Effects of fiber ellipticity and orientation on dynamic stress concentrations in porous fiber-reinforced composites. (English) Zbl 1178.74073
Summary: Interaction of time harmonic fast longitudinal and shear incident plane waves with an elliptical fiber embedded in a porous elastic matrix is studied. The novel features of Biot dynamic theory of poroelasticity along with the classical method of eigen-function expansion and the pertinent boundary conditions are employed to develop a closed form series solution involving Mathieu and modified Mathieu functions of complex arguments. The complications arising due to the non-orthogonality of angular Mathieu functions corresponding to distinct wave numbers in addition to the problems associated with appearance of additional angular dependent terms in the boundary conditions are all avoided by expansion of the angular Mathieu functions in terms of transcendental functions and subsequent integration, leading to a linear set of independent equations in terms of the unknown scattering coefficients. A MATHEMATICA code is developed for computing the Mathieu functions in terms of complex Fourier coefficients which are themselves calculated by numerically solving appropriate sets of eigen-systems. The analytical results are illustrated with numerical examples in which an elastic fiber of elliptic cross section is insonified by a plane fast compressional or shear wave at normal incidence. The effects of fiber cross sectional ellipticity, angle of incidence (fiber two-dimensional orientation), and incident wave polarization (P, SV, SH) on dynamic stress concentrations are studied in a relatively wide frequency range. Limiting cases are considered and fair agreements with well-known solutions are established.
MSC:
| 74H35 | Singularities, blow-up, stress concentrations for dynamical problems in solid mechanics |
| 74E30 | Composite and mixture properties |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74J10 | Bulk waves in solid mechanics |
Software:
MathematicaReferences:
| [1] | Zok FW, Levi CG, Shield RT (2001) Mechanical properties of porous-matrix ceramic composites. Adv Eng Mater 3:15–23 · doi:10.1002/1527-2648(200101)3:1/2<15::AID-ADEM15>3.0.CO;2-A |
| [2] | Marshall DB, Davis JB (2001) Ceramics for future power generation technology: fiber reinforced oxide composites. Curr Opin Solid State Mater Sci 5:283–289 · doi:10.1016/S1359-0286(01)00017-1 |
| [3] | Cinibulk MK, Keller KA, Mah TI (2004) Effect of yttrium aluminum garnet additions on alumina-fiber-reinforced porous-alumina-matrix composites. J Am Ceram Soc 87:881–887 · doi:10.1111/j.1551-2916.2004.00881.x |
| [4] | Kanka B, Scneider H, Mah TI (2000) Aluminosilicate fiber/mullite matrix composites with favorable high-temperature properties. J Euro Ceram Soc 20:619–623 · doi:10.1016/S0955-2219(99)00260-5 |
| [5] | Haslam JJ, Berroth KE, Lange FF (2000) Processing and properties of an all-oxide composite with a porous matrix. J Euro Ceram Soc 20:607–618 · doi:10.1016/S0955-2219(99)00259-9 |
| [6] | Kanka BJ, Goring J, Schmucker M, Schneider H (2001) Processing, microstructure and properties of Nextel [Trademark] 610, 650 and 720 fiber/porous mullite matrix composites. In: Ceramic Engineering and Science Proceedings, 25th annual conference on composites, Advanced Ceramics, Materials and Structures 22:703–710 |
| [7] | Schmucker M, Kanka B, Schneider H (2000) Temperature-induced fiber/matrix interaction in porous alumino silicate ceramic matrix composites. J Euro Ceram Soc 20: 2491–2497 · doi:10.1016/S0955-2219(00)00150-3 |
| [8] | Tsukrov I, Piat R, Novak J, Schnack E (2003) Influence of pores on effective elastic properties of unidirectional carbon/carbon composites. Key Eng Mater 251–252, 339–344 |
