# Pure measure of bending for soft plates

Epifanio G. Virga<EMAIL_ADDRESS>Dipartimento di Matematica, Università di
Pavia, Via Ferrata 5, 27100 Pavia, Italy

(August 28, 2024)

###### Abstract

This paper, originally motivated by a question raised by Wood and Hanna [_Soft
Matter_ , 15, 2411 (2019)], shows that pure measures of bending for soft
plates can be defined by introducing the class of _bending-neutral_
deformations, finite incremental changes of the plate’s shape bearing _no_
further bending. This class of deformations is subject to a geometric
compatibility condition, which is fully characterized. A tensorial pure
measure of bending, which is accordingly invariant under bending-neutral
deformations, is described in details. As shown by an illustrative class of
examples, the general notion of pure measure of bending could be of use to
formulate direct theories for soft plates, where stretching and bending
energies are kept separate.

##  I Introduction

It has been remarked that a _pure_ measure of bending for extensible rods
_cannot_ be provided by the curvature of its centerline, as this can easily
been shown to be affected by a superimposed dilation [1]. Actually, within the
nonlinear theory of extensible elastic rods, this issue has long be known, at
least since the work of Antman [2] and others who soon followed in his
footsteps [3, 4]. It has also consequences on the choice of the appropriate
form of the bending content of the strain energy, when in the direct approach
one wishes that it be nicely separated from the stretching content.

This was indeed the criterion advocated in [2] for a planar extensible
_elastica_ : if $\vartheta$ denotes the centerline’s deflection angle
(relative to a fixed direction in the plane) of a naturally straight rod, the
appropriate quadratic bending energy independent of stretching would be
proportional to $(\partial_{x}\vartheta)^{2}$, and not
$(\partial_{s}\vartheta)^{2}$, where $x$ and $s$ denote the arc-length
coordinate in the reference and present configurations, respectively. As also
remarked in [2], a theory measuring the bending energy of an extensible rod by
$(\partial_{s}\vartheta)^{2}$, such as that proposed in [5], would in general
be more complicated.111It seems indeed simplicity the major advantage of
Antman’s proposed constitutive law, as witnessed by the neat analysis that
ensued in [2].

Other attempts to derive the balance equations for an extensible elastic rod
have been based in [6] and [7] on Biot’s _nominal_ strain tensor [8] (see also
[9] and [10]); they corroborated the approach purported in [2]. A similar
corroboration came from [11], which followed yet another avenue, designed
according to the canons of modern continuum mechanics.

Similarly, for a plate whose planar midsurface $S$ in the reference
configuration gets deformed into the surface $\mathscr{S}$ embedded in three-
dimensional space, the invariants of the curvature tensor
$\nabla\\!_{\mathrm{s}}\bm{\nu}$, where $\bm{\nu}$ is the outer unit normal to
$\mathscr{S}$ and $\nabla\\!_{\mathrm{s}}$ denotes the surface gradient,
cannot be pure measures of bending, as also neatly illustrated in [12].

In analogy with the theory of stretchable rods, proposals for the bending
energy of plates and shells (i.e., plates curved in their natural state) have
been advanced that are not affected by one or another form of stretching. The
lines of thought followed to achieve this goal are disparate, as are the
conclusions reached; controversies abound, especially for shells, which I
deliberately leave out of the scope of this paper.

In an alternative approach to geometric elasticity as formulated in [13], a
theory for deformable, geometrically incompatible sheets was derived in [7]
from a three-dimensional model phrased in terms of Biot’s strain measures. The
outcome is in contrast with [13], also in the rod-limit case, where the
alternative energy is closer to Antman’s. The theory of [7] has been applied
to a soft matter system in [14]; it effectively employs a measure of bending
for plates that has a distant antecedent in [15].

More recently, a dimension reduction method following the general lines
proposed in [16, 17, 18, 19] was applied in [20] to a three-dimensional
isotropic quadratic strain energy [21] formulated in terms of Biot’s strain
tensor to justify a model for shells. In particular, in the limiting case of
plates, the one that concerns us here, two two-dimensional bending measures
arose naturally from the method employed in [20], one invariant under
arbitrary superimposed stretchings, and the other invariant only under
superimposed dilations, both reducing to Antman’s in the rod-limit.

The questions addressed in this paper will be of a pure kinematic nature.
First, we need to identify a criterion of _bending-neutrality_ for soft
(highly deformable) plates: we must single out the incremental deformations
that, although changing the shape of $\mathscr{S}$, do so without any further
bending. Thus, _pure_ measures of bending will be defined as the ones
_invariant_ under bending-neutral deformations. I shall focus on one isotropic
such measure, $\mathbf{A}$, tensorial in character, alongside with its scalar
invariants and contrast them with both similar and dissimilar measures from
the literature. It will be shown that $\mathbf{A}$ reduces to Antman’s measure
for extensible rods.

Identifying the pure measure of bending appropriate to a class of materials
should prelude to a proposal for a direct theory of plates for which
stretching and bending modes are kept separate from one another.

This paper is organized as follows. In Sec. II, we define the class of
bending-neutral deformations and present an isotropic measure of pure bending
associated with them. Section III 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 [22] and [23]. Finally, in Sec. IV, we
summarize the conclusions of this study and comment briefly about its possible
extensions.

##  II Bending-Neutral Deformations

Consider a surface $S$ lying on the $(x_{1},x_{2})$ plane of a Cartesian frame
$(\bm{e}_{1},\bm{e}_{2},\bm{e}_{3})$, so that $\bm{e}_{3}$ is one of its unit
normals. Let $\bm{x}$ denote the position vector in $S$ and let $\bm{y}$ be
the deformation that maps $S$ onto $\mathscr{S}$ in three-dimensional
Euclidean space $\mathscr{E}$. We shall assume that $\bm{y}$ is at least of
class $C^{2}$ and we shall denote by $\bm{\nu}$ the unit normal to
$\mathscr{S}$ oriented coherently with the orientation of $\bm{e}_{3}$ (see
Fig. 1).

Figure 1:  A flat surface $S$ in the $(x_{1},x_{2})$ plane of a fixed
Cartesian frame $(\bm{e}_{1},\bm{e}_{2},\bm{e}_{3})$ is deformed by the
mapping $\bm{y}$ into a smooth surface $\mathscr{S}$ embedded in three-
dimensional Euclidean space $\mathscr{E}$. The unit vectors
$(\bm{r}_{1},\bm{r}_{2})$ are the right principal directions of stretching in
the reference configuration, while $(\bm{l}_{1},\bm{l}_{2})$ are the left
principal directions of stretching in the present configuration; $\bm{e}_{3}$
is the outer unit normal to $S$, while $\bm{\nu}$ is the outer unit normal to
$\mathscr{S}$; they are oriented so that
$\bm{e}_{3}=\bm{r}_{1}\times\bm{r}_{2}$ and
$\bm{\nu}=\bm{l}_{1}\times\bm{l}_{2}$, respectively. Surface
$\mathscr{S}^{\ast}$ is obtained by combining $\bm{y}$ with the incremental
deformation $\bm{y}^{\prime}$, so that
$\bm{y}^{\ast}=\bm{y}^{\prime}\circ\bm{y}$. The outer unit normal to
$\mathscr{S}^{\ast}$ is $\bm{\nu}^{\ast}$ and the left principal directions of
stretching associated with $\bm{y}^{\ast}$ are $\bm{l}_{1}^{\ast}$ and
$\bm{l}_{2}^{\ast}$, with
$\bm{\nu}^{\ast}=\bm{l}_{1}^{\ast}\times\bm{l}_{2}^{\ast}$.

Extending to the present geometric setting results well-known from three-
dimensional kinematics (see, e.g., [24, Ch. 6]), the deformation gradient
$\nabla\bm{y}$ can be represented as

$\nabla\bm{y}=\lambda_{1}\bm{l}_{1}\otimes\bm{r}_{1}+\lambda_{2}\bm{l}_{2}\otimes\bm{r}_{2},$
(1)

where $\nabla$ denotes the gradient in $\bm{x}$, the positive scalars
$\lambda_{1}$, $\lambda_{2}$ are the principal stretches and the unit vectors
$\bm{r}_{i}$, $\bm{l}_{i}$ are the corresponding _right_ and _left_ principal
directions of stretching. While $\bm{r}_{1}(\bm{x})$ and $\bm{r}_{2}(\bm{x})$
live on the $(x_{1},x_{2})$ plane for all $\bm{x}\in S$, $\bm{l}_{1}(\bm{x})$
and $\bm{l}_{2}(\bm{x})$ live on the tangent plane to $\mathscr{S}$ at
$\bm{y}(\bm{x})$. The right and left Cauchy-Green tensors are correspondingly
defined and represented by

$\displaystyle\mathbf{C}:=(\nabla\bm{y})^{\mathsf{T}}(\nabla\bm{y})$
$\displaystyle=\lambda_{1}^{2}\bm{r}_{1}\otimes\bm{r}_{1}+\lambda_{2}^{2}\bm{r}_{2}\otimes\bm{r}_{2},$
(2a) $\displaystyle\mathbf{B}:=(\nabla\bm{y})(\nabla\bm{y})^{\mathsf{T}}$
$\displaystyle=\lambda_{1}^{2}\bm{l}_{1}\otimes\bm{l}_{1}+\lambda_{2}^{2}\bm{l}_{2}\otimes\bm{l}_{2}.$
(2b)

I wish to introduce a _bending-neutral_ deformation as an incremental (finite)
deformation $\bm{y}^{\prime}$ that maps $\mathscr{S}$ into a surface
$\mathscr{S}^{\ast}$ _without_ further bending it. To make this notion
precise, we first consider the composed deformation
$\bm{y}^{\ast}:=\bm{y}^{\prime}\circ\bm{y}$, where $\bm{y}^{\prime}$ is a
general, smooth (at least $C^{2}$) mapping that changes $\mathscr{S}$ into
$\mathscr{S}^{\ast}$ and is characterized by having

$\nabla\bm{y}^{\prime}=\mathbf{R}^{\prime}\mathbf{U}^{\prime},$ (3)

where $\mathbf{R}^{\prime}$ belongs to the proper orthogonal group in three
dimensions, $\mathsf{SO}(3)$, and

$\mathbf{U}^{\prime}=\lambda_{1}^{\prime}\bm{u}^{\prime}_{1}\otimes\bm{u}^{\prime}_{1}+\lambda_{2}^{\prime}\bm{u}^{\prime}_{2}\otimes\bm{u}^{\prime}_{2}$
(4)

is a _stretching_ tensor with principal directions of stretching
$\bm{u}^{\prime}_{1}$, $\bm{u}^{\prime}_{2}$ tangent to $\mathscr{S}$ at
$\bm{y}(\bm{x})$ and principal stretches $\lambda_{1}^{\prime}$,
$\lambda_{2}^{\prime}>0$. The tensor $\mathbf{U}^{\prime}$ is to be regarded
as a (linear) mapping of the tangent plane to $\mathscr{S}$ at
$\bm{y}(\bm{x})$ into itself.

