Abstract
This paper is concerned with the effect of finite pure homogeneous biaxial stretch together with simple shear deformation on the reflection from a plane boundary of elastic waves propagating in a half-space of incompressible isotropic elastic material. This generalizes the previous work in which, separately, pure homogeneous strain and simple shear were considered. For a special class of constitutive law, it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave for every angle of incidence. The amplitude of the surface wave may vanish at certain discrete angles depending on the state of stress, biaxial stretch and simple shear deformation and then specialized to recover results obtained previously. The amplitude of the reflected wave is independent of the pre-stress but does depend upon the magnitude of deformation under consideration. The dependence of the reflected and surface wave behavior on the angle of incidence, amount of shear strain, biaxial stretch and the state of stress is illustrated graphically.
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References
Biot M.A.: Mechanics of Incremental Deformations. Wiley, New York (1965)
Connor P., Ogden R.W.: The effect of shear on the propagation of elastic surface waves. Int. J. Eng. Sci. 33, 973–982 (1995)
Connor P., Ogden R.W.: The influence of shear strain and hydrostatic stress on stability and elastic waves in a layer. Int. J. Eng. Sci. 34, 375–397 (1996)
Destrade M., Ogden R.W.: Surface waves in a stretched and sheared incompressible elastic material. Int. J. Nonlinear Mech. 40, 241–253 (2005)
Dowaikh M.A., Ogden R.W.: On surface waves and deformations in a pre-stressed incompressible elastic solid. IMA J. Appl. Math. 44, 261–284 (1990)
Dowaikh M.A., Ogden R.W.: On surface waves and deformations in a compressible elastic half-space. Stab. Appl. Anal. Contin Media 1, 27–45 (1991)
Hayes M.A., Rivlin R.S.: Surface waves in deformed elastic materials. Arch. Rat. Mech. Anal. 8, 358–380 (1961)
Hussain W., Ogden R.W.: On the reflection of plane waves at the boundary of an elastic half-space subject to simple shear. Int. J. Eng. Sci. 37, 1549–1576 (1999)
Hussain W., Ogden R.W.: Reflection and transmission of plane waves at a shear-twin interface. Int. J. Eng. Sci. 38, 1789–1810 (2000)
Hussain W.: Asymptotic analysis of the dispersion relation of an incompressible elastic layer of uniform thickness. Math Track 2, 84–97 (2006)
Ogden, R.W.: (1997) Non-linear Elastic Deformations. Dover Publications, Mineola
Ogden R.W., Sotiropoulos D.A.: The effect of pre-stress on the propagation and reflection of plane waves in incompressible elastic solids. IMA J. Appl. Math. 59, 95–121 (1997)
Ogden R.W., Sotiropoulos D.A.: Reflection of plane waves from the boundary of pre-stressed compressible elastic half-space. IMA J. Appl. Math. 61, 61–90 (1998)
Rogerson G.A., Sandiford K.J.: On small amplitude vibrations of pre-stressed laminates. Int. J. Eng. Sci. 34, 853–872 (1996)
Rogerson G.A., Sandiford K.J.: Flexural waves in incompressible pre-stressed elastic composites. Q. J. Mech. Appl. Math. 50, 597–624 (1997)
Wolfram, S.: Mathematica, version 5, Wolfram Research, Champaign, Illinois (2003)
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Hussain, W. Reflection of plane waves at the boundary of a pre-stressed elastic half-space subject to biaxial stretch and simple shear. Acta Mech 203, 63–75 (2009). https://doi.org/10.1007/s00707-008-0021-y
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DOI: https://doi.org/10.1007/s00707-008-0021-y