| [9] | Naplocha K, Janus A, Kaczmar JW, Samsonowicz Z (2000) Technology and mechanical properties of ceramic preforms for composites material. J Mater Process Technol 106: 119–122 · doi:10.1016/S0924-0136(00)00601-4 |
| [10] | Roerden AM, Herakovich CT (2000) The inelastic response of porous, hybrid-fiber composites. Compos Sci Technol 60:2443–2454 · doi:10.1016/S0266-3538(00)00038-5 |
| [11] | Neithalath N, Weiss WJ, Olek J (2004) Acoustic performance and damping behavior of cellulose-cement composites. Cement Concr Compos 26:359–370 · doi:10.1016/S0958-9465(03)00020-9 |
| [12] | Olek J, Weiss WJ, Neithalath N (2004) Concrete mixtures that incorporate inclusions to reduce the sound generated in portland cement concrete pavements. Center for Advanced Cement-Based Materials 26:359–370 |
| [13] | Ulm FJ, Constantinides G, Heukamp FH (2004) Is concrete a poromechanics material? A multiscale investigation of poroelastic properties. Mater Struct/Materiaux et Constructions 37:43–58 |
| [14] | Reynolds WN, Wilkinson SJ (1978) Analysis of fiber-reinforced porous composite materials by the measurement of ultrasonic wave velocities. Ultrasonics 16:159–163 · doi:10.1016/0041-624X(78)90071-9 |
| [15] | Blodgett ED, Thomas LJ, Miller JG (1986) Effects of porosity on polar backscatter from fiber reinforced composites. Rev Progress Quantit Nondestruct Eval 5B:1267–1274 |
| [16] | Widrig JE, McCabe DD, Conner RL (1988) Nondestructive evaluation of fiber FP reinforced metal matrix composites. ASTM special technical publication, STP(964), pp 227–247 |
| [17] | Qu J, Achenbach JD, Roberts RA (1988) Backscatter from porosity in fiber-reinforced composites. AMD (Symposia Series) (ASME Appl. Mech. Div.) 92:69–77 |
| [18] | Kunerth DC, Lott LA, Walter JB (1990) Nondestructive evaluation of ceramic matrix composites. In: Ceramic Engineering and Science Proceedings, 14th annual conference on composites and advanced ceramic materials 11:1685–1688 |
| [19] | Generazlo ER (1990) Theory and experimental technique for nondestructive evaluation of ceramic composites. NASA technical memorandum 192561 |
| [20] | Jeong H, Hsu DK (1995) Experimental analysis of porosity-induced ultrasonic attenuation and velocity change in carbon composites. Ultrasonics 33:195–203 · doi:10.1016/0041-624X(95)00023-V |
| [21] | Zhou X, You H, Cheng Y (1997) Ultrasonic attenuation model of void contained carbon-fiber reinforced plastics. Fuhe Cailiao Xuebao/Acta Materiae Compositae Sinica 14:99–106 |
| [22] | Ono T, Miyakoshi S, Watanabe U (2002) Acoustic characteristics of unidirectionally fiber-reinforced polyurethane foam composites for musical instrument sound boards. Acoust Sci Technol 23:135–142 · doi:10.1250/ast.23.135 |
| [23] | Abdul-Aziz A, Ghosn A, Baaklini LJ, Bhatt RA (2003) Combined NDT/finite element technique to study the effects of matrix porosity on the behavior of ceramic matrix composites. Mater Eval 61:1217–1221 |
| [24] | Tournat V, Pagneux V, Lafarge D, Jaouen L (2004) Multiple scattering of acoustic waves and porous absorbing media. Phys Rev E 70:1–10 |
| [25] | Pao YH, Mow CC (1972) Diffraction of elastic waves and dynamic stress concentrations. Crane Russak, New York |
| [26] | Waterman PC (1978) Matrix theory of elastic wave scattering II. A new conservation law. J Acoust Soc Am 63: 1320–1325 · Zbl 0379.73031 |
| [27] | Varadan VK, Varadan VV, Poa YH (1978) Multiple scattering of elastic waves by cylinders of arbitrary cross section I. SH Waves. J Acoust Soc Am 63:1310–1319 · Zbl 0381.73028 · doi:10.1121/1.381883 |
| [28] | Twersky V (1983) Scattering and nonscattering obstacles. SIAM J Appl Math 43:711–725 · Zbl 0546.76097 · doi:10.1137/0143049 |
| [29] | Datta SK (1974) Diffraction of SH-waves by an elliptic elastic cylinder. Int J Solids Struct 10:123–133 · Zbl 0272.73023 · doi:10.1016/0020-7683(74)90105-X |
