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

In 1978, Murdoch presented a direct second-gradient hyperelastic theory for thin shells in which the strain-energy density associated with a deformation $\mathit{\eta }$ $\boldsymbol{\eta }$ of a surface $\mathcal{S}$ $\mathcal{S}$ is allowed to depend constitutively on the three kinematical descriptors $\mathit{C}$ $\boldsymbol{C}$ , $\mathit{H}$ $\boldsymbol{H}$ , and ${\mathit{F}}^{\mathrm{\top }}\mathit{G}$ $\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}$ , where $\mathit{F}={\text{Grad}}_{\mathcal{S}}\mathit{\eta }$ $\boldsymbol{F}=\text{Grad} _{\scriptscriptstyle \mathcal{S}} \boldsymbol{\eta }$ , $\mathit{C}={\mathit{F}}^{\mathrm{\top }}\mathit{F}$ $\boldsymbol{C}=\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{F}$ , $\mathit{H}={\mathit{F}}^{\mathrm{\top }}{\mathit{L}}_{{\mathcal{S}}^{\prime }}\mathit{F}$ $\boldsymbol{H}=\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{L}_{ \scriptscriptstyle \mathcal{S}'}\boldsymbol{F}$ is the covariant pullback of the curvature tensor ${\mathit{L}}_{{\mathcal{S}}^{\prime }}$ $\boldsymbol{L}_{\scriptscriptstyle \mathcal{S}'}$ of the deformed surface ${\mathcal{S}}^{\prime }$ $\mathcal{S}'$ , and $\mathit{G}={\text{Grad}}_{\mathcal{S}}\mathit{F}$ $\boldsymbol{G}=\text{Grad} _{\scriptscriptstyle \mathcal{S}} \boldsymbol{F}$ . On the other hand, in Koiter’s direct thin-shell theory, the strain-energy density depends constitutively on only $\mathit{C}$ $\boldsymbol{C}$ and $\mathit{H}$ $\boldsymbol{H}$ . Due to the popularity of Koiter’s theory, the second-order tensors $\mathit{C}$ $\boldsymbol{C}$ and $\mathit{H}$ $\boldsymbol{H}$ are well understood and have been extensively characterized. However, the third-order tensor ${\mathit{F}}^{\mathrm{\top }}\mathit{G}$ $\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}$ in Murdoch’s theory is largely overlooked in the literature. We address this gap, providing a detailed characterization of ${\mathit{F}}^{\mathrm{\top }}\mathit{G}$ $\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}$ . We show that for $\mathit{\eta }$ $\boldsymbol{\eta }$ twice continuously differentiable, ${\mathit{F}}^{\mathrm{\top }}\mathit{G}$ $\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}$ depends solely on $\mathit{C}$ $\boldsymbol{C}$ and its surface gradient ${\text{Grad}}_{\mathcal{S}}\mathit{C}$ $\text{Grad} _{\scriptscriptstyle \mathcal{S}}\boldsymbol{C}$ and does not depend on ${\mathit{L}}_{{\mathcal{S}}^{\prime }}$ $\boldsymbol{L}_{\scriptscriptstyle \mathcal{S}'}$ . For the special case of a conformal deformation, we find that a suitably defined strain measure corresponding to ${\mathit{F}}^{\mathrm{\top }}\mathit{G}$ $\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}$ 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 ${\mathit{F}}^{\mathrm{\top }}\mathit{G}$ $\boldsymbol{F}^{\scriptscriptstyle \top }\boldsymbol{G}$ 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 ${\mathcal{S}}^{\prime }$ $\mathcal{S}'$ and the other to the curvature of $\mathcal{S}$ $\mathcal{S}$ .