Sóng rayleigh trong các bán không gian đàn hồi không tự do đối với ứng suất

ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN  Trịnh Thị Thanh Huệ SĨNG RAYLEIGH TRONG CÁC BÁN KHƠNG GIAN ĐÀN HỒI KHƠNG TỰ DO ĐỐI VỚI ỨNG SUẤT LUẬN ÁN TIẾN SĨ CƠ HỌC Hà Nội - 2017 ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN  Trịnh Thị Thanh Huệ SÓNG RAYLEIGH TRONG CÁC BÁN KHÔNG GIAN ĐÀN HỒI KHÔNG TỰ DO ĐỐI VỚI ỨNG SUẤT Chuyên ngành: Cơ học vật thể rắn Mã số: 62 44 01 07 LUẬN ÁN TIẾN SĨ CƠ HỌC Chủ tịch Hội đồng Người hướng dẫn khoa học GS TSKH Nguyễn Đơng Anh GS TS Phạm Chí Vĩnh Hà Nội - 2017 L˝I CAM OAN Tæi xin cam oan Ơy l cổng trnh nghiản cứu ca riảng tổi C¡c sŁ li»u v k‚t qu£ ÷ỉc tr…nh b y lu“n ¡n l trung thüc v ch÷a tłng ÷ỉc cỉng bŁ b§t ký cỉng tr…nh n o kh¡c Nghi¶n cøu sinh Trành Thà Thanh Hu» L˝IC MÌN Lu“n ¡n n y ÷ỉc thüc hi»n v ho n th nh dữợi sỹ hữợng dÔn khoa hồc ca GS TS Phm Ch Vắnh, ngữới  tn tnh giúp ù tổi trản ữớng khoa hồc Thy  du dt tổi trản ữớng l m cỡ hồc, luổn to nhœng thß th¡ch gióp tỉi tü håc häi, t…m tỈi v s¡ng t⁄o Tỉi xin b y tä lỈng bi‚t ìn vỉ cịng s¥u s›c ‚n Thƒy Tỉi muŁn b y tä sü c£m ìn ch¥n th nh ‚n ban GiĂm hiằu Trữớng i hồc XƠy dỹng, ban ch nhi»m Khoa X¥y Düng D¥n dưng v Cỉng nghi»p v °c bi»t l c¡c thƒy cỉ Bº mỉn Cì håc lỵ thuyt trữớng i hồc XƠy dỹng  ng viản, khuy‚n kh‰ch,t⁄o måi i•u ki»n cho tỉi ho n th nh lu“n ¡n Tỉi xin ch¥n th nh c£m ìn c¡c thƒy cỉ Bº mỉn Cì håc, Khoa To¡n - Cì - Tin håc, Tr÷íng ⁄i håc Khoa håc Tü nhi¶n, ⁄i håc QuŁc gia H Nºi, c¡c anh ch nhõm sermina ca thy Phm Ch Vắnh  hữợng dÔn, chia sà kinh nghiằm, to mt mổi trữớng nghiản cứu khoa hồc tt nhĐt cho bÊn thƠn tổi CuŁi cịng, tỉi xin b y tä lỈng bi‚t ìn sƠu sc n gia nh tổi  luổn luổn giúp ï, ºng vi¶n v ıng hº tỉi suŁt qu¡ tr…nh l m lu“n ¡n Nghi¶n cøu sinh Trành Thà Thanh Hu» Möc löc M— U T˚NG QUAN 1.1 Sâng Rayleigh tü øn 1.2Sâng Rayleigh khæng tü 1.2.1 1.2.2 1.2.3 1.2.4 1.3K‚t lu“n S´NG RAYLEIGH TRONG B N KH˘NG GIAN NH˙ICHÀU I UKI NBI NTR—KH NG13 2.1 H» thøc cì b£n 2.2 Sõng Rayleigh bĂn khổng gian n hỗ nn ữổc chu iãu kiằn biản tr khĂng 2.2.1 2.2.2 2.2.3 2.3 Sâng Rayleigh b¡n khæng gian n hỗ vt liằu monoclinic vợi mt phflng i xứng x3 ữổc chu iãu kiằn biản tr khĂng 2.3.1 2.3.2 2.3.3 2.3.4 2.4 Sõng Rayleigh bĂn khổng n hỗi trỹc nn ữổc chu iãu kiằn biản tr khĂng 2.4.1 2.4.2 2.4.3 2.5 Sâng Rayleigh truy•n b¡n khỉng gian nn ữổc ữổc to bi vt liằu monoclinic vợ i xứng x3 = 0, khổng nn ữổc chu iãu k kh¡ng 2.5.1 2.5.2 2.5.3 2.5.4 2.6 K‚t lu“n S´NG RAYLEIGH TRONG B N KH˘NG GIAN N H˙I C´ ÙNG SU T TRײC CHÀU I U KI NBI NTR—KH NG 3.1 3.2 3.3 3.4 Sõng Rayleigh bĂn khổng gian n hỗi ứng suĐt trữợc chu iãu kiằn biản tr khĂng 3.1.1 3.1.2 3.1.3 Sõng Rayleigh bĂn khổng gian n hỗ ữổc cõ ứng suĐt trữợc chu iãu kiằn biản tr 3.2.1 3.2.2 3.2.3 Sõng Rayleigh bĂn khổng gian n hỗ ữổc, cõ bin dng trữợc: chu ỗng thới ko (n chu iãu kiằn biản tr khĂng 3.3.1 3.3.2 3.3.3 3.3.4 K‚t lu“n