The effect of constitutive law perturbations on finite antiplane shear deformations of a semi-infinite strip
Authors:
C. O. Horgan and L. E. Payne
Journal:
Quart. Appl. Math. 51 (1993), 441-465
MSC:
Primary 73C10; Secondary 73B99, 73C50, 73G05
DOI:
https://doi.org/10.1090/qam/1233524
MathSciNet review:
MR1233524
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Abstract: This paper is concerned with assessing the effects of small perturbations in the constitutive laws on antiplane shear deformation fields arising in the theory of nonlinear elasticity. The mathematical problem is governed by a second-order quasilinear partial differential equation in divergence form. Dirichlet (or Neumann) boundary-value problems on a semi-infinite strip, with nonzero data on one end only, are considered. Such problems arise in investigation of Saint-Venant end effects in elasticity theory. The main result provides a comparison between two solutions, one of which is a solution to a simpler equation, for example Laplace’s equation. Three examples involving perturbations of power-law material models are used to illustrate the results.
- James K. Knowles, On finite anti-plane shear for incompressible elastic materials, J. Austral. Math. Soc. Ser. B 19 (1975/76), no. 4, 400–415. MR 475116, DOI https://doi.org/10.1017/S0334270000001272
- James K. Knowles, On finite anti-plane shear for incompressible elastic materials, J. Austral. Math. Soc. Ser. B 19 (1975/76), no. 4, 400–415. MR 475116, DOI https://doi.org/10.1017/S0334270000001272
- Morton E. Gurtin, Topics in finite elasticity, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 35, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. MR 599913
- Qing Jiang and James K. Knowles, A class of compressible elastic materials capable of sustaining finite anti-plane shear, J. Elasticity 25 (1991), no. 3, 193–201. MR 1115552, DOI https://doi.org/10.1007/BF00040926
- C. O. Horgan and J. K. Knowles, The effect of nonlinearity on a principle of Saint-Venant type, J. Elasticity 11 (1981), no. 3, 271–291. MR 625953, DOI https://doi.org/10.1007/BF00041940
- C. O. Horgan and R. Abeyaratne, Finite anti-plane shear of a semi-infinite strip subject to a self-equilibrated end traction, Quart. Appl. Math. 40 (1982/83), no. 4, 407–417. MR 693875, DOI https://doi.org/10.1090/S0033-569X-1983-0693875-6
- C. O. Horgan and L. E. Payne, Decay estimates for second-order quasilinear partial differential equations, Adv. in Appl. Math. 5 (1984), no. 3, 309–332. MR 755383, DOI https://doi.org/10.1016/0196-8858%2884%2990012-5
- C. O. Horgan and L. E. Payne, On Saint-Venant’s principle in finite anti-plane shear: an energy approach, Arch. Rational Mech. Anal. 109 (1990), no. 2, 107–137. MR 1022511, DOI https://doi.org/10.1007/BF00405239
- Cornelius O. Horgan and James K. Knowles, Recent developments concerning Saint-Venant’s principle, Adv. in Appl. Mech. 23 (1983), 179–269. MR 889288
- Cornelius O. Horgan, Recent developments concerning Saint-Venant’s principle: an update, AMR 42 (1989), no. 11, 295–303. MR 1021553, DOI https://doi.org/10.1115/1.3152414
- C. O. Horgan and L. E. Payne, A Saint-Venant principle for a theory of nonlinear plane elasticity, Quart. Appl. Math. 50 (1992), no. 4, 641–675. MR 1193661, DOI https://doi.org/10.1090/qam/1193661
- Shlomo Breuer and Joseph J. Roseman, Phragmén-Lindelöf decay theorems for classes of nonlinear Dirichlet problems in a circular cylinder, J. Math. Anal. Appl. 113 (1986), no. 1, 59–77. MR 826658, DOI https://doi.org/10.1016/0022-247X%2886%2990332-X
- C. O. Horgan and L. E. Payne, Decay estimates for a class of nonlinear boundary value problems in two dimensions, SIAM J. Math. Anal. 20 (1989), no. 4, 782–788. MR 1000722, DOI https://doi.org/10.1137/0520055
