Appendix A

Material constants

A.I Elastic isotropic materials

Linear elastic isotropic materials possess two independent elastic constants. Depending on specific problems one can prefer, for instance, the pairs A, J.I" E, // or J.I" a. The pairs E, // and J.I" a are very often used in the shell and rod theories. For dynamic problems we also use the dimensionless ratios 'Y and e. Relations between the elastic constants are given in the following table:

Constant A,J.I, E, // J.I"a

A A Ev ~ (1+v)(l-2v) l-u

J.I, J.I, E J.I, 2(1+0

E JL(3A+2JL) E 2JL(1+2u) >'+tL l+u

// >.

// u

2(A+;) l+u

a >. v a >'+2/J I-v

'Y ~ 2v 2u tL I-2v l-u

e J J.I,/(A + 2J.1,) J(l- 2//)/(2 - 2//) J(l - a)/2

406 APPENDIX A. MATERIAL CONSTANTS

A.2 Piezoelectric crystals

Electroelastic moduli for the 32 crystal classes

A.2. PIEZOELECTRIC CRYSTALS

Electroelastic moduli for the 32 crystal classes (continued)

I T~IClINIC SYSTEM

II 1oI0H0CI.1N1C

S'($TEIoi

llt TETRAGOW.AL

SY$TEIoI

.....

(QI

Cz (Z) ..

•••• ••• • •••• • • •

CI III -•••••• ••• •••••• ••• •••••• ••• •••••• ••• •••••• ••• ••••••••• ......... ~ .•....... ••••••••• 4\

• ••••• •••••• ••• • • • • •• • •• • •••• • •••••• : :: : 13 ••• • • • • • • 13 .... .: ... ~ • •• -fE ••• • • eTr

... ••• ...

• • •

Czv 12mml •

::: 1 :

•

CI IiI

•••••• • ••••• • ••••• • ••••• •••••• •••••• • •• 21

••• 0 . .. ~ CzI,

121011 ••• • • • ••• • ••• • • •• • • • •• .13 . ~

• • 17

~ cUm Vm 2Iml

••• • •• • •• • • .... ! ..

• • I--....,..-.~.,...--=-!. 9 • • • '9 • • 9

~ • • ~ 0 ~ • • L ___ "'·L-.......:J· 15 • • • • i7L.. ___ ...J.._.:J0 i2

X t X

•

C4V (4 mml ..,

:--: I t ..... •

...... • • 7~-~'-_~O~---1 7

X '4 , g • . I~ ° 9

°4h (4/ ... 21m Z".I

..... '- ....

• • 61----.!·~---1 6 (bl '- \, , / ~ __ ....,._O~",,--1 0

\, ,f /, L... ___ ..l..._.::J. i ..... 0 '- '2 '0 .1 OIP;OZL-__ ..-JL-..;0:Ji

Sf.E C/iART I ICONT) FOIl KEY

407

408 APPENDIX A. MATERIAL CONSTANTS

A.3 Piezoceramic materials

Piezoceramic materials are transversely isotropic and possess therefore five elastic, three piezoelectric, and two dielectric constants. Depending on spe­cific problems one can prefer, for instance, the constants eE , e, e;T or SE, d, e;s. Relations between these constants, written in the abbreviated indicial nota­tion, are

E 1 d15 c 66 = E' e15 = E'

8 66 8 55

Ef3 = Ef3 - 2d31 e31 - d33 e 33

The following table presents experimental data [9] for some piezoceramics (EO = 8.854· 1O-12C2/Nm2 is the dielectric constant of vacuum):

Quantity PZT-4 PZT-5 BaTi03

8 E 1O-12m2/N 11' 12.3 16.4 9.1

8f; -5.31 -7.22 -2.9

8f3 15.5 18.8 9.5

E 8 55 39.0 47.5 22.8

E 8 66 32.7 44.3 23.6

d15 , 1O-12C/N 496 584 260

d31 -123 -172 -78

d33 -289 -374 -190

Efl/Eo 1475 1730 1450

Ef3/EO 635 830 1260

p, 103kg/m3 7.5 7.75 5.7

Appendix B

List of notations

General

Time - t Dimensionless time - T

Three-dimensional Euclidean space - £ Cartesian co-ordinates - zi, i = 1, 2, 3 Curvilinear co-ordinates - xa, a = 1,2,3 Vectors and tensors - u, ua , t, tab, ...

3-D metrics - gab, gab

Kronecker delta - 8ij Gradient - Vu, U~b Time derivative of u - it 3-D domain - U, B Boundary of B - aB Volume element - dv 2-D surface - S 2-D co-ordinates - xa, a = 1,2 First and second fundamental forms of a surface - aa!:}, ba/3

Mean and Gaussian curvatures of a surface - H, K Covariant derivative on a surface - u:/J Area element - da 1-D curve - c(x) Curvature and torsion of a curve - K., W

Length element - ds or dx Functional - I[w] Variation of w - 8w Stationary point - w Frequency and dimensionless frequency - w, {)

410 APPENDIX B. LIST OF NOTATIONS

Dimensionless co-ordinates - ("', ( Derivatives with respect to the dimensionless co-ordinates - ulP Wave number and dimensionless wave number - k, K.

3-D elasticity

Displacements - Wa

Strains - f.ab

Stresses - (Yab

Mass density - p Elastic moduli - Cabcd

Elastic moduli for isotropic bodies - >., /-L or E, v Elastic energy density - W(f.ab)

Kinetic energy density - T( tVa)

3-D piezoelectricity

Electric potential - cp Electric field - Ea

Electric induction - Da

Electroelastic moduli - cabcd ecab cab or Cabcd hcab (.lab E, , S D, , iJS

Electric enthalpy - W(f.ab, Ea)

Internal energy - U(f.ab, Da)

Complementary energy (Gibbs function) - G((Yab, Ea)

Elastic enthalpy - F((Yab, Da)

Elastic shells

Middle surface - S Thickness - h Characteristic radius of curvature - R Characteristic scale of change of the deformation pattern - l Displacements of the middle surface - u"" u Internal degrees of freedom - 'lj;"" 'lj;, Va

Measures of extension and bending - A a,8, Ba,8 or A a,8, Pa,8

Membrane stresses, bending moments - Na,8, Ma,8 or na,8, ma,8

2-D kinetic energy density - e 2-D elastic energy - <l>

Elastic rods

Central line - c(x) Cross section - S Diameter of the cross section - h

Curvatures and torsion - W a , W

Length - L Displacements of the central line - U a , U

Internal degrees of freedom - ,¢, '¢1, v Rotation of the cross section - cp Measures of elongation, bending and twist - ,}" Oa, 0 Tension, bending and twisting moments - T, M a , M I-D kinetic energy density - e I-D elastic energy - <P

Piezoelectric shells and rods

2-D and I-D electric potential - '¢ 2-D and I-D electric field - Fa, F 2-D and I-D electric induction - Ga , G 2-D and I-D kinetic energy density - e 2-D and I-D electric enthalpy - <Po, <P1

411

Bibliography

[1] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions. Dover Publications, New York, 1965.

[2] A. E. Armenakas, D. G. Gazis, and G. Herrman. Free vibrations of circular cylindrical shells. Pergamon Press, New York, 1969.

[3] R. C. Batra and J. S. Yang. Saint-Venant's principle in linear piezoelec­tricity. J. Elasticity, 38:209-218, 1995.

[4] V. L. Berdichevsky. On the proof of the Saint-Venant principle for bodies of arbitrary shape. J. Appl. Math. Mech. (PMM) , 38:851-864, 1974.

[5] V. L. Berdichevsky. Variational-asymptotic method for constructing shell theory. J. Appl. Math. Mech. (PMM), 43:664-687, 1979.

[6] V. L. Berdichevsky. Variational principles of continuum mechanics. Nauka, Moscow, 1983.

[7] V. L. Berdichevsky and K. C. Le. High frequency, long wave shell vi­bration. J. Appl. Math. Mech. (PMM), 44:520-525, 1980.

[8] V. L. Berdichevsky and K. C. Le. High frequency vibration of shells. Soviet Physics Doklady, 27:988-990, 1982.

[9] D. A. Berlincourt, D. R. Curran, and H. Jaffe. Piezoelectric and piezo­magnetic materials and their function in transducers. In Physical acous­tics, volume 1A. Academic Press, New York, 1964.

[10] V. A. Boriseiko, V. S. Martynenko, and A. F. Dlitko. On the theory of vibrations ofpiezoceramic shells. Mathematical Physics, 21:71-76, 1977.

[11] W. G. Cady. Piezoelectricity. Dover Publications, New York, 1964.

