The refraction phenomenon of singularities in thin elastic shells with developable mid-surface in presence of rigid folds: Case of parabolic shells

• We consider two thin elastic shells for which the middle surfaces are developable.
• The shells are joined together along a fold which is not a characteristic of PDEs system.
• Due to PDEs system, singularities cross folds and they propagate on characteristic curves keeping the degree of singularity.
• We call this, “phenomenon refraction by a rigid fold”.
• We also provide numerical simulations to exhibit the associated numerical refraction phenomenon.

This paper deals with the refraction phenomenon that appears in the theory of thin shells of which the mid-surface is developable in presence of rigid folds when the thickness ε tends to zero. Roughly speaking we talk about thin parabolic shells. On each side of the fold, the nature of the mid-surface of the shell is developable and boundary conditions ensure the geometric rigidity. The limit problem (ε = 0) which is also parabolic has some peculiarities that induce singularities. These singularities propagate along the asymptotic curves also called characteristics. When these singularities encounter a fold on which transmission conditions are given, they pass through the fold and they continue to propagate along the characteristic curves of the opposite part of the shell corresponding to the adjacent parts of the fold. A theoretical approach is proposed in order to study the refraction phenomenon in thin parabolic shells. Numerical simulations have been constructed to illustrate and observe the refraction phenomenon. The main result is that, once the singularities have crossed the fold, they continue to propagate along the characteristic curves of the other side of the fold without any loss or gain of regularity but with a variation in amplitude.