Equation (3) is the form of polar decomposition fit to describe the
deformation of material surfaces; it was established in [25] within the
general coordinate-free theory introduced in [26], which will also be followed
here (see also [27]). A tensor $\mathbf{V}^{\prime}$ that maps the tangent
plane to $\mathscr{S}^{\ast}$ at
$\bm{y}^{\ast}(\bm{x})=\bm{y}^{\prime}(\bm{y}(\bm{x}))$ into itself, can also
be introduced via the following equation,

$\mathbf{R}^{\prime}\mathbf{U}^{\prime}=\mathbf{V}^{\prime}\mathbf{R}^{\prime}.$
(5)

It follows from (5) that $\mathbf{V}^{\prime}$ possesses the same eigenvalues
$\lambda_{1}^{\prime}$, $\lambda_{2}^{\prime}$ as $\mathbf{U}^{\prime}$, while
its eigenvectors are $\mathbf{R}^{\prime}\bm{u}^{\prime}_{1}$,
$\mathbf{R}^{\prime}\bm{u}^{\prime}_{2}$, which generally differ from the left
principal directions of stretching $\bm{l}_{1}^{\ast}$, $\bm{l}_{2}^{\ast}$
associated with $\bm{y}^{\ast}$ (and shown Fig. 1). By the chain rule,

$\nabla\bm{y}^{\ast}(\bm{x})=\nabla\bm{y}^{\prime}(\bm{y}(\bm{x}))\nabla\bm{y}(\bm{x})\quad\forall\
\bm{x}\in S,$ (6)

(3) and (5) imply that

$\mathbf{B}^{\ast}:=(\nabla\bm{y}^{\ast})(\nabla\bm{y}^{\ast})^{\mathsf{T}}=\mathbf{V}^{\prime}(\mathbf{R}^{\prime}\mathbf{B}\mathbf{R}^{\prime\mathsf{T}})\mathbf{V}^{\prime},$
(7)

which shows that the eigenframes of $\mathbf{B}^{\ast}$ and
$\mathbf{V}^{\prime}$ coincide whenever $\mathbf{B}$ and $\mathbf{U}^{\prime}$
commute.222This follows easily from applying both sides of (7) to the
eigenvectors $\mathbf{R}^{\prime}\bm{u}^{\prime}_{i}$ of
$\mathbf{V}^{\prime}$.

It should also be noted that since $\bm{y}^{\prime}$ is defined on
$\mathscr{S}$, its gradient $\nabla\bm{y}^{\prime}$ featuring in (3) is to be
properly regarded as a _surface_ gradient on $\mathscr{S}$,
$\nabla\\!_{\mathrm{s}}\bm{y}^{\prime}$, according to the definition given in
[26] (see also [28]).

We say that $\bm{y}^{\prime}$ is a _bending-neutral_ deformation if

$\mathbf{R}^{\prime}=\mathbf{R}_{0}\mathbf{R}_{\bm{\nu}},$ (8)

where $\mathbf{R}_{0}\in\mathsf{SO}(3)$ is a _uniform_ rotation, i.e.,
independent of position, and $\mathbf{R}_{\bm{\nu}}\in\mathsf{SO}(\bm{\nu})$,
where $\mathsf{SO}(\bm{\nu})$ is the group of rotations about the unit normal
$\bm{\nu}$. While $\mathbf{R}_{0}$ is a global _rigid motion_ ,
$\mathbf{R}_{\bm{\nu}}$ is a local _drill_ rotation. For $\mathbf{R}^{\prime}$
as in (8), (3) represents an incremental deformation that entails local
_stretching_ of $\mathscr{S}$, described by $\mathbf{U}^{\prime}$, possibly
followed by a _twist_ distortion about $\bm{\nu}$, described by
$\mathbf{R}_{\bm{\nu}}$, and an overall undistorting rotation
$\mathbf{R}_{0}$.

We shall discuss in Sec. III the compatibility condition that a bending-
neutral deformation $\bm{y}^{\prime}$ must obey to exist, at least locally, on
$\mathscr{S}$. Here we only heed that this class of incremental deformations
includes both pure stretching deformations, for which $\mathbf{R}_{\bm{\nu}}$
is the identity, and pure drill rotations, for which $\mathbf{U}^{\prime}$ is
the projection $\mathbf{P}(\bm{\nu}):=\mathbf{I}-\bm{\nu}\otimes\bm{\nu}$ on
the plane orthogonal to $\bm{\nu}$.333$\mathbf{I}$ is the identity tensor in
three space dimensions. As shown in [22] and [23], respectively, neither of
these subclasses is empty.

###  II.1 Pure Measure of Bending

We say that a deformation measure $\mathbb{A}(\bm{y})$, of whatever nature
(scalar or tensor), is a _pure_ measure of bending if it obeys the following
invariance condition,

$\mathbb{A}(\bm{y}^{\ast})=\mathbb{A}(\bm{y}),$ (9)

for all deformations $\bm{y}^{\ast}=\bm{y}^{\prime}\circ\bm{y}$ such that
$\bm{y}^{\prime}$ is bending-neutral.444A different definition for a _natural_
measure of bending was proposed in [29] in the case of infinitesimal
incremental deformations of $\mathscr{S}$.

Since bending is intuitively associated with curvature, to flesh out this
definition we first need determine how $\bm{\nu}^{\ast}$, the outer unit
normal to $\mathscr{S}^{\ast}$, is related to $\bm{\nu}$ when
$\bm{y}^{\prime}$ is a bending-neutral deformation (and then also how
$\nabla\bm{\nu}^{\ast}$ is related to $\nabla\bm{\nu}$). To this end, we first
recall that

$\bm{\nu}(\bm{x})=\frac{(\nabla\bm{y})\bm{e}_{1}\times(\nabla\bm{y})\bm{e}_{2}}{|(\nabla\bm{y})\bm{e}_{1}\times(\nabla\bm{y})\bm{e}_{2}|}=\frac{\operatorname{co}(\nabla\bm{y})\bm{e}_{3}}{|\operatorname{co}(\nabla\bm{y})\bm{e}_{3}|},$
(10)

where $\operatorname{co}(\cdot)$ denotes the cofactor of a tensor (according
to the definition in [24, $\S$ 2.11]). Since for any two tensors, $\mathbf{L}$
and $\mathbf{M}$,
$\operatorname{co}(\mathbf{L}\mathbf{M})=\operatorname{co}(\mathbf{L})\operatorname{co}(\mathbf{M})$,
by (6) and (10), we have that555Use is also made here of the fact that
$\operatorname{co}(\mathbf{R})=\mathbf{R}$, for any rotation
$\mathbf{R}\in\mathsf{SO}(3)$.

$\bm{\nu}^{\ast}=\frac{\operatorname{co}(\nabla\bm{y}^{\ast})\bm{e}_{3}}{|\operatorname{co}(\nabla\bm{y}^{\ast})\bm{e}_{3}|}=\frac{\operatorname{co}(\nabla\bm{y}^{\prime})\bm{\nu}}{|\operatorname{co}(\nabla\bm{y}^{\prime})\bm{\nu}|}=\mathbf{R}^{\prime}\frac{\operatorname{co}(\mathbf{U}^{\prime})\bm{\nu}}{|\operatorname{co}(\mathbf{U}^{\prime})\bm{\nu}|}.$
(11)

Now, if $\bm{y}^{\prime}$ is a bending-neutral deformation, since by (4)
$\operatorname{co}(\mathbf{U}^{\prime})=\lambda_{1}^{\prime}\lambda_{2}^{\prime}\bm{\nu}\otimes\bm{\nu}=(\det\mathbf{U}^{\prime})\bm{\nu}\otimes\bm{\nu}$
and $\mathbf{R}_{\bm{\nu}}\bm{\nu}=\bm{\nu}$, we learn from (11) and (8) that

$\bm{\nu}^{\ast}=\mathbf{R}_{0}\bm{\nu}.$ (12)

Moreover, since $\nabla\mathbf{R}_{0}=\bm{0}$, we see that

$\nabla\bm{\nu}^{\ast}=\mathbf{R}_{0}\nabla\bm{\nu}.$ (13)

Thus, if we set

$\mathbf{A}(\bm{y}):=(\nabla\bm{\nu})^{\mathsf{T}}(\nabla\bm{\nu}),$ (14)

we readily see from (13) that this is a pure (tensorial) measure of bending,
as

$\mathbf{A}(\bm{y}^{\ast})=(\nabla\bm{\nu}^{\ast})^{\mathsf{T}}(\nabla\bm{\nu}^{\ast})=(\nabla\bm{\nu})^{\mathsf{T}}(\nabla\bm{\nu})=\mathbf{A}(\bm{y}).$
(15)

In the rest of the section, we shall focus on this measure. As defined in
(14), $\mathbf{A}$ is a linear mapping of the plane hosting the reference
configuration $S$ into itself, precisely as $\mathbf{C}$ in (2a). Like the
latter tensor, $\mathbf{A}$ is both symmetric and positive definite. According
ot our definition, its scalar invariants are pure measures of bending as well.

It is perhaps worth noting that the tensor
$\mathbf{Z}(\bm{y}):=(\nabla\bm{\nu})(\nabla\bm{\nu})^{\mathsf{T}}$, similar
in structure to $\mathbf{B}$, which maps the plane tangent to $\mathscr{S}$
into itself, is _not_ a pure measure of bending, although it has the same
invariants as $\mathbf{A}$: $\mathbf{Z}$ transforms as follows,
$\mathbf{Z}(\bm{y}^{\ast})=\mathbf{R}_{0}\mathbf{Z}(\bm{y})\mathbf{R}_{0}^{\mathsf{T}}$.

