An anti-plane shear problem

Journal of Elasticity - Tập 33 - Trang 213-231 - 1993
Jean-Pierre Raymond1
1Laboratoire d’Analyse Numérique, Université Paul Sabatier, Toulouse Cedex, France

Tóm tắt

We consider an anti-plane shear of an elastic cylinder with a non-convex stored energy function. So, we look for solutions of a non-convex problem of the calculus of variations with Dirichlet boundary conditions. We give sufficient conditions on the boundary data to get existence or non-existence results for this non-convex problem. We also prove some uniqueness results for the relaxed problem associated with the initial problem.

Tài liệu tham khảo

G. Aubert and R. Tahraoui, Théorèmes d'existence en optimisation non convexe. Applicable Anal. (1984) 75–100. J.M. Ball and R. James, Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 15–52. P. Bauman and D. Phillips, A non-convex variational problem related to change of phase. J. Applied Math. Opt. 21 (1990) 113–138. L.A. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems. Comm. in Partial Diff. Eq. 7 (1982) 1337–1379. F.H. Clarke, Optimization and Nonsmooth Analysis. New-York: Wiley Interscience (1983). I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels. Paris: Dunod (1974). J.L. Ericksen, Equilibrium of bars. J. Elasticity 5 (1975) 191–201. H. Federer, Geometric Measure Theory. Berlin: Springer (1969). R.L. Fosdick and G. MacSithigh, Helical shear of an elastic, circular tube with a nonconvex stored energy. Arch. Rat. Mech. Anal. 84 (1983) 31–53. D.A. French, On the convergence of finite element approximations of a relaxed variational problem. S.I.A.M.J. Numer. Anal. 27 (1990) 419–436. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn. Berlin: Springer (1983). E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Basel: Birkhäuser (1984). M.E. Gurtin, Topics in Finite Elasticity CBMS Regional Conference Series in Applied Mathematics, Vol. 35. Philadelphia: S.I.A.M. (1981). M.E. Gurtin, Two-phase deformations of elastic solids. Arch. Rat. Mech. Anal. 84 (1983) 1–29. M.E. Gurtin and R. Temam, On the anti-plane shear problem in finite elasticity. J. Elasticity 11 (1981) 197–206. P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations. Acta Math. 115 (1966) 271–310. R.D. James, Co-existent phases in the one-dimensional static theory of elastic bars. Arch. Rat. Mech. Anal. 72 (1979) 99–140. B. Kawohl, Regularity, uniqueness and numerical experiments for a relaxed optimal design problem. In: International Series of Numerical Mathematics, Vol. 95. Basel: Birkhäuser (1990) pp. 85–100. B. Kawohl, J. Stara and G. Wittum, Analysis and numerical studies of a shape design problem. Arch. Rat. Mech. Anal. 114 (1991) 349–363. J.K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Rat. Mech. Anal. 63 (1977) 221–236. P. Marcellini, A relation between existence of minima for non-convex integrals and uniqueness for non strictly convex integrals of the calculus of variations. In Mathematical Theories of Optimization, Springer, Lect. Notes Math., Vol. 979 (1983) pp. 216–232. P. Marcellini, Some remarks on uniqueness in calculus of variations. In: Non-linear P.D.E., Collège de France Seminar, Vol. IV. Brezis, Lions Eds., Pitman (1983). P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions. Arch. Rat. Mech. Anal. 105 (1989) 267–284. E. Mascolo, Existence results for non-convex problems of the calculus of variations. In: S. Hildebrandt, D. Kinderlehrer and M. Miranda (eds), Calculus of Variations and Partial Differential Equations, Trento 1986. Berlin: Springer (1988) pp. 201–207. E. Mascolo and R. Schianchi, Existence theorems for non convex problems. J. Math. Pures Appl. 62 (1983) 349–359. E. Mascolo and R. Schianchi, Existence theorems in the calculus of variations. J. Diff. Eq. 67 (1987) 185–198. J.P. Raymond, Théorème d'existence pour des problèmes variationnels non convexes. Proc. royal Soc. Ed. 107 A (1987) 43–64. J.P. Raymond, Existence of minimizers for vector problems without quasiconvexity condition. Nonlinear Analysis T.M.A. 18 (1992) 815–828.