# A Critical Study of Efrati _et al._ ’s _Elastic Theory of Unconstrained non-
Euclidean Plates_

Kavinda Jayawardana<EMAIL_ADDRESS>
###### Abstract

In our analysis, we show that Efrati _et al._ ’s publication [1] is
inconsistent with the mathematics of plate theory. However it is more
consistent with the mathematics of shell theory, but with an incorrect strain
tensor. Thus, the authors’ numerical results imply that a thin object can be
stretched substantially with very little force, which is physically
unrealistic and mathematically disprovable. All the theoretical work of the
authors, i.e. nonlinear plate equations in curvilinear coordinates, can easily
be rectified with the inclusion of both a sufficiently differentiable
diffeomorphism and a set of external loadings, such as an external strain
field.

###### keywords:

Finite Deformation , Mathematical Elasticity , Plate Theory , Shell Theory

##  1 Introduction

A _plate_ is a structural element with planar dimensions that are large
compared to its thickness. Thus, plate theories are derived from the three-
dimensional elastic theory by making suitable assumptions concerning the
kinematics of deformation or the state stress through the thickness of the
lamina, thereby reducing three-dimensional elasticity problem into a two-
dimensional one.

Efrati _et al._ [1] present a model, defined as the _elastic theory of
unconstrained non-Euclidean plates_ , for modelling deformation of thin
objects. The main application of the authors’ work is in the study of natural
growth of tissue such as growth of leaves and other natural slender bodies.
Some numerical results are present, which is based on an example of a
hemispherical plate, and they imply the occurrence of buckling transition,
from a stretching-dominated configuration to a bending-dominated
configuration, under variation of the plate thickness.

However, we show that what the authors present is not a plate theory model; It
is, in fact, a shell theory model, but with an incorrect strain tensor. Thus,
the authors numerical results imply that a thin object can be stretched
substantially with very little force, which is physically unrealistic and
mathematically disprovable. All the theoretical work of the authors, i.e.
nonlinear plate equations in curvilinear coordinates, can easily be rectified
with the inclusion of both a sufficiently differentiable
$\mathbb{R}^{2}\to\textbf{E}^{2}$ diffeomorphism and a set of external
loadings, such as an external strain field.

##  2 Efrati _et al._ ’s Work

Given that a growing leaf can be modelled by a plate, Efrati _et al._ [1]111
http://www.ma.huji.ac.il/$\sim$razk/iWeb/My_Site/Publications_files/ESK08.pdf
focus on the elastic response of the plate after its planar (i.e. rest)
configuration is modified, either by growth or by a plastic deformation. Thus,
the goal is to derive a thin plate theory, as a generalisation of existing
elastic plate theories, that it is valid for large displacements and small
strains in arbitrary intrinsic geometries. The authors ignore the
thermodynamic limitations on plastic deformations as they are considered to be
not relevant when modelling naturally growing tissue, and further assume that
the reference configuration is a known quantity. Their main postulate is that
a non-Euclidean plate cannot assume a rest configuration, i.e. no stress-free
configuration can exist, and thus, one faces a nontrivial problem that always
exhibits residual stress. Note that by _non-Euclidean_ the authors mean ‘the
internal geometry of the plate is not immersible in 3D Euclidean space’ [1].
Also, the authors define ‘a metric is immersible in $\textbf{E}^{3}$’ if Ricci
curvature tensor with respect to the implicit coordinate system is identically
zero, i.e. for a given immersion
$\boldsymbol{\varphi}:\mathbb{R}^{n}\to\textbf{E}^{3}$, where $1\leq n\leq 3$,
the metric $\bar{\boldsymbol{g}}$ on $\mathbb{R}^{n}$ induced by the immersion
$\boldsymbol{\varphi}$ results in $\boldsymbol{Ric}=\boldsymbol{0}$ in
$\mathbb{R}^{n}$, where $\mathbb{R}^{n}$ and $\textbf{E}^{n}$ are curvilinear
and Euclidean spaces respectively.

The authors define a plate as an elastic medium for which there exists a
curvilinear set of coordinates
$\boldsymbol{x}=\boldsymbol{(}x^{1},x^{2},x^{3}\boldsymbol{)}$, in which the
‘reference metric’, $\bar{g}_{ij}=\bar{g}_{ij}(x^{1},x^{2})$, takes the form
$\bar{g}_{\alpha\beta}=\bar{g}_{\beta\alpha}$, $\bar{g}_{\alpha 3}=0$,
$\bar{g}_{33}=1$. The reference metric is a symmetric positive-definite tensor
and considered to be a known quantity. The plate is considered to be ‘even’,
i.e. the domain $\mathscr{D}\subset\mathbb{R}^{3}$ of curvilinear coordinates
can be decomposed into
$\mathscr{D}=\mathscr{S}\times[-\frac{h}{2},\frac{h}{2}]$, where
$\mathscr{S}\subset\mathbb{R}^{2}$ and $h$ is the thickness of the plate.
Thus, it is given that an even plate is fully characterised by the metric of
its mid-surface, i.e. at where $x^{3}=0$.

Although thin plates are three-dimensional bodies, the authors took advantage
of the large aspect ratio by modelling the plates as two-dimensional surfaces,
and thus, reducing the dimensionality of the problem. To achieve this the
authors assume ‘Kirchhoff-Love assumptions:’ (i) ‘the body is in a state of
plane-stress (the stress is parallel to the deformed mid-surface)’, and (ii)
‘points which are located in the undeformed configuration on the normal to the
mid-surface at a point $p$, remain in the deformed state on the normal to the
mid-surface at $p$, and their distance to $p$ remains unchanged’ [1].

