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Wave motion in relaxation-testing of nonlinear elastic media. (English) Zbl 1139.74423
Summary: Relaxation testing is a fundamental tool for mechanical characterization of viscoelastic materials. Inertial effects are usually neglected when analysing these tests. However, relaxation tests involve sudden stretching of specimens, which causes propagation of waves whose effects may be significant. We study wave motion in a nonlinear elastic model specimen and derive expressions for the conditions under which loading may be considered to be quasi-static. Additionally, we derive expressions for wave properties such as wave speed and the time needed to reach a steady-state wave pattern. These expressions can be used to deduce nonlinear elastic material properties from dynamic experiments.

74J30 Nonlinear waves in solid mechanics
74B20 Nonlinear elasticity
74D10 Nonlinear constitutive equations for materials with memory
74S20 Finite difference methods applied to problems in solid mechanics
Full Text: DOI
[1] Whitham, G.B. 1974 Linear and nonlinear waves. New York: Wiley. · Zbl 0373.76001
[2] Achenbach, J.D. 1973 Wave propagation in elastic solids. North-Holland: Amsterdam. · Zbl 0268.73005
[3] Armen, H. & Garnet, H. 1976 Finite element analysis of elastic-plastic wave propagation effects. <i>Comput. Struct.</i>&nbsp;<b>6</b>, 45–53. · Zbl 0332.73076
[4] Barclay, D.W. 2004 Shock calculations for axially symmetric shear wave propagation in a hyperelastic incompressible solid. <i>Int. J. Nonlinear Mech.</i>&nbsp;<b>39</b>, 101–121. · Zbl 1348.74170
[5] Bukiet, B., Pelesko, J., Li, X.L. & Sachdev, P.L. 1996 A characteristic based numerical method with tracking for nonlinear wave equation. <i>Comput. Math. Appl.</i>&nbsp;<b>31</b>, 75–99. · Zbl 0849.65071
[6] Cohen, R.E., Hooley, C.J. & McCrum, N.G. 1976 Viscoelastic creep of collagenous tissue. <i>J. Biomech.</i>&nbsp;<b>9</b>, 175–184.
[7] Eringen, A.C. & Suhubi, E.S. 1975 <i>Elastodynamics</i> New York: Academic.
[8] Flügge, W. 1975 Viscoelasticity. New York: Springer.
[9] Fox, P.A. 1955 Perturbation theory of wave propagation based on method of characteristic. <i>J. Math. Phys.</i>&nbsp;<b>34</b>, 133–151.
[10] Giacaglia, G.E.O. 1972 Perturbation methods in nonlinear systems. New York: Springer. · Zbl 0282.34001
[11] Ginsberg, J. 2001 Mechanical and structural vibrations. New York: Wiley.
[12] Jeffrey, A. & Taniuti, T. 1964 Nonlinear wave propagation with applications to physics and magnetohydrodynamics, New York: Academic, pp. 3–110. · Zbl 0117.21103
[13] Kakutani, T., Ono, H., Taniuti, T. & Wei, C.C. 1968 Reductive perturbation method in nonlinear wave propagation. II. Application to hydromagnetic waves in cold plasmas. <i>J. Phys. Soc. Jpn</i>&nbsp;<b>24</b>, 1159–1166.
[14] Moodie, T.B. & Barclay, D.W. 1991 Pade-extended wavefront expansions and nonlinear dissipative waves. <i>Int. J. Nonlinear Mech.</i>&nbsp;<b>26</b>, 25–39. · Zbl 0733.73027
[15] Moodie, T.B., He, Y. & Barclay, D.W. 1991 Wavefront expansions and nonlinear hyperbolic waves. <i>Wave Motion</i>&nbsp;<b>14</b>, 347–367. · Zbl 0770.35041
[16] Nayfeh, A.H. & Mook, D.T. 1979 Nonlinear oscillations, New York: Wiley, pp. 544–555. · Zbl 0418.70001
[17] Oncu, T.S. & Moodie, T.B. 1993 An asymptotic analysis of a hyperbolic history problem in one-dimensional viscoelasticity. <i>Stud. Appl. Math.</i>&nbsp;<b>90</b>, 91–117.
[18] Press, W.H., Teukolsky, S.A., Vetterling, W.T. & Flannery, B.P. 1992 Numerical recipes in Fortran: the art of scientific computing. 2nd edn. Cambridge: Cambridge University Press. · Zbl 0778.65002
[19] Pryse, K.M., Nekouzadeh, A.N., Genin, G.M., Elson, E.L. & Zahalak, G.I. 2003 Incremental mechanics of collagen gels: new experiments and a new viscoelastic model. <i>Ann. Biomed. Eng.</i>&nbsp;<b>31</b>, 1287–1296.
[20] Shamardan, A.B. 1990 Central finite-difference scheme for nonlinear dispersive waves. <i>Comput. Math. Appl.</i>&nbsp;<b>19</b>, 9–15. · Zbl 0728.65095
[21] Skorokhod, A.V., Hoppensteadt, F.C. & Salehi, H.D. 2002 Random perturbation methods with applications in science and engineering. New York: Springer. · Zbl 0998.37001
[22] Suf, S.G. & Farris, T.N. 1994 Generalized characteristic method of elastodynamics. <i>Int. J. Solids Struct.</i>&nbsp;<b>31</b>, 109–126. · Zbl 0848.73079
[23] Thoo, J.B. & Hunter, J.K. 2003 Nonlinear hyperbolic wave propagation in a one-dimensional random medium. <i>Wave Motion</i>&nbsp;<b>37</b>, 381–405. · Zbl 1163.74450
[24] Van Den Abeele, K.E. 1996 Elastic pulsed wave propagation in media with second or higher order nonlinearity. <i>J. Acoust. Soc. Am.</i>&nbsp;<b>99</b>, 3334–3345.
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