We denote by $\nabla\\!_{\mathrm{s}}\bm{\nu}$ the _curvature_ tensor of
$\mathscr{S}$, for which we adopt a sign convention opposite to the one
customary in differential geometry.666Here we follow [30] and [31] in the
desire to represent as
$\nabla\\!_{\mathrm{s}}\bm{\nu}=\frac{1}{R}\mathbf{P}(\bm{\nu})$ the curvature
tensor of a _sphere_ of radius $R$. $\nabla\\!_{\mathrm{s}}\bm{\nu}$ is a
symmetric tensor mapping the tangent plane to $\mathscr{S}$ into itself; in
its eigenframe, it is represented as

$\nabla\\!_{\mathrm{s}}\bm{\nu}=\kappa_{1}\bm{n}_{1}\otimes\bm{n}_{1}+\kappa_{2}\bm{n}_{2}\otimes\bm{n}_{2},$
(16)

where $\bm{n}_{1}$, $\bm{n}_{2}$ are the principal directions of curvature of
$\mathscr{S}$ and $\kappa_{1}$, $\kappa_{2}$ are the corresponding principal
curvatures.

By the chain rule,

$\nabla\bm{\nu}(\bm{x})=\nabla\\!_{\mathrm{s}}\bm{\nu}(\bm{y}(\bm{x}))\nabla\bm{y}(\bm{x}),$
(17)

and so it follows from (14) that $\mathbf{A}$ can also be written as

$\mathbf{A}=(\nabla\bm{y})^{\mathsf{T}}(\nabla\\!_{\mathrm{s}}\bm{\nu})^{2}(\nabla\bm{y}).$
(18)

Here and in the following, for brevity, we drop the argument from
$\mathbf{A}(\bm{y})$ and denote
$\mathbf{A}^{\ast}:=\mathbf{A}(\bm{y}^{\ast})$.

An interesting interpretation for $\mathbf{A}$ and its invariance under
bending-neutral deformations can be given starting from (18). This can indeed
be seen as the referential representation for the _third_ fundamental form of
$\mathscr{S}$ (see, for example [32, p. 205]). Given two arbitrary vectors
$\bm{v}_{1}$, $\bm{v}_{2}$ on $S$,

$\bm{v}_{1}\cdot\mathbf{A}\bm{v}_{2}=\bm{v}_{1}^{\prime}\cdot(\nabla\\!_{\mathrm{s}}\bm{\nu})^{2}\bm{v}_{2}^{\prime}=\bm{v}_{1}\cdot\mathbf{A}^{\ast}\bm{v}_{2}=\bm{v}_{1}^{\ast}\cdot(\nabla\\!_{\mathrm{s}}\bm{\nu}^{\ast})^{2}\bm{v}_{2}^{\ast},$
(19)

where $\bm{v}_{i}^{\prime}=(\nabla\bm{y})\bm{v}_{i}$ are vectors tangent to
$\mathscr{S}$, $\bm{v}_{i}^{\ast}=(\nabla\bm{y}^{\ast})\bm{v}_{i}$ are vectors
tangent to $\mathscr{S}^{\ast}$, and $\nabla\\!_{\mathrm{s}}\bm{\nu}^{\ast}$
is the curvature tensor of $\mathscr{S}^{\ast}$. The identity (19) simply says
that the third fundamental form is _invariant_ under a bending-neutral
deformation.

As pointed out in [33], the tensorial deformation measure for shells
considered in the works of Koiter [34], Sanders [31], and Budiansky [30] can
be given a coordinate-free expression, which in the case of plates reads
simply as

$\mathbf{K}:=(\nabla\bm{y})^{\mathsf{T}}(\nabla\\!_{\mathrm{s}}\bm{\nu})(\nabla\bm{y}).$
(20)

This appears to be the referential representation of the _second_ fundamental
form of $\mathscr{S}$. In contrast with $\mathbf{A}$, $\mathbf{K}$ is not a
pure measure of bending.

The linear invariant of $\mathbf{A}$, $\operatorname{tr}\mathbf{A}$ (which is
quadratic in the principal curvatures of $\mathscr{S}$), has already featured
in the literature; it emerged as a strain energy density from a discretized
model for membranes and plates [35] and, more recently, also from a dimension
reduction to plates performed in [20]. In the latter, the reduced strain
energy density also involves the scalar invariants of the tensor
$\bm{\Pi}:=\frac{1}{2}(\mathbf{V}(\nabla\\!_{\mathrm{s}}\bm{\nu})+(\nabla\\!_{\mathrm{s}}\bm{\nu})\mathbf{V})$,
where $\mathbf{V}$ is the stretching tensor of $\bm{y}$ in the present
configuration, defined as the positive definite root of
$\mathbf{V}^{2}=\mathbf{B}$. It is a simple exercise to show by example that
neither $\bm{\Pi}$ nor its invariants are pure measures of bending according
to the definition proposed in this paper.

Finally, the shell model advanced in [36] employs strains that reduce to the
eigenvalues of $\mathbf{A}$ in the case of plates, under the further
hypothesis that $\mathbf{V}$ and $\nabla\\!_{\mathrm{s}}\bm{\nu}$ commute, a
hypothesis implicit in the symmetric class of deformations adopted in
[36].777This hypothesis, however, fails to make $\bm{\Pi}$ a pure measure of
bending for plates.

###  II.2 Impure Measures of Bending

It is interesting to link the invariants of $\mathbf{A}$ to other invariant,
but impure measures of bending, namely, the mean curvature $H$ and the
Gaussian curvature $K$ of $\mathscr{S}$, defined as

$H:=\frac{1}{2}\operatorname{tr}(\nabla\\!_{\mathrm{s}}\bm{\nu})\quad\text{and}\quad
K:=\det(\nabla\\!_{\mathrm{s}}\bm{\nu}).$ (21)

We begin by computing $\operatorname{tr}\mathbf{A}$. It readily follows from
(18) that

$\operatorname{tr}\mathbf{A}=\operatorname{tr}(\mathbf{B}(\nabla\\!_{\mathrm{s}}\bm{\nu})^{2})=2H\operatorname{tr}(\mathbf{B}\nabla\\!_{\mathrm{s}}\bm{\nu})-K\operatorname{tr}\mathbf{B},$
(22)

where the last equality follows from an identity proven in [37] (see their
equation (14)).

To obtain $\det\mathbf{A}$, we start from $\operatorname{tr}\mathbf{A}^{2}$,
which by (18) also reads as

$\operatorname{tr}\mathbf{A}^{2}=(\nabla\\!_{\mathrm{s}}\bm{\nu})^{2}\mathbf{B}\cdot\mathbf{B}(\nabla\\!_{\mathrm{s}}\bm{\nu})^{2},$
(23)

where $\cdot$ denotes the inner product of tensors.888For generic tensors
$\mathbf{L}$ and $\mathbf{M}$,
$\mathbf{L}\cdot\mathbf{M}:=\operatorname{tr}(\mathbf{L}^{\mathsf{T}}\mathbf{M})$.
Letting $\mathbf{B}$ be represented as

$\displaystyle\mathbf{B}$
$\displaystyle=B_{11}\bm{n}_{1}\otimes\bm{n}_{1}+B_{12}(\bm{n}_{1}\otimes\bm{n}_{2}+\bm{n}_{2}\otimes\bm{n}_{1})+B_{22}\bm{n}_{2}\otimes\bm{n}_{2}$
(24)

in the eigenframe of $\nabla\\!_{\mathrm{s}}\bm{\nu}$ in (16), we easily
obtain from (23) that

$\operatorname{tr}\mathbf{A}^{2}=\kappa_{1}^{4}B_{11}^{2}+\kappa_{2}^{4}B_{22}^{2}+2\kappa_{1}^{2}\kappa_{2}^{2}B_{12}^{2}=(\operatorname{tr}\mathbf{A})^{2}-2K^{2}\det\mathbf{B},$
(25)

which by the Cayley-Hamilton theorem applied to $\mathbf{A}$ implies that

$\det\mathbf{A}=K^{2}\det\mathbf{B}=K^{2}\det\mathbf{C}.$ (26)

Thus, equations (22) and (26) show how pure scalar measures of bending may
result from combining measures of stretching and impure measures of bending.

In the special case of an isometry, for which
$\mathbf{C}=\mathbf{I}-\bm{e}_{3}\otimes\bm{e}_{3}$ and
$\mathbf{B}=\mathbf{P}(\bm{\nu})$,

$\operatorname{tr}\mathbf{A}=\kappa_{1}^{2}+\kappa_{2}^{2}=2C\quad\text{and}\quad\det\mathbf{A}=(\kappa_{1}\kappa_{2})^{2}=K^{2},$
(27)

where $C$ is the Casorati total curvature [38].

###  II.3 Pure Rod-Bending Measure

To connect $\mathbf{A}$ to Antman’s measure of pure bending for rods [2], we
consider a class of deformations of $S$ that produce a cylindrical surface, as
shown in Fig. 2.

Figure 2: Cylindrical surface produced by the deformation $\bm{y}$ described
in (28).

More precisely, we take

$\bm{y}(x_{1},x_{2})=y_{1}(x_{1})\bm{e}_{1}+x_{2}\bm{e}_{2}+y_{3}(x_{1})\bm{e}_{3},$
(28)

where $y_{1}$, $y_{3}$ are scalar functions of class $C^{2}$. From (28) we
easily arrive at

$\nabla\bm{y}=y_{1}^{\prime}\bm{e}_{1}\otimes\bm{e}_{1}+\bm{e}_{2}\otimes\bm{e}_{2}+y_{3}^{\prime}\bm{e}_{3}\otimes\bm{e}_{1},$
(29)

where a prime ′ now denotes differentiation with respect to $x_{1}$. Thus, by
applying (10), we see that

$\bm{\nu}=\cos\vartheta\bm{e}_{3}-\sin\vartheta\bm{e}_{1}\quad\text{with}\quad\cos\vartheta=\frac{y_{1}^{\prime}}{\sqrt{y_{1}^{\prime
2}+y_{3}^{\prime
2}}}\quad\text{and}\quad\sin\vartheta=\frac{y_{3}^{\prime}}{\sqrt{y_{1}^{\prime
2}+y_{3}^{\prime 2}}}.$ (30)

With this choice of the angle $\vartheta$, the unit tangent $\bm{t}$ to the
curve in space that sweeps $\mathscr{S}$ along the generatrix $\bm{e}_{2}$ is
represented as

$\bm{t}=\frac{(\nabla\bm{y})\bm{e}_{1}}{|(\nabla\bm{y})\bm{e}_{1}|}=\cos\vartheta\bm{e}_{1}+\sin\vartheta\bm{e}_{3},$
(31)

and it follows from (30) that

$\mathbf{A}=\vartheta^{\prime 2}\bm{e}_{1}\otimes\bm{e}_{1}.$ (32)

The only non-vanishing invariant of $\mathbf{A}$ is
$\operatorname{tr}\mathbf{A}=\vartheta^{\prime 2}$, which, to within an
elastic modulus, is precisely Antamn’s bending energy for an extensible
elastic rod.

