Summary
We consider axially symmetric shear waves propagating in an incompressible hyperelastic thick-walled cylindrical shell, whose strain energy function is expressible as a truncated power series in terms of the basic strain invariants. A continuous pulse is initiated at the interior boundary of the cylinder by surface tractions of finite duration. The pulse propagates away from the interior boundary, then reflects from the outer boundary, and subsequently reflects back and forth between the two boundaries of the cylinder. We analyze shock development of the first incident and first reflected wave. The incident pulse can break before it reaches the outer boundary. Using Whitham's nonlinearization technique, we determine conditions under which the incident wave breaks and which shock waves can subsequently occur. Similar calculations are carried out for the first reflection. The formulas obtained for the incident pulse provide accurate estimates of the breaking distance and time, and the location of the shock paths, for any incident shock waves that occur. Results obtained for the reflected wave cannot be used to make similar estimates, but they do reveal that once the pulse has completely left the outer boundary, the possible shock that can occur is the same as for the incident wave. Our analysis is carried out for axial shear waves. A similar analysis can be done for torsional shear waves, but not for combined axial and torsional shear wave propagation. We illustrate the conclusions of our shock analysis with numerical solutions obtained using a relaxation scheme for systems of conservation laws. Numerical results are obtained for axial shear and for combined axial and torsional shear. These results indicate that the shock behavior indicated by our analysis of axial shear is also valid for combined axial and torsional shear wave propagation.
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References
Haddow J. B., Lorimer S. A. and Tait R. J. (1987). Non-linear combined axial and torsional shear wave propagation in an incompressible hyperelastic solid. Int. J. Non-Linear Mech. 22: 297–306
Haddow J. B., Lorimer S. A. and Tait R. J. (1987). Nonlinear axial shear wave propagation in a hyperelastic incompressible solid. Acta Mech. 66: 205–216
Tait R. J., Lorimer S. A. and Haddow J. B. (1989). Finite amplitude elastic wave propagation. Wave Motion 11: 251–260
Barclay D. W. (1999). Shock front analysis for axial shear wave propagation in a hyperelastic compressible solid. Acta Mech. 133: 105–129
Barclay D. W. (2004). Shock calculations for axially symmetric shear wave propagation in a hyperelastic incompressible solid. Int. J. Non-Linear Mech. 39: 101–121
Barclay D. W. (2004). The shock pattern for axially symmetric shear wave propagation in a hyperelastic incompressible solid. Int. J. Solids Struct. 41: 5265–5284
Fu Y. B. and Scott N. H. (1991). Transverse cylindrical simple waves in elastic non-conductors. Int. J. Solids Struct. 27: 547–563
Ogden R. W. (1984). Non-linear elastic deformations. Ellis Horwood, Chichester
Alexander H. (1968). A constitutive relation for rubber-like materials. Int. J. Engng. Sci. 6: 549–563
Tschoegl N. W. (1971). Constitutive equations for elastomers. J. Polym. Sci. A-1 9: 1959–1970
James A. G., Green A. and Simpson G. M. (1975). Strain energy functions of rubber. I. Characterization of gum vulcanizates. J. Appl. Polym. Sci. 19: 2033–2058
Haines D. W. and Wilson W. D. (1979). Strain-energy density function for rubber-like materials. J. Mech. Phys. Solids 27: 345–360
Whitham G. B. (1974). Linear and nonlinear waves. John Wiley & Sons, New York
Jin J. and Xin Z. (1995). The relaxation schemes for systems of conservation laws in arbitrary space dimension. Comm. Pure Appl. Math. 48: 235–276
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Barclay, D.W. The propagation and reflection of finite amplitude cylindrically symmetric shear waves in a hyperelastic incompressible solid. Acta Mechanica 193, 17–42 (2007). https://doi.org/10.1007/s00707-007-0441-0
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DOI: https://doi.org/10.1007/s00707-007-0441-0