S´NG RAYLEIGH TRONG B N KH˘NG GIAN N H˙I QUAY CHÀU I U KI N BI N TR— KH NG 4.1 Sõng Rayleigh bĂn khổng gian n hỗ x3 = quay, nn ữổc chu iãu kiằn biản tr 4.1.1 4.1.2 4.1.3 Sâng Rayleigh b¡n khæng gian n hỗ ct sổi, khổng nn ữổc, quay chu iãu kiằn 4.2.1 4.2.2 4.2.3 K‚t lu“n 4.2 4.3 S´NG RAYLEIGH TRONG B N KH˘NG GIAN N HI MONOCLINIC C M T PH NG ăI XÙNG x3 = ×ĐC PHÕ L˛P M˜NG 101 5.1 Sõng Rayleigh bĂn khổng gian n hỗi monoclinic câ m°t phflng Łi xøng x3 = n†n ÷ỉc ph lợp mọng n hỗi monoclinic cõ mt phflng i xøng x3 = n†n ÷ỉc102 5.1.1 5.1.2 5.1.3 5.1.4 5.2 Sõng Rayleigh bĂn khổng gian n hỗi monoclinic câ m°t phflng Łi xøng x3 = khæng n†n ữổc ph lợp mọng n hỗi monoclinic cõ mt phflng Łi xøng x3 = khỉng n†n ÷ỉc 5.2.1 5.2.2 5.2.3 5.2.4 5.3 K‚t lu“n K TLU NV KI NNGHÀ DANH MÖC C˘NG TR NH KHOA H¯C CÕA T C GI LI NQUAN NLU N N T i li»u tham khÊo M U Lỵ chồn ã t i nghiản cøu C¡c b i to¡n truy•n sâng c¡c mỉi trữớng n hỗi (xem, chflng hn [3], [7], [11], [26]), ni bt l sõng mt Rayleigh, l cỡ s lỵ thuy‚t cho nhi•u øng dưng kh¡c khoa håc cổng nghằ Sõng mt Rayleigh truyãn mổi trữớng n hỗi flng hữợng nn ữổc, m Rayleigh [52] tm hỡn mt trôm nôm trữợc, v vÔn ang ữổc nghiản cøu mºt c¡ch m⁄nh m‡ v… nhœng øng döng to lợn ca nõ nhiãu lắnh vỹc khĂc ca khoa hồc v cổng nghằ nhữ a chĐn hồc, Ơm hồc, a vy lỵ, cổng nghằ truyãn thổng v khoa håc v“t li»u Câ th” nâi r‹ng nhœng nghi¶n cøu cıa Rayleigh v• sâng m°t truy•n b¡n khỉng gian n hỗi cõ Ênh hững sƠu rng n cuc sng hiằn i Nõ ữổc sò dửng nghiản cứu ng §t, thi‚t k‚ mobile phone v nhi•u thi‚t bà i»n tò cỹc nhọ, nhữ Adams v cĂc cng sỹ [4]  nhĐn mnh  cõ mt s lữổng nghiản cứu rĐt lợn vã sõng mt Rayleigh Nhữ  vit [92], Google.Scholar, mºt nhœng cỉng cư t…m ki‚m mnh nhĐt vã khoa hồc, cho hỡn mt triằu ữớng links cho yảu cu tm kim "Rayleigh waves" K‚t qu£ t…m ki‚m thu ÷ỉc th“t ¡ng kinh ng⁄c! Nâ ch¿ r‹ng, sâng m°t Rayleigh câ tr‰ cao khoa hồc,  v ang ữổc sỹ quan tƠm rĐt lợn ca cĂc nh khoa hồc v ngo i nữợc Tuy nhiản, hu ht cĂc nghiản cứu trữợc Ơy vã sõng Rayleigh, bĂn khổng gian n hỗi ữổc giÊ thit l tỹ i vợi ứng suĐt Cõ rĐt t nghiản cứu d nh cho bĂn khổng gian n hỗi khổng tỹ i vợi ứng suĐt Chnh v lỵ n y m lun Ăn i nghiản cứu cĂc b i toĂn truyãn sõng Rayleigh cĂc bĂn khổng gian n hỗi khổng tỹ i vợi ứng suĐt Mửc ch ca lun Ăn Mửc tiảu thứ nhĐt ca lun Ăn l phĂt trin phữỡng phĂp vectỡ phƠn cỹc cho trữớng hổp ma trn Stroh l ma tr“n phøc ( ÷ỉc gåi l "ph÷ìng phĂp vectỡ phƠn cỹc phức") Mửc tiảu thứ hai ca lu“n ¡n l t…m c¡c ph÷ìng tr…nh t¡n s›c dng hiằn (dng tữớng minh) ca sõng Rayleigh truyãn cĂc bĂn khổng gian n hỗi khổng tỹ i vợi ứng suĐt i tữổng nghiản cứu Sõng Rayleigh cĂc bĂn khổng gian n hỗi khổng tỹ i vợi ứng suĐt nhữ bĂn khổng gian chu iãu kiằn biản tr khĂng, bĂn khổng gian ph lợp mọng Phm vi nghiản cứu Sõng Rayleigh cĂc mổi trữớng n hỗi tuyn tnh Phữỡng phĂp nghiản cứu Phữỡng phĂp iãu kiằn biản hiằu dửng (xem t i liằu [88]) ữa cĂc b i toĂn