###  II.4 An Illustration: Axisymmetric Deformations

To illustrate the theory proposed in this paper, we consider now the general
class of axisymmetric deformations of a _disk_ of radius $R$. The geometric
setting is shown in Fig. 3, where $S$ is the undeformed disk and $\mathscr{S}$
is an axisymmetric _nightcap_ generated by the deformation $\bm{y}$
parametrized as

$\bm{y}(s,\varphi)=r(s)\bm{e}_{r}+z(s)\bm{e}_{3},$ (33)

where $(s,\varphi)$, with $0\leqq s\leqq R$ and $0\leqq\varphi\leqq 2\pi$, are
polar coordinates in the plane $(x_{1},x_{2})$, so that $x_{1}=s\cos\varphi$
and $x_{2}=s\sin\varphi$, $\bm{e}_{r}$ is the radial unit vector and
$\bm{e}_{\varphi}$ the azimuthal unit vector, $r(s)$ and $z(s)$ are smooth
functions representing radial and vertical components of the image under
$\bm{y}$ of the generic point $\bm{x}=s\bm{e}_{\varphi}$ of $S$.

Figure 3: Sketch of an axisymmetric _nightcap_ $\mathscr{S}$ obtained by
deforming a flat disk $S$ of radius $R$ via the mapping $\bm{y}$ described by
(33). The unit vector $\bm{\nu}$ is the outward normal to $\mathscr{S}$, while
$\bm{\tau}$ is the tangent unit vector along the meridians of $\mathscr{S}$,
so that $(\bm{\tau},\bm{e}_{\varphi},\bm{\nu})$ is a mobile frame oriented
like the fixed frame $(\bm{e}_{1},\bm{e}_{2},\bm{e}_{3})$. The angle
$\vartheta$, defined in (44), designates the relative inclination of the
normals $\bm{\nu}$ and $\bm{e}_{3}$, to $\mathscr{S}$ and $S$, respectively.

Let $\bm{x}(t)=s(t)\bm{e}_{r}(\varphi(t))$ be a generic curve on $S$
parameterized in $t$. Denoting by a superimposed dot differentiation with
respect to $t$, we readily obtain that

$\dot{\bm{x}}=\dot{s}\bm{e}_{r}+s\dot{\varphi}\bm{e}_{\varphi}\quad\text{and}\quad\dot{\bm{y}}=\dot{s}\left(r^{\prime}\bm{e}_{r}+z^{\prime}\bm{e}_{3}\right)+r\dot{\varphi}\bm{e}_{\varphi},$
(34)

where a prime ′ denotes differentiation with respect to $s$ and $\dot{\bm{y}}$
is the tangent to the curve $\bm{y}(t)$ resulting from composing $\bm{y}$ with
$\bm{x}(t)$. The deformation gradient $\nabla\bm{y}$ at a point $\bm{x}_{0}$
must satisfy the identity $\dot{\bm{y}}=(\nabla\bm{y})\dot{\bm{x}}$ for every
curve $\bm{x}(t)$ through $\bm{x}_{0}$. The following formula is then an easy
consequence of (34),

$\nabla\bm{y}=r^{\prime}\bm{e}_{r}\otimes\bm{e}_{r}+\frac{r}{s}\bm{e}_{\varphi}\otimes\bm{e}_{\varphi}+z^{\prime}\bm{e}_{3}\otimes\bm{e}_{r}.$
(35)

It follow from (35) that

$\mathbf{C}=(\nabla\bm{y})^{\mathsf{T}}(\nabla\bm{y})=\lambda_{1}^{2}\bm{e}_{r}\otimes\bm{e}_{r}+\lambda_{2}^{2}\bm{e}_{3}\otimes\bm{e}_{3},$
(36)

where

$\lambda_{1}:=\sqrt{r^{\prime 2}+z^{\prime
2}}\quad\text{and}\quad\lambda_{2}:=\frac{r}{s}$ (37)

are the principal stretches of $\mathbf{U}=\sqrt{\mathbf{C}}$.

If the top of the nightcap in Fig. 3 is regular, that is, with a well defined
tangent plane, symmetry requires the latter to be parallel to the
$(x_{1},x_{2})$ plane, and so $\nabla\bm{y}$ must be the identity in that
plane. From (35), this implies that $r^{\prime}(0)=\lim_{s\to 0}r(s)/s=1$ and
$z^{\prime}(0)=0$, meaning that $\lambda_{1}(0)=\lambda_{2}(0)=1$.

A simple computation yields

$\operatorname{co}(\nabla\bm{y})=\frac{r}{s}\left(r^{\prime}\bm{e}_{3}\otimes\bm{e}_{3}-z^{\prime}\bm{e}_{r}\otimes\bm{e}_{3}\right),$
(38)

which by (10) leads us to

$\bm{\nu}=\frac{1}{\lambda_{1}}\left(r^{\prime}\bm{e}_{3}-z^{\prime}\bm{e}_{r}\right).$
(39)

Defining by $\bm{\tau}:=\bm{e}_{\varphi}\times\bm{\nu}$ a tangent unit vector
to $\mathscr{S}$ oriented along its meridians, so that the mobile frame
$(\bm{\tau},\bm{e}_{\varphi},\bm{\nu})$ is oriented like
$(\bm{e}_{1},\bm{e}_{2},\bm{e}_{3})$, we can write

$\bm{\tau}=\frac{1}{\lambda_{1}}\left(r^{\prime}\bm{e}_{r}+z^{\prime}\bm{e}_{3}\right),$
(40)

so that (35) becomes

$\nabla\bm{y}=\lambda_{1}\bm{\tau}\otimes\bm{e}_{r}+\lambda_{2}\bm{e}_{\varphi}\otimes\bm{e}_{\varphi}.$
(41)

From this latter we easily compute

$\mathbf{B}=(\nabla\bm{y})(\nabla\bm{y})^{\mathsf{T}}=\lambda_{1}^{2}\bm{\tau}\otimes\bm{\tau}+\lambda_{2}^{2}\bm{e}_{\varphi}\otimes\bm{e}_{\varphi}$
(42)

and

$(\nabla\bm{y})^{-1}=\frac{1}{\lambda_{1}}\bm{e}_{r}\otimes\bm{\tau}+\frac{1}{\lambda_{2}}\bm{e}_{\varphi}\otimes\bm{e}_{\varphi}.$
(43)

By introducing the angle

$\vartheta(s):=-\arctan\frac{z^{\prime}}{r^{\prime}},$ (44)

we give a more compact representation of $\bm{\nu}$ and $\bm{\tau}$,

$\bm{\nu}=\cos\vartheta\bm{e}_{3}+\sin\vartheta\bm{e}_{r},\quad\bm{\tau}=\cos\vartheta\bm{e}_{r}-\sin\vartheta\bm{e}_{3}.$
(45)

As a consequence of (45) and (41), we can write

$\nabla\bm{\nu}=\vartheta^{\prime}\bm{\tau}\otimes\bm{e}_{r}+\frac{1}{s}\sin\vartheta\bm{e}_{\varphi}\otimes\bm{e}_{\varphi},$
(46)

from which it follows that the pure measure of bending $\mathbf{A}$ in (14)
acquires here the explicit representation

$\mathbf{A}=\vartheta^{\prime
2}\bm{e}_{r}\otimes\bm{e}_{r}+\frac{1}{s^{2}}\sin^{2}\vartheta\bm{e}_{\varphi}\otimes\bm{e}_{\varphi},$
(47)

so that a quadratic pure bending invariant would be

$\operatorname{tr}\mathbf{A}=\vartheta^{\prime
2}+\frac{1}{s^{2}}\sin^{2}\vartheta.$ (48)

For a comparison, we also compute the quadratic invariants of the curvature
tensor $\nabla\\!_{\mathrm{s}}\bm{\nu}$. By (17), (43), and (46), we obtain
that

$\nabla\\!_{\mathrm{s}}\bm{\nu}=(\nabla\bm{\nu})(\nabla\bm{y})^{-1}=\frac{1}{\lambda_{1}}\vartheta^{\prime}\bm{\tau}\otimes\bm{\tau}+\frac{1}{\lambda_{2}s}\sin\vartheta\bm{e}_{\varphi}\otimes\bm{e}_{\varphi},$
(49)

which, compared with (16), yields

$\kappa_{1}=\frac{\vartheta^{\prime}}{\sqrt{r^{\prime 2}+z^{\prime
2}}}\quad\text{and}\quad\kappa_{2}=\frac{\sin\vartheta}{r}.$ (50)

The quadratic impure bending measures would then read as

$H^{2}=\frac{1}{4}\left(\frac{\vartheta^{\prime}}{\sqrt{r^{\prime 2}+z^{\prime
2}}}+\frac{\sin\vartheta}{r}\right)^{2}\quad\text{and}\quad
K=\frac{\vartheta^{\prime}\sin\vartheta}{r\sqrt{r^{\prime 2}+z^{\prime 2}}}.$
(51)

By combining together (42), (47), and the expression for $K$ in (51), one
easily sees that identity (26) is indeed satisfied.

A simple direct theory of isotropic soft plates developed according to the
principles purported in this paper would be based on the following strain
energy density (per unit reference area),

$\displaystyle W$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\alpha\operatorname{tr}\mathbf{A}+\frac{1}{2}\beta[\operatorname{tr}\mathbf{U}^{2}+2(1-\operatorname{tr}\mathbf{U})]$
(52) $\displaystyle=$ $\displaystyle\frac{1}{2}\alpha\left(\vartheta^{\prime
2}+\frac{1}{s^{2}}\sin^{2}\vartheta\right)+\frac{1}{2}\beta[(\lambda_{1}-1)^{2}+(\lambda_{2}-1)^{2}],$

where $\alpha$ and $\beta$ are positive elastic moduli. A similar (but not
identical) expression for $W$ was proposed for axisymmetric shells in [39]
(see also [40, p. 642]); it was amenable to a formal Hamiltonian treatment.
The strain energy functional resulting from $W$ is then

$F[r,z]:=\int_{S}W\mathrm{d}A=\pi\int_{0}^{R}\left\\{\alpha\left(\vartheta^{\prime
2}+\frac{1}{s^{2}}\sin^{2}\vartheta\right)+\beta[(\lambda_{1}-1)^{2}+(\lambda_{2}-1)^{2}]\right\\}s\mathrm{d}s,$
(53)

subject to appropriate boundary conditions on the edge of $S$.

