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Il Nuovo Cimento B (1971-1996)

, Volume 83, Issue 1, pp 61–70 | Cite as

Discontinuity waves in a viscoelastic solid saturated with an inviscid fluid

  • A. Borrelli
  • M. C. Patria
Article

Summary

In this paper we linearize the system of Szefer describing the mechanics of a viscoelastic isotropic solid saturated with an inviscid incompressible fluid and we study by means of the singular-surface theory the propagation of discontinuity waves of any order through the continuum characterized by the linear equations. Under suitable hypotheses (conditions (7)), we obtain the normal speeds of propagation of the wave front and the evolution law along the corresponding normal trajectories for transverse and longitudinal propagation.

PACS. 47.55.

Nonhomogeneous flows 

Разрывы непрерывности в упруговязком твердом теле, насыщенном невязкой жидкостью

Резюме

В этой работе линеаризуется система Шефера, описывающая механику упруговязкого изотропного твердого тела, насыщенной невязкой несжимаемой жидкостью. С помощью теории сингулярных поверхностей исследуется распространение разрывов непрерывности любого порядка через континуум, описываемый полученными, линейными уравнениями. Используя определенные гипотезы, мы получаем нормальные скорости распространения волнового фронта и закон эволюции вдоль соответствующих нормальных траекторий в случаях поперечного и продольного распространения.

Riassunto

In questo articolo, dopo aver linearizzato il sistema di equazioni di Szefer che governa la meccanica di un solido viscoelastico isotropo saturato con un fluido non viscoso e incomprimibile, si studia, mediante la teoria delle superfici singolari, la propagazione di onde di discontinuità di ogni ordine attraverso il continuo descritto dalle equazioni così ottenute. Imponendo opportune ipotesi alle costanti materiali e ai valori iniziali delle funzioni che caratterizzano il comportamento viscoelastico del solido, si ottengono le velocità normali di avanzamento del fronte d'onda e la legge di evoluzione lungo le corrispondenti traiettorie, normali nel caso sia di propagazione trasversale sia di propagazione longitudinale.

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References

  1. (1).
    Szefer's theory is exhaustively developed inG. Szefer:Symposium Franco-Polonais, Problèmes non linéaires de mécanique (Cracovie, 1977), p. 585.Google Scholar
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    For a complete review of the theory of mixtures and its applications seeR. J. Atkin andR. E. Craine:Q. J. Mech. Appl. Math.,29, 209 (1976);J. Inst. Math. Its Appl.,7, 153 (1976).MathSciNetCrossRefMATHGoogle Scholar
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    G. Szefer andG. Pallotti:Biomechanics (in press).Google Scholar
  5. (5).
    The summation convention is understood to apply to repeated, indices. Small Roman indices take the values 1, 2, 3, capital Roman indices take the values 1, 2, …, 13. Small Greek indices take the values 1, 2. The components of\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{t} \) in (2) are given by\(l_{ij} (x,t) = \int\limits_0^t {C_{ijkl} } (t - \tau )\dot e_{kl} (x,\tau )d\tau + \int\limits_0^t {\frac{{A(t - \tau )}}{R}} \dot p(x,\tau )d\tau \delta _{ij} \).Google Scholar
  6. (6).
    IfZ suffers a jump discontinuity acrossS t, we denote by\(\left[\kern-0.15em\left[ Z \right]\kern-0.15em\right]\) the difference between the values ofZ just behind and in front of the wave.Google Scholar
  7. (7).
    G. M. Fisher andM. E. Gurtin:Q. Appl. Math.,23, 257 (1965).MathSciNetMATHGoogle Scholar
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    T. Y. Thomas:Int. J. Eng. Sci.,4, 207 (1966);P. Chadwick andB. Powdrill:Int. J. Eng. Sci.,3, 561 (1965).CrossRefMATHGoogle Scholar
  9. (9).
    δ/δt=Thomas δ-time derivative,a αβ=controvariant components of the first fundamental tensor ofS t; a comma followed by Greek index denotes covariant differentiation with respect to {n α}.Google Scholar
  10. (10).
    T. Y. Thomas:J. Math. Mech.,6, 455 (1957).MathSciNetMATHGoogle Scholar

Copyright information

© Società Italiana di Fisica 1984

Authors and Affiliations

  • A. Borrelli
    • 1
  • M. C. Patria
    • 1
  1. 1.Istituto Matematico dell'UniversitàFerraraItalia

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