cn nghiản cứu vã b i to¡n truy•n sâng Rayleigh c¡c b¡n khỉng gian khổng tỹ i vợi ứng suĐt Phữỡng phĂp vectỡ phƠn cỹc phức ( ữổc trnh b y mưc 2.1 H» thøc cì b£n) ” t…m c¡c ph÷ìng tr…nh t¡n s›c d⁄ng t÷íng minh cıa sâng Ngo i ra, lun Ăn cặn sò dửng phữỡng phĂp truy•n thŁng (tham kh£o t i li»u [1]) ” thi‚t l“p c¡c ph÷ìng tr…nh t¡n s›c cıa sâng - Nhœng õng gõp mợi ca lun Ăn PhĂt trin phữỡng phĂp vectỡ phƠn cỹc Tm ữổc phữỡng trnh tĂn s›c ch‰nh x¡c d⁄ng t÷íng minh cıa sâng Rayleigh bĂn khổng gian n hỗi d hữợng (trỹc hữợng v monoclinic vỵi m°t phflng Łi xøng x3 = 0) n†n ữổc v khổng nn ữổc chu iãu kiằn biản tr khĂng XƠy dỹng ữổc phữỡng trnh tĂn sc chnh x¡c d⁄ng hi»n cıa sâng Rayleigh b¡n khæng gian n hỗi cõ ứng suĐt trữợc (chu ko nn thun túy v ỗng thới chu ko nn v ct) chu iãu kiằn biản tr khĂng Thit lp ữổc phữỡng tr…nh t¡n s›c ch‰nh x¡c d⁄ng hi»n cıa sâng Rayleigh bĂn khổng gian n hỗi monoclinic vợi mt phflng i xứng x3 = quay chu iãu kiằn biản trð kh¡ng v sâng Rayleigh b¡n khæng gian n hỗi khổng nn ữổc quay cõ gia c ct sổi chu iãu kiằn biản tr khĂng DÔn ữổc phữỡng trnh tĂn sc xĐp x ca sõng Rayleigh bĂn khổng gian n hỗi d hữợng (nn ữổc v khổng nn ữổc) ữổc ph lợp mọng n hỗi d hữợng (nn ữổc v khổng nn ữổc) Phữỡng trnh tĂn sc tm ữổc cõ dng bc hai i vợi d y cıa lỵp mäng Mºt sŁ k‚t qu£ ch‰nh ca lun Ăn  ữổc cổng b trản cĂc ch‰ quŁc t‚ (SCI: hai b i), hai b¡o c¡o t⁄i Hºi nghà Cì håc v“t r›n bi‚n d⁄ng to n quŁc DANH MÖC C˘NG TR NH KHOA H¯C CÕAT CGI LI NQUAN NLU N N Ph⁄m Ch‰ Vắnh, Trnh Th Thanh Huằ (2013), "Phữỡng trnh tĂn sc x§p x¿ cıa sâng Rayleigh b¡n khỉng gian n hỗi monoclinic x = ữổc ph lợp mọng n hỗi trỹc hữợng", Hi ngh Khoa hồc to n quŁc Cì håc V“t r›n bi‚n d⁄ng lƒn thø XI, pp 1387-1394 Pham Chi Vinh, Trinh Thi Thanh Hue (2014), "Rayleigh waves with impedance boundary conditions in anisotropic solids", Wave Motion (51), pp 1082-1092 Pham Chi Vinh, Trinh Thi Thanh Hue (2014), "Rayleigh waves with impedance boundary conditions in incompressible anisotropic halfspace", International Journal of Engineering Science (85), pp 175-185 Ph⁄m Ch‰ V¾nh, Trành Thà Thanh Hu» (2015), "Phữỡng trnh tĂn sc xĐp x ca sõng Rayleigh bĂn khổng gian n hỗi monoclinic x = ữổc ph lợp mọng n hỗi monoclinic x3=0 khổng n†n ÷ỉc", Hºi nghà Khoa håc to n quŁc Cì håc V“t r›n bi‚n d⁄ng lƒn thø XII, pp 1685-1691 123 T i li»u tham kh£o [1] Ph⁄m Ch‰ V¾nh (2015), C¡c ph÷ìng ph¡p t…m ph÷ìng tr… nh t¡n s›c d⁄ng hi»n cıa sâng Rayleigh v øng dưng, Nh xu§t b£n ⁄i håc QuŁc gia H Nºi, H Nºi [2] Achenbach J D and Keshava S P (1967), Free waves in a plate supported by a semi-infinite continuum , J Appl Mech 34, pp 397-404 [3] Achenbach J D (1973), Wave propagation in Elastic Solids, North-Holland, Sterdam [4] Adam S D M et al (2007), Rayleigh Waves Guided by Topogra-Phy , Proc R Soc London, Ser A 463, pp 531-550 [5] Antipov Y A (2002), "Diffraction of a plane wave by a circular cone with an impedance