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Abstracts

XVIII conference

Stable forms of plate shells

Grachev V., Nayshtut Yu.

Russia, 443001, Samara, Str. Molodogvardeyskaya, 194

The article continues looking for elastic systems that can move from one shape to another without bulk force. Concurrently the surface load in the starting and final position equals to zero. Based on Ericsson’s theorem [1] about no general solutions in non-linear three-dimensional theory of elasticity, similar examples should be looked for among thin wall plates and shells[2].

The article [3] reviews a net of strips generated with hinged parallelograms. While unwrapping the package, the surface that is close to a plane is formed. The article [4] studies a different situation when a package assembled from linear strips (the strips are formed by hinge merge of the trapeziums) is first unwrapped into the shell with main curvatures and later is converted into the surface with new curvatures .

The objective of this report is to demonstrate that the flat surface of parallelograms [3] can be converted into the latticed shell surface of negative Gauss curvature.

The study of shell dimensional behavior is based on the movements of Bricard’s 6R links [5] and the possibility to introduce “new continuous variables” [3] while analyzing the nets assembled from trapeziums (slightly different than rectangles). The last assumption is tantamount to Cartan’s moving frame method for latticed shells.

The proposed mechanical model explains different stable forms of latticed shells based on variation of the ideal rigid plasticity theory [6] for hinged connections of plates. Besides, it is assumed that during transformation of the trapezium plate shell only elastic twist and bending are possible along the plane with minimum rigidity.

References.

1. Lurie A.I. Non-linear theory of elasticity. М.: Science, 1980, 512 p.

2. Norman A. D., Golabchi M. R., Seffen K.A.,Guest S. D. Multistable Textured Shell Structures

//Advanced Science and Technology, vol. 54: (2008), 168-173

3. Grachev V.A., Nayshtut Yu. S. Transformed systems based on 6R links //Mathematics, Computer, Education, 15 (2), 2008, 131-139

4. Grachev V.A., Nayshtut Yu. S. Stable forms of transformed shells // Bulletin of Samara Technical Univ, Matheatics. , № 2(8): (2009)

5. Phillips J. Freedom in machinery. Cambridge University Press. 2006. 253 p.

6. Kachanov L.M. Fundamentals of plasticity theory. М.: Science, 1969. 420 p.

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