Now, consider the deformed plate in the Euclidean space, which is defined as a
compact domain $\Omega\subset\mathbb{R}^{3}$ endowed with a regular set of
material curvilinear coordinates. Define the mapping
$\boldsymbol{r}:\mathscr{D}\subset\mathbb{R}^{3}\mapsto\Omega$, from the
domain of parameterisation $\mathscr{D}$ into $\Omega$, as the configuration
of the body endowed with the metric tensor $\boldsymbol{g}$, which is defined
as $g_{ij}=\partial_{i}\boldsymbol{r}\cdot\partial_{j}\boldsymbol{r}$, where
$\cdot$ is the Euclidean dot-product. It is given that Kirchhoff-Love second
assumption implies that $g_{\alpha 3}=0$. Thus, when defined more precisely,
one finds that
$\boldsymbol{r}(x^{1},x^{2},x^{3})=\boldsymbol{R}(x^{1},x^{2})+x^{3}\hat{\boldsymbol{N}}(x^{1},x^{2})$,
$g_{\alpha\beta}=a_{\alpha\beta}-2x^{3}b_{\alpha\beta}+(x^{3})^{2}c_{\alpha\beta}$,
$g_{\alpha 3}=0$ and $g_{33}=1$, where $\boldsymbol{R}$ is the mid-surface,
$\hat{\boldsymbol{N}}$ is the unit normal to the mid-surface, and
$a_{\alpha\beta}$, $b_{\alpha\beta}$ and $c_{\alpha\beta}$ are the first, the
second and the third fundamental form tensors respectively. With further
inspection one finds that
$a_{\alpha\beta}=\partial_{\alpha}\boldsymbol{R}\cdot\partial_{\beta}\boldsymbol{R}$,
$b_{\alpha\beta}=(\partial_{\alpha}\partial_{\beta}\boldsymbol{R})\cdot\hat{\boldsymbol{N}}$
and $c_{\alpha\beta}=(a^{-1})^{\gamma\delta}b_{\alpha\gamma}b_{\beta\delta}$.
The ultimate goal is to find the metric tensor $\boldsymbol{g}$, and the
authors state that the metric tensor $\boldsymbol{g}$ is immersed in
$\mathbb{R}^{3}$, and thus, the metric tensor uniquely defines the physical
configuration of a three-dimensional body. It is also the case that one needs
to find equations to six unknowns which make up the metric tensor for the
general case, where $\bar{\boldsymbol{g}}$ is not defined by $\boldsymbol{r}$.
For the general case, the authors describe one approach to this problem via
the use of ‘the modified version of the hyper-elasticity principle … the
elastic energy stored within a deformed elastic body can be written as a
volume integral of a local elastic energy density, which depends only on (i)
the local value of the metric tensor and (ii) local material properties that
are independent of the configuration’ [1]. It is unclear what the authors mean
by this definition; thus, for a more precise definition of hyperelasticity, we
refer the reader to Ball [2] or Ciarlet [3].

The authors define the strain tensor as follows,

$\displaystyle\epsilon_{ij}$
$\displaystyle=\frac{1}{2}(g_{ij}-\bar{g}_{ij})~{},$ (1)

and thus, the energy functional is expressed as follows,

$\displaystyle E(g)$
$\displaystyle=\int_{\mathscr{D}}w(g)\sqrt{\bar{g}}~{}dx^{1}dx^{2}dx^{3}~{},$
(2)

where $w=\frac{1}{2}A^{ijkl}\epsilon_{ij}\epsilon_{kl}$ is the energy density
and
$A^{ijkl}=\lambda\bar{g}^{ij}\bar{g}^{kl}+\mu(\bar{g}^{ik}\bar{g}^{jl}+\bar{g}^{il}\bar{g}^{jk})$
is the elasticity tensor. With the use of the energy functional (2), the
configuration $\boldsymbol{r}$ is varied to find the three constraints that
$g_{\alpha\beta}$ must satisfy, i.e. ‘the fundamental model for three-
dimensional elasticity’ [1]. Note that _Einstein’s summation notation_ is
assumed throughout, and we regard the indices $i,j,k,l\in\\{1,2,3\\}$ and
$\alpha,\beta,\gamma,\delta\in\\{1,2\\}$, unless it is strictly states
otherwise. Also note that the authors define the symmetric Ricci curvature
tensor of the metric $\boldsymbol{g}$ as follows,

$\displaystyle
Ric_{li}=\frac{1}{2}\left(g^{-1})^{kj}(\partial_{k}\partial_{i}g_{lj}-\partial_{k}\partial_{j}g_{li}+\partial_{j}\partial_{l}g_{ki}-\partial_{i}\partial_{l}g_{kj}\right)$
$\displaystyle+(g^{-1})^{kj}g_{pq}(\Gamma^{p}_{\\!lj}\Gamma^{q}_{\\!ki}-\Gamma^{p}_{\\!kj}\Gamma^{q}_{\\!li})$
$\displaystyle~{}.$

As the elastic body is immersed in $\mathbb{R}^{3}$, the variational principle
implies that the six independent components of the symmetric Ricci curvature
tensor must all vanish, i.e. $Ric_{ij}=0$. However, $Ric_{ij}=0$ and the three
equations obtained by varying the configuration $\boldsymbol{r}$ in equation
(2) imply that the system is over-determined. Thus, the authors postulate that
there are two possible ways to resolve this ‘seemingly over-determination’.
The first is by noticing that the six independent components of Ricci
curvature tensor’s derivatives are related through second Bianchi identity.
The second way of resolving this issue is by identifying the immersion
$\boldsymbol{r}$ as the three unknown functions (as defined previously), in
which case the six equations that form the Ricci tensor are the solvability
conditions for the partial differential equation. However, as the equations in
$\boldsymbol{r}$ are of the higher order, one needs to supply additional
conditions, namely to set the position and the orientation of the body, in
order to obtain a unique solution for $\boldsymbol{r}$.

