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**Modeling of a folded plate.**
*(English)*
Zbl 0741.73026

Contrary to the preceding studies by Ciarlet and Ciarlet et al. [e.g.: Ph. G. Ciarlet, the author, and R. Nzengwa, J. Math. Pures Appl., IX. Ser. 68, No. 3, 261-295 (1989, Zbl 0661.73013)], the junction problem considered here is that of two plates folded at a right angle, while the preceding ones were that of the junction of structures of different dimensions, such as a bulk and a plate. In both cases the same ideas are utilized, which consist of first to start from a three- dimensional modelling of the plates with thickness of order of \(\varepsilon\) and to replace the integration domain by a fixed one, and the integrands by quantities function of \(\varepsilon\) through a change of scaling. And second, to consider separately the junction integrals in both domains of the two plates.

In a preceding article the author considered already folded plates with both parts clamped somewhere on their edges [the author, C. R. Acad. Sci., Paris, Sér. I 304, 571-573 (1987; Zbl 0634.73047)]. This restriction is overcome here. Recalling the change of scaling which affects the coordinates, the displacement unknowns, the external forces and the elasticity data altogether, the author shows that the non clamped plate, where finite rigid motion displacements are allowed, would prevent the functional from coercivity. But boundedness is recovered by substracting infinitesimal displacements in the new scaled motions.

Then a Kirchhoff-Love type of solution is proved by a central theorem (as in preceding papers), when \(\varepsilon\to 0\) which shows: a) that the clamped plate “blocks” the free one, b) that the free plate stiffens the clamped one, and c) that the two plates remain perpendicular to each other. The proof is limited to flexural displacements only for brevity. From this results and the resulting variational principle, existence and uniqueness are shown for the plate problem.

The author, in a preceding part [ibid. 5, No. 5, 345-365 (1989; Zbl 0741.73025)], treats the problem of plates folded at an arbitrary angle strictly different from the square one. The same method is applied through a simple change of coordinates, but then modified integration domains are considered, obtained by slight restriction or extension of the preceding ones relative to the rectangular case. The same results are then obtained.

In the following sections corner cases are considered as well and also extensions to other geometries. All the proofs are detailed with care and appear rather, but necessarily, lengthy, the paper being incidentally self-contained.

We would not fail to remark that, if the junction problems clearly pose real and mathematical difficulties for structures of different dimensions, it was not expected that folded plates would present such pitfalls as they did. These unforseable difficulties are now solved, at least in the linear case.

In a preceding article the author considered already folded plates with both parts clamped somewhere on their edges [the author, C. R. Acad. Sci., Paris, Sér. I 304, 571-573 (1987; Zbl 0634.73047)]. This restriction is overcome here. Recalling the change of scaling which affects the coordinates, the displacement unknowns, the external forces and the elasticity data altogether, the author shows that the non clamped plate, where finite rigid motion displacements are allowed, would prevent the functional from coercivity. But boundedness is recovered by substracting infinitesimal displacements in the new scaled motions.

Then a Kirchhoff-Love type of solution is proved by a central theorem (as in preceding papers), when \(\varepsilon\to 0\) which shows: a) that the clamped plate “blocks” the free one, b) that the free plate stiffens the clamped one, and c) that the two plates remain perpendicular to each other. The proof is limited to flexural displacements only for brevity. From this results and the resulting variational principle, existence and uniqueness are shown for the plate problem.

The author, in a preceding part [ibid. 5, No. 5, 345-365 (1989; Zbl 0741.73025)], treats the problem of plates folded at an arbitrary angle strictly different from the square one. The same method is applied through a simple change of coordinates, but then modified integration domains are considered, obtained by slight restriction or extension of the preceding ones relative to the rectangular case. The same results are then obtained.

In the following sections corner cases are considered as well and also extensions to other geometries. All the proofs are detailed with care and appear rather, but necessarily, lengthy, the paper being incidentally self-contained.

We would not fail to remark that, if the junction problems clearly pose real and mathematical difficulties for structures of different dimensions, it was not expected that folded plates would present such pitfalls as they did. These unforseable difficulties are now solved, at least in the linear case.

Reviewer: R.Valid (Paris)

Full Text:
DOI

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