[12] I. Ekeland and R. Temam. Analyze convexe et problemes variationnels. Dunod, Paris, 1974.

414 BIBLIOGRAPHY

[13] A. Erdelyi. Higher transcendental functions, volume 1 and 2. McGraw­Hill, New York, 1953.

[14] J. N. Flavin, R. J. Knops, and L. E. Payne. Decay estimates for the constrained elastic cylinder of variable cross section. Quart. Appl. Math., 47:325-350, 1989.

[15] D. C. Gazis. Three-dimensional investigation ofthe propagation of waves in hollow circular cylinders. 1. Analytical foundation. J. Acoust. Soc. Am., 31:568-573, 1959.

[16] A. 1. Goldenveizer. Theory of thin elastic shells. Nauka, Moscow, 1976.

[17] V. T. Grinchenko and V. V. Meleshko. Harmonic vibrations and waves in elastic bodies. Naukova Dumka, Kiev, 1981.

[18] M. E. Gurtin. The linear theory of elasticity, volume VIa/2 of Handbuch der Physik. Springer, Berlin, 1972.

[19] B. Jaffe, W. R. Cook, and H. Jaffe. Piezoelectric ceramics. Academic Press, New York, 1971.

[20] J. D. Kaplunov. High-frequency stress-strain states. Izv. Akad. Nauk S.S.S.R, MTT, 25:147-157, 1990.

[21] J. D. Kaplunov, L. Y. Kossovich, and E. V. Nolde. Dynamics of thin walled elastic bodies. Academic Press, New York, 1998.

[22] W. T. Koiter. A consistent first approximation in the general theory of thin elastic shells. In Proc. IUTAM Symp. Theory of Thin Elastic Shells, pages 12-33, 1960.

[23] W. T. Koiter. On the mathematical foundation of shell theory. In Proc. Intern. Congress of Math. Nice, 1970, volume 3, pages 123-130, Paris, 1971. Gauthier-Villars.

[24] S. S. Kvashnina. High-frequency long-wave vibrations of elastic rods. Applied Mathematics and Mechanics, 43:335-341, 1979.

[25] V.1. Krylov L. V. Kantorovich. Approximate method of higher analysis. Wiley, New York, 1964.

[26] K. C. Le. High frequency, long wave vibration of piezoelectric ceramic plates. Soviet Physics Doklady, 27:422-423, 1982.

BIBLIOGRAPHY 415

[27] K. C. Le. Fundamental relations of the theory of anisotropic piezoelectric shells. Mechanical Bulletin of Moscow University, Math.-Mech., (5):57-60, 1984.

[28] K. C. Le. On the edge resonance of the semi-infinite elastic plate. Bul­letin of Moscow University, Math.-Mech., (5):57-60, 1984.

[29] K. C. Le. High frequency vibrations of piezoelectric ceramic shells. Soviet Physics Doklady, 30:899-900, 1985.

[30] K. C. Le. High frequency longitudinal vibrations of elastic rods. J. Appl. Math. Mech. (PMM) , 50:335-341, 1986.

[31] K. C. Le. The theory of piezoelectric shells. J. Appl. Math. Mech. (PMM) , 50:98-105, 1986.

[32] K. C. Le. High frequency vibrations and wave propagation in elas­tic shells: variational-asymptotic approach. Int. J. Solids Structures, 34:3923-3939, 1997.

[33] A. W. Leissa. Vibration of shells. NASA, Washington, 1973.

[34] A. E. H. Love. A treatise on the mathematical theory of elasticity. Cam­bridge University Press, Cambridge, 4th edition, 1927.

[35] W. P. Mason. Piezoelectric crystals and their applications to ultrasonics. D. Van Nostrand Company, New York, 1950.

[36] G. A. Maugin. Continuum mechanics of electromagnetic solids. North­Holland, Amsterdam, 1988.

[37] G. A. Maugin and A. C. Eringen. Electrodynamics of continua. I -Foundations and solid media. Springer-Verlag, New York, 1989.

[38] J. C. Maxwell. Electricity and magnetism. Clarendon Press, Oxford, 1892.

[39] A. J. McConnell. Applications of tensor analysis. Dover publications, New York, 1957.

[40] R. D. Mindlin. The collected papers. Springer, Berlin, 1989.

[41] V. V. Novozhilov. Thin shell theory. Wolters-Noordhoff Publishing, Groningen, 2th edition, 1970.

