Abstract
We study incremental wave propagation for what is seemingly the simplest boundary value problem, namely that constitued by the plane interface of a semi-infinite solid. With a view to model loaded elastomers and soft tissues, we focus on incompressible solids, subjected to large homogeneous static deformations. The resulting strain-induced anisotropy complicates matters for the incremental boundary value problem, but we transpose and take advantage of powerful techniques and results from the linear anisotropic elastodynamics theory. In particular we cover several situations where fully explicit secular equations can be derived, including Rayleigh and Stoneley waves in principal directions, and Rayleigh waves polarized in a principal plane or propagating in any direction in a principal plane. We also discuss the merits of polynomial secular equations with respect to more robust, but less transparent, exact secular equations.
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Bibliography
D.M. Barnett. Bulk, surface, and interfacial waves in anisotropic linear elastic solids. International Journal of Solids and Structures, 37:45–54, 2000.
D.M. Barnett and J. Lothe. Free surface (Rayleigh) waves in anisotropic elastic half-spaces: the surface impedance method. Proceedings of the Royal Society of London, Series A, 402:135–152, 1985.
M.A. Biot. Surface instability of rubber in compression. Applied Science Research, Series A, 12:168–182, 1963.
Ph. Boulanger and M. Hayes. Finite-amplitude waves in deformed Mooney-Rivlin material. Quarterly Journal of Mechanics and Applied Mathematics, 45:575–593, 1992.
P. Chadwick. Continuum Mechanics. Allen & Unwin, 1976.
P. Chadwick. Interfacial and surface waves in pre-strained isotropic elastic media. ZAMP, 46:S51–S71, 1995.
P. Chadwick. The application of the Stroh formalism to prestressed elastic media. Mathematics and Mechanics of Solids, 2:379–403, 1997.
P. Chadwick and D.A. Jarvis. Surface waves in a pre-stressed elastic body. Proceedings of the Royal Society of London, Series A, 366:517–536, 1979a.
P. Chadwick and D.A. Jarvis. Interfacial waves in a pre-strained neo-Hookean body. Quarterly Journal of Mechanics and Applied Mathematics, 32:387–399, 1979b.
B. Collet and M. Destrade. Explicit secular equations for piezoacoustic surface waves: Shear-horizontal modes. Journal of the Acoustical Society of America, 116:3432–3442, 2004.
B. Collet and M. Destrade. Explicit secular equations for piezoacoustic surface waves: Rayleigh modes. Journal of Applied Physics, 98:054903, 2005.
P. Connor and R.W. Ogden. The effect of shear on the propagation of elastic surface waves. International Journal of Engineering Science, 33:3432–3442, 1995.
P. Connor and R.W. Ogden. The influence of shear strain and hydrostatic stress on stability and elastic waves in a layer. International Journal of Engineering Science, 34:375–397, 1996.
P.K. Currie. The secular equation for Rayleigh waves on elastic crystals. Quarterly Journal of Mechanics and Applied Mathematics, 32:163–173, 1979.
M. Destrade. Finite-amplitude inhomogeneous plane waves in a deformed Mooney-Rivlin material. Quarterly Journal of Mechanics and Applied Mathematics, 53:343–361, 2000.
M. Destrade. Small-amplitude inhomogeneous plane waves in a deformed Mooney-Rivlin material. Quarterly Journal of Mechanics and Applied Mathematics, 55:109–126, 2002.
M. Destrade. Elastic interface acoustic waves in twinned crystals. International Journal of Solids and Structures, 40:7375–7383, 2003.
M. Destrade. On interface waves in misoriented pre-stressed incompressible elastic solids. IMA Journal of Applied Mathematics, 70:3–14, 2005.
M. Destrade and Y.B. Fu. The speed of interfacial waves polarized in a symmetry plane. International Journal of Engineering Science, 44:26–36, 2006.
M. Destrade and R.W. Ogden. Surface waves in a stretched and sheared incompressible elastic material. International Journal of Non Linear Mechanics, 40:241–253, 2005.
M. Destrade, M. Otténio, A.V. Pichugin, and G.A. Rogerson. Non-principal surface waves in deformed incompressible materials. International Journal of Engineering Science, 43:1092–1106, 2005.
M.A. Dowaikh and R.W. Ogden. On surface waves and deformations in a pre-stressed incompressible elastic solid. IMA Journal of Applied Mathematics, 44:261–284, 1990.
M.A. Dowaikh and R.W. Ogden. Interfacial waves and deformations in pre-stressed elastic media. Proceedings of the Royal Society of London, Series A, 433:313–328, 1991.
J.N. Flavin. Surface waves in pre-stressed Mooney material. Quarterly Journal of Mechanics and Applied Mathematics, 16:441–449, 1963.
Y.B. Fu. Existence and uniqueness of edge waves in a generally anisotropic elastic plate. Quarterly Journal of Mechanics and Applied Mathematics, 56:605–616, 2003.
Y.B. Fu. An explicit expression for the surface-impedance matrix of a generally anisotropic incompressible elastic material in a state of plane strain. International Journal of Non Linear Mechanics, 40:229–239, 2005a.
Y.B. Fu. An integral representation of the surface-impedance tensor for incompressible elastic materials. Journal of Elasticity, 81:75–90, 2005b.
Y.B. Fu and D.W. Brookes. An explicit expression for the surface-impedance tensor of a compressible monoclinic material in a state of plane strain. IMA Journal of Applied Mathematics, 71:434–445, 2006.
Y.B. Fu and A. Mielke. A new identity for the surface impedance matrix and its application to the determination of surface-wave speeds. Proceedings of the Royal Society of London, Series A, 458:2523–2543, 2002.
A.N. Gent. A new constitutive relation for rubber. Rubber Chemistry and Technology, 69:59–61, 1996.
