##
**Plates and junctions in elastic multi-structures. An asymptotic analysis.**
*(English)*
Zbl 0706.73046

Collection Recherches en Mathématiques Appliquées, 14. Paris etc.: Masson; Berlin etc.: Springer-Verlag. vi, 215 p. FF 200.00 (1990).

This monograph presents a lucid, state-of-the-art account of the so- called asymptotic approach to plate theory. This approach consists in deriving various two-dimensional plate models from more accurate three- dimensional elasticity (or otherwise) models, by letting the thickness \(\epsilon\) of the plate go to zero. The main ingredients of this approach are the geometrical scaling of the plate in its thin direction, which maps it onto a domain that does not depend on \(\epsilon\), and the associated scalings of the various unknowns involved in powers of \(\epsilon\). Depending on the situation, either formal asymptotic expansions or rigorous convergence proofs then supply the desired two- dimensional models.

The book is divided into five chapters. The first chapter is devoted to the derivation of two-dimensional equations for a nonlinearly elastic clamped plate, starting from a family of nonlinearly elastic three- dimensional plates of thickness \(\epsilon\) made of a Saint Venant- Kirchhoff material. In this case, only formal asymptotic expansions are available since there is no existence result of the three-dimensional problem. Two competing approaches are presented: The displacement approach, in which the displacement is considered as the only main unknown and is subjected to an asymptotic expansion Ansatz, and the displacement-stress approach, in which a mixed Hellinger- Reissner formulation is used and both displacements and stresses are formally expanded. It is shown that both approaches yield the same result, a Kirchhoff-Love nonlinear plate model. The displacement approach is conceptually more satisfactory than the mixed approach, as it is more economic in terms of formal assumptions. The older displacement-stress approach is however quite simpler [the author, Ph. Destuynder, Comput. Math. Appl. Mech. Eng. 17-18, 227-258 (1979; Zbl 0405.73050)].

Chapter 2 deals with the derivation of the von Kármán equations. Once again, the starting point is three-dimensional elasticity with a Saint Venant-Kirchhoff material, but the boundary conditions are not of clamping but allow for some freedom in the horizontal directions. Accordingly, horizontal resultance forces are applied on the lateral surface of the plate. It is shown by using the same analysis as in chapter 1, that the first nonzero term in the asymptotic expansion of the scaled displacements is the solution of a certain two-dimensional nonlinear plate model. This model is in turn shown to be equivalent to the well-known von Kármán equations, which involve the Airy stress function. This approach thus provides adequate boundary conditions for the three-dimensional problem in order that the von Kármán equations be acceptable, as well as the corresponding correct boundary conditions for the von Kármán equations themselves [the author, Arch. Ration. Mech. Anal. 73, 349-389 (1980; Zbl 0443.73034)].

In chapter 3, the case of a linearly elastic clamped plate is considered. In this case, it is shown that the scaled displacements strongly converge in the \(H^ 1\)-sense toward Kirchhoff-Love displacements that are solutions of the classical linear Kirchhoff-Love plate equations. This is achieved by using a standard procedure of a priori estimates, weak convergence, identification of the weak limit of the scaled displacements with appropriate test-functions in the variational equilibrium equations and finally, proof that the convergence is in fact strong. Convergence of the stresses in weaker spaces \((L^ 2(\Omega)\) for the in-plane components, \(H^ 1(]-1,1[;H^{-1}(\omega))\) for the shear components and \(H^ 2(]-1,1[;H^{-2}(\omega))\) for the normal component) is also established.

Chapter 4 concerns the more general situation of a multi-structure, in this case the junction between a three-dimensional linearly elastic body with a linearly elastic plate, as is considered by the author, H. Le Dret and R. Nzengwa [J. Math. Pure Appl. 68, 261-295 (1989; Zbl 0661.73013)]. The same approach as in chapter 3 is used, with a few new ideas to deal with the difficulties caused by the junction. Strong \(H^ 1\)-convergence of the scaled displacements toward the solution of a 3d-2d model that couples three-dimensional elasticity equations with two- dimensional plate equations via a set of appropriate junction conditions, is established. In the final fifth chapter, eigenvalue and time-dependent problems are briefly discussed for plates and 3d-2d multi-structures. All these results are largely commented upon and a comprehensive bibliography closes the book.