| [30] | Datta SK (1975) Propagation of SH-waves through a fiber-reinforced composite-elliptic cylindrical fibers. J Appl Mech 75:165–170 · Zbl 0348.73017 · doi:10.1115/1.3423510 |
| [31] | Varadan VV (1978) Scattering matrix for elastic waves II. Application to elliptic cylinders. J Acoust Soc Am 63:1014–1025 · Zbl 0377.73024 · doi:10.1121/1.381832 |
| [32] | Varadan VK, Varadan VV, Pao YH (1978) Mutiple scattering of elastic waves by cylinders of arbitrary section I. SH waves. J Acoust Soc Am 63:1310–1319 · Zbl 0381.73028 · doi:10.1121/1.381883 |
| [33] | Varadan VK, Varadan VV (1979) Frequency dependence of elastic (SH-) wave velocity and attenuation in anisotropic two phase media. Wave Motion 1:53–63 · Zbl 0418.73023 · doi:10.1016/0165-2125(79)90025-8 |
| [34] | Jain DL, Kanwal RP (1979) Scattering of elastic P and SV waves by a rigid elliptic cylindrical inclusion. Int J Solids Struct 17:941–954 · Zbl 0423.73025 |
| [35] | Goel GC, Jain DL (1981) Scattering of plane waves by a penetrable elliptic cylinder. J Acoust Soc Am 69:371–379 · Zbl 0459.76060 · doi:10.1121/1.385463 |
| [36] | Fu SL (1983) Scatter of elastic waves by a thin flat elliptical inhomogenity. NASA contractor reports 17:17–17 |
| [37] | Talbot DRS, Willis JR (1983) Variational estimates for dispersion and attenuation of waves in random composites. Int J Solids Struct 19:793–811 · Zbl 0516.73027 · doi:10.1016/0020-7683(83)90073-2 |
| [38] | Coussy O (1986) Scattering of SH-waves by a rigid elliptic fiber partially debonded from its surrounding matrix. Mech Res Commun 13:39–45 · Zbl 0611.73030 · doi:10.1016/0093-6413(86)90009-1 |
| [39] | Aleshin NP, Gusarov VR, Mogil’ner, Yu L (1989) Quantitative investigation of the scattering of longitudinal and transverse waves by elliptical cylinders. Sov J Nondestruct Test 24:810–815 |
| [40] | Radlinski RP, Simon MM (1993) Acoustic and elastic wave scattering from ellipti-cylindrical shells. J Acoust Soc Am 93:2443–2452 · doi:10.1121/1.405865 |
| [41] | Gurevich B, Sadavnichaja AP, Lapatnikov SL (1993) Elastic wave scattering by an elliptical inclusion in a fluid-saturated porous medium. J Acoust Soc Am 94:1766–1767 · doi:10.1121/1.408036 |
| [42] | Wang YS, Qui ZY, Yu GL (2002) Influence of microstructure on scattering of plane elastic waves by a distribution of partially debonded elliptical inclusions. Mech Mater 34:401–409 · doi:10.1016/S0167-6636(02)00142-4 |
| [43] | Sato H, Shindo Y (2002) Influence of microstructure of plane elastic by a distribution of partially elliptical inclusion. Mech Mater 7:401–409 · doi:10.1016/S0167-6636(02)00142-4 |
| [44] | Wang JH, Zhou XL, Lu JF (2005) Dynamic stress concentration around elliptic cavities in saturated poroelastic soil under harmonic plane waves. Mech Mater 42:4295–4310 · Zbl 1120.74521 |
| [45] | Fan Y, Sinclair AN, Honarvar F (2005) Scattering of a plane acoustic wave from a transversely isotropic cylinder encased in a solid elastic medium. J Acoust Soc Am 42:4295–4310 |
| [46] | Cai LW (2004) Multiple seattering in single scatterers. J Acoust Soc Am 115:986–995 · doi:10.1121/1.1643362 |
| [47] | Hasheminejad SM, Safari N (2005) Acoustic scattering from viscoelastically coated spheres and cylinders in viscous fluids. J Sound Vib 280:101–125 · doi:10.1016/j.jsv.2003.12.027 |
| [48] | Abramowitz, M, Stegun IA (1964) Handbook of mathematical functions. National Bureau of Standards, Washington DC · Zbl 0171.38503 |
| [49] | Achenbach J.D (1976) Wave propagation in elastic solids. North-Holland, New York · Zbl 0347.73020 |
| [50] | Allard JF (1993) Propagation of sound in porous media: modeling sound absorbing materials. Elsevier, London |
| [51] | Johnson DL, Koplik J, Dashen R (1987) Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J Fluid Mech 379:402–427 · Zbl 0612.76101 |
| [52] | Schneider M, Marqurdt J (1999) Fast computation of modified Mathieu functions applied to elliptical waveguide problems. IEEE Trans Microw Theory Tech 47:513–515 · doi:10.1109/22.754887 |
