## Saint-Venant’s problem and semi-inverse solutions in nonlinear elasticity.(English)Zbl 0651.73006

The author addresses Saint Venant’s Problem in an infinite prism $$\Omega=\Sigma\times\mathbb{R}$$ in $$\mathbb{R}^3$$ with cross-section $$\Sigma$$. The prism is a reference configuration for a nonlinearly elastic body which is homogeneous and isotropic, and this configuration is natural. Saint Venant’s Problem consists in finding the solutions of the equilibrium problem for this body under zero loading that correspond to a given resultant force $$\mathcal F$$ and moment $$\mathcal M$$ over each cross-section. The author considers the set of such solutions for any $$(\mathcal F,\mathcal M)$$ which in addition have a uniformly ”small” strain. He shows by means of a center manifold reduction that all these solutions lie on a six- dimensional manifold. This is done by introducing a pair of new vector unknowns $$(u,v)$$ which describe the internal displacements responsible for the stress within a cross-section and from which the actual deformation phi of the body can be recovered. The equilibrium equations are then reformulated as a differential equation of the form: $\frac d{dt} \binom uv - \mathcal L \binom uv = F(u,v), \tag{1}$ where $$t$$ is the axial variable of the prism, while $$u$$ and $$v$$ belong to Sobolev spaces of fractional order over the cross-section $$\Sigma$$. The linear operator $$\mathcal L$$ is unbounded between such spaces. The functional $$F$$ is obtained by use of the implicit function theorem and it is smooth. To apply the center manifold reduction technique given by the author in Math. Methods Appl. Sci. 10, 51-66 (1988; Zbl 0647.35034), the author studies the spectral properties of the operator $$\mathcal L$$ (eigenvalues, generalized kernel and decay of resolvent at infinity) by using the close relationship of $$\mathcal L$$ with the operator of linearized three-dimensional elasticity. Saint Venant’s Principle for linearized elasticity plays an important rôle in the proofs. Equation (1) admits $$\mathcal F$$ and $$\mathcal M$$ as first integrals and by expressing these in terms of $$(u,v)$$, the author finds a remarkable connection with the classical rod equations [cf. S.S. Antman, Arch. Ration. Mech. Anal. 84, 293- 305 (1984; Zbl 0533.73042)] which are in some sense equivalent to the equation obtained by reducing equation (1) to the center manifold. Semi-inverse solutions (i.e., solutions whose strain does not depend on $$t$$) are then further investigated when the cross-section $$\Sigma$$ has two axes of symmetry. The set of semi-inverse solutions is shown to consist of three two-dimensional manifolds in this case.
Reviewer: H.Le Dret

### MSC:

 74B20 Nonlinear elasticity 74G50 Saint-Venant’s principle 47A50 Equations and inequalities involving linear operators, with vector unknowns 74B99 Elastic materials 47A10 Spectrum, resolvent 34C45 Invariant manifolds for ordinary differential equations

### Citations:

Zbl 0601.35018; Zbl 0647.35034; Zbl 0533.73042
Full Text:

### References:

  H. Amann: Parabolic Evolution Equations with Nonlinear Boundary Conditions, Proceedings of Symposia in Pure Mathematics 45, 17–27 (1986), pt. 1. · Zbl 0611.35043  S. S. Antman: Ordinary Differential Equations of Nonlinear Elasticity I, Arch. Rational Mech. Anal. 61, 306–351 (1976).  S. S. Antman: Large Lateral Buckling of Nonlinearly Elastic Beams, Arch. Rational Mech. and Anal. 84, 293–305 (1984). · Zbl 0533.73042  J. M. Ball: Constitutive Inequalities and Existence Theorems in Nonlinear Elastostatics, in ”Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. I”, R. J. knops ed., Pitman London 1977, 187–241.  P. G. Ciarlet: ”Mathematical Elasticity Vol. I” North-Holland, Amsterdam 1987. · Zbl 0612.73060  J. L. Ericksen: On the Formulation of Saint-Venant’s Problem, in ”Nonlinear Analysis and Mechanics: Heriot-Watt Symposium Vol. I”, R. J. knops ed., Pitman London 1977, 158–186.  J. L. Ericksen: Problems for Infinite Elastic Prisms – St.-Venant’s Problem for Elastic Prisms, in ”Systems of Nonlinear Partial Differential Equations”, J. M. Ball ed., NATO ASI Series C 111, Reidel Publ. Comp. 1983.  P. J. Holmes & A. Mielke: Complex Equilibria of Buckled Rods, Arch. Rational Mech. Anal. 101, 319–348 (1988). · Zbl 0655.73029  T. J. R. Hughes & J. E. Marsden: Topics in the Mathematical Foundations of Elasticity, in ”Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. II.” R. J. Knops ed., Pitman London 1978, 30–285.  D. Kinderlehrer: Remarks about Saint-Venant’s Solutions in Finite Elasticity, Proceedings of Symposia in Pure Mathematics 45, 37–50 (1986), pt. 2.  J. L. Lions & E. Magenes: ”Problèmes aux limites non homogènes et applications, Vol. 1”, Dunod Paris 1968. · Zbl 0165.10801  A. Mielke: Reduction of Quasilinear Elliptic Equations in Cylindrical Domains with Applications, Math. Meth. Appl. Sciences 10, 51–67 (1988). · Zbl 0647.35034  A. Mielke: On Saint-Venant’s Problem for an Elastic Strip, Proc. Roy. Soc. Edinburgh, to appear. · Zbl 0657.73003  R. G. Muncaster: Saint-Venant’s Problem in Nonlinear Elasticity: A Study of Cross-Sections, in ”Nonlinear Analysis and Mechanics: Heriot-Watt Symp. Vol. IV”, R. J. knops ed., Pitman London 1979, 17–75.  R. G. Muncaster: Saint-Venant’s Problem for Slender Prisms, Utilitas mathematica 23, 75–101 (1983).  J. Nečas: ”Les Méthodes Directes en Théorie des Equations Elliptiques,” Masson, Paris 1967.  O. A. Oleinik & G. A. Yosifian: On the Asymptotic Behavior at Infinity of Solutions in Linear Elasticity, Arch. Rational Mech. Anal. 78, 29–53 (1983).
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