To find the reduced energy density the authors integrate the energy density
(2) over the thin dimension as
$w_{2D}=\int^{\frac{h}{2}}_{-\frac{h}{2}}w~{}dx^{3}$ to obtain the equation,

$\displaystyle w_{2D}$ $\displaystyle=hw_{S}+h^{3}w_{B}~{},$

where

$\displaystyle w_{S}$
$\displaystyle=\frac{Y}{8(1+\nu)}\left(\frac{\nu}{1-\nu}\bar{g}^{\alpha\beta}\bar{g}^{\gamma\delta}+\bar{g}^{\alpha\gamma}\bar{g}^{\beta\delta}\right)(a_{\alpha\beta}-\bar{g}_{\alpha\beta})(a_{\gamma\delta}-\bar{g}_{\gamma\delta})~{}~{}\text{and}$
$\displaystyle w_{B}$
$\displaystyle=\frac{Y}{24(1+\nu)}\left(\frac{\nu}{1-\nu}\bar{g}^{\alpha\beta}\bar{g}^{\gamma\delta}+\bar{g}^{\alpha\gamma}\bar{g}^{\beta\delta}\right)b_{\alpha\beta}b_{\gamma\delta}~{},$
(3)

which are defined as stretching and bending densities respectively. Note that
$Y$ is the Young’s modulus and $\nu$ is the Poisson’s ratio of the plate.

It is stated that with the use of Cayley-Hamilton theorem, the density of the
bending content can be written in the following form,

$\displaystyle
w_{B}=\frac{Y}{8(1+\nu)}\big{(}\frac{1}{1-\nu}(\bar{g}^{\alpha\beta}b_{\alpha\beta})^{2}-2\frac{|b|}{|\bar{g}|}\big{)}~{}.$

It is also stated that, if $a_{\alpha\beta}=\bar{g}_{\alpha\beta}$ (i.e. the
two-dimensional configuration has zero-stretching energy), then the density of
the bending content can be expressed as the density of Willmore functional [4]
as follows,

$\displaystyle w_{W}$
$\displaystyle=\frac{Y}{24(1+\nu)}\left(\frac{4H^{2}}{1-\nu}-2K\right)~{},$

where $K$ and $H$ are Gaussian and the mean curvatures of the mid-surface
respectively.

With the vanishing of the Ricci tensor the authors obtain Gaussian curvature
and Gauss-Mainardi-Peterson-Codazzi equations, which are respectively defined
as

$\displaystyle K=\frac{|b|}{|a|}$
$\displaystyle=\frac{1}{2}(a^{-1})^{\alpha\beta}(\partial_{\gamma}\Gamma^{\gamma}_{\\!\alpha\beta}-\partial_{\beta}\Gamma^{\gamma}_{\\!\alpha\gamma}+\Gamma^{\gamma}_{\\!\gamma\delta}\Gamma^{\delta}_{\\!\alpha\beta}-\Gamma^{\gamma}_{\\!\beta\delta}\Gamma^{\delta}_{\\!\alpha\gamma})~{}~{}\text{and}$
(4) $\displaystyle\partial_{2}b_{\alpha 1}+\Gamma^{\beta}_{\\!\alpha
1}b_{\beta 2}$ $\displaystyle=\partial_{1}b_{\alpha
2}+\Gamma^{\beta}_{\\!\alpha 2}b_{\beta 1}~{},$ (5)

where equations (4) and (5) given to provide sufficient conditions for
immersiblility of the metric tensor $\boldsymbol{g}$ in $\mathbb{R}^{3}$.

It may appear to the uninitiated in the study mathematical elasticity that
Efrati _et al._ ’s publication [1] is a coherent piece of work, but it is, in
fact, flawed. To illustrate this matter in detail, we direct the reader’s
attention to section 3.4 of Efrati _et al._ [1]. Upon examining the governing
equations and the boundary conditions, one can see that the governing
equations are defined for zero-external loadings, and the boundary conditions
are defined for zero-tractions, zero-boundary moments and there are no
descriptions of any Dirichlet conditions. Thus, it is mathematically
impossible to obtain a non-zero solution (excluding any rigid motions).
Furthermore, there is no evidence in the authors’ publication for proof of the
existence of solutions, either via rigorous mathematics ($\Gamma$-limit or
otherwise) or via numerical analysis (only some conjectures regarding
solvability conditions are given by the authors, as we previously discussed).

Figure 1: ‘A schematic illustration of an unconstrained plate exhibiting
residual stress. (a) The two elements composing the plate are shows side by
side. (b) As the red trapezoid is too large to fit into the square opening, it
is compressed. (c) For a plate sufficiently thin, the induced compression
exceeds the buckling threshold, and the trapezoid buckles out of plane. Note
that there are many shapes that preserve all lengths along the faces of the
plate, yet they cannot be planar’ [1].

Efrati _et al._ ’s [1] erroneous work arises from not fully understanding how
to model the given problem. Consider figure 1 (c) (see figure 1 from Efrati
_et al._ [1]): the very reason the _red trapezoid_ is deformed is because it
is compressed at the boundary, i.e. it is deformed as it is subjected to a
Dirichlet boundary condition. Thus, if one attempts to model this problem with
mathematical rigour, then one can derive the actual energy functional for this
problem. To do so, consider the map

$\displaystyle\boldsymbol{r}(\boldsymbol{u})$
$\displaystyle=\boldsymbol{R}(\boldsymbol{u})+z\frac{\partial_{x}\boldsymbol{R}(\boldsymbol{u})\times\partial_{y}\boldsymbol{R}(\boldsymbol{u})}{||\partial_{x}\boldsymbol{R}(\boldsymbol{u})\times\partial_{y}\boldsymbol{R}(\boldsymbol{u})||}~{},$
(6)

which is assumed to be a sufficiently differentiable
$\textbf{E}^{3}\to\textbf{E}^{3}$ diffeomorphism for an appropriate
$\boldsymbol{u}\in\textbf{E}^{3}$, where

$\boldsymbol{R}(\boldsymbol{u})=\boldsymbol{(}x,~{}y,~{}0\boldsymbol{)}_{\text{E}}+\boldsymbol{(}u^{\bar{1}}(x,y),~{}u^{\bar{2}}(x,y),~{}u^{\bar{3}}(x,y)\boldsymbol{)}_{\text{E}}$

is the deformed mid-surface of the plate, $\boldsymbol{u}$ is the displacement
field that describes a vector displacement in the three-dimensional Euclidean
space and $\times$ is the Euclidean cross-product. The metric with respect to
map (6) in the Euclidean space is
$g_{\bar{\alpha}\bar{\beta}}(\boldsymbol{u})=\partial_{\bar{\alpha}}r_{\bar{i}}(\boldsymbol{u})\partial_{\bar{\beta}}r^{\bar{i}}(\boldsymbol{u})$,
where the over-bar in the indices highlights the fact that one is using
Euclidean coordinates. Now, the strain tensor of a plate can be expressed as
follows