| [53] | Dyott RB (1995) Elliptical fiber waveguides. Artech House, Norwood |
| [54] | Sebak AR (2000) Scattering from dielectric coated impedance elliptic cylinder. IEEE Trans Antenna Prog 48:1574–1580 · doi:10.1109/8.899674 |
| [55] | Sato K (1974) Free vibration analysis of a composite elliptical membrane consisting of confocal elliptical parts. J Sound Vib 34:161–171 · Zbl 0291.73043 · doi:10.1016/S0022-460X(74)80301-9 |
| [56] | Chen G, Morris PJ, Zhou J (1994) Visualization of special eigenmode shape of a vibrating elliptical membrane. SIAM Rev 36:453–469 · Zbl 0806.73032 · doi:10.1137/1036101 |
| [57] | Hong K, Kim J (1995) Natural mode analysis of hollow and annular elliptical cylindrical cavities. J Sound Vib 183:327–351 · Zbl 0982.76550 · doi:10.1006/jsvi.1995.0257 |
| [58] | Chinnery PA, Humphrey VF (1998) Fluid column resonances of water-filled cylindrical shells of elliptical cross section. J Acoust Soc Am 103:1296–1305 · doi:10.1121/1.421270 |
| [59] | Harumi K (1961) Scattering of plane waves by a rigid ribbon in a solid. J Appl Phys 32:1488–1497 · Zbl 0108.38201 · doi:10.1063/1.1728383 |
| [60] | Yeh C (1963) The Diffraction of plane waves by an elliptic cylinder. J Math Phys 4:65–71 · Zbl 0114.20504 · doi:10.1063/1.1703890 |
| [61] | Barakat R (1963) Diffraction of plane waves by an elliptic cylinder. J Acoust Soc Am 35:1990–1996 · doi:10.1121/1.1918878 |
| [62] | Zorumski WE (1971) Acoustic scattering and absorption by a rigid porous elliptic cylindrical shell. Langley Research Center, Washington |
| [63] | Wang YS, Qiu ZY, Yu GL (2001) Scattering of SH waves from a partially debonded rigid elliptic cylinder. Soil Dyn Earthq Eng 21:139–149 · doi:10.1016/S0267-7261(00)00097-X |
| [64] | Toyama N, Shogen K, Yu GL (1984) Computation of the value of the even and odd mathieu functions of order N for a given parameter S and Argument X. IEEE Trans Antennas Propag AP-32:537–539 · Zbl 0954.33501 · doi:10.1109/TAP.1984.1143362 |
| [65] | Alhargan FA (1984) A complete method for the computation of mathieu characteristic numbers of integer order. SIAM Rev 38:239–255 · Zbl 0858.33018 · doi:10.1137/1038040 |
| [66] | Alhargan FA (2000) Algorithms of all mathieu functions of integer order. ACM Trans Math Softw 38:390–407 · doi:10.1145/358407.358420 |
| [67] | Wang YS, Qiu ZY, Yu GL (2001) Scattering of SH waves from a partially debonded rigid elliptic cylinder. Soil Dyn Earthq Eng 21:139–149 · doi:10.1016/S0267-7261(00)00097-X |
| [68] | Zhang S, Jin J (1996) Computation of special functions. Wiley, New York · Zbl 0865.33001 |
| [69] | Hamid AK, Hussein MI, Ragheb AA, Hamid M (2000) Mathieu functions of complex argument and their applications to the scattering by lossy elliptic cylinders. Appl CompElec Soc J 17:209–217 |
| [70] | Available in http://www.mech.iust.ac.ir/files/3/document/dr. hasheminejad.pdf · Zbl 1085.90500 |
| [71] | Berryman JG (1981) Elastic waves in fluid-saturated porous media. In: Burridge R et\(\sim\)al (eds) Macroscopic properties of disordered media. Lecture Notes in Physics, vol 154. Springer, Berlin, pp 39–49 |
| [72] | Hasheminejad SM, Hosseini H (2002) Radiation loading of a cylindrical source in a fluid-filled cylindrical cavity embedded within a fluid-saturated poroelastic medium. J Appl Mech 69:675–683 · Zbl 1110.74472 · doi:10.1115/1.1488664 |
| [73] | Leon F, Chati F, Conoir JM (2004) Modal theory applied to the acoustic scattering by elastic cylinders of arbitrary cross section. J Acoust Soc Am 116:686–692 · doi:10.1121/1.1771592 |
| [74] | Hasheminejad SM, Badsar SA (2005) Elastic wave scattering by two spherical inclusions in a poroelastic medium. J Eng Mech 131:953–965 · doi:10.1061/(ASCE)0733-9399(2005)131:9(953) |
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