- C. O. Horgan and L. E. Payne, On the asymptotic behavior of solutions of inhomogeneous second-order quasilinear partial differential equations, Quart. Appl. Math. 47 (1989), no. 4, 753–771. MR 1031690, DOI https://doi.org/10.1090/qam/1031690
- C. O. Horgan and L. E. Payne, Exponential decay estimates for capillary surfaces and extensible films, Stability Appl. Anal. Contin. Media 1 (1991), no. 3, 261–282. MR 1166584
- D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. MR 0274686
- L. E. Payne and G. A. Philippin, On some maximum principles involving harmonic functions and their derivatives, SIAM J. Math. Anal. 10 (1979), no. 1, 96–104. MR 516755, DOI https://doi.org/10.1137/0510012
L. E. Payne, Isoperimetric inequalities, maximum principles and their applications, Report of lectures given at the University, Newcastle-upon-Tyne (1972)
J. K. Knowles, On finite anti-plane shear for incompressible elastic materials, J. Austral. Math. Soc. Ser. B 19, 400–415 (1976)
J. K. Knowles, A note on anti-plane shear for compressible materials in finite elastostatics, J. Austral. Math. Soc. Ser. B 20, 1–7 (1977)
M. E. Gurtin, Topics in finite elasticity, NSF-CBMS Regional Conference Series in Appl. Math. vol. 35, SIAM, Philadelphia, 1981
Q. Jiang and J. K. Knowles, A class of compressible elastic materials capable of sustaining finite anti-plane shear, J. Elasticity 25, 193–201 (1991)
C. O. Horgan and J. K. Knowles, The effect of nonlinearity on a principle of Saint-Venant type, J. Elasticity 11, 271–291 (1981)
C. O. Horgan and R. Abeyaratne, Finite anti-plane shear of a semi-infinite strip subject to a self-equilibrated end traction, Quart. Appl. Math. 40, 407–417 (1983)
C. O. Horgan and L. E. Payne, Decay estimates for second-order quasilinear partial differential equations, Adv. in Appl. Math. 5, 309–332 (1984)
C. O. Horgan and L. E. Payne, On Saint-Venant’s principle in finite anti-plane shear: An energy approach, Arch. Rational Mech. Anal. 109, 107–137 (1990)
C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant’s principle, Advances in Applied Mechanics (J. W. Hutchinson, ed.), vol. 23, Academic Press, New York, 1983, pp. 179–269
C. O. Horgan, Recent developments concerning Saint-Venant’s principle: An update, Appl. Mech. Rev. 42, 295–303 (1989)
C. O. Horgan and L. E. Payne, A Saint-Venant principle for a theory of nonlinear plane elasticity, Quart. Appl Math. 50, 641–675 (1992)
S. Breuer and J. J. Roseman, Phragmén-Lindelöf decay theorems for classes of nonlinear Dirichlet problems in a circular cylinder, J. Math. Anal. Appl. 113, 59–77 (1986)
C. O. Horgan and L. E. Payne, Decay estimates for a class of nonlinear boundary value problems in two dimensions, SIAM J. Math. Anal. 20, 782–788 (1989)
C. O. Horgan and L. E. Payne, On the asymptotic behavior of solutions of inhomogeneous second-order quasilinear partial differential equations, Quart. Appl. Math. 47, 753–771 (1989)
C. O. Horgan and L. E. Payne, Exponential decay estimates for capillary surfaces and extensible films, Stability Appl. Anal. Contin. Media 1, 261–282 (1991)
D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, Berlin, 1970
L. E. Payne and G. A. Phillipin, On some maximum principles involving harmonic functions and their derivatives, SIAM J. Math. Anal. 10, 96–104 (1979)
L. E. Payne, Isoperimetric inequalities, maximum principles and their applications, Report of lectures given at the University, Newcastle-upon-Tyne (1972)
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Article copyright:
© Copyright 1993
American Mathematical Society
