Saint-Venant’s problem and semi-inverse solutions in nonlinear elasticity.

*(English)*Zbl 0651.73006The 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 |

##### Keywords:

linear unbounded operator; smooth functional; uniformly ”small” strain; center manifold reduction; solutions lie on a six-dimensional manifold; Sobolev spaces of fractional order; implicit function theorem; linearized three-dimensional elasticity; classical rod equations; set of semi-inverse solutions; three two-dimensional manifolds.
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\textit{A. Mielke}, Arch. Ration. Mech. Anal. 102, No. 3, 205--229 (1988; Zbl 0651.73006)

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##### References:

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