## 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.
Reviewer: R.Valid (Paris)

### MSC:

 74K20 Plates 74E30 Composite and mixture properties

### Citations:

Zbl 0741.73025; Zbl 0712.73044; Zbl 0634.73047; Zbl 0661.73013
Full Text:

### References:

 [1] Adams, R. A. (1975): Sobolev spaces. New York: Academic Press · Zbl 0314.46030 [2] Aganovi?, I.; Tutek, Z. (1986): A justification of the one-dimensional model of elastic beam. Math. Meth. Appl. Sci. 8, 1-14 · Zbl 0617.35136 [3] Bermudez, A.; Viaño, J. M. (1984): Une justification des équations de la thermoélasticité des poutres à section variable par des máthodes asymptotiques. R. A. I. R. O. Analyse Numérique 18, 347-376 · Zbl 0572.73053 [4] Ciarlet, P. G.; Destuynder, P. (1979): A justification of a nonlinear model in plate theory. Comp. Meth. Appl. Mech. Engrg. 17/18, 227-258 · Zbl 0405.73050 [5] Ciarlet, P. G. (1980): A justification of the von Kármán equations. Arch. Rat. Mech. Anal. 73, 349-389 · Zbl 0443.73034 [6] Ciarlet, P. G. (1987): Recent progresses in the two-dimensional approximation of three-dimensional plate models in nonlinear elasticity. In: Ortiz, E. L. (Ed.): Numerical approximation of partial differential equations, pp. 3-19. Amsterdam: North-Holland [7] Ciarlet, P. G. (1988 a): Mathematical elasticity. Amsterdam: North-Holland · Zbl 0648.73014 [8] Ciarlet, P. G. (1988 b): Junctions between plates and rods (in preparation) [9] Ciarlet, P. G.; Le Dret, H.; Nzengwa, R. (1988): Junctions between three-dimensional and two-dimensional linearly elastic structures. To appear in J. Math. Pures Appl. · Zbl 0661.73013 [10] Cimetière, A.; Geymonat, G.; Le Dret, H.; Raoult, A.; Tutek, Z. (1988): Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods. J. Elasticity 19, 111-161 · Zbl 0653.73010 [11] Colson, A. (1984): Modélisation des conditions aux limites de liaisons et d’assemblages en mécanique des structures métalliques. Doctoral Dissertation, Université Paris 6 [12] Destuynder, P. (1986): Une théorie asymptotique des plaques minces en élasticité linéaire. Paris: R. M. A. Masson [13] Lions, J. L.; Magenes, E. (1968 a): Problèmes aux limites non homogenes et applications. Vol. 1 Paris: Dunod · Zbl 0165.10801 [14] Lions, J. L.; Magenes, E. (1968 b): Problèmes aux limites non homogènes et applications. Vol. 2 Paris: Dunod · Zbl 0165.10801 [15] Marsden, J. E.; Hughes, T. J. R. (1983): Mathematical foundations of elasticity. Englewood Cliffs: Prentice-Hall · Zbl 0545.73031 [16] Rigolot, A. (1976): Sur une théorie asymptotique des poutres. Doctoral Dissertation, Université Paris 6 [17] Wang, C. C.; Truesdell, C. (1975): Introduction to rational elasticity. Groningen: Noordhoff · Zbl 0308.73001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.