×

zbMATH — the first resource for mathematics

The effect of different scalings in the modelling of nonlinearly elastic plates with rapidly varying thickness. (English) Zbl 0759.73032
Summary: We deal with the bending of a nonlinearly elastic plate with rapidly varying thickness, assuming it to obey a Saint-Venant-Kirchhoff’s constitutive law. Our analysis is centered on the case when the plate mean thickness and the periodic variation length scale are of different order. The associated limiting model is a fourth order system posed on the plate’s midplane, and with coefficients, determinated by the geometry of the plate, depending on the velocity of the mean thickness variation. The two-dimensional limiting problem generalizes nonlinear models already known in the literature for constant thickness plates. Finally, numerical results for plates with rib-like stiffeners are presented.

MSC:
74K20 Plates
74B20 Nonlinear elasticity
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bensoussan, A.; Lions, J.L.; Papanicolaou, G., Asymptotic analysis for periodic structures, () · Zbl 0411.60078
[2] Sanchez-Palencia, E., Non-homegeneous media and vibration theory, ()
[3] Kohn, R.V.; Vogelius, M., A new model for thin plates with rapidly varying thickness, Internat. J. solids and structures, 20, 333-350, (1984) · Zbl 0532.73055
[4] Kohn, R.V.; Vogelius, M., A new model for thin plates with rapidly varying thickness II, Quart. appl. math., 43, 1-22, (1985) · Zbl 0565.73046
[5] Kohn, R.V.; Vogelius, M., A new model for thin plates with rapidly varying thickness III, Quart. appl. math., 44, 35-48, (1986) · Zbl 0605.73048
[6] Quintela-Estevez, P., A new model for nonlinear elastic plates with rapidly varying thickness, Applicable anal., 32, 107-127, (1989) · Zbl 0683.73027
[7] Quintela-Estevez, P., Thin plates with rapidly varying thickness II: the effect of the behaviour of the forces when the thickness approaches zero, Applicable anal., 39, 151-164, (1990) · Zbl 0687.73061
[8] Davet, J.L., Justification de modeles de plaques non lineaires pour des lois de comportement generales, Model. math. anal. numer., 20, 225-249, (1986) · Zbl 0634.73048
[9] Ciarlet, P.G., Mathematical elasticity, () · Zbl 0542.73046
[10] Ciarlet, P.G.; Destuynder, P., A justification of a nonlinear model in plate theory, Comput. methods appl. mech. engrg., 17/18, 227-258, (1979) · Zbl 0405.73050
[11] Ciarlet, P.G.; Rabier, P., LES equations de von karman, () · Zbl 0433.73019
[12] Ciarlet, P.G., Recent progresses in the two-dimensional approximation of three-dimensional plate models in nonlinear elasticity, () · Zbl 0612.73060
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.