$\displaystyle\epsilon_{\bar{\alpha}\bar{\beta}}(\boldsymbol{u})$
$\displaystyle=\frac{1}{2}\left(g_{\bar{\alpha}\bar{\beta}}(\boldsymbol{u})-\delta_{\bar{\alpha}\bar{\beta}}\right)~{},$

where $x^{\bar{1}}=x$ and $x^{\bar{2}}=y$. Thus, the energy functional can be
expressed as follows,

$\displaystyle
J(\boldsymbol{u})=\int\limits_{\mathscr{S}}\Bigg{[}\frac{1}{8}hA^{\bar{\alpha}\bar{\beta}\bar{\gamma}\bar{\delta}}$
$\displaystyle\left(\partial_{\bar{\alpha}}R_{\bar{i}}(\boldsymbol{u})\partial_{\bar{\beta}}R^{\bar{i}}(\boldsymbol{u})-\delta_{\bar{\alpha}\bar{\beta}}\right)\left(\partial_{\bar{\gamma}}R_{\bar{k}}(\boldsymbol{u})\partial_{\bar{\delta}}R^{\bar{k}}(\boldsymbol{u})-\delta_{\bar{\gamma}\bar{\delta}}\right)$
$\displaystyle+\frac{1}{24}h^{3}A^{\bar{\alpha}\bar{\beta}\bar{\gamma}\bar{\delta}}$
$\displaystyle\left(\partial_{\bar{\alpha}\bar{\beta}}R_{\bar{i}}(\boldsymbol{u})\frac{\left(\partial_{x}\boldsymbol{R}(\boldsymbol{u})\times\partial_{y}\boldsymbol{R}(\boldsymbol{u})\right)^{\bar{i}}}{||\partial_{x}\boldsymbol{R}(\boldsymbol{u})\times\partial_{y}\boldsymbol{R}(\boldsymbol{u})||}\right)$
$\displaystyle\left(\partial_{\bar{\gamma}\bar{\delta}}R_{\bar{k}}(\boldsymbol{u})\frac{\left(\partial_{x}\boldsymbol{R}(\boldsymbol{u})\times\partial_{y}\boldsymbol{R}(\boldsymbol{u})\right)^{\bar{k}}}{||\partial_{x}\boldsymbol{R}(\boldsymbol{u})\times\partial_{y}\boldsymbol{R}(\boldsymbol{u})||}\right)\Bigg{]}dxdy~{},$
(7) $\displaystyle\boldsymbol{u}\in$
$\displaystyle\\{\boldsymbol{v}\in\boldsymbol{W}(\mathscr{S})\mid\boldsymbol{v}|_{\partial\mathscr{S}}=\boldsymbol{u}_{0}\\}~{},$
(8)

where $\mathscr{S}\subset\textbf{E}^{2}$ is the mid-plane of the unstrained
plate, $\boldsymbol{W}(\mathscr{S})$ is an appropriate Sobolev space,
$\boldsymbol{u}_{0}\in L^{2}(\partial\mathscr{S})$ is a Dirichlet boundary
condition, and
$A^{\bar{\alpha}\beta\bar{\gamma}\bar{\delta}}=\frac{1}{2}(1+\nu)^{-1}Y(2(1-\nu)^{-1}\nu\delta^{\bar{\alpha}\bar{\beta}}\delta^{\bar{\gamma}\bar{\delta}}+\delta^{\bar{\alpha}\bar{\gamma}}\delta^{\bar{\beta}\bar{\delta}}+\delta^{\bar{\alpha}\bar{\delta}}\delta^{\bar{\beta}\bar{\gamma}})$
is the elasticity tensor of a plate. As $\boldsymbol{R}(\boldsymbol{u})$
describes a surface, one may argue that with an appropriate coordinate
transform, one can express equation (7) in the same form as the authors’
energy functional (see equation 3.7 of Efrati _et al._ [1]), which is true if
one is using an appropriate
$\boldsymbol{(}x,y\boldsymbol{)}_{\text{E}}:\mathbb{R}^{2}\to\textbf{E}^{2}$
coordinate transform (the authors’ erroneous coordinate transform leads to a
reference metric of a shell at $x^{3}=0$; please see equation 4.1 and see the
definition of $\Phi(r)$ from section 4.2 of Efrati _et al._ [1]). However,
without the Dirichlet boundary condition from equation (8) or some other
external loading, which is exactly what the authors are considering (see
respectively equation 2.5 and section 3.2 of Efrati _et al._ [1]), one gets
the trivial zero-displacement solution, i.e. $\boldsymbol{u}=\boldsymbol{0}$
in $\mathscr{S}$. The fact that the authors are claiming that nonzero
solutions are possible without tractions, Dirichlet boundary conditions,
external loadings or boundary moments imply that they are doing something
fundamentally flawed.

Now, consider the sufficiently differentiable diffeomorphism
$\boldsymbol{\varphi}:\omega\subset\mathbb{R}^{2}\to\boldsymbol{\varphi}(\omega)\subset\textbf{E}^{2}$
with the property $\det(\boldsymbol{J})>0$, where
$\boldsymbol{(}x,y\boldsymbol{)}_{\text{E}}=\boldsymbol{(}\varphi^{\bar{1}}(x^{1},x^{2}),$
$\varphi^{\bar{2}}(x^{1},x^{2})\boldsymbol{)}_{\text{E}}$ and
$J^{\bar{\beta}}_{\alpha}=\partial_{\alpha}\varphi^{\bar{\beta}}$ is the
Jacobian matrix of the map $\boldsymbol{\varphi}$. Note that we reserve the
vector brackets $\boldsymbol{(}\cdot\boldsymbol{)}_{\text{E}}$ for vectors in
the Euclidean space and $\boldsymbol{(}\cdot\boldsymbol{)}$ for vectors in the
curvilinear space. Now, consider the mapping
$\boldsymbol{R}(\boldsymbol{u})\circ\boldsymbol{\varphi}:\omega\subset\mathbb{R}^{2}\to\boldsymbol{R}(\boldsymbol{u})(\mathscr{S})\subset\textbf{E}^{3}$.
As $\boldsymbol{\varphi}$ is a diffeomorphism,
$\boldsymbol{R}(\boldsymbol{u})\circ\boldsymbol{\varphi}$ is a well defined
surface for a suitable displacement field
$\boldsymbol{u}\in\boldsymbol{W}(\mathscr{S})$, and as
$\det(\boldsymbol{J})>0$, the unit normal to the surface
$\boldsymbol{R}(\boldsymbol{u})$ is equal to the unit normal to the surface
$\boldsymbol{R}(\boldsymbol{u})\circ\boldsymbol{\varphi}$. Thus, with respect
to $\boldsymbol{\varphi}$, equation (7) reduces to the following,