boundary condition", SIAM Journal on Applied Mathematics 62, pp 1122-1152 [6] Asghar S., Zahid G H (1986), "Field in an open-ended waveg-uide satisfying impedance boundary conditions", Journal of Applied Mathematics and Physics (ZAMP) 37, pp 194-205 [7] Ben-Menahem, A and Singh, S J (2000), Seismic waves and Sources, Springer-Verlag New York Inc., Second edition [8] Bergmann L (1948), Ultrasonics and their scientific and technical applications, Jonh Wiley Sons, New York [9] Bovik P (1996), A comparison between the Tiersten model and O(H) boundary conditions for elastic surface waves guided by thin layers , Journal of Applied Mechanics, 63, pp 162-167 [10] Brekhovskikh L M (1990), "Acoustics of layered media: plane and quasi-plane waves", Springer-Verlag, Berlin [11] Brekhovskikh, L M and Goncharov, V (1994), Mechanics of con-tinua and wave dynamics, Springer-Verlag, New York [12] Briggs G A D (1992), Acoustic microscopy, Clarendon Press, Ox-ford [13] Castro L P., Kapanadze D (2008), "The impedance boundary-value problem of diffraction by a strip ", Journal of Mathematical Analysis and Applications 337(2), pp 1031-1040 [14] Chadwick P (1976), The existence of pure surface modes in elastic materials with orthorhombic symmetry, J Sound Vib 47, pp 39-52 [15] Clarke N.S., Burdess J.S (1994), A rotation rate sensor based upon a Rayleigh resonator , J Appl Mech 61, pp 139 143 [16] Clarke N.S., Burdess J.S (1994), Rayleigh waves on a rotating sur-face , J Appl Mech 61, pp 724 726 [17] Collet B., Destrade M (2004), Explicit secular equations for piezoa-coustic surface waves: Shear - horizontal modes , J Acoust Soc Am 116(6), pp 3432-3442 [18] Currie P K (1979), "The secular equation for Rayleigh waves on elastic crystal", Q J Mech Appl Math 32, pp 163173 [19] Destrade M (2001), The explicit secular equation for surface acous-tic waves in monoclinic elastic crystals , J Acoust Soc Am 109(4), pp 1398 - 1402 [20] Destrade M (2003), Rayleigh waves in symmetry planes crystals: explicit secular equation and some explicit wave speeds , Mech Mat 35, pp 931-939 [21] Destrade M (2003), Surface acoustic waves in rotating orthorhom-bic crystals , Proc R Soc London 460, pp 653 665 [22] Destrade M (2004), Rayleigh waves in anisotropic crystals rotat-ing about the normal to a symmetry plane , Jourmal of Applied Mechanics 71(4), pp 516-520 [23] M Destrade, R W Ogden (2005), "Surface waves in a stretched and sheared incompressible elastic material", International Journal of Non-Linear Mechanics 40, pp 241 - 253 [24] Dowaikh M A., Ogden R W (1991), "On surface waves and defor-mations in a compressible elastic half-space", Stability and Applied of Continuous Media 1(1), pp 27 - 45 125 [25] Every A G (2002), Measurement of the near-surface elastic prop-erties of solids and thin supported films , Meas Sci Technol 13, pp 21-39 [26] Ewing, W M., Jardetzky, W S., and