416 BIBLIOGRAPHY

[42] J. F. Nye. Physical Properties of Crystals. Clarendon Press, London and New York, 1957.

[43J I. G. Petrovsky. Lectures on Partial Differential Equations. Dover Pub­lications, New York, 1991.

[44] T. Poston and I. Stewart. Catastrophe Theory and its Applications. Pitman, Boston, 1978.

[45J W. Prager and J. L. Synge. Approximation in elasticity based on the concepts of function space. Quart. Appl. Math., 5:1-21, 1947.

[46J D. Hilbert R. Courant. Methods of mathematical physics. Wiley­Interscience, New York, 1966.

[47J J. W. Rayleigh. The theory of sound. Dover Publications, New York, second edition, 1945.

[48J E. Reissner. Selected works in applied mechanics and mathematics. Jones & Bartlett Publishers, Boston, 1996.

[49J K. Rektorys. Variational methods in mathematics, science and engi­neering. D. Reidel Publishing Company, Dordrecht, 1980.

[50J N. N. Rogacheva. The theory of piezoelectric shells and plates. CRC Press, London, 1994.

[51 J M. Yu. Ryazantseva. Flexural vibrations of symmetrical sandwich plates. Mechanics of Solids (MTT), 20:153-159, 1985.

[52] J. L. Sanders. An improved first order approximation theory of thin shells. In NASA Report, volume 24, 1959.

[53] E. A. G. Shaw. On the resonant vibrations of thick barium titanate disks. J. Acoust. Soc. America, 30:979-984, 1956.

[54J M. Spivak. Differential Geometry. Publish or Perish, Berkeley, 1975.

[55J S. Sternberg. Lectures on Differential Geometry. Chelsea, New York, 1983.

[56J H. F. Tiersten. Linear piezoelectric plate vibrations. Plenum Press, New York,1969.

[57J S. Timoshenko. On the transverse vibrations of bars of uniform cross section. Philosophical Magazine, 43:125-131, 1922.

BIBLIOGRAPHY 417

[58] S. Timoshenko. Vibration problems in engineering. D. Van Nostrand Company, Princeton, 1955.

[59] R. A. Toupin. Saint-Venant's principle. Arch. Rational Mech. Anal., 18:83-96, 1965.

[60] I. A. Vekovisheva. Variational principles in the theory of electroelasticity. Appl. Mech., 7:129-133, 1971.

[61] G. B. Whitham. Linear and Nonlinear Waves. Wiley Interscience, New York, 1974.

[62] L. C. Young. Lectures on the calculus of variations and optimal control theory. Chelsea, New York, 1969.

Index

2-D moduli, 174, 218 3-D equations

of motion, 32, 77, 83, 103

Abbreviated indicial notation, 39, 185, 190, 192, 352

Acceleration, 30, 38, 62, 127, 211 Action functional

1-D, 127, 211, 311, 381 2-D, 62, 165, 251, 349 3-D, 33, 40, 70, 132, 216, 259,

316,352 Antiresonance, 44 Area element, 23, 27 Asymptotic equivalency, 68

Balance equations, 30, 38 Basis, 19, 61, 124, 125, 132 Bending

measures of, 61, 97, 118, 126, 210, 350

of beams, 49 Bending moments, 64, 98, 128, 167,

212 Bernouli-Euler beam, 49 Bessel functions, 104, 151 Bessel's equation, 90 Body

elastic, 30 piezoelectric, 37

Boundary conditions, 32, 65, 129, 168,213,255,314,351,383

Boundary layer, 177, 221

Boundary-value problems, 32, 39, 65, 129, 255, 314, 350, 382

Characteristic length, 74, 136, 177, 179, 221,

223, 257, 316, 385 radius of curvature, 70, 172 radius of curvatures and tor-

sion, 131, 215 Characteristic scale, 47 Characteristics, 242 Charge, 40, 196, 198,204, 207,236,

239 Christoffel symbols, 22, 120

of a surface, 28 Clamped edge, 63, 65, 90, 94, 113,

128, 129, 259, 353 Co-ordinates

cartesian, 17, 82, 149 curvilinear, 18 cylindrical, 23, 103, 150 elliptical, 333 of a rod, 130 of a shell, 68 polar, 89, 286 spherical, 23, 120