$\displaystyle
J(\boldsymbol{u})=\int\limits_{\omega}\Bigg{[}\frac{1}{2}A^{\alpha\beta\gamma\delta}\left(ha_{\alpha\beta}(\boldsymbol{u})a_{\gamma\delta}(\boldsymbol{u})+\frac{1}{12}h^{3}b_{\alpha\beta}(\boldsymbol{u})b_{\gamma\delta}(\boldsymbol{u})\right)$
$\displaystyle\sqrt{\det(\bar{g})}$ $\displaystyle\Bigg{]}dx^{1}dx^{2}~{},$
(9)
$\displaystyle\boldsymbol{u}\in\\{\boldsymbol{v}\in\boldsymbol{W}(\mathscr{S})\mid\boldsymbol{v}|_{\partial\mathscr{S}}=\boldsymbol{u}_{0}\\}$
$\displaystyle~{},$

where

$\displaystyle a_{\alpha\beta}(\boldsymbol{u})$
$\displaystyle=\frac{1}{2}\left((\partial_{\alpha}R_{\bar{i}}(\boldsymbol{u})\circ\boldsymbol{\varphi})(\partial_{\beta}R^{\bar{i}}(\boldsymbol{u})\circ\boldsymbol{\varphi})-\bar{g}_{\alpha\beta}\right)~{},$
$\displaystyle b_{\alpha\beta}(\boldsymbol{u})$
$\displaystyle=(\partial_{\alpha\beta}R_{\bar{i}}(\boldsymbol{u})\circ\boldsymbol{\varphi})\frac{\left((\partial_{1}\boldsymbol{R}(\boldsymbol{u})\circ\boldsymbol{\varphi})\times(\partial_{2}\boldsymbol{R}(\boldsymbol{u})\circ\boldsymbol{\varphi})\right)^{\bar{i}}}{||(\partial_{1}\boldsymbol{R}(\boldsymbol{u})\circ\boldsymbol{\varphi})\times(\partial_{2}\boldsymbol{R}(\boldsymbol{u})\circ\boldsymbol{\varphi})||}~{},$
$\displaystyle\bar{g}_{\alpha\beta}$
$\displaystyle=\partial_{\alpha}\varphi^{\bar{\gamma}}\partial_{\beta}\varphi_{\bar{\gamma}}~{},$
$\displaystyle A^{\alpha\beta\gamma\delta}$
$\displaystyle=\frac{Y}{2(1+\nu)}\left(\frac{2\nu}{1-\nu}\bar{g}^{\alpha\beta}\bar{g}^{\gamma\delta}+\bar{g}^{\alpha\gamma}\bar{g}^{\beta\delta}+\bar{g}^{\alpha\delta}\bar{g}^{\beta\gamma}\right)~{}.$

Now, equation (9) is exactly the same form as the nonlinear plates equations
in curvilinear coordinates put forward by the authors, excluding the Dirichlet
boundary condition. However, equation (9) is derived from the plate equations
in Euclidean coordinates (7) with the use of the map $\boldsymbol{\varphi}$
which is a $\mathbb{R}^{2}\to\textbf{E}^{2}$ diffeomorphism, and thus, the
reference metric $\bar{\boldsymbol{g}}$ is immersible in $\textbf{E}^{2}$. To
be more precise, consider Ricci curvature tensor in the two-dimensional
Euclidean space,
$Ric^{\text{Euclidean}}_{\bar{\alpha}\bar{\beta}}=Ric_{\gamma\delta}(J^{-1})^{\gamma}_{\bar{\alpha}}(J^{-1})^{\delta}_{\bar{\beta}}$.
As Ricci curvature tensor is identically zero in the Euclidean space
(clearly!) and the map $\boldsymbol{\varphi}$ is a diffeomorphism, we have the
vanishing of Ricci curvature tensor in $\omega\subset\mathbb{R}^{2}$, i.e.
$\boldsymbol{Ric}=\boldsymbol{0}$. Now, consider the map
$\boldsymbol{\varphi}\times\boldsymbol{(}z\boldsymbol{)}_{\text{E}}:\\{\omega\times[-\frac{1}{2},\frac{1}{2}]\\}\subset\mathbb{R}^{2}\times\textbf{E}\to\\{\boldsymbol{\varphi}(\omega)\times[-\frac{1}{2},\frac{1}{2}]\\}\subset\textbf{E}^{3}$.
As $\boldsymbol{\varphi}$ is a diffeomorphism, the map
$\boldsymbol{\varphi}\times\boldsymbol{(}z\boldsymbol{)}_{\text{E}}$ is also a
diffeomorphism, and thus, the metric generated by the map
$\boldsymbol{\varphi}\times\boldsymbol{(}z\boldsymbol{)}_{\text{E}}$ is
immersible in $\textbf{E}^{3}$. Furthermore, as
$\boldsymbol{r}(\boldsymbol{u}):\mathscr{S}\times[-\frac{1}{2},\frac{1}{2}]\subset\textbf{E}^{3}\to\boldsymbol{r}(\boldsymbol{u})(\mathscr{S}\times[-\frac{1}{2},\frac{1}{2}])\subset\textbf{E}^{3}$
is a diffeomorphism, the metric on $\textbf{E}^{3}$ induced by
$\boldsymbol{r}(\boldsymbol{u})$ (i.e. the metric of deformation) is
immersible in $\textbf{E}^{3}$. Thus, the metric on
$\mathbb{R}^{2}\\!\times\\!\textbf{E}$ generated by the map
$r(\boldsymbol{u})\circ\boldsymbol{(}\boldsymbol{\varphi}\times\boldsymbol{(}z\boldsymbol{)}_{\text{E}}\boldsymbol{)}:\\{\omega\times[-\frac{1}{2},\frac{1}{2}]\\}\subset\mathbb{R}^{2}\\!\times\\!\textbf{E}\to\boldsymbol{r}(\boldsymbol{u})(\mathscr{S}\times[-\frac{1}{2},\frac{1}{2}])\subset\textbf{E}^{3}$
(i.e. the metric $\boldsymbol{g}$) is also immersible in $\textbf{E}^{3}$.
However, the authors assert that their reference metric is not immersible in
$\textbf{E}^{3}$. But it is mathematically impossible to derive equation (9)
from equation (7) without the use of a sufficiently differentiable
$\mathbb{R}^{2}\to\textbf{E}^{2}$ diffeomorphism; thus, the fact that the
authors claiming that their reference metric is not immersible in the three-
dimensional Euclidean space (while their metric $\boldsymbol{g}$ is immersible
in $\textbf{E}^{3}$) means that the authors are attempting something
fundamentally flawed.

