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On Saint-Venant’s Principle in Finite Anti-Plane Shear: An Energy Approach

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Mechanics and Thermodynamics of Continua

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Abstract

Some time ago a version of Saint-Venant’s principle was formulated and established for finite elastostatics [1]. As was discussed in [1], the issues of concern in connection with Saint-Venant’s principle in the nonlinear theory of elasticity are considerably more involved then those arising in the linear theory. (For a survey of results on Saint-Venant’s principle, primarily for linear theories, see e.g. [2–5].) One difficulty is that the appropriate Saint-Venant solutions need to be carefully characterized (see e.g. [6–13] and the references cited therein). Secondly, in the absence of superposition, consideration of self-equilibrated end loads is no longer sufficient. Furthermore, instabilities may have to be taken into account. Also the decay rate for end effects, even if exponential, might depend on the overall loading as well as on geometry and material characteristics. Several of these issues have been considered in recent studies in the nonlinear elasticity context [1, 14–23] as well as in investigations of spatial decay of solutions of nonlinear elliptic partial differential equations [24–32].

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References

  1. Horgan, C. O., & Knowles, J. K., The effect of nonlinearity on a principle of Saint-Venant type. Journal of Elasticity 11, 271–291 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  2. Horgan, C. O., & Knowles, J. K., Recent developments concerning Saint-Venant’s principle, Advances in Applied Mechanics, J. W. Hutchinson, ed., Vol. 23, pp. 179–269. Academic Press, New York, 1983.

    Google Scholar 

  3. Horgan, C. O., Recent developments concerning Saint-Venant’s principle: an update. Applied Mechanics Reviews 42 (1989) (in press).

    Google Scholar 

  4. Fichera, G., Il principio di Saint-Venant: Intuizione dell’ingegnere e rigore del matematico. Rend. di Mat. Serie VI 10, 1–24 (1977).

    MathSciNet  MATH  Google Scholar 

  5. Fichera, G., Remarks on Saint-Venant’s principle. Rend. di Mat. Serie VI 12, 181–200 (1979).

    MathSciNet  ADS  MATH  Google Scholar 

  6. Ericksen, J. L., On the formulation of St. Venant’s problem, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium (R. J. Knops, ed.) Vol. I, pp. 158–186. Pitman, London, 1977.

    Google Scholar 

  7. Muncaster, R. G., Saint-Venant’s problem in nonlinear elasticity: a study of cross-sections, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium (R. J. Knops, ed.), Vol. IV, pp. 17–75. Pitman, London, 1979.

    Google Scholar 

  8. Muncaster, R. G., Saint-Venant’s problem for slender prisms. Utilitas Math. 23, 75–101 (1983).

    MathSciNet  MATH  Google Scholar 

  9. Ericksen, J. L., Saint-Venant’s problem for elastic prisms, in Systems of Nonlinear Partial Differential Equations (J. M. Ball, ed.), pp. 87–93. D. Reidel, Dordrecht, 1983.

    Google Scholar 

  10. Ericksen, J. L., Problems for infinite elastic prisms, in Systems of Nonlinear Partial Differential Equations (J. M. Ball, ed.), pp. 80–86. D. Reidel, Dordrecht, 1983.

    Google Scholar 

  11. Kinderlehrer, D., A relation between semi-inverse and Saint-Venant solutions for prisms. SIAM J. Math. Anal. 17, 626–640 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  12. Kinderlehrer, D., Remarks about Saint-Venant solutions in finite elasticity, in Nonlinear Functional Analysis and its Applications: Proceedings of Symposia in Pure Mathematics (F. E. Browder, ed.), Vol. 45, Part 2, pp. 37–50. American Mathematical Society, Providence, Rhode Island, 1986.

    Google Scholar 

  13. Mielke, A., Saint-Venant’s problem and semi-inverse solutions in nonlinear elasticity. Arch. Rational Mech. Anal. 102, 205–229 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Breuer, S., & Roseman, J. J., On Saint-Venant’s principle in three-dimensional nonlinear elasticity. Arch. Rational Mech. Anal. 63, 191–203 (1977).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Breuer, S., & Roseman, J. J., Saint-Venant’s principle in nonlinear plane elasticity with sufficiently small strains. Arch. Rational Mech. Anal. 80, 19–37 (1982).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Knops, R. J., & Payne, L. E., A Saint-Venant principle for nonlinear elasticity. Arch. Rational Mech. Anal. 81, 1–12 (1983).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Horgan, C. O., & Abeyaratne, R., Finite anti-plane shear of a semi-infinite strip subject to a self-equilibrated end traction. Q. Appl. Math. 40, 407–417 (1983).