The book is divided into five chapters. The first chapter is devoted to the derivation of two-dimensional equations for a nonlinearly elastic clamped plate, starting from a family of nonlinearly elastic three- dimensional plates of thickness \(\epsilon\) made of a Saint Venant- Kirchhoff material. In this case, only formal asymptotic expansions are available since there is no existence result of the three-dimensional problem. Two competing approaches are presented: The displacement approach, in which the displacement is considered as the only main unknown and is subjected to an asymptotic expansion Ansatz, and the displacement-stress approach, in which a mixed Hellinger- Reissner formulation is used and both displacements and stresses are formally expanded. It is shown that both approaches yield the same result, a Kirchhoff-Love nonlinear plate model. The displacement approach is conceptually more satisfactory than the mixed approach, as it is more economic in terms of formal assumptions. The older displacement-stress approach is however quite simpler [the author, Ph. Destuynder, Comput. Math. Appl. Mech. Eng. 17-18, 227-258 (1979; Zbl 0405.73050)].

Chapter 2 deals with the derivation of the von Kármán equations. Once again, the starting point is three-dimensional elasticity with a Saint Venant-Kirchhoff material, but the boundary conditions are not of clamping but allow for some freedom in the horizontal directions. Accordingly, horizontal resultance forces are applied on the lateral surface of the plate. It is shown by using the same analysis as in chapter 1, that the first nonzero term in the asymptotic expansion of the scaled displacements is the solution of a certain two-dimensional nonlinear plate model. This model is in turn shown to be equivalent to the well-known von Kármán equations, which involve the Airy stress function. This approach thus provides adequate boundary conditions for the three-dimensional problem in order that the von Kármán equations be acceptable, as well as the corresponding correct boundary conditions for the von Kármán equations themselves [the author, Arch. Ration. Mech. Anal. 73, 349-389 (1980; Zbl 0443.73034)].

In chapter 3, the case of a linearly elastic clamped plate is considered. In this case, it is shown that the scaled displacements strongly converge in the \(H^ 1\)-sense toward Kirchhoff-Love displacements that are solutions of the classical linear Kirchhoff-Love plate equations. This is achieved by using a standard procedure of a priori estimates, weak convergence, identification of the weak limit of the scaled displacements with appropriate test-functions in the variational equilibrium equations and finally, proof that the convergence is in fact strong. Convergence of the stresses in weaker spaces \((L^ 2(\Omega)\) for the in-plane components, \(H^ 1(]-1,1[;H^{-1}(\omega))\) for the shear components and \(H^ 2(]-1,1[;H^{-2}(\omega))\) for the normal component) is also established.

Chapter 4 concerns the more general situation of a multi-structure, in this case the junction between a three-dimensional linearly elastic body with a linearly elastic plate, as is considered by the author, H. Le Dret and R. Nzengwa [J. Math. Pure Appl. 68, 261-295 (1989; Zbl 0661.73013)]. The same approach as in chapter 3 is used, with a few new ideas to deal with the difficulties caused by the junction. Strong \(H^ 1\)-convergence of the scaled displacements toward the solution of a 3d-2d model that couples three-dimensional elasticity equations with two- dimensional plate equations via a set of appropriate junction conditions, is established. In the final fifth chapter, eigenvalue and time-dependent problems are briefly discussed for plates and 3d-2d multi-structures. All these results are largely commented upon and a comprehensive bibliography closes the book.

Reviewer: H.Le Dret

### MSC:

74K20 | Plates |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74B20 | Nonlinear elasticity |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |

74B05 | Classical linear elasticity |

35B45 | A priori estimates in context of PDEs |

49J45 | Methods involving semicontinuity and convergence; relaxation |