Note that, if equation (7) linearised and along with the Dirichlet boundary
condition, then one gets the following,

$\displaystyle
J(\boldsymbol{u})=\int\limits_{\mathscr{S}}\frac{1}{2}A^{\bar{\alpha}\bar{\beta}\bar{\gamma}\bar{\delta}}\left(ha_{\bar{\alpha}\bar{\beta}}(\boldsymbol{u})a_{\bar{\gamma}\bar{\delta}}(\boldsymbol{u})+\frac{1}{12}h^{3}b_{\bar{\alpha}\bar{\beta}}(\boldsymbol{u})b_{\bar{\gamma}\bar{\delta}}(\boldsymbol{u})\right)dxdy$
$\displaystyle~{},$ $\displaystyle\boldsymbol{u}\in\\{\boldsymbol{v}\in
H^{1}(\mathscr{S})\\!\times\\!H^{1}(\mathscr{S})\\!\times\\!H^{2}(\mathscr{S})\mid\boldsymbol{v}|_{\partial\mathscr{S}_{0}}=\boldsymbol{0},$
$\displaystyle~{}n^{\bar{\alpha}}\partial_{\bar{\alpha}}v^{3}|_{\partial\mathscr{S}_{0}}=0,~{}\boldsymbol{v}|_{\\{\partial\mathscr{S}\setminus\partial\mathscr{S}_{0}\\}}=\boldsymbol{u}_{0}$
$\displaystyle\\}~{},$

where
$a_{\bar{\alpha}\bar{\beta}}(\boldsymbol{u})=\frac{1}{2}(\partial_{\bar{\alpha}}u_{\bar{\beta}}+\partial_{\bar{\beta}}u_{\bar{\alpha}})$,
$b_{\bar{\alpha}\bar{\beta}}(\boldsymbol{u})=\partial_{\bar{\alpha}\bar{\beta}}u^{\bar{3}}$,
$\boldsymbol{n}$ is the unit outward normal to the boundary
$\partial\mathscr{S}$ and $\partial\mathscr{S}_{0}\subset\partial\mathscr{S}$
with $\mathrm{meas}(\partial\mathscr{S}_{0};\mathbb{R})>0$, where
$\mathrm{meas}(\cdot;\mathbb{R}^{n})$ is the standard Lebesgue measure in
$\mathbb{R}^{n}$ and $H^{n}(\cdot)$ are the standard $W^{n,2}(\cdot)$-Sobolev
spaces (see section 5.2.1 of Evans [5]). Such problems can be solved by
consulting the literature that are specialised in the study of linear plate
theory (see Ciarlet [6] and Reddy [7]).

For numerical results, instead of finding the unknown metric tensor
$\boldsymbol{g}$ (which is the goal of the publication), the authors attempt
to analyse the stretching and the bending densities, i.e. $w_{s}$ and $w_{b}$
respectively, for a predetermined reference metric $\bar{\boldsymbol{g}}$ and
a predetermined deformed mid-surface $\boldsymbol{R}$, and thus, a
predetermined deformed metric $\boldsymbol{g}$ (see section 4 of Efrati _et
al._ [1]). The authors give numerical results for an ‘annular hemispherical
plate’, i.e. annular plate deformed in to a hemispherical shape, and state
that numerical results demonstrate that in the general case there is no
‘equipartition’ between bending and stretching energies. The authors conclude
by saying their numerical findings support treating very thin bodies as
inextensible, and ‘it also shows that not only in the equilibrium 3D
configuration dominated by the minimisation of the bending energy term, but
the total elastic energy is dominated by it also’ [1]. The reader must
understand that the authors’ numerical results do not imply the existence of a
solutions (i.e. the existence of the deformation metric $\boldsymbol{g}$ not
proven), as the numerical results are obtained for a predetermined metric
$\boldsymbol{g}$.

