Normal hyperbolicity of center manifolds and Saint-Venant's principle

AbstractThe concept of normal hyperbolicity of center manifolds is generalized to infinite-dimensional differential equations, in particular, to elliptic problems in cylindrical domains. It is shown that all solutions u staying close to the center manifold for t ∈ (−l,l) satisfy an estimate of the form $$\left\| {u(t) - \tilde u(t)} \right\| \leqslant Ce^{ - \alpha (l - |t|)} $$ where C and α are independent of l, and ũ is a solution on the center manifold. These results are applied to Saint-Venant's principle for the static deformation of nonlinearly elastic prismatic bodies. The use of the center manifold permits the effective treatment of the general case of non-zero resultant forces and moments acting on each cross-section.

[1]  R. Knops,et al.  A Saint-Venant principle for nonlinear elasticity , 1983 .

[2]  K. Kirchgässner.  Wave-solutions of reversible systems and applications , 1982 .

[3]  A. Mielke.  A reduction principle for nonautonomous systems in infinite-dimensional spaces , 1986 .

[4]  J. J. Roseman,et al.  On Saint-Venant's Principle in three-dimensional nonlinear elasticity , 1977 .

[5]  J. J. Roseman.  The principle of Saint-Venant in linear and non-linear plane elasticity , 1967 .

[6]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[7]  Charles S. Kenney,et al.  Large buckled states of nonlinearly elastic rods under torsion, thrust, and gravity , 1981 .

[8]  Alexander Mielke,et al.  Reduction of quasilinear elliptic equations in cylindrical domains with applications , 1988 .

[9]  G. Kirchhoff,et al.  Ueber das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. , 1859 .

[10]  A. Mielke.  Saint-Venant's problem and semi-inverse solutions in nonlinear elasticity , 1988 .

[11]  B. B. Orazov.  On the asymptotic behaviour at infinity of solutions of the traction boundary value problem , 1989 .

[12]  O. Oleinik,et al.  On the asymptotic behaviour at infinity of solutions in linear elasticity , 1982 .

[13]  K. Palmer.  Linearization near an integral manifold , 1975 .

[14]  R. Toupin,et al.  Saint-Venant's Principle , 1965 .

[15]  James K. Knowles,et al.  Recent Developments Concerning Saint-Venant's Principle , 1983 .

[16]  J. Hale,et al.  Smoothness of bounded solutions of nonlinear evolution equations , 1985 .

[17]  G. Fischer.  Zentrumsmannigfaltigkeiten bei elliptischen Differentialgleichungen , 1984 .

[18]  Philip Holmes,et al.  Spatially complex equilibria of buckled rods , 1988 .