Une théorie asymptotique des plaques minces en élasticité linéaire. (Asymptotic theory of thin plates in linear elasticity).

*(French)*Zbl 0627.73064
Recherches en Mathématiques Appliquées, 2. Paris etc.: Masson. 175 p. (1986).

This book is a contribution to the linear theory of thin plates in linear elasticity. After having asserted that all the existing theories suffered of lack of justifications, the author proposes a pure mathematical justification from the three-dimensional theory. The method utilized in the treatise is the asymptotic one applied through the mixed variational principle of Hellinger-Reissner.

Chapter 0 and 1 are brief recalls of theoretical mechanics. Chapter 2 presents in a perhaps to much concise way the method of asymptotic expansion with respect to a small parameter, namely the plate thickness \(\epsilon\), (Zoom method). Chapters 3 and 4 give important results on the fundamental solution. The author shows very clearly that this solution satisfies the Kirchhoff-Love conditions, gives an expression of the shear and normal resulting stresses, together with results of convergence and error estimates whose order are in \(\sqrt{\epsilon}\). But these last results relating to the shear and normal stresses are submitted to regularity assumptions the sense of which do not appear very clearly. These results would be weakened without them as indicated.

Chapter 5 studies a certain type of crack propagation in an anisotropic plate in bending and calculates the rate of energy release by a direct analytical method. It is observed that the shear stress will play an important role. The 6th and last chapter is an attempt to calculate the next corrective term due to the boundary layer. The same zoom method with local coordinates is applied which takes account of the vicinity of the edge, and by an analytical calculus which shows that the problem separates into two uncoupled ones, namely a torsion and a bending- extension problems, both with exponential decrease of the corrective terms. The calculation is tedious though approximate and the author suggests that a numerical approach could not be avoided for multilayered plates.

In conclusion, this treatise is interesting and important on many aspects. The theoretical proofs are particularly clear for any results which concern the fundamental solution. The latter incidentally remains by no means without shear strains, which could be a drawback for many applications. The contribution relating to edge effects is less convincing, but the problem is doubtless difficult and the method does not appear to be the most efficient to that aim.

One may express some reproach to the author, who does homage to some pioneers of this type of researches, as far as he does not justice to the numerous specialists of theoretical mechanics whose contributions in that field are eminent.

Chapter 0 and 1 are brief recalls of theoretical mechanics. Chapter 2 presents in a perhaps to much concise way the method of asymptotic expansion with respect to a small parameter, namely the plate thickness \(\epsilon\), (Zoom method). Chapters 3 and 4 give important results on the fundamental solution. The author shows very clearly that this solution satisfies the Kirchhoff-Love conditions, gives an expression of the shear and normal resulting stresses, together with results of convergence and error estimates whose order are in \(\sqrt{\epsilon}\). But these last results relating to the shear and normal stresses are submitted to regularity assumptions the sense of which do not appear very clearly. These results would be weakened without them as indicated.

Chapter 5 studies a certain type of crack propagation in an anisotropic plate in bending and calculates the rate of energy release by a direct analytical method. It is observed that the shear stress will play an important role. The 6th and last chapter is an attempt to calculate the next corrective term due to the boundary layer. The same zoom method with local coordinates is applied which takes account of the vicinity of the edge, and by an analytical calculus which shows that the problem separates into two uncoupled ones, namely a torsion and a bending- extension problems, both with exponential decrease of the corrective terms. The calculation is tedious though approximate and the author suggests that a numerical approach could not be avoided for multilayered plates.

In conclusion, this treatise is interesting and important on many aspects. The theoretical proofs are particularly clear for any results which concern the fundamental solution. The latter incidentally remains by no means without shear strains, which could be a drawback for many applications. The contribution relating to edge effects is less convincing, but the problem is doubtless difficult and the method does not appear to be the most efficient to that aim.

One may express some reproach to the author, who does homage to some pioneers of this type of researches, as far as he does not justice to the numerous specialists of theoretical mechanics whose contributions in that field are eminent.

Reviewer: R.Valid

##### MSC:

74K20 | Plates |

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

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

74R05 | Brittle damage |

74E10 | Anisotropy in solid mechanics |

46N99 | Miscellaneous applications of functional analysis |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |