Abstract
In the asymptotic theory of thin elastic shells the rigidity of the mid-surface with kinematic boundary conditions plays an important role. Rigidity is understood in the sense of infinitesimal (linearized) rigidity, i.e., the displacements vanish provided the variation of the first fundamental form vanishes. In this case the surface is also called “stiff”, as it cannot undergo pure bendings. A stiff surface is imperfectly stiff or perfectly stiff when the origin respectively does or does not belong to the essential spectrum of the boundary-value problem. These questions are investigated in the framework of Douglis-Nirenberg elliptic systems, with boundary conditions and transmission conditions at the folds. The index properties ensures quasi-stiffness, i.e. stiffness up to a finite number of degrees of freedom. The concept of perfect stiffness is linked with estimates for the rigidity system at an appropriate level of regularity for the data and the solution. It is proved that surfaces with folds are never perfectly stiff. It is also shown that the transmission conditions at the folds contain more conditions than those satisfying the Shapiro-Lopatinskii property. This leads to certain rigidity properties of the folds. Some examples are given.
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Geymonat, G., Sanchez-Palencia, E. On the rigidity of certain surfaces with folds and applications to shell theory. Arch. Rational Mech. Anal. 129, 11–45 (1995). https://doi.org/10.1007/BF00375125
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DOI: https://doi.org/10.1007/BF00375125