4)/27 + (δ 3 + 12δ 2 + 12δ + 136)/27 , δ ∈ [0 , 1). (3) x r =1−     3  26 − 9δ 27 +  (δ + 11)(δ 2 + 4) 27 − 8 + 3δ 9 3  26−9δ 27 +  (δ+11)(δ 2 +4) 27 − 1 3     2 . (4) 4 Approximate formulas for the velocity of Rayleigh waves By applying the method of least squares, we drive approximate formulas for the velocity of Rayleigh waves: x 1 = B − √ B 2 − 4AC 2A , (5) with A = −(5.1311 + 2δ), B = −(21.2576 + 8δ + δ 2 ), C = −(15.1266 + 8δ). x 2 = 1 −  −(2.9475724 + δ) + √ δ 2 + 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 di- mensionless velocity x r (δ) is given by Eqs. (3) or (4). (iii) The squared dimensionless Rayleigh wave velocity x r is a strictly monotonously increasing function in the interval [0 , 1), from x 0 to 1 (but not equal to 1), where: x 0 = 1 −    26 27 + 2 3  11 3  1/3 − 8 9  26 27 + 2 3  11 3  −1/3 − 1 3   2 . (7) 2.1.4 Conclusion In this paper, the exact and highly accurate approximate formulas for the velocity of Rayleigh waves in an incompressible isotropic elastic half-space under gravity are derived. These formulas are useful tools for evaluating the effect of gravity on propagation of Rayleigh waves and for solving the inverse problem as well. They are new results and have been published in Acta Mechanica, Vol. 223, 1537-1544, 2012. 5 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) occupying the half-space x 2 ≥ 0, its principal material axes are x 1 , x 2 and x 3 axis (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 secular equation, namely: (B 66 − ρc 2 )[B 2 12 − B 22 (B 11 − ρc 2 )] +ρc 2  B 22 B 66  (B 11 − ρc 2 )(B 66 − ρc 2 ) = 0 (8) where: B ij are material (stiffness) coefficients which can be expressed in terms of the engineering constants (Young’s and shear moduli, Poisson’s ratios) as: B 11 = E 1 1 − ν 12 ν 21 , B 22 = E 2 1 − ν 12 ν 21 , B 12 = ν 21 E 1 1 − ν 12 ν 21 = ν 12 E 2 1 − ν 12 ν 21 , B 66 = G 12 , (9) Remark: The secular equation (8) is much more simple than the sec- ular equations of Cerv and valid for any orthotropic elastic materials. 2.2.1.2 Formulas for the velocity Exact formula 6 Following the same procedure carried out in [Pham Chi Vinh and Ogden, R. W., Ach. Mech., 56 (3) (2004), 247-265], formula for the velocity of Rayleigh waves is derived: ρc 2 /B 66 = √ b 1 b 2 b 3 /  ( √ b 1 /3)(b 2 b 3 +2)+ 3  R+ √ D+ 3  R − √ D  (10) where b 1 = B 22 /B 11 , b 2 = 1 −B 2 12 /(B 11 B 22 ), b 3 = B 11 /B 66 , R and D are given by: R = − 1 54 h(b 1 , b 2 , b 3 ), D = − 1 108  2 √ b 1 (1 − b 2 ) h(b 1 , b 2 , b 3 ) + 27b 1 (1 − b 2 ) 2 + b 1 (1 − b 2 b 3 ) 2 + 4  , h(b 1 , b 2 , b 3 ) = √ b 1 [2b 1 (1 − b 2 b 3 ) 3 + 9(3b 2 − b 2 b 3 − 2)] (11) and the roots in (10) taking their principal values. Three dimensionless parameters b k are expressed in terms E 1 , E 2 , G 12 , ν 12 as: b 1 = E 2 E 1 , b 2 = 1 − E 2 ν 2 12 E 1 , b 3 = E 2 1 G 12 (E 1 ν 2 12 − E 2 5) (12) Approximate formulas Using the method of least squares, we obtain approximate formulas: x 1 = B 1 −  B 2 1 − 4A 1 C 1 2A 1 , A 1 =b 1 b 3 [b 3 (1 + 0.5b 1 − 2b 1 b 2 b 3 ) − 1.5], B 1 =b 1 b 3 [0.6(b 1 b 3 − 1) − b 1 b 2 b 2 3 (b 2 b 3 + 2)], C 1 =0.05b 1 b 3 (b 1 b 3 − 1) − b 2 1 b 2 2 b 4 3 (13) x 2 = B 2 −  B 2 2 − 4A 2 C 2 2A 2 , A 2 =b 1 b 3 [b 3 (1 + 0.5b 1 − 2b 1 b 2 b 3 ) − 1.5], B 2 =b 1 b 3 [0.5625(b 1 b 3 − 1) − b 1 b 2 b 2 3 (b 2 b 3 + 2)], C 2 =0.03125b 1 b 3 (b 1 b 3 − 1) − b 2 1 b 2 2 b 4 3 (14) Remark: Approximate formulas for the velocity (13), (14) are highly accurate. 