Characterization of an Overlooked Kinematical Descriptor in the Second-Gradient Hyperelastic Theory for Thin Shells

Abstract

In 1978, Murdoch presented a direct second-gradient hyperelastic theory for thin shells in which the strain-energy density associated with a deformation \(\boldsymbol{\eta }\) of a surface \(\mathcal{S}\) is allowed to depend constitutively on the three kinematical descriptors \(\boldsymbol{C}\), \(\boldsymbol{H}\), and \(\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}\), where \(\boldsymbol{F}=\text{Grad} _{\scriptscriptstyle \mathcal{S}} \boldsymbol{\eta }\), \(\boldsymbol{C}=\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{F}\), \(\boldsymbol{H}=\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{L}_{ \scriptscriptstyle \mathcal{S}'}\boldsymbol{F}\) is the covariant pullback of the curvature tensor \(\boldsymbol{L}_{\scriptscriptstyle \mathcal{S}'}\) of the deformed surface \(\mathcal{S}'\), and \(\boldsymbol{G}=\text{Grad} _{\scriptscriptstyle \mathcal{S}} \boldsymbol{F}\). On the other hand, in Koiter’s direct thin-shell theory, the strain-energy density depends constitutively on only \(\boldsymbol{C}\) and \(\boldsymbol{H}\). Due to the popularity of Koiter’s theory, the second-order tensors \(\boldsymbol{C}\) and \(\boldsymbol{H}\) are well understood and have been extensively characterized. However, the third-order tensor \(\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}\) in Murdoch’s theory is largely overlooked in the literature. We address this gap, providing a detailed characterization of \(\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}\). We show that for \(\boldsymbol{\eta }\) twice continuously differentiable, \(\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}\) depends solely on \(\boldsymbol{C}\) and its surface gradient \(\text{Grad} _{\scriptscriptstyle \mathcal{S}}\boldsymbol{C}\) and does not depend on \(\boldsymbol{L}_{\scriptscriptstyle \mathcal{S}'}\). For the special case of a conformal deformation, we find that a suitably defined strain measure corresponding to \(\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}\) depends only the conformal stretch and its surface gradient. For the further specialized case of an isometric deformation, this strain measure vanishes. An orthogonal decomposition of \(\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}\) reveals that it belongs to a ten-dimensional subspace of the space of third-order tensors and embodies two independent types of non-local phenomena: one related to the spatial variations in the stretching of \(\mathcal{S}'\) and the other to the curvature of \(\mathcal{S}\).