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