Press, F (1957), Elastic Waves in Layered Media, McGraw-Hill, New York [27] Fang H., Yang J., Jiang Q (2000), Rotation perturbed surface acoustic waves propagating in piezoelectric crystals , Int J Solids Struct 37, pp 4933-4947 [28] Fu Y B and Mielke A (2002), A new identity for the surface-impedance matrix and its application to the determination of surface-wave speeds , Proc R Soc Lond A 458, pp 25232543 [29] Godoy E., Dur¡n M., N†d†lec J-C (2012), On the existence of sur-face waves in an elastic half-space with impedance boundary condi-tions , Wave Motion 49, pp 585594 [30] Grigorevskii V.I., Gulyaev Yu.V., Yu.V Kozlov Yu.V (2000), Acoustic waves in rotating elastic medium , Acoust Phys 46, pp 236-238 [31] Hiptmair R., Lopez-Fernander M and Paganini A (2014), "Fast convolution quadrature based impedance boundary conditions", Journal of Computational and Applied Mathematics 263, pp 500-517 [32] Jahangir E., Howe R.M (1993), Time-optimal attitude control scheme for spinning missile , J Guidance Contr Dyn 16, pp 346-353 [33] Jose K.A., Suh W.D., Xavier P.B., Varadan V.K., Varadan V.V (2002), Surface acoustic wave MEMS gyroscope , Wave Motion 36, pp 367-381 [34] Kawasaki K., Sekiguchi M., Matsuhisa T (1991), Detecting flaws formed in surfaces of rotating members with ultrasonic waves , J Acoust Soc Am 90, pp 3386-3386 [35] Lao B.Y (1980), Gyroscopic effect in surface acoustic waves , IEEE Ultras Symp pp 687-690 [36] Li X-F (2006), On approximate analytic expressions for the veloc-ity of Rayleigh waves , Wave Motion 44, pp 120-127 [37] Makarov S., Chilla E and Frohlich H J (1995), Determination of elastic constants of thin films from phase velocity dispersion of 126 different surface acoustic wave modes , J Appl Phys 78, pp 5028-5034 [38] Malischewsky P.G (1987), Surface Waves and Discontinuities, Else-vier, Amsterdam [39] Malischewsky P G (2000), Comment to "A new formula for ve-locity of Rayleigh waves" by D.Nkemzi [Wave Motion 26 (1997) pp 199-205] , Wave Motion 31, pp 93-96 [40] Mathews I C., Jeans R A (2007), "An acoustic boundary integral formulation for open shells allowing different impedance conditions, top and bottom surfaces", Journal of Sound and Vibration 300, pp 580-588 [41] Mozhaev V G (1995), Some new ideas in the theory of surface acoustic waves in anisotropic media, in D F Parker and A H., England (eds.), IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, Kluwer Academic Pub, Dor-drecht, The Netherlands, pp 455-462 [42] Murty G S (1975), A theoretical model for the attenuation and dispersion of stoneley waves at the loosely bonded interface of elastic half spaces , Phys Earth Planet Interiors 11, pp 65-79 [43] Murty G S (1975), Wave propagation at an unbounded interface between two elastic half-spaces , J Acoust Soc Am 58, pp 1094-1095 [44] Nesvijski E G (2001), On Rayleigh Equation and Accuracy of Its Real Roots Calculations , J Thermo Plast Compt Mater 14, pp 356-364 [45] Niklasson A J., Datta S K., Dunn M L (2000), On approximat-ing guided waves in thin anisotropic coatings by means of effective boundary conditions , J Acoust Soc Am 108, pp 924-933 [46] Nkemzi D (1997), A new formula for the velocity of Rayleigh waves , Wave Motion 26, pp 199-205 [47] Ogden R W (1984), "Non-linear Elastic Deformations", Ellis Hor-wood, Chichester [48] Ogden R W and Pham Chi Vinh (2004), On Raylegh waves in in-compressible orthotropic elastic solids , J Acoust Soc Am 115(2), pp 530-533 127 [49] Pohl A., Ostermayer G., Reindl L., Seifert F (1997), Monitoring the tire pressure at cars using passive SAW sensors , IEEE Ultras Symp 1, pp 471-474 [50] Qin H-H., Colton D (2012), "The inverse scattering problem for cavities with impedance boundary condition", Advances in Compu-tational Mathematics 36(2), pp 157-174 [51] Rahman M., Barber J.R (1995), Exact expression for the roots of the secular equation for Rayleigh waves , ASME J Appl Mech 62, pp 250-252 [52] Rayleigh, L (1885), "On waves propagating along the plane surface of an elastic solid", Proc R Soc Lond A17, pp 4-11 [53] Rokhlin S.I and Huang W (1993), Ultrasonic wave interaction with a thin anisotropic layer between two anisotropic solids II Second-order asymptotic boundary conditions , J Acoust Soc Am 94, pp 3405-3420 [54] Schoenberg M., Censor D (1973), Elastic waves in rotating media , Quart Appl Math 31, pp 115-125 [55] Senior T B A (1960), "Impedance boundary conditions for imper-fectly conducting surfaces", Applied Scientific Research, Section B 8, pp 418-436 [56] Baljeet Singh (2015), "Rayleigh wave in an incompressible fibre-reinforced elastic solid with impedance boundary conditions", Jour-nal of the Mechanical Behavior of Materials 24(5-6), pp 183186 [57] Steigmann, D.J (2007), "Surface waves supported by thin-film/substrate interactions", IMA J Appl Math 72, pp 730747 [58] Stoneley R (1924), Elastic waves at the surface of separation of two solids , Proceedings of the Royal Society of London Series A, Con-taining Papers of a Mathematical and Physical Character, 106(738), pp 416-428 [59] Stroh A N (1958), Dislocations and cracks in anisotropic elasticity, Philos Mag , pp 625- 646 [60] Stroh A.N (1962), Steady state problems in anisotropic elasticity , J Math Phys 41, pp 77-103 [61] Stupfel B., Poget D (2011), "Sufficient uniqueness conditions for the solution of the time harmonic Maxwell’s equations associated with surface impedance boundary conditions", Journal of Computational Physics 230, pp 45714587 128 [62] Taylor D B., Currie P K (1981), "The secular equation for Rayleigh waves on elastic crystals II: corrections and additions", textitQ J Mech Appl Math 34, pp 231-234 [63] Taziev R.M (1989), Dispersion relation for acoustic waves in an anisotropic elastic half-space , Sov Phys Acous 35(5), pp 535-538 [64] Tiersten