Co-vector, 20 Compatibility, 60, 68 Completeness, 35 Constants

dielectric, 38 elastic, 38 piezoelectric, 38

Constitutive equations

420

for elastic rods, 128 for elastic shells, 67, 254 for piezoelectric rods, 212 for piezoelectric shells, 171,351

Constraint, 34 Coupling, 38, 214, 235, 236, 243 Covariant derivative, 28 Cross section

centrally symmetric, 228 circular, 147, 332, 381 elliptical, 142, 225, 333 rectangular, 144, 229, 231, 333

Cross section problems, 140, 222, 224, 329, 395

Curvature Gaussian, 28 mean, 28

Curve binormal to, 24 curvature of, 24 moving triad of, 24 normal to, 24 tangent to, 24 torsion of, 25

Dispersion, 52, 79, 100, 107, 108, 147,149,280,285,300,336, 339

non-linear, 54 Displacement, 30, 37 Donnel-Mushtari operator, 99 Dual problem, 141, 229, 233

Edge mode, 343 Edge resonance, 291 Eigenfunction, 34 Eigenvalue, 34 Eigenvalue problem, 33 Elastic enthalpy, 42 Electric enthalpy

I-D, 211, 214, 227

INDEX

2-D, 165, 170 3-D, 39, 173, 216, 352, 384 longitudinal, 174, 217 transverse, 174, 217

Electric field, 38, 173, 216 1-D,211 2-D, 165

Electric induction, 38 1-D,212 2-D, 167

Electric potential, 37 1-D,21O 2-D, 164

Electroded faces, 164, 169, 179, 195, 202,349

Electroded side, 210, 213, 223, 227 Electrodes, 39, 163 Electroelastic state, 181 Elongation, 125, 210 Energy

1-D kinetic, 127, 211, 312 1-D strain, 127, 312 2-D kinetic, 62, 165, 252 2-D strain, 62, 67, 252 3-D internal, 41 3-D kinetic, 33, 41, 70, 132, 172,

216, 352, 384 3-D strain, 31, 70, 132 balance, 36, 44 longitudinal, 72, 133 shear, 133 transverse, 72, 133

Equation of electrostatics, 38, 183, 186

Equations of equilibrium, 187

Equations of motion for elastic rods, 129, 313 for elastic shells, 66, 255 for piezoelectric rods, 213, 382 for piezoelectric shells, 167, 350

INDEX

Error estimate, 183 Euclidean point space, 17 Euler equation, 53, 226 Extension

measures of, 60, 97, 118, 350

Fast variable, 53 Field

kinematically admissible, 185 statically admissible, 187

First approximation, 47 Fixed edge, 63, 65, 113, 129 Force, 30 Free edge, 63, 65, 91, 95, 113, 129,

200, 213, 255, 314 Frenet formulae, 25 Frequency, 33, 52, 79

antiresonant, 196,205,207,235, 236, 240

cut-off, 101, 258, 301, 318, 355, 386

resonant, 196, 204, 207, 235, 236,239

Frequency equation, 90, 92, 94, 95, 114, 122, 158, 160, 161, 194, 196,199,202,204,207,235, 236,239,287,289,342,344, 348, 371, 377, 402

Frequency spectra of plates, 89, 193, 285, 369 of rods, 155,234,340,400 of shells, 109, 118, 200, 305

Gauge invariance, 103 Gauss' formula, 29 Gauss' theorem, 23 Gibbs function, 42 Greek indices, 26 Group velocity, 80, 276

Hamilton's principle, 33, 41 for elastic rods, 128, 311

421

for elastic shells, 62, 63, 251 for piezoelectric rods, 212, 381 for piezoelectric shells, 166, 349

Helmholtz's decomposition, 93, 103, 150

Hooke's law, 31 Hyperbolicity, 36, 273

Impact interval, 245 Initial conditions, 32, 40, 65, 129,

256, 314, 351, 383 Internal degrees of freedom, 251,

312, 381

Jacobian, 19 Jump conditions, 170, 214, 241

Kronecker delta, 20

Lagrangian, 45, 316 average, 76, 138, 179, 181,263,

319, 357, 387 Lame constants, 32 Laplace's equation, 143 Legendre transformation, 41, 141,

191, 229 Legendre's functions, 121 Legendre's polynomials, 121 Length element, 29 Levi-Civita tensor, 103 Longitudinal impact, 240 Lower bound, 230

