On in-plane drill rotations for Cosserat surfaces

Abstract

We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a C 2 -smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations are fixed at a portion of the boundary. Put otherwise, if the tangent vectors of the new surface are obtained locally by only rotating the given tangent vectors, and if these rotations have a rotation axis which coincides everywhere with the normal of the initial surface, then the two surfaces are equal provided they coincide at a portion of the boundary. In the language of differential geometry of surfaces, we show that any isometry which leaves normals invariant and which coincides with the given surface at a portion of the boundary is the identity mapping.

Pure measures of bending for soft plates

“…For λ ′ = 1, this result was already proved in ref. 23 ¶¶¶¶As appropriately remarked, 23 the same result had also been proved, albeit with different methods, in the differential geometry literature, 51–53 but it was unknown to plates and shells practitioners.…”

“…Section 3 is devoted to the compatibility condition that a bending-neutral deformation must obey; we shall see that this class contains both pure tangential stretching deformations and planar drill rotations, characterized separately in ref. 22 and 23. Finally, in Section 4, we summarize the conclusions of this study and comment briefly about its possible extensions.…”

“…§ I represents the identity tensor in three space dimensions. As shown in ref. 22 and 23, respectively, neither of these subclasses is empty.…”

“…The condition of compatibility arises from the requirement that the curvature tensor ∇ s ν * of be symmetric, as it is expected to be. 23 By writing (17) for ν * (and y *) instead of ν (and y ), we have that∇ ν * = (∇ s ν *)(∇ y *).On the other hand, using (12) and (13), we can also write∇ ν * = R 0 ∇ ν = R 0 (∇ s ν )(∇ y ),which, combined with (56), leads us to∇ s ν * = R 0 (∇ s ν )( U ′) −1 R T ν R T 0 ,once use is also made of (6), (3), and (8). It follows from (58) that ∇ s ν * is symmetric if and only if (∇ s ν )( U ′) −1 R T ν is so.…”