H F (1969), Elastic Surface Waves Guided by Thin Films , J Appl Phys 46, pp 770-789 [65] Ting T.C.T., 1996 Anisotropic Elasticity: Theory and Applications Oxford University Press, New York [66] T C T Ting (2002), Explicit secular equations for surface waves in monoclinic materials with the symmetry plane at x = 0, x2 = or x3 = 0, Proc R Soc Lond A 458, 1017-1031 [67] Ting T.C.T (2004), The polarization vector and secular equation for surface waves in an anisotropic elastic half-space , Int J solids and Struct 41, pp 2065-2083 [68] Ting T.C.T (2004), Surface waves in a rotating anisotropic elastic half-space , Wave Motion 40, pp 329-346 [69] Ting T.C.T (2005), The polarization vectors at the interface and the secular equation for stoneley waves in monoclinic bimaterials , Proc R Soc A, 461, pp 711-731 [70] Ting T.C.T (2013), A new secular equation for slip waves along the interface of two dissimilar anisotropic elastic half-spaces in sliding contact , Wave Motion 50(8), pp 1262 1270 [71] Tuan T.T (2008), The ellipticity (H/V -ratio) of Rayleigh surface waves , PhD thesis, Friedrich-Schiller University Jena [72] Viktorov I A (1967), Rayleigh and Lamb waves: Physical theory and applications, Plenum Press, New York [73] Vinh P.C and Ogden R.W (2004), On formulas for the Rayleigh wave speed , Wave Motion 39, pp 191-197 [74] Vinh P.C and Ogden R.W (2004), Formulas for the Rayleigh wave speed in orthotropic elastic solids , Ach Mech 56 (3), pp 247-265 [75] Vinh P.C and Ogden R.W (2005), On a general formula for the Rayleigh wave speed in orthotropic elastic solids , Meccanica 40, pp 147-161 129 [76] Vinh P.C and Malischewsky P (2006), Explanation for Malis-chewsky’s approximate expression for the Rayleigh wave velocity , Ultrasonics 45, pp 77-81 [77] Vinh P.C and Malischewsky P (2007), An improved approximation of Bergmann’s form for the Rayleigh wave velocity , Ultrasonic 47, pp 49-54 [78] Vinh P.C and Malischewsky P (2007), An approach for obtaining approximate formulas for the Rayleigh wave velocity , Wave Motion 44, pp 549-562 [79] Vinh P.C and Malischewsky P (2008), Improved Approximations of the Rayleigh Wave Velocity , J Thermoplast Comp Mater 21, pp 337-352 [80] Vinh P.C and Malischewsky P (2008), Improved Approximations for the Rayleigh Wave Velocity in [-1 0.5] , Vietnam Journal of Mechanics 30, pp 347-358 [81] Pham Chi Vinh and Geza Seriani (2010), Explicit secular equations of Stoneley waves in a non-homogeneous orthotropic elastic medium under the influence of gravity , Appl Math Compt 215, pp 3515-3525 [82] Vinh P.C., Pham Thi Ha Giang (2010), On formulas for the Rayleigh wave velocity in pre-strained elastic materials subject to an isotropic internal constraint , Int J Eng Sci 48, pp 275289 [83] Pham Chi Vinh (2010), On Formulas for the velocity of Rayleigh waves in prestrained incompressible elastic solids , Trans ASME, J