Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-28T01:17:05.479Z Has data issue: false hasContentIssue false
  • English
  • Français

Antiplane Wave Scattering from a Cylindrical Void in a Pre-Stressed Incompressible Neo-Hookean Material

Published online by Cambridge University Press:  20 August 2015

William J. Parnell  and
I. David Abrahams
William J. Parnell*
Affiliation:
School of Mathematics, Alan Turing Building, University of Manchester, Manchester, M13 9PL, UK
I. David Abrahams*
Affiliation:
School of Mathematics, Alan Turing Building, University of Manchester, Manchester, M13 9PL, UK
*
Corresponding author.Email:william.parnell@manchester.ac.uk
Email address:david.abrahams@manchester.ac.uk
Get access

Abstract

An isolated cylindrical void is located inside an incompressible nonlinear-elastic medium whose constitutive behaviour is governed by a neo-Hookean strain energy function. In-plane hydrostatic pressure is applied in the far-field so that the void changes its radius and an inhomogeneous region of deformation arises in the vicinity of the void. We consider scattering from the void in the deformed configuration due to an incident field (of small amplitude) generated by a horizontally polarized shear (SH) line source, a distance r0 (R0) away from the centre of the void in the deformed (undeformed) configuration. We show that the scattering coefficients of this scattered field are unaffected by the pre-stress (initial deformation). In particular, they depend not on the deformed void radius a or distance r0, but instead on the original void size A and original distance R0.

Keywords

74B1574B2074J2074Q15Pre-stressincompressibleneo-Hookeanrubberincremental SH wavesscatteringline source

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 367 - 382
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover, New York, 1965.Google Scholar
[2]Bertoldi, K. and Boyce, M. C., Wave propagation and instabilities in monolithic and periodically structured elastomeric materials undergoing large deformations, Phys. Rev. B, 78 (2008), 184107.CrossRefGoogle Scholar
[3]Brun, M., Lopez-Pamies, O. and Ponte-Castaneda, P., Homogenization estimates for fiber reinforced elastomers with periodic microstructures, Int. J. Solids Struct., 44 (2007), 5953–5979.CrossRefGoogle Scholar
[4]deBotton, G., Hariton, I. and Socolsky, E. A., Neo-Hookean fiber-reinforced composites in finite elasticity, J. Mech. Phys. Solids, 54 (2006), 533–559.CrossRefGoogle Scholar
[5]Destrade, M. and Scott, N., Surface waves in a deformed isotropic hyperelastic material subject to an isotropic internal constraint, Wave Motion, 40 (2004), 347–357.CrossRefGoogle Scholar
[6]Degtyar, A. D., Huang, W. and Rokhlin, S.I., Wave propagation in stressed composites, J. Acoust. Soc. Am., 104 (1998), 2192–2199.Google Scholar
[7]Gei, M., Movchan, A. B. and Bigoni, D., Band-gap shift and defect-induced annihilation in prestressed elastic structures, J. Appl. Phys., 105 (2009), 063507.Google Scholar
[8]Graff, K. F., Wave Motion in Elastic Solids, Dover, New York, 1975.Google Scholar
[9]Green, A. E. and Zerna, W., Theoretical Elasticity, Dover, New York, 1992.Google Scholar
[10]Kaplunov, J. D. and Rogerson, G. A., An asymptotically consistent model for long-wave high frequency motion in a pre-stressed elastic plate, Math. Mech. Solids, 7 (2002), 581–606.CrossRefGoogle Scholar
[11]Leungvichcharoen, S. and Wijeyewickrema, A. C., Stress concentration factors and scattering cross-section for plane SH wave scattering by a circular cavity in a pre-stressed elastic medium, J. Appl. Mech. JSCE, 7 (2004), 15–20.CrossRefGoogle Scholar
[12]Martin, P. A., Multiple Scattering, Interaction of Time-Harmonic Waves with N Obstacles, Cambridge University Press, Cambridge, 2006.Google Scholar
[13]Ogden, R. W., Elastic deformations of rubberlike solids, in Mechanics of Solids, the Rodney Hill 60th Anniversary Volume, Hopkins, H. G. and Sewell, M. J. eds, Oxford, Pergamon Press, (1982), 499–537.Google Scholar
[14]Ogden, R. W., Recent advances in the phenomenological theory of rubber elasticity, Rubber Chem. Tech., 59 (1986), 361–383.Google Scholar
[15]Ogden, R. W., Nonlinear Elasticity, Dover, New York, 1997.Google Scholar
[16]Pao, Y.-H. and Mow, C.-C., Diffraction of Elastic Waves and Dynamic Stress Concentrations, Rand Coorporation, New York, 1973.CrossRefGoogle Scholar
[17]Parnell, W. J., Effective wave propagation in a pre-stressed nonlinear elastic composite bar, IMA J. Appl. Math., 72 (2007), 223–244.Google Scholar
[18]Waterman, P. C. and Truell, R., Multiple scattering of waves, J. Math. Phys., 2 (1961), 512–537.Google Scholar
[19]Yang, R.-B. and Mal, A. K., Elastic waves in a composite containing inhomogeneous fibers, Int. J. Eng. Sci., 34 (1996), 67–79.CrossRefGoogle Scholar