By contrast, a quadratic theory building on the impure measures in (51), would
be based on an Helfrich-type functional,

$\displaystyle F_{\mathrm{H}}[r,z]$ $\displaystyle:=$
$\displaystyle\int_{\mathscr{S}}(\alpha H^{2}+\beta K)\mathrm{d}a$ (54)
$\displaystyle=$
$\displaystyle\int_{S}\lambda_{1}\lambda_{2}\left[\frac{1}{4}\alpha\left(\frac{\vartheta^{\prime}}{\lambda_{1}}+\frac{\sin\vartheta}{r}\right)^{2}+\beta\frac{\vartheta^{\prime}}{\lambda_{1}}\frac{\sin\vartheta}{r}\right]\mathrm{d}A$
$\displaystyle=$
$\displaystyle\pi\int_{0}^{R}\left[\frac{1}{2}\alpha\left(\frac{r}{\sqrt{r^{\prime
2}+z^{\prime 2}}}\vartheta^{\prime 2}+\frac{\sqrt{r^{\prime 2}+z^{\prime
2}}}{r}\sin^{2}\vartheta\right)+(2\beta+\alpha)\vartheta^{\prime}\sin\vartheta\right]\mathrm{d}s,$

where the area element $\mathrm{d}a$ on the present shape $\mathscr{S}$ has
been related to the area element $\mathrm{d}A=s\mathrm{d}s\mathrm{d}\varphi$
on the reference shape $S$ through the equation
$\mathrm{d}a=\lambda_{1}\lambda_{2}\mathrm{d}A$.

Two features deserve notice in (54). First, as appropriate for a liquid
membrane, $F_{\mathrm{H}}$ is formulated as an integral on the present
configuration $\mathscr{S}$. Second, the Gaussian curvature $K$ reveals itself
as a _null Lagrangian_ , as its contribution to the total strain energy
depends only on the boundary value of $\vartheta$,

$\int_{\mathscr{S}}K\mathrm{d}a=2\pi[1-\cos\vartheta(R)],$ (55)

thus becoming an effective _edge_ energy.999The interplay between
$F_{\mathrm{H}}$ and edge strain energies has been explored in a series of
papers, some also recent [41, 42, 43, 44, 45, 46, 47, 48, 49].

##  III Bending-Neutrality Compatibility

For $\bm{y}^{\prime}$ to be a bending-neutral deformation, neither
$\mathbf{U}^{\prime}$ nor $\mathbf{R}_{\bm{\nu}}$ can be arbitrary. They must
be subject to a compatibility condition, which we shall discuss in some detail
here along with its solutions. The special case where
$\mathbf{U}^{\prime}=\mathbf{P}(\bm{\nu})$ has been considered and fully
solved in [23].

Compatibility arises from requiring that the curvature tensor
$\nabla\\!_{\mathrm{s}}\bm{\nu}^{\ast}$ of $\mathscr{S}^{\ast}$ be symmetric,
as it should (see also [23]). By writing (17) for $\bm{\nu}^{\ast}$ (and
$\bm{y}^{\ast}$) instead of $\bm{\nu}$ (and $\bm{y}$), we have that

$\nabla\bm{\nu}^{\ast}=(\nabla\\!_{\mathrm{s}}\bm{\nu}^{\ast})(\nabla\bm{y}^{\ast}).$
(56)

On the other hand, by (12) and (13), we can also write

$\nabla\bm{\nu}^{\ast}=\mathbf{R}_{0}\nabla\bm{\nu}=\mathbf{R}_{0}(\nabla\\!_{\mathrm{s}}\bm{\nu})(\nabla\bm{y}),$
(57)

which, combined with (56), leads us to

$\nabla\\!_{\mathrm{s}}\bm{\nu}^{\ast}=\mathbf{R}_{0}(\nabla\\!_{\mathrm{s}}\bm{\nu})(\mathbf{U}^{\prime})^{-1}\mathbf{R}_{\bm{\nu}}^{\mathsf{T}}\mathbf{R}_{0}^{\mathsf{T}},$
(58)

once use is also made of (6), (3), and (8). It follows from (58) that
$\nabla\\!_{\mathrm{s}}\bm{\nu}^{\ast}$ is symmetric if and only if
$(\nabla\\!_{\mathrm{s}}\bm{\nu})(\mathbf{U}^{\prime})^{-1}\mathbf{R}_{\bm{\nu}}^{\mathsf{T}}$
is so. This latter condition will be written in the following equivalent way,

$\nabla\\!_{\mathrm{s}}\bm{\nu}=\mathbf{R}_{\bm{\nu}}(\mathbf{U}^{\prime})^{-1}(\nabla\\!_{\mathrm{s}}\bm{\nu})\mathbf{R}_{\bm{\nu}}\mathbf{U}^{\prime}.$
(59)

Tensors on both sides of (59) act on the local tangent plane of $\mathscr{S}$.
It is clear that every solution $\mathbf{U}^{\prime}$ of (59) is defined to
within a multiplicative surface dilation,
$\lambda^{\prime}\mathbf{P}(\bm{\nu})$, with arbitrary $\lambda^{\prime}>0$.

Two special classes of solutions of (59) deserve notice. They are better
illustrated by use of the classical Euler-Rodriguez formula to represent
$\mathsf{SO}(\bm{\nu})$ (see also [50] for an updated account),

$\mathbf{R}(\alpha):=\mathbf{I}+\sin\alpha\mathbf{W}(\bm{\nu})-(1-\cos\alpha)\mathbf{P}(\bm{\nu}),\quad\alpha\in]-\pi,\pi],$
(60)

where $\mathbf{W}(\bm{\nu})$ is the skew-symmetric tensor associated with
$\bm{\nu}$, so that $\mathbf{W}(\bm{\nu})\bm{v}=\bm{\nu}\times\bm{v}$, for all
vectors $\bm{v}$.

First, if either $\mathbf{R}_{\bm{\nu}}=\mathbf{R}(0)=\mathbf{I}$ or
$\mathbf{R}_{\bm{\nu}}=\mathbf{R}(\pi)=-\mathbf{I}+2\bm{\nu}\otimes\bm{\nu}$,
the latter acting as $-\mathbf{I}$ on the plane tangent to $\mathscr{S}$, then
(59) reduces to requiring that $\mathbf{U}^{\prime}$ commute with
$\nabla\\!_{\mathrm{s}}\bm{\nu}$, which just amounts to say that
$\nabla\\!_{\mathrm{s}}\bm{\nu}$ and $\mathbf{U}^{\prime}$ must have the same
eigenframe (see also [22]).

Second, for $\mathbf{U}^{\prime}=\lambda^{\prime}\mathbf{P}(\bm{\nu})$, with
$\lambda^{\prime}>0$, (59) becomes

$\nabla\\!_{\mathrm{s}}\bm{\nu}=\mathbf{R}_{\bm{\nu}}(\nabla\\!_{\mathrm{s}}\bm{\nu})\mathbf{R}_{\bm{\nu}}.$
(61)

Taking the trace of both sides of (61), and using both (16) and (60), we
arrive at

$2H(1-\cos 2\alpha)=0,$ (62)

which for $2H=\operatorname{tr}(\nabla\\!_{\mathrm{s}}\bm{\nu})\neq 0$ reduces
to either $\alpha=0$ or $\alpha=\pi$, delivering again the first special
solution of (59) considered above. For $H=0$, which is the case of _minimal_
surfaces, (62) is identically satisfied and direct inspection of (61) with the
aid of (60) shows that it is solved by all
$\mathbf{R}_{\bm{\nu}}\in\mathsf{SO}(\bm{\nu})$. For $\lambda^{\prime}=1$,
this result was already proved in [23].101010As appropriately remarked in
[23], the same result had also been proved, albeit with different methods, in
the differential geometry literature (see [51, 52, 53]), but it was unknown to
plates and shells practitioners.

To find the general solution of (59), we represent $\mathbf{U}^{\prime}$ in
(4) in the eigenframe of $\nabla\\!_{\mathrm{s}}\bm{\nu}$ in (16), by writing

$\mathbf{U}^{\prime}=\mathbf{R}(\chi)\mathbf{U}_{0}^{\prime}\mathbf{R}(-\chi),$
(63)

where $\mathbf{R}(-\chi)=\mathbf{R}(\chi)^{\mathsf{T}}$,

$\mathbf{U}_{0}^{\prime}:=\lambda_{1}^{\prime}\bm{n}_{1}\otimes\bm{n}_{1}+\lambda_{2}^{\prime}\bm{n}_{2}\otimes\bm{n}_{2},$
(64)

and $\chi$ is the angle that $\bm{u}_{1}^{\prime}$ makes with $\bm{n}_{1}$
(the same as the angle that $\bm{u}_{2}^{\prime}$ makes with $\bm{n}_{2}$). By
use of these representation formulae and (60), we finally give (59) the
following expanded form,

$\nabla\\!_{\mathrm{s}}\bm{\nu}=\mathbf{R}(\alpha+\chi)(\mathbf{U}_{0}^{\prime})^{-1}\mathbf{R}(-\chi)(\nabla\\!_{\mathrm{s}}\bm{\nu})\mathbf{R}(\alpha+\chi)\mathbf{U}_{0}^{\prime}\mathbf{R}(-\chi),$
(65)

which, for given $\kappa_{1}$, $\kappa_{2}$, is meant to be an equation for
$\lambda_{2}^{\prime}/\lambda_{1}^{\prime}$, $\alpha$, and $\chi$.