The authors’ numerical analysis implies that a thin object can be stretched
substantially with very little force. To examine this in more detail, consider
the following simple example in accordance with the authors’ numerical
analysis. Consider two circular plates: plate $c$ and plate $s$, with same
Young’s modulus $Y$, Poisson’s ratio $\nu$, thickness $h$ and radius $r$, and
assume that $h/r=\varepsilon\ll 1$. Now, take plate $c$ and deform it into the
shape of a semi-cylinder with a radius $\frac{2}{\pi}r$ (an area preserving
deformation). Following the authors’ publication, one finds that the mid-
surface can be express by the following map,

$\boldsymbol{R}(x^{1},x^{2})=\boldsymbol{(}x^{1},~{}\frac{2}{\pi}r\sin(x^{2}),~{}\frac{2}{\pi}r\cos(x^{2})\boldsymbol{)}_{\text{E}}~{},$

and as one knows the deformed configuration in advance, one finds that the
reference metric has the form
$\bar{\boldsymbol{g}}(x^{1},x^{2})=\mathrm{diag}(1,~{}(\frac{2}{\pi}r)^{2},~{}1)$.
Thus, the stored energy of a circular plate that is being deformed into a
semi-cylindrical shape can be expressed as follows,

$\displaystyle E_{c}$
$\displaystyle=\frac{1}{2}\frac{Y}{1-\nu^{2}}\int_{-\frac{1}{2}h}^{\frac{1}{2}h}\iint_{\\{(x^{1})^{2}+(\frac{2r}{\pi}x^{2})^{2}\leq
r^{2}\\}}\left(\frac{\pi
x^{3}}{2r}\right)^{2}\left(\frac{2r}{\pi}\right)dx^{1}dx^{2}dx^{3}$
$\displaystyle=\frac{1}{96}\pi^{3}\frac{Y}{1-\nu^{2}}h^{3}~{}.$ (10)

Now, take plate $s$ and deform it into a shape of a hemisphere with a radius
$\frac{1}{2}\sqrt{2}r$ (an area preserving deformation). Following the
authors’ publication, one finds that the mid-surface can be express by the
following map,

$\boldsymbol{R}(x^{1},x^{2})=\frac{1}{2}\sqrt{2}r\boldsymbol{(}\sin(x^{1})\cos(x^{2}),~{}\sin(x^{1})\sin(x^{2}),~{}\cos(x^{1})\boldsymbol{)}_{\text{E}}$

(see the definition of $\boldsymbol{R}(r,\theta)$ from section 4.1 of Efrati
_et al._ [1]), and as one knows the deformed configuration in advance, one
finds that the reference metric has the form
$\bar{\boldsymbol{g}}(x^{1},x^{2})=\mathrm{diag}(\frac{1}{2}r^{2},~{}\frac{1}{2}r^{2}\sin^{2}(x^{1}),~{}1)$
(see equation 4.1 and the definition of $\Phi(r)$ from section 4.2 of Efrati
_et al._ [1]), where this configuration is defined as the ‘stretch-free
configuration’ (see section 4.1 of Efrati _et al._ [1]). Thus, the stored
energy of a circular plate that is being deformed into a hemisphere can be
expressed as follows,

$\displaystyle E_{s}$
$\displaystyle=\frac{Y}{1-\nu}\int_{-\frac{1}{2}h}^{\frac{1}{2}h}\int_{0}^{\frac{1}{2}\pi}\int_{-\pi}^{\pi}\left(\frac{\sqrt{2}~{}x^{3}}{r}\right)^{2}\left(\frac{\sqrt{2}~{}r}{2}\right)\sin(x^{2})~{}dx^{1}dx^{2}dx^{3}$
$\displaystyle=\frac{1}{3}\pi\frac{Y}{1-\nu}h^{3}~{}.$ (11)

Equations (10) and (11), therefore, imply that, if one deforms a circular
plate into a semi-cylinder with a radius $\frac{2}{\pi}r$ and deform a
circular plate into a hemisphere with radius $\frac{1}{2}\sqrt{2}r$, then one
gets the very similar respective energy densities
$\frac{1}{96}\pi^{2}(1-\nu^{2})^{-1}Y\varepsilon^{2}$J$\text{m}^{-3}$ and
$\frac{1}{3}Y(1-\nu)^{-1}\varepsilon^{2}$J$\text{m}^{-3}$, i.e. both
deformations’ internal energies are of
$\mathcal{O}(\varepsilon^{2})$J$\text{m}^{-3}$. Which in turn implies that
both deformations require force of $\mathcal{O}(\varepsilon)$N, given that one
is applying the forces to the boundaries of the each respective plates. Thus,
the authors’ work asserts that it take approximately the same amount of force
to bend a plate into a semi-cylindrical shape or to stretch a plate into a
hemispherical shape with a similar radius. The reader may try this one’s self:
find a piece of aluminium foil (i.e. kitchen foil) and try to bend it over
one’s water bottle. This is a very simple process and the reader will able to
accomplish this with a minimum of effort. In fact, the force of gravity is
alone may even be sufficient to deform the piece of aluminium foil over the
bottle without much interference. Now, try to stretch that same piece of
aluminium foil smoothly over a rigid sphere with a similar radius, e.g. over a
cricket ball. Can the reader do this without tearing or crumpling, and with
the same force as one applied in the previous case?

To attempt this problem with mathematical regiour, consider the set
$\mathscr{S}=\\{\boldsymbol{(}x,y,0\boldsymbol{)}_{\text{E}}\in\textbf{E}^{3}\mid
x^{2}+y^{2}\leq r^{2}\\}$, which describes the mid-plane of the unstrained
plates $c$ and $s$. Now, if one deforms plate $c$ is into a semi-cylindrical
shape with a radius $\frac{2}{\pi}r$, then one finds that the map of the
deformed mid-surface has the following form,

$\boldsymbol{R}(x,y)=\boldsymbol{(}x,~{}\frac{2}{\pi}r\sin\left(\frac{\pi}{2r}y\right),~{}\frac{2}{\pi}r\cos\left(\frac{\pi}{2r}y\right)\boldsymbol{)}_{\text{E}}~{},$

and thus, the total stored energy of a circular plate of radius $r$ that is
being deformed into a semi-cylindrical shape with a radius $\frac{2}{\pi}r$
can be expressed as follows,

$\displaystyle E_{c}$
$\displaystyle=\frac{1}{2}\frac{Y}{1-\nu^{2}}r\int_{-\frac{1}{2}h}^{\frac{1}{2}h}\iint_{\mathscr{S}}\left(\frac{\pi
z}{2r}\right)^{2}dxdydz$
$\displaystyle=\frac{1}{96}\pi^{3}\frac{Y}{1-\nu^{2}}h^{3}~{}.$ (12)

Now, if one deforms plate $s$ is in to a hemisphere with a radius
$\frac{1}{2}\sqrt{2}r$, then one finds that the map of the deformed mid-
surface has the following form,