A.N. Guz. Elastic waves in bodies with initial (residual) stresses. International Applied Mechanics, 38:23–59, 2002.
M.A. Hayes and R.S. Rivlin. Surface waves in deformed elastic materials. Archives for Rational Mechanics and Analysis, 8:358–380, 1961.
C.O. Horgan and G. Saccomandi. A description of arterial wall mechanics using limiting chain extensibility constitutive models. Biomechanics Modeling in Mechanobiology, 1:251–266, 2003.
G. Huet. Fronts d’Ondes Ultrasonores à la Surface d’un Milieu Semi-Infini Anisotrope: Théorie des Rayons Réels et Complexes. PhD Thesis, Université de Bordeaux 1, 2006.
W. Hussain and R.W. Ogden. Reflection and transmission of plane waves at a shear-twin interface. International Journal of Engineering Science, 38:1789–1810, 2000.
K.A. Ingebrigsten and A. Tonning. Elastic surface waves in crystals. Physical Review, 184:942–951, 1969.
A.R. Karduna, H.R. Halerpin, and F.C.P. Yin. Experimental and numerical analyses of indentation in finite-sized isotropic and anisotropic rubber-like materials. Annals of Biomedical Engineering, 25:1009–1016, 1997.
J. Merodio and R.W. Ogden. Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation. Archives of Mechanics, 54:525–552, 2002.
V.G. Mozhaev. Approximate analytical expressions for the velocity of Rayleigh waves in isotropic media and on the basal plane in high-symmetry crystals. Soviet Physics Acoustics, 37:1009–1016, 1991.
V.G. Mozhaev. Some new ideas in the theory of surface acoustic waves in anisotropic media. In D.F. Parker and A.H. England, editors, IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, pages 455–462. Kluwer, 1995.
R.W. Ogden. Elements of the theory of finite elasticity. In Y.B. Fu and R.W. Ogden, editors, Nonlinear Elasticity: Theory and Applications, pages 1–58. Cambridge University Press, 2001.
R.W. Ogden. List of publications. Mathematics and Mechanics of Solids, 8,9:449–450, 3–4, 442–443, 2003, 2004.
Y.-H. Pao, W. Sachse, and H. Fukuoka. Acoustoelasticity and ultrasonic measurements of residual stresses. In W.P. Mason and R.N. Thurston, editors, Physical Acoustics, Vol. 17, pages 61–143. Academic Press, 1984.
A.V. Pichugin. Asymptotic Models for Long Wave Motion in a Pre-Stressed Incompressible Elastic Plate. PhD Thesis, University of Salford, 2001.
M.L. Raghavan and D.A. Vorp. Toward a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: identification of a finite strain constitutive model and evaluation of its applicability. Journal of Biomechanics, 33:475–482, 2000.
Lord Rayleigh. On waves propagated along the plane surface of an elastic solid. Proceedings of the London Mathematical Society, 17:4–11, 1885.
G.A. Rogerson and K.J. Sandiford. Harmonic wave propagation along a non-principal direction in a pre-stressed elastic plate. International Journal of Engineering Science, 37:1663–1691, 1999.
G. Saccomandi. Phenomenology of rubber-like materials, CISM Lecture Notes 452. In G. Saccomandi and R.W. Ogden, editors, Mechanics and Thermomechanics of Rubberlike Solids, pages 91–134. Springer, 2004.
R. Stoneley. Elastic waves at the surface of separation of two solids. Proceedings of the Royal Society of London, 106:416–428, 1924.
A.N. Stroh. Some analytic solutions for rayleigh waves in cubic crystals. Journal of Mathematics and Physics, 41:77–103, 1962.
D.B. Taylor. Surface waves in anisotropic media: the secular equation and its numerical solution. Proceedings Royal Society of London, Series A, 376:265–300, 1981.
D.B. Taylor and P.K. Currie. The secular equation for Rayleigh waves on elastic crystals ii: corrections and additions. Quarterly Journal of Mechanics and Applied Mathematics, 34:231–234, 1981.
R.M. Taziev. Bipartial surface acoustic waves. Soviet Physics Acoustics, 33:100–103, 1987.
R.M. Taziev. Dispersion relation for acoustic waves in an anisotropic elastic half-space. Soviet Physics Acoustics, 35:535–538, 1989.
I. Thompson, I.D. Abrahams, and A.N. Norris. On the existence of flexural edge waves on thin orthotropic plates. Journal of the Acoustical Society of America, 112:1756–1765, 2002.
T.C.T. Ting. Anisotropic Elasticity: Theory and Applications. University Press, 1996.
T.C.T. Ting. The polarization vector and secular equation for surface waves in an anisotropic half-space. International Journal of Solids and Structures, 41:2065–2083, 2004.
T.C.T. Ting. The polarization vectors at the interface and the secular equation for Stoneley waves in monoclinic bimaterials. Proceedings of the Royal Society of London, Series A, 461:711–731, 2005.
L.R.G. Treloar. The Physics of Rubber Elasticity. Clarendon Press, 1949.
A.J. Willson. Surface and plate waves in biaxially-stressed elastic media. Pure and Applied Geophysics, 102:182–192, 1973a.
A.J. Willson. Surface waves in uniaxially-stressed Mooney material. Pure and Applied Geophysics, 112:352–364, 1973b.
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Destrade, M. (2007). Interface Waves in Pre-Stressed Incompressible Solids. In: Destrade, M., Saccomandi, G. (eds) Waves in Nonlinear Pre-Stressed Materials. CISM Courses and Lectures, vol 495. Springer, Vienna. https://doi.org/10.1007/978-3-211-73572-5_3
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DOI: https://doi.org/10.1007/978-3-211-73572-5_3
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