Actually, it is not difficult to show that (65) amounts to four scalar
equations, whose solution is

$\frac{\lambda_{2}^{\prime}}{\lambda_{1}^{\prime}}=1-\frac{(\kappa_{1}+\kappa_{2})\sin\alpha}{\kappa_{2}\sin\alpha+(\kappa_{1}-\kappa_{2})\cos\chi\sin(\alpha+\chi)}.$
(66)

Requiring the ratio $\lambda_{2}^{\prime}/\lambda_{1}^{\prime}$ as delivered
by (66) to be positive, with little labour we arrive at the inequalities

$\alpha_{\mathrm{m}}<\alpha<\alpha_{\mathrm{M}},$ (67)

where $\alpha_{\mathrm{m}}(\chi):=\min\\{\alpha_{1},\alpha_{2}\\}$ and
$\alpha_{\mathrm{M}}(\chi):=\max\\{\alpha_{1},\alpha_{2}\\}$, with

$\alpha_{1}:=\arctan\left(\frac{(\kappa_{2}-\kappa_{1})\cos\chi\sin\chi}{\kappa_{1}\sin^{2}\chi+\kappa_{2}\cos^{2}\chi}\right)\quad\text{and}\quad\alpha_{2}:=\arctan\left(\frac{(\kappa_{2}-\kappa_{1})\cos\chi\sin\chi}{\kappa_{1}\cos^{2}\chi+\kappa_{2}\sin^{2}\chi}\right).$
(68)

Thus, for given $\chi\in]-\pi,\pi[$ and
$\alpha\in]\alpha_{\mathrm{m}},\alpha_{\mathrm{M}}[$, $\mathbf{U}^{\prime}$ is
delivered by (63) to within a surface dilation
$\lambda^{\prime}\mathbf{P}(\bm{\nu})$; in general, there is a whole three-
dimensional set of solutions to (59), parameterized by
$(\chi,\alpha,\lambda^{\prime})$. A singular case arises at umbilical points
of $\mathscr{S}$, where $\kappa_{1}=\kappa_{2}$. There, (66) would be
incompatible for any $\alpha\notin\\{0,\pi\\}$ since it requires
$\lambda_{2}^{\prime}/\lambda_{1}^{\prime}=-1$, whereas, by (68),
$\alpha_{\mathrm{m}}=\alpha_{\mathrm{M}}=0$, so that the standard solution
$\mathbf{U}^{\prime}=\lambda^{\prime}\mathbf{P}(\bm{\nu})$,
$\mathbf{R}_{\bm{\nu}}=\mathbf{I}$ is recovered by continuity.

A particular two-dimensional family of solutions deserves attention. It is
obtained from (66) by setting $\chi=0$, under the assumption that
$\alpha\notin\\{0,\pi\\}$ (an equivalent family would be obtained by setting
$\chi=\frac{\pi}{2}$). If $\mathscr{S}$ is a _hyperbolic_ surface, that is, if
$K<0$, (66) readily gives

$\frac{\lambda_{2}^{\prime}}{\lambda_{1}^{\prime}}=-\frac{\kappa_{2}}{\kappa_{1}},$
(69)

while $\alpha$ is arbitrary. What makes this solution special is the curvature
tensor of the surface $\mathscr{S}^{\ast}$ delivered by $\bm{y}^{\prime}$.
Letting $\mathbf{R}_{0}=\mathbf{I}$, with no prejudice to generality, we
obtain from (58) that

$\nabla\\!_{\mathrm{s}}\bm{\nu}^{\ast}=\frac{\kappa_{1}}{\lambda_{1}^{\prime}}\bm{n}_{1}\otimes\mathbf{R}(\alpha)\bm{n}_{1}+\frac{\kappa_{2}}{\lambda_{2}^{\prime}}\bm{n}_{2}\otimes\mathbf{R}(\alpha)\bm{n}_{2},$
(70)

whence, also by (69), it follows that

$\displaystyle 2H^{\ast}$
$\displaystyle:=\operatorname{tr}(\nabla\\!_{\mathrm{s}}\bm{\nu}^{\ast})=\cos\alpha\left(\frac{\kappa_{1}}{\lambda_{1}^{\prime}}+\frac{\kappa_{2}}{\lambda_{2}^{\prime}}\right)=0,$
(71a) $\displaystyle K^{\ast}$
$\displaystyle:=\det(\nabla\\!_{\mathrm{s}}\bm{\nu}^{\ast})=-\left(\frac{\kappa_{1}}{\lambda_{1}^{\prime}}\right)^{2}=-\left(\frac{\kappa_{2}}{\lambda_{2}^{\prime}}\right)^{2}.$
(71b)

In particular, (71a) implies that $\mathscr{S}^{\ast}$ is a minimal surface.

Clearly, this conclusion relies on the very existence of $\mathscr{S}^{\ast}$,
which would be guaranteed, at least locally, if the surface tensor field
defined on $\mathscr{S}$ as

$\mathbf{S}:=\lambda^{\prime}\mathbf{R}(\alpha)\mathbf{U}^{\prime}$ (72)

were a surface _gradient_. This is an integrability requirement that poses
restrictions on both $\lambda^{\prime}$ and $\alpha$, which remain the only
unknown scalar surface fields, once (69) is enforced. The study of these
restrictions exceeds the scope of this paper and will be the subject of a
future one, as will also be the integrability of $\mathbf{S}$ for the general
class of incremental deformations that obey (66).

We have considered two distinct conditions that must be met for
$\bm{y}^{\prime}$ to be a bending-neutral deformation, one compatibility
condition and one integrability condition. Whenever $\mathscr{S}$ is such that
either one or the other condition is not met, any incremental
$\bm{y}^{\prime}$ will necessarily bring additional bending.

##  IV Conclusions

The primary object of this paper is answering a question posed in [1]:“How do
we wish to define a measure of bending, and by extension a bending energy and
the concept of pure stretching deformation, for a thin structure”?

We (the patient reader and I) caracolled our way toward a possible answer
through the notion of bending-neutral deformation, an incremental deformation
that does not affect bending, but it is more than a pure stretching as it may
include normal twisting.

Quite naturally, a pure measure of bending was then introduced as a
deformation measure unaffected by bending-neutral deformations. An explicit,
tensorial example of such a measure was given and contrasted against other
measures known for plates.

Common wisdom has it that plates (and shells, for that matter) are rather
unresponsive to twists exciting local drill rotations (see, for example,
[22]). This may well be the case for conventional, _hard_ plates, but it is
likely not to be so for _soft_ , polymeric plates, especially those with an
activable internal structure, such as nematic elastomers (described in the
landmark textbook [54]).

These latter material surfaces are _anisotropic_ , as the nematic director
field $\bm{n}$ on $\mathscr{S}$ breaks the rotational symmetry about the
normal $\bm{\nu}$. It might be interesting to find anisotropic pure measure of
bending fit for these systems; they might be suggestive of direct theories
where stretching and bending energy contents are kept separate, oblivious to
whether they can be derived by dimension reduction of established three-
dimensional strain energies or not.111111The role of bending energy in the
theory for nematic elastomer plates has been highlighted in [55]; applications
are presented in [56, 57] and [58].

Whether the answer presented in this paper to the opening question is
satisfactory or not is not for me to tell. In the second part of the paper, my
interest shifts toward the legitimacy of bending-neutral deformations. They
have been characterized by the fulfillment of a necessary kinematic
compatibility condition, which generates a three-dimensional set of solutions.
Of course, this only represents the admissible bending-neutral deformations;
which ones do actually exist, given a surface $\mathscr{S}$ they are meant to
transform, is decided, at least locally, by the fulfillment of an
integrability condition. The study of this condition and its solutions will be
the subject of a future paper.

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# An information retrieval and extraction tool for covid-19 related papers

Marcos V. L. Pivetta
Universidade Federal do Rio Grande do Sul (UFRGS)
Rio Grande do Sul, Brazil
<EMAIL_ADDRESS>

###### Abstract

Background: The COVID-19 pandemic has caused severe impacts on health systems
worldwide. Its critical nature and the increased interest of individuals and
organizations to develop countermeasures to the problem has led to a surge of
new studies in scientific journals. Objetive: We sought to develop a tool that
incorporates, in a novel way, aspects of Information Retrieval (IR) and
Extraction (IE) applied to the COVID-19 Open Research Dataset (CORD-19). The
main focus of this paper is to provide researchers with a better search tool
for COVID-19 related papers, helping them find reference papers and hightlight
relevant entities in text. Method: We applied Latent Dirichlet Allocation
(LDA) to model, based on research aspects, the topics of all English abstracts
in CORD-19. Relevant named entities of each abstract were extracted and linked
to the corresponding UMLS concept. Regular expressions and the K-Nearest
Neighbors algorithm were used to rank relevant papers. Results: Our tool has
shown the potential to assist researchers by automating a topic-based search
of CORD-19 papers. Nonetheless, we identified that more fine-tuned topic
modeling parameters and increased accuracy of the research aspect classifier
model could lead to a more accurate and reliable tool. Conclusion: We
emphasize the need of new automated tools to help researchers find relevant
COVID-19 documents, in addition to automatically extracting useful information
contained in them. Our work suggests that combining different algorithms and
models could lead to new ways of browsing COVID-19 paper data.

_K_ eywords Health Information Management $\cdot$ Information Storage $\cdot$
Retrieval $\cdot$ Healthcare $\cdot$ COVID-19 $\cdot$ Unified Medical Language
System

##  1 Introduction

The disease caused by the new coronavirus, named COVID-19 (Coronavirus Disease
2019), has led to a worldwide public health crisis. The World Health
Organization (WHO), in its strategic plan to combat COVID-19, highlights the
urgent need for research, innovation, and knowledge sharing about the issue
[44]. Because of the rapidly increasing number of scientific papers about the
topic, scientists report not having time to identify the core claims and
values added of every COVID-19 related paper that is being published [9].

Automated tools to help researchers of COVID-19 find relevant documents and
information within them are a growing necessity and could solve the previously
described hindrances. We introduce a tool, named CORD-19 Knowledge Tool
(CORD-19 KTool) 111https://github.com/pivettamarcos/CORD-19_KTool, that can
help researchers in this task by means of (i) recommending CORD-19 papers
based on LDA topic modeling combined with research aspect classification and
(ii) extracting named entities and linking them to the respective UMLS
concept.

In the following introductory sections, we give the reader an insight into the
existing sources of curated COVID-19 paper data. We also describe the current
situation of COVID-19 paper recommendation systems and the usefulness of Named
Entity Recognition (NER) systems for COVID-19 studies. These sections also
provide contextualization to our own approach to both of these tasks.

###  1.1 Data sources

Initiatives have been taken to create a reliable source of information about
COVID-19 related studies. Resources such as the COVID-19 Open Research Dataset
(CORD-19) [41] and LitCovid [13] aim to gather an ever-increasing number of
papers about COVID-19 and related coronavirus diseases for data mining
purposes.

Released in March 2020, CORD-19 is one of the main curated sources of articles
about COVID-19 and related coronaviruses. The data set was created from a
collaboration between the Allen Institute for AI (AI2) and organizations such
as The White House Office of Science and Technology Policy (OSTP) and National
Library of Medicine (NLM).

CORD-19 contains over 1 million articles from its last update in June 2022. It
is structured in a way that facilitates the access by automated tools. To
achieve that, it adopts some guiding principles:

  * •

standardization of paper’s metadata

  * •

elimination of duplicate papers

  * •

machine readability

The broader topics presented in CORD-19 provide a diverse but complex source
for text mining purposes.