Mass density, 30 Mathieu's equation, 334 Membrane stresses, 64, 98, 167 Metrics, 21, 28, 120

of a rod, 131 of a shell, 69

Minimizer, 50 Mirror plane, 175, 219 Mixed faces, 169, 196

422

Moments of inertia, 139, 222

Neumann problem, 141 Newton's rule, 48, 87 Normalization, 257, 354, 385 Null Lagrangian, 270, 326, 394

Orthogonality, 35, 263

Permutation symbols, 124 Phase, 52 Phase velocity, 79 Plane wave, 36 Plate, 78

circular, 89, 193, 369 elastic, 59 piezoelectric, 171

Pochhammer equation, 153, 337 Poisson's equation, 142 Poisson's ratio, 37 Prager-Synge's identity, 183 Principal curvatures, 27 Principal terms, 50, 71, 72

Rayleigh velocity, 87 Rayleigh's formula, 37 Reciprocal basis, 21 Reduction, 274 Resonance, 44 Rod

central line of, 123, 130, 209 cross section of, 123, 209, 316 curvatures of, 124 diameter, 131 displacements, 124, 210 elastic, 123, 311, 316 kinematics, 124, 210 piezoceramic, 219, 381 piezoelectric, 209 torsion of, 124

Rotation, 126, 210 Rotation axis, 176, 219

INDEX

Scalar product, 17 Self-adjointness, 35 Separation of variables, 89, 93 Series

asymptotic, 47 Shear diaphragms, 115, 308 Shell

cylindrical, 96, 109, 200, 296, 305, 374

displacements, 60, 164 elastic, 59, 251 face surface, 163 kinematics, 60, 164 middle surface of, 59, 163 piezoceramic, 176, 349 piezoelectric, 163 spherical, 68, 118 thickness, 59, 163 thickness parameter, 99

Shifter, 69 Short-wave extrapolation, 267, 322,

360,390 Slow variables, 53 Small parameter, 46 Small terms, 46, 73, 75, 177 Spherical harmonics, 121 Spiral, 29 Stationary point, 33, 46 Strain, 30, 37, 71, 132, 173, 216 Stress, 30 Summation convention, 18 Superposition, 267 Surface

basicforms of, 26, 27, 97 co-ordinates, 25 divergence theorem, 28 moving triad of, 26 normal to, 26 tensors, 28

Symmetrization, 30 Symmetry properties, 31, 39

INDEX

Tangential polarization, 191, 381 Tension, 128, 212 Tensor, 20

contravariant, 20 covariant, 20 field, 20, 22 lowering index of, 21 product, 20 raising index of, 21 transpose of, 30

Tensor field divergence of, 23 gradient of, 22

Tensors contraction of, 21

Thickness polarization, 189, 193, 200, 237, 349

Timoshenko beam, 50 Torsional rigidity, 145 Traction, 32 Transverse isotropy, 176, 219 Triad, 124, 209 Twist

measure of, 126, 210 Twisting moment, 128, 212

Unelectroded faces, 163, 165, 176, 193, 200

Unelectroded side, 209, 211, 220, 224, 381

Upper bound, 230

Variation, 33 Variational principle

dual, 42 Variational-asymptotic

method, 45 procedure, 47, 72, 134, 176, 220,

260, 316, 353, 384 Vector, 18

field, 18

product, 17 Velocity, 30, 38, 62, 127, 211 Vibrations

axial-radial, 112

423

flexural, 89, 157, 285, 307, 312, 345

free, 32 longitudinal, 93, 193, 234, 288,

307, 311, 340 low-frequency, 72, 134, 173, 216 plane, 159, 237 radial-tangential, 120 tangential, 120 thickness-shear, 258, 317, 354,

385 thickness-stretch, 258,317,354,

385 torsional, 111, 156

Voltage, 164, 195, 197, 234, 237 Volume element, 23

of a shell, 69

Wave equation, 103 Wave number, 52, 79 Waves

amplitude of, 52 axial-radial, 101, 108, 304 dilatational, 37 flexural, 78, 148, 275, 304, 339 harmonic, 52, 79, 147, 148 longitudinal, 81, 148, 153, 277,

336 reflection of, 292 shear, 37 standing, 116 torsional, 101, 107, 148, 152,

303 Wedge product, 27 Weingarten's formula, 29 Whitham's method, 52

Young's modulus, 37