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On the membrane approximation for thin elastic shells in the hyperbolic case. (English) Zbl 0809.73042

Variational formulation of the problem of thin elastic shells in the membrane approximation, in the case of the hyperbolic middle surface, is presented. The corresponding bilinear form behaves elastically without shear rigidity. Two different asymptotic processes can describe the limit behavior, according to the fact that the middle surface either admits pure bendings or such pure bendings are not allowed. In the first case the middle surfaces are called “noninhibited”, and in the second one “inhibited” or “rigid” surfaces, which leads to the membrane approximation. The mathematical and physical reasons for this behavior are explained, and consequences are thrown concerning the admissible applied forces and the behavior of the described solutions. The proposed approach gives a description of the problem for somewhat general boundary conditions, whereas the classical approach only works in the case when the boundary conditions lead to either Cauchy or Goursat problems.
In conclusion, the lack of rigidity of the hyperbolic shell in the membrane approximation with respect to shear implies that the corresponding rigidity must be furnished by flexion terms. This fact is relevant to stability and buckling of the shell.

MSC:

74K15 Membranes
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
74K25 Shells
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