Figure 1: Diagram describing the tool’s use cases and interaction with CORD-19
abstracts and the topic models generated with LDA

###  1.2 Paper recommendation

Many different studies have presented Information Retrieval (IR) systems for
COVID-19 paper recommendation. All these systems tackle the problem in a
different way. For example, [38] combined machine learning algorithms such as
K-means, DBSCAN, and HAC to group COVID-19 papers based on the predefined
research question they answer. [21] use a combination of Siamese-BERT model
and two keyword models to better score retrieved documents in the context of
COVID-19 research.

The application of Latent Dirichlet Allocation (LDA) [7] to model the topics
of COVID-19 papers has also shown positive results for retrieval tasks. LDA
can help researchers organize textual information by modeling the relevant
topics from a set of documents. An LDA model generates topics inherent to the
documents, using a probability distribution that ensures that all topics obey
a Dirichlet polynomial prior distribution [45]. The works of [39], [43] and
[23] describe systems that provide document exploration based on salient
topics in COVID-19 studies and social media posts.

Topic probability distributions from LDA can be used to aid document search.
Determining the most similar documents to a document based on a query text can
be done programmatically. Algorithms such as K-Nearest Neighbors (KNN) can
classify this new query text based on the class of the nearest previously
classified objects [15], using topic probability as the feature.

We could go more in depth with the LDA strategy by combining it to a
particular text segment. This local topic modeling strategy increases search
granularity and has shown measurable benefits, like presented in the work of
[2]. To achieve this, we need to classify the sentences or paragraphs in the
paper and assign them a role. After that, we can provide topic modeling
results to each local text segment, e.g., the introduction, methods or
results.

####  1.2.1 CODA-19

The COVID-19 Open Research Dataset (CODA-19) [25] is a data set constructed by
crowd workers that annotated useful textual structure from 10,966 CORD-19
abstracts. These units, or research aspects, include: background, purpose,
method, finding/contribution, and other. The research aspects reflect the role
of a particular sentence based on the author’s intention. The workers followed
a strict annotation scheme that was adapted from the works of [10] and [24].

A great number of studies have shown the benefits of structured abstracts
since their introduction in medical research journals [22]. Although well
received by researchers, many abstracts still lack defined structural units.
Automatic annotation of research structure in abstracts has been the subject
of many previous works [16, 5, 17] that also suggest its usefulness in
information retrieval tasks.

The authors of CODA-19 have publicly released a classifier model that uses
SciBERT. SciBERT is a language model trained on 1.14 million biomedical texts
from Semantic Scholar and is based on the Bidirectional Encoder
Representations from Transformers (BERT) [19] architecture. Bidirectional
networks are state-of the art in several NLP tasks, which involve, for
example, textual understanding and question answering [19].

Based on experiments made by [25], CODA-19 classifier’s achieves an overall
accuracy score of 0.774. It’s precision, recall and F-measure for each
research aspect can be seen on Table 1.

Research aspect |  P |  R |  F1
---|---|---|---
Background |  .733 |  .768 |  .750
Purpose |  .616 |  .636 |  .626
Method |  .715 |  .636 |  .673
Finding |  .783 |  .775 |  .779
Other |  .794 |  .852 |  .822

Table 1: Performance of the pre-trained Sci-Bert model, as described by Huang
et al. citeHuang2020CODA-19:Crowd

We use LDA topic modeling to identify latent topics for each individual
research aspect identified by CODA-19 in CORD-19 abstracts.

###  1.3 Named Entity Recognition (NER)

Named Entity Recognition (NER), a subtask of Information Extraction (IE) [31],
aims to automatically extract named entities (names of organizations,
locations, quantities, dates, etc.) from text. Scientific papers about
COVID-19 bring about many different words from the fields of virology and
biomedicine that are important to the understanding of the text, such as the
name of: viruses, chemical compounds, diseases, viral components, genes, among
others. These words must be extracted with some computational method, such as
a NER system.

####  1.3.1 Unified Medical Language System (UMLS).

Works from [29, 40] show that linking named entities to a controlled
vocabulary improves the usefulness of IE and IR tasks. A controlled vocabulary
system, like the Unified Medical Language System (UMLS) [8] provides a
standardized means of identifying biomedical concepts and their relation to
other concepts. We can use this knowledge to improve the IE aspect of our
tool.

###  1.4 Aim of the study

In this paper, we describe a researcher-driven tool to aid the process of new
hypothesis proposal and the search of reference articles by the researcher.
The tool feeds on data provided by CORD-19 and can identify the topics of the
abstracts from the data set to aid the search of relevant papers. To achieve
more detailed results, we not only identify topics for the whole text, but for
each research aspect (introduction, purpose, method and finding/contribution)
of the study, using the COVID-19 Research Aspect Dataset (CODA-19) [25] pre-
trained classifier model. The tool also gives researchers a readily way to
visualize named entities and their descriptions according to the UMLS
Metathesaurus.

##  2 Methods

###  2.1 Tool definition and prototype creation

CORD-19 KTool provides the use cases shown in Figure 1. The diagram shows the
extraction of COVID-19 abstracts from CORD-19, their segmentation into
research aspects, and the respective training of topic models for each
segment. It also shows all possible actions the researcher can do.

###  2.2 Algorithms

####  2.2.1 Extracting abstracts from CORD-19

The abstract of every publication was extracted from the CORD-19’s CSV
metadata file (version 2020-10-01), and served as input to the algorithms
described in the next sections. Only the abstracts were extracted. This is
purposefully done to fit the research aspect classification model, which was
trained only on abstracts. Title and publication date were also extracted from
the metadata file.

####  2.2.2 Preprocessing

Prior to the application of the Natural Language Processing (NLP) algorithms,
The CORD-19 documents went through 4 preprocessing steps:

  1. 1.

Only keep abstracts in the English language. Since the integrated NLP tools
used were trained on English texts, only English papers are available in the
tool.

  2. 2.

Segment the abstracts into sentences. We segmented the abstract into sentences
using the en_core_sci_lg SciSpacy model
222https://allenai.github.io/scispacy/.

  3. 3.

If the abstract is absent, use the title of the paper as the abstract

  4. 4.

Remove stop words. Words such as “method", “introduction", “conclusion" and
other English stop words were removed as they provide no useful information
for the topic modeling process.

####  2.2.3 Categorizing research aspects

We used the CODA-19’s baseline SciBERT [25] pretrained model
333https://github.com/windx0303/CODA-19/tree/master/classification to
categorize the content of every abstract based on research aspects. With this
classifier model, we could assign each sentence in the abstracts a role:
background, purpose, method, finding/contribution or other. The SciBERT model
was used as it outperformed other models, as presented in the CODA-19 paper.

####  2.2.4 Modeling topics

For this task, we used the ktrain [28] wrapper for the Python 3.9.0
programming language. The wrapper offers a readily available set of methods to
deal with LDA topic modeling. Separate LDA models were trained for each
research aspect of all abstracts. We also trained LDA models that comprise the
entirety of all abstracts. Papers that include “COVID-19” or “COVID” in the
abstract were trained to separate models to discriminate them from papers
related to other coronaviruses.

The statistical model resultant of LDA can give us an idea of the underlying
topics of a collection of documents, based on research aspects and the
abstract as a whole. It is also used as a way to cluster documents that cover
the same (or similar) topic in an unsupervised way.

We linked the topic modeling process to the research aspects with the purpose
of making the search functionality of our tool more fine-grained, as the
researcher can focus their search attempt on one of the research aspects or
the whole abstract, like previously described.

The ktrain parameters used for all topic models are:

  1. 1.

Minimum document frequency (min_df) = 0.001

  2. 2.

Maximum document frequency (max_df) = 0.65

  3. 3.

Number of maximum LDA iterations (lda_max_iter) = 15

  4. 4.

Number of features = 1000000

The maximum number of topics for each model (whole text and for each research
aspect) is then defined using the default heuristic formula 1 set by ktrain:

$\min{(400,\sqrt{\frac{\textrm{Number of documents in research aspect}}{2}})}$
(1)

The topic modeling done by LDA can yield a lower number of topics than that
assigned, meaning some models will differ from the formula.

####  2.2.5 Finding similar documents to a query using K-Nearest Neighbors
(KNN)

Using ktrain, we were able to train a recommender model that can find similar
documents to a user query, based on the topic probability distribution of the
text provided by the user. The ktrain wrapper uses KNN for this purpose.
Default parameters were used.

To find documents similar to a search query, we had to:

  1. 1.

Train the recommender model

In the training phase, previously classified sample points are stored as
multidimensional vectors in a vector space. In our tool, the topic probability
distribution is used as the feature that constructs the vectors. This vector
space is then saved as a model for later use.

  2. 2.

Assess the similarity to CORD-19 documents

To determine the most similar documents to the query, we first determine the
topic probability distribution of the user provided text. We do this by using
the appropriate topic model and predicting the topic probabilities of the
query text. Secondly, we find the k-nearest neighbors to that text using the
Euclidean distance metric. The k-nearest neighbors will be the output of this
step, meaning they are the most similar documents to that which the user
provided.

####  2.2.6 Searching papers with query string.

Regular expressions were used to search for documents based on a predefined
query of words or sentences. After conversion of the query string to lower
case, the expression 2 was used for this purpose.

$(\verb/r"\b%s\b" & query/)$ (2)

The predefined query terms were adapted from the key scientific questions
posed on the kaggle CORD-19 challenge [26] in the form of a “What is known
about [topic]?” question.

####  2.2.7 Graphical representation of a topic model.

The ktrain library was used to generate a 2D graphical representation of the
topic models. It uses a T-distributed Stochastic Neighbor Embedding (t-SNE) to
the topic distribution vector of each paper, reducing it to two dimensions.
This way, each paper can be plotted on a 2D graph representing how close each
data point (paper) is to another in the topic space.

####  2.2.8 Named Entity Recognition and UMLS linking.

Named Entity Recognition (NER) was achieved using the SciSpacy [33] NER model
trained on the CRAFT corpus. CRAFT’s annotation can identify terms from nine
important biomedical conceptual systems [4]:

  * •

Cell Type Ontology (CL) [6]

  * •

chemical Entities of Biological Interest ontology (ChEBI) [18]

  * •

NCBI Taxonomy (NCBITaxon) [36]

  * •

Protein Ontology (PRO) [32]

  * •

Sequence Ontology (SO) [20]

  * •

Entrez Gene database (EG) [27]

  * •

Gene Ontology (GO) [3]

The NER model has an F1 of 75.02. The step of linking the named entity to UMLS
[8] concepts was done using the Entity Linker in SciSpacy.