7 2.2.2 Non-principal Rayleigh waves 2.2.2.1. Secular equation Consider a thin homogeneous orthotropic elastic panel occupying the half-space x 2 ≥ 0 whose principal material axes are X, Y, Z (hình 2.9). Suppose that the Z axis coincides with the x 3 axis and (x 1 , x 2 ) Figure 2.9. The model of non-principal Rayleigh waves is the rotated one from (X, Y ) by counter clockwise angle θ. Suppose that the panel is subjected to the plane stress state. Using the method of first integrals, we derive the secular equation: F (X, θ) ≡dX 2 [(d + d 2 )X −d 3 ][d 2 2 − Q 66 (dX −d 3 )] + (dX −d 3 )[(d+d 2 )X −d 3 ][Q 22 dX 2 −(d 2 +d 2 1 +Q 22 d 3 )X +dd 3 ] − 2d 1 X 2 (dX −d 3 )[Q 26 (dX −d 3 ) − d 1 d 2 ] = 0 (15) where X = ρc 2 and Q 11 = B 11 c 4 θ + 2(B 12 + 2B 66 )c 2 θ s 2 θ + B 22 s 4 θ , Q 22 = B 11 s 4 θ + 2(B 12 + 2B 66 )c 2 θ s 2 θ + B 22 c 4 θ , Q 12 = (B 11 + B 22 − 4B 66 )c 2 θ s 2 θ + B 12 (c 4 θ + s 4 θ ), (16) Q 66 = (B 11 + B 22 − 2B 12 − 2B 66 )c 2 θ s 2 θ + B 66 (c 4 θ + s 4 θ ), Q 16 = −(B 11 − B 12 − 2B 66 )c 3 θ s θ − (B 12 − B 22 + 2B 66 )c θ s 3 θ , Q 26 = −(B 11 − B 12 − 2B 66 )c θ s 3 θ − (B 12 − B 22 + 2B 66 )c 3 θ s θ , d = Q 22 Q 66 − Q 2 26 , d 1 = Q 12 Q 26 − Q 22 Q 16 d 2 = Q 12 Q 66 − Q 16 Q 26 , d 3 = Q 11 d + Q 16 d 1 − Q 12 d 2 . with c θ := cosθ, s θ := sinθ (0 ≤ θ ≤ π). 8 2.2.3 Conclusion In this chapter we obtain the secular equation for principal Rayleigh waves that is valid for all orthotropic elastic materials and much more simple than the ones obtained recently by Cerv. Exact and approx- imate formulas for the velocity of principal Rayleigh-edge waves are also established and they are a powerful tool for analyzing the effect of material parameters on the Rayleigh wave velocity. For non-principal Rayleigh waves a secular equation in explicit form is obtained by us- ing the method of first integrals. They are new results and have been published "Vietnam Journal of Mechanics, 34 (2) (2012), 123 – 134" Chapter 3. Rayleigh waves in elastic half- spaces underlying a water layer 3.1 The exact secular equation Consider an incompressible isotropic elastic half-space x 3 ≥ 0 that is overlaid with a layer of incompressible non-viscous water occupying the domain 0 ≤ x 3 ≤ h (see Figure 3.1). Both the elastic half-space and the water layer are assumed to be under the gravity Figure 3.1. The problem model From the basic equations, the boundary conditions at x 3 = h and the continuity conditions at x 3 = 0, we obtain the secular equation: (2 − x) 2 − 4 √ 1 − x − δ x + r δ x − r f (x, δ, ε)x 2 = 0, 0 < x < 1 (17) where: x = c 2 /c 2 2 , c 2 =  µ/ρ , δ = g/kc 2 2 , ε = kh, r = ρ  ρ , f (x, δ, ε) = (δ −xthε)/(x −δthε), c is velocity, µ and ρ are Lame contants and the mass density of the medium, g is the acceleration due to gravity, k is the wave number, ρ  is the mass density of the water [...]... Khanh Linh, New results on Rayleigh waves in incompressible elastic media subjected to gravity, Acta Mechanica, Vol 223, pp 1537-1544, 2012 4) Pham Chi Vinh and Nguyen Thi Khanh Linh, An approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space coated by a thin orthotropic elastic layer, Wave Motion, Vol 49, pp 681-689, 2012 5) Pham Chi Vinh and Nguyen Thi Khanh... Published papers related the thesis 1) Pham Chi Vinh, Nguyen Thi Khanh Linh, The approximate secular equation of Rayleigh waves in an orthotropic elastic half-space coated by a thin orthotropic elastic layer, Procceding of 9th the National Congress on Mechanics, Hanoi, December 8-9, 2012, Vol 2, pp 1263-1270 2) Pham Chi Vinh and Nguyen Thi Khanh Linh, Rayleigh waves in an incompressible elastic half-space... approximate secular equation of generalized Rayleigh waves in pre-stressed compressible elastic solids, International Journal of Non-Linear Mechanics, Vol 50, pp 91–96, 2013 6) Pham Chi Vinh, Nguyen Thi Khanh Linh, An explicit secular equation of Rayleigh waves propagating along an obliquely cut surface in a directional fiber-reinforced composite, Vietnam Journal of Mechanics, Vol 34 (2) , pp 123 – 134, . VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MECHANICS ———– * ———– NGUYEN THỊ KHANH LINH SURFACE WAVES AND WAVES IN THIN STRUCTURES Major: Engineering mechanics Code: 62 52 01