A two-dimensional inhomogeneous isotropic elastic material is considered in the arch-like region a ≤ r ≤ b,0 ≤ θ ≤ α, where (r,θ) denotes plane polar coordinates. It is envisaged that three of the edges r = a, r = b, θ = α are traction-free, while the edge θ = 0 is subjected to an (in-plane) self-equilibrated load. An appropriate energy-like measure E(θ) of the Airy stress function φ in the region between arbitrary θ and θ = α is defined, and it is proved to be positive definite provided that some appropriate assumptions are satisfied by the material and geometric characteristics of the arch-like region. Then, a version of Saint-Venant's Principle for the curvilinear strip is established.
Knowles, J. K.On Saint-Venant's principle in the two-dimensional linear theory of elasticity. Archive for Rational Mechanics and Analysis, 21, 1–22 (1966).
2.
Knowles, J. K.An energy estimate for the biharmonic equation and its application to Saint-Venant's principle in plane elastostatics. Indian Journal on Pure and Applied Mathematics, 14, 791–805 (1983).
3.
Flavin, J. N.On Knowles' version of Saint-Venant's principle in two-dimensional elastostatics. Archive for Rational Mechanics and Analysis, 53, 366–375 (1974).
4.
Horgan, C. O.Decay estimates for the biharmonic equation with applications to Saint-Venant principles in plane elasticity and Stokes flows. Quarterly of Applied Mathematics, 47, 147–157 (1989).
5.
Vafeades, P. and Horgan, C. O.Exponential decay estimates for solutions of the von Karman equations on a semi-infinite strip. Archive for Rational Mechanics and Analysis, 104, 1–25 (1988).
6.
Payne, L. E. and Schaefer, P. W.Phragmèn–Lindelöf type results for the biharmonic equation. Zeitschrift fur Angewandte Mathematik und Physik (ZAMP), 45, 414–432 (1994).
7.
Horgan, C. O. and Knowles, J. K.Recent developments concerning Saint-Venant's principle, in Advances in Applied Mechanics, Vol. 23, ed. T. Y. Wu and J. W. Hutchinson, Academic, New York, pp. 179–269, 1983.
8.
Horgan, C. O.Recent developments concerning Saint-Venant's principle: an update. Applied Mechanics Reviews, 42, 295–303 (1989).
9.
Horgan, C. O.Recent developments concerning Saint-Venant's principle: A second update. Applied Mechanics Reviews, 49, S101–S111 (1996).
10.
Flavin, J. N.Convexity considerations for the biharmonic equation in plane polars with applications to elasticity. Quarterly Journal of Mechanics and Applied Mathematics, 45, 555–566 (1992).
11.
Flavin, J. N. and Gleeson, B.On Saint-Venant's principle for a curvilinear rectangle in linear elastostatics. Mathematics and Mechanics of Solids, 8, 337–344 (2003).
12.
Flavin, J. N.Saint-Venant decay estimates for inhomogeneous isotropic two-dimensional elastostatics, in Symposium on Trends in the Application of Mathematics to Mechanics, ed. P. E. O'Donoghue and J. N. Flavin, Elsevier, New York, pp. 72–79, 2000.
13.
Flavin, J. N.Spatial-decay estimates for a generalized biharmonic equation in inhomogeneous elasticity. Journal of Engineering Mathematics, 46, 241–252 (2003).
14.
Horgan, C. O.Recent mathematical developments for functionally graded linearly elastic materials. Invited Chapter in Integral Methods in Science and Engineering, ed. P. Schiavone, C. Constanda and A. Mioduchowski, Birkhauser, Boston, pp. 123–130, 2002.
15.
Horgan, C. O. and Payne, L. E.On the asymptotic behaviour of solutions of linear second-order boundary-value problems on a semi-infinite strip, Archive for Rational Mechanics and Analysis, 124, 277–303 (1993).
16.
Chan, A. M. and Horgan, C. O.End effects in anti-plane shear for an inhomogeneous isotropic linearly elastic semi-infinite strip. Journal of Elasticity, 51, 227–242 (1998).
17.
Flavin, J. N.Qualitative estimates for laminate-like elastic materials, in Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, ed. D. F. Parker and A. H. England, Kluwer, Dordrecht, pp. 339–344, 1995.
18.
Horgan, C. O. and Quintanilla, R.Saint-Venant end effects in anti-plane shear for functionally graded linearly elastic materials. Mathematics and Mechanics of Solids, 6, 115–132 (2001).
19.
Horgan, C. O. and Quintanilla, R.Spatial decay of transient end effects in functionally graded heat conducting materials. Quarterly of Applied Mathematics, 59, 529–542 (2001).
20.
Borrelli, A., Horgan, C. O. and Patria, M. C.Exponential decay of end effects in anti-plane shear for functionally graded piezoelectric materials. Proceedings of the Royal Society of London A, 460, 1193–1212 (2004).