##  3 Results

###  3.1 Research aspect classification and topic modeling

CORD-19 1 October 2020 update has a total of 268,170 English abstracts, 72,920
of which mention the words “COVID-19” or “COVID”. We present the number of
abstracts that contain each research aspect in Table 2.

|  All papers |  Only COVID-19 papers
---|---|---
Background |  166,160 |  63,243
Purpose |  94,896 |  37,653
Method |  76,590 |  27,092
Finding/Contribution |  175,780 |  65,065
Whole text |  268,170 |  72,920

Table 2: Number of abstracts that contain the respective research aspect

The topic count for every research aspect and the whole text is shown in Table
3. Topic count is based on the number of unique words present in the available
documents for training.

|  All papers |  COVID-19 only papers
---|---|---
Background |  146 |  96
Purpose |  130 |  83
Method |  103 |  84
Finding/Contribution |  152 |  103
Whole text |  176 |  101

Table 3: Topic count for each research aspect

###  3.2 List of topics

This listing gives the researcher the topics LDA identified for the selected
research aspect. The list is ordered by the number of documents in CORD-19
that fall into the respective topic. The list can comprise all of CORD-19
abstracts or be narrowed down to only those related to COVID-19. Figure 2
shows a case study for this functionality. The researcher is looking for the
main topics contained in the finding/contribution segment of abstracts about
COVID-19. The overall idea of each topic can be deduced from the list of
words. For example, one might suggest that topic 148, represented by the
words: “patients, days, median, hospital, time, admission”, indicates findings
related to patient admission to hospital, time spent in hospital, etc. Topic
169, made up of the words “pandemic, impact, response, economic, current,
government, future, work, policies and situation” could indicate findings
related to the pandemic’s economic impact, government response to the
pandemic, etc.

Figure 2: List of topics addressed in the finding/contribution section of
COVID-19 abstract Figure 3: List of COVID-19 abstracts whose
finding/contribution addresses the topic “patients, day, median, hospital,
iqr, admission, range, died and years”, ordered by date of publication Figure
4: List of COVID-19 abstracts whose finding/contribution addresses the topic
“human, protein, screening, ace2 expression, compounds, spike, active,
proteins and screened”, ordered by score Figure 5: List of recommended papers
similar to query text

###  3.3 List papers in a topic

The researcher can obtain the papers that best reflect a selected topic. Like
the previous use case, filtering by research aspects and COVID-19 only papers
is possible. Additionally, the researcher can filter topics that contain a
query keyword. The score shown is the affinity of each paper to the topic in
question.

The case study shown in Figure 3 shows a search for the 19 most relevant
documents about COVID-19 in a topic related to the finding/contribution
research aspect. The researcher used the “filter topic” text box and entered
the word “patients admission time”, meaning only topics containing these words
will be available for selection.

In this case, the researcher opted to order the document list by publication
date, meaning the most recent paper that addresses the topic is shown at the
top. The earliest published paper does show a study about a treatment method
that resulted in a reduction of the number of days the patients were
hospitalized. This finding is arguably related to the selected topic,
comprised of the words: “patients, day, median, hospital, iqr, admission,
range, died and years”.

Another example of this search is presented in Figure 4. The term “spike
protein” is used as a filter. The researcher is interested in knowing all the
studies done on this protein, which is being used in candidate vaccines [34].
The topic comprised the words: “human, protein, screening, ace2 expression,
compounds, spike, active, proteins and screened” was selected. The abstracts
from the list all describe, in their methods, techniques like spike protein
binding, expressing spike proteins and protein-protein interactions.

###  3.4 Recommend reference papers based on a query text

The researcher can search for papers that are the most similar to a query
sentence given by them. As with previous use cases, the researcher can choose
a specific aspect or the whole abstract for searching and narrow their search
to only COVID-19 papers.

The researcher may seek prospective observational studies about COVID-19 and
related coronavirus. He can provide a query text for that: “The aim of this
prospective observational cohort study…”. Figure 5 shows that this search
could return the documents of their interest.

###  3.5 Search papers based on a research question

A list of predefined terms are given to the user to select from. This search
returns papers where the term selected is present. As with previous use cases,
the researcher can choose a specific aspect or the whole abstract for
searching and narrow their search to only COVID-19 papers.

Figure 6 outlines a search about the “environmental stability” of the virus.

###  3.6 Plot documents in topic model space

The plot in Figure 7 expresses each abstract as a point in 2D space with an
assigned color representing a specific topic. The researcher can choose a
specific aspect or the whole abstract for plotting and narrowing down their
visualization to only COVID-19 papers. Titles and topics are visible when a
point is hovered with the mouse cursor.

###  3.7 Single abstract visualization

Figure 8 shows a case where the researcher looks at an individual abstract. He
can see the individual role of each sentence (background, purpose, method,
finding and other) based on its color. Hovering over an underlined entity
shows the UMLS concept type, ID, and description of the corresponding entity.

Figure 6: List abstracts whose background addresses the topic of environmental
stability of the virus Figure 7: Interactive visualization of documents in
research aspect Figure 8: Individual abstract visualization

##  4 Discussion

###  4.1 Dataset choice

CORD-19 is an open-access, curated, and constantly updated data set. Another
important characteristic that it possesses is its broad set of topics [14].
Given its consistency in providing good-quality papers on the topic, the data
set fits the core purpose of this work, which is to filter information. To
discern between abstracts about COVID-19 and abstracts about all coronavirus-
related studies in CORD-19, we had to find those abstracts that contained the
words “COVID-19” or “COVID”. This approach can return false positive results
simply because the authors can refer to one of these terms without them being
the central theme of the study. This can be solved if we use a COVID-19
specific data set like LitCovid [12]

It is important to note that CORD-19 does not include other data sources
besides scientific articles. CORD-19 does not contain social media posts,
forums, or articles. It also does not include preprint papers.

###  4.2 LDA topic modeling

Topic quality and count is very dependent on CODA-19 classifier’s ability to
recognize a particular research aspect. Research aspects with higher accuracy
scores, such as findings and purpose, had more documents for training and led
to more coherent topics.

A certain degree of arbitrariness was present in the parametrization step of
the LDA topic modeling. The number of topics chosen for each research aspect,
for example, was chosen using the default value of the ktrain tool. The number
of topics often depends on what the tool is trying to achieve. A greater
number leads to less general topics and the splitting of similar documents to
slightly more specific topics. A lower topic count yields more general
groupings.

Methods that measure the coherence of the topic modeling to better determine
the number of topics could be used, such as the one described by [30] and
[35]. Visualization tools like LDAvis [37] can be useful in manually analyzing
topic coherence but require expert opinion. For the pruning parameters (min_df
and max_df) the default value was used. Changing these two parameters to more
conservative values does not decrease the quality of the model, but it can
speed up the topic modeling process. We recognize that fine-tuning the
parameters, such as the number of topics and iterations, could lead to better
topic models and, thus a more accurate tool.

###  4.3 Research aspect classification

The performance of the CODA-19 SciBert pre-trained model for identifying
research aspects has an average accuracy of 0.774, an overall satisfactory
result. The model is slightly better at identifying findings and background.

The model has the potential to be improved. It is important to note that
volunteers annotated the research aspects without prior knowledge of the
subject matter. The agreement between annotators was considerable (Cohen’s
kappa of 0.677), but indicates the possibility of improvements in the model.
The search functionality of the tool might become more reliable and accurate
with future improvements to the model’s accuracy.

Considering only abstract text for analysis and not the full text can lead to
problems. There are instances where research aspects are left out from the
abstract by the study’s author. These missing research aspects could be
extracted from the full text. We also noted that, when explicitly written, the
abstract sections (BACKGROUND, OBJECTIVES, METHOD, DISCUSSION, and
CONCLUSIONS) could be used with CODA-19’s classifier to improve its accuracy.

###  4.4 Named Entity Recognition and UMLS linking

The NER approach used in the tool can recognize terms from various biomedical
conceptual systems, describe them, and link them to UMLS. We realize that new
terms are being created to express new findings in COVID-19 research.
SciSpacy’s NER might not recognize said terms. Because of this, more up-to-
date solutions are important. Not only that, datasets focused on COVID-19
studies, such as CORD-NER [42], achieved a higher F1 score compared to
SciSpacy’s annotation method, and show an improvement over broader NER
solutions. Further development of the tool could leverage such resources.

###  4.5 Documents visualization

The graphical visualization allows researchers to explore the hidden relations
identified by the topic model. Although less obvious to the end user than the
other features in the tool, the graph can be used to discover unforeseen
patterns relevant to that research aspect.

###  4.6 Relevant document search

It was thought from the start that modeling the topics based on research
aspects would make the search of relevant documents more fine-grained and as a
result, more accurate. While this statement might be true, it requires further
investigation. We noticed that this strategy is heavily dependent on the topic
model quality. Exploration of COVID-19 only papers yields more coherent
results than searching all CORD-19 papers. This might be caused by the
extremely broad topics addressed in coronavirus-related diseases.

The cases presented in the result section represent ideal search situations.
We identified a couple of problems in our approach. For instance, the topics
shown in the search functionality are represented as a collection of words.
This collection can be confusing to the researcher and hard to interpret. We
know that topic label generators [1] can solve this problem by building a
coherent sentence that explains the topic better. The K-Nearest Neighbor
search based on topic probability distribution is also questionable. Other
representations could be used to measure textual content similarity, such as
word embeddings or sentence embeddings from BioSentVec [11].

The search with predefined terms from Kaggle, although simple, can return
papers that address the terms chosen by the researcher by keyword matching.
New research questions may arise, making it necessary to update this list.

###  4.7 Single abstract visualization

Highlighting the research aspects of the abstract speeds up the identification
of the most important sections for the researcher, such as scientific finding,
method etc. The description of the named entities simplifies the acquisition
of knowledge about the a particular entity in the text. Linking the named
entity to an UMLS entry allows the researcher to access the term directly in
UMLS.

##  5 Conclusion

The worldwide crisis the new coronavirus has caused, in recent years, very
negative effects to the world’s health systems and areas such as economy and
education. The disease is the subject of an increasing number of articles that
describe new findings about the topic. The need for automated tools to support
the researcher has been deemed important to tackle this increase in COVID-19
paper data. Such tools can apply machine learning algorithms to improve the
search of COVID-19 documents and extract relevant information. The solution
developed in this study was oriented towards the solution to this problem.

We can use several techniques to add value to such an application. The
literature demonstrates that topic modeling, extraction of named entities,
extraction of relations, detection of contradictory text, models for question
answering, and other PLN algorithms have the potential to fill the gaps
present in this type of solution. We demonstrated that combining different
algorithms, libraries, and models can enhance the potential of information
retrieval and extraction from articles about COVID-19.

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