Appl, Mech 77(2), pp 1-9 [84] Pham Chi Vinh (2011), On formulas for the Rayleigh wave velocity in pre-stressed compressible solids , Wave Motion 48, pp 614-625 [85] Pham Chi Vinh and Pham Thi Ha Giang (2011), On formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two elastic half-spaces , Wave Motion 48, pp 646-656 [86] Pham Chi Vinh and Pham Thi Ha Giang (2012), Uniqueness of Stoneley waves in pre-stressed incompressible elastic media, Int J Non-Liner Mech 47, pp.128-134 [87] Pham Chi Vinh (2013), Scholte-wave velocity formulae, Wave Mo-tion 50, pp 180-190 130 [88] Pham Chi Vinh, Nguyen Thi Khanh Linh (2013), An approximate secular equation of generalized Rayleigh waves in prestressed com-pressible elastic solids , Int J Non-Linear Mech 50, pp 91-96 [89] Pham Chi Vinh, Nguyen Thi Khanh Linh (2013), Rayleigh waves in an incompressible elastic half-space overlaid with a water layer under the effect of gravity , Meccanica 48(8), pp 20512060 [90] Pham Chi Vinh, Trinh Thi Thanh Hue (2014), "Rayleigh waves with impedance boundary conditions in anisotropic solids", Wave Motion 51(7), pp 1082-1092 [91] Pham Chi Vinh, Trinh Thi Thanh Hue (2014), "Rayleigh waves with impedance boundary conditions in incompressible anisotropic half-spaces", International Journal of Engineering Science 85, pp 175-185 [92] Voloshin V (2010), Moving load on elastic structures: passage through the wave speed barriers , PhD thesis, Brunel University [93] Wauer J (1999), Waves in rotating conducting piezoelectric media , J Acoust Soc Am 106, pp 626-636 [94] Wang J et al (2006), Exact and approximate analysis of surface acoustic waves in an infinite elastic plate with a thin metal layer , Ultrasonics 44, pp 941-945 [95] Yla-Oijala P., Jarvenppa S (2006), "Iterative solution of high-order boundary element method for acoustic impedance boundary value problems", Journal of Sound and Vibration 291, pp 824-843 [96] Zakharov D D (2006), "Surface and internal waves in a stratified layer of liquid and an analysis of the impedance boundary condi-tions", Journal of Applied Mathematics and Mechanics 70, pp 573-581 [97] Zhou Y.H., Jiang Q (2001), Effects of Coriolis force and centrifugal force on acoustic waves propagating along the surface of a piezoelec-tric half-space , Z Angew Math Phys 52, pp 950-965 131 ... QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN  Trịnh Thị Thanh Huệ SĨNG RAYLEIGH TRONG CÁC BÁN KHƠNG GIAN ĐÀN HỒI KHƠNG TỰ DO ĐỐI VỚI ỨNG SUẤT Chuyên ngành: Cơ học vật thể rắn Mã... vctỡ ứng suĐt bng khổng ữổc gồi l bĂn khổng gian tỹ i vợi ứng suĐt Sõng Rayleigh truyãn bĂn khổng gian n y ữổc gåi l sâng Rayleigh tü øng su§t hay sâng Rayleigh thổng thữớng Mt bĂn khổng gian. .. Sõng Rayleigh khổng tỹ ứng suĐt Nhữ  nh nghắa trản, sõng Rayleigh khổng tỹ ứng suĐt l sâng Rayleigh truy•n c¡c b¡n khỉng gian n hỗi khổng tỹ i vợi ứng suĐt Ngo i cĐu trúc gỗm ch mt bĂn khổng gian