    MathSciNet  MATH  Google Scholar 

  18. Galdi, G. P., Knops, R. J., & Rionero, S., Asymptotic behavior in the nonlinear elastic beam. Arch. Rational Mech. Anal. 87, 305–318 (1985).

    MathSciNet  ADS  MATH  Google Scholar 

  19. Abeyaratne, R., Horgan, C. O., & Chung, D.-T., Saint-Venant end effects for incremental plane deformations of incompressible nonlinearly elastic materials. J. Appl. Mech. 52, 847–852 (1985).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Mielke, A., Normal hyperbolicity and Saint-Venant’s principle, Preprint 10, Mathematical Problems in Nonlinear Mechanics, Univ. Stuttgart (1988).

    Google Scholar 

  21. Durban, D., & Stronge, W. J., Diffusion of self-equilibrating end loads in elastic solids. J. Appl. Mech. 55, 492–495 (1988).

    Article  ADS  Google Scholar 

  22. Vafeades, P., & Horgan, C. O., Exponential decay estimates for solutions of the von Kármán equations on a semi-infinite strip. Arch. Rational Mech. Anal. 104, 1–25 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Quintanilla, R., Asymptotic behavior of solutions in elasticity (preprint, Univ. Polit. de Catalunya, Barcelona, Spain, 1987).

    Google Scholar 

  24. Horgan, C. O., & Payne, L. E., Decay estimates for second-order quasilinear partial differential equations. Advances in Appl. Math. 5, 309–332 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  25. Horgan, C. O., & Payne, L. E., Decay estimates for a class of second-order quasi-linear equations in three dimensions. Arch. Rational Mech. Anal. 86, 279–289 (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Horgan, C. O., A note on the spatial decay of a three-dimensional minimal surface over a semi-infinite cylinder. J. Math. Anal. Appl. 107, 285–290 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  27. Breuer, S., & Roseman, J. J., Phragmén-Lindelöf decay theorems for classes of nonlinear Dirichlet problems in a circular cylinder. J. Math. Anal. Appl. 113, 59–77 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  28. Breuer, S., & Roseman, J. J., Decay theorems for nonlinear Dirichlet problems in semi-infinite cylinders. Arch. Rational Mech. Anal. 94, 363–371 (1986).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Horgan, C. O., Some applications of maximum principles in linear and nonlinear elasticity, in Maximum Principles and Eigenvalue Problems in Partial Differential Equations, pp. 49–67, ed. P. W. Schaefer, Pitman Research Notes in Mathematics Series, 175, Longman, New York, 1988.

    Google Scholar 

  30. Horgan, C. O., & Payne, L. E., Decay estimates for a class of nonlinear boundary value problems in two dimensions. SIAM J. Math. Anal. 20, 782–788 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  31. Horgan, C. O., & Payne, L. E., On the asymptotic behavior of solutions of inhomogeneous second-order quasilinear partial differential equations. Q. Appl. Math. 47 (1989).

    Google Scholar 

  32. Horgan, C. O., & Siegel, D., On the asymptotic behavior of a minimal surface over a semi-infinite strip. J. Math. Anal. Appl. (in press).

    Google Scholar 

  33. Knowles, J. K., On finite anti-plane shear for incompressible elastic materials. J. Austral. Math. Soc. Series B 19, 400–415 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  34. Knowles, J. K., The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids. Int. J. Fracture 13, 611–639 (1977).

    Article  MathSciNet  Google Scholar 

  35. Gurtin, M. E., Topics in Finite Elasticity, SIAM Regional Conference Series in Applied Mathematics, No. 35, SIAM, Philadelphia, 1981.

    Google Scholar 

  36. Mitrinovic, D. S., Analytic Inequalities, Springer-Verlag, Berlin, 1970.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Virginia, Charlottesville, USA

    C. O. Horgan & L. E. Payne

  2. Department of Mathematics, Cornell University, Ithaca, New York, USA

    C. O. Horgan & L. E. Payne

Authors
  1. C. O. Horgan
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  2. L. E. Payne
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Editors and Affiliations

  1. 21 Disraeli St., 92222, Jerusalem, Israel

    Hershel Markovitz

  2. Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Victor J. Mizel  & David R. Owen  & 

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Dedicated to Bernard Coleman on the occasion of his 60th birthday

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© 1991 Springer-Verlag Berlin Heidelberg

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Horgan, C.O., Payne, L.E. (1991). On Saint-Venant’s Principle in Finite Anti-Plane Shear: An Energy Approach. In: Markovitz, H., Mizel, V.J., Owen, D.R. (eds) Mechanics and Thermodynamics of Continua. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75975-8_14

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  • DOI: https://doi.org/10.1007/978-3-642-75975-8_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52999-6

  • Online ISBN: 978-3-642-75975-8

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