$\displaystyle\boldsymbol{R}(x,y)=\frac{1}{2}\sqrt{2}\boldsymbol{(}x\sin\left(\frac{\pi}{2r}\sqrt{x^{2}+y^{2}}\right),~{}y\sin\left(\frac{\pi}{2r}\sqrt{x^{2}+y^{2}}\right),$
$\displaystyle r\cos\left(\frac{\pi}{2r}\sqrt{x^{2}+y^{2}}\right)$
$\displaystyle\boldsymbol{)}_{\text{E}}~{},$

and thus, the total stored energy of a circular plate with a radius $r$ that
is being deformed into a hemisphere with a radius $\frac{1}{2}\sqrt{2}r$ can
be expressed as follows

$\displaystyle E_{s}$
$\displaystyle=\frac{1}{2}\int_{-\frac{1}{2}h}^{\frac{1}{2}h}\iint_{\mathscr{S}}A^{\alpha\beta\gamma\delta}\epsilon_{\alpha\beta}(x,y)\epsilon_{\gamma\delta}(x,y)~{}dxdydz+\mathcal{O}(h^{3})$
$\displaystyle=ChYr^{2}+\mathcal{O}(h^{3})~{},$ (13)

where $C$ is an order-one positive constant that is independent of $h$, $Y$
and $r$, and

$\displaystyle\boldsymbol{\epsilon}(x,y,0)=~{}$
$\displaystyle\left(\frac{r}{2}\right)^{2}\frac{\sin^{2}\left(\frac{\pi}{2r}\sqrt{x^{2}+y^{2}}\right)}{(x^{2}+y^{2})^{2}}\begin{pmatrix}y^{2}&-xy\\\
-xy&x^{2}\end{pmatrix}$
$\displaystyle+\left(\frac{\pi}{4}\right)^{2}\frac{1}{x^{2}+y^{2}}\begin{pmatrix}x^{2}&xy\\\
xy&y^{2}\end{pmatrix}-\frac{1}{2}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}~{},$

is the strain tensor of the plate at $z=0$. As the reader can see from
equations (12) and (13) that if one deforms a circular plate into a semi-
cylinder with a radius $\frac{2}{\pi}r$ and deform a circular plate into a
hemisphere with a radius $\frac{1}{2}\sqrt{2}r$, then one get the respective
energy densities $\mathcal{O}(\varepsilon^{2})$J$\text{m}^{-3}$ and
$\mathcal{O}(1)$J$\text{m}^{-3}$. Thus, one can see that it takes
significantly higher amount of energy to deform plate in to a hemisphere than
to simply bend it in to a semi-cylinder, as the former deformation requires a
significant amount of stretching and compression, while the latter requires no
such in-plane deformations, which is far more realistic than results obtained
by Efrati _et al._ ’s approach [1]. Note that the both deformations conserve
area.

As further analysis, consider the deformed plate $s$ in curvilinear
coordinates $\boldsymbol{(}x^{1},x^{2}\boldsymbol{)}$, where $0\leq
x^{1}\leq\frac{1}{2}\pi$ and $|x^{2}|\leq\pi$. Now, the first and the second
fundamental form tensors of the deformed configuration can be expressed
respectively as
$\boldsymbol{F}_{\\!\text{[I]}}(x^{1},x^{2})=\frac{1}{2}r^{2}\mathrm{diag}(1,~{}\sin^{2}(x^{1}))$
and
$\boldsymbol{F}_{\\!\text{[II]}}(x^{1},x^{2})=-\frac{1}{2}\sqrt{2}r\mathrm{diag}(1,~{}\sin^{2}(x^{1}))$.
If one follows the authors’ publication, then one finds that the reference
metric tensor has the form

$\bar{\boldsymbol{g}}(x^{1},x^{2})=\mathrm{diag}(\frac{1}{2}r^{2},~{}\frac{1}{2}r^{2}\sin^{2}(x^{1}),~{}1)~{},$

and this can only be derived by doing the following,

$\bar{g}_{ij}(x^{1},x^{2})=\partial_{i}r_{\bar{k}}(x^{1},x^{2},x^{3})\partial_{j}r^{\bar{k}}(x^{1},x^{2},x^{3})|_{x^{3}=0}~{},$

where
$\boldsymbol{r}(x^{1},x^{2})=(\frac{1}{2}\sqrt{2}r+x^{3})\boldsymbol{(}\sin(x^{1})\cos(x^{2}),~{}\sin(x^{1})\sin(x^{2}),~{}\cos(x^{1})\boldsymbol{)}_{\text{E}}$.
This implies that $\bar{\boldsymbol{g}}$ is the reference metric of a shell at
$x^{3}=0$, and thus, $\bar{\boldsymbol{g}}$ is clearly not immersible in
$\textbf{E}^{3}$ as Ricci tensor is not identically zero, i.e.

$\boldsymbol{Ric}=\frac{1}{2}r^{2}\mathrm{diag}(1,~{}\sin^{2}(x^{1}))\neq\boldsymbol{0}~{},~{}\text{for}~{}x^{1}\neq
0~{}.$

Thus, the authors’ erroneous reference metric implies that
$a_{\alpha\beta}-\bar{g}_{\alpha\beta}=0$,
$\forall~{}\alpha,\beta\in\\{1,2\\}$, i.e. zero-planar strain, which in turn
implies the existence of a ‘stretch-free configuration’ for a substantially
deformed plate.

Now, if one attempts this same problem with mathematical precision, then one
finds that the reference metric tensor can be expressed as follows,

$\bar{\boldsymbol{g}}(x^{1},x^{2})=4\left(\frac{r}{\pi}\right)^{2}\mathrm{diag}(1,~{}(x^{1})^{2})~{},$

where
$\bar{g}_{\alpha\beta}(x^{1},x^{2})=\partial_{\alpha}x\partial_{\beta}x+\partial_{\alpha}y\partial_{\beta}y$
with $x^{1}=\frac{1}{2}r^{-1}\pi\sqrt{x^{2}+y^{2}}$ and $x^{2}=\arctan(y/x)$.
The coordinate transform
$\boldsymbol{(}x(x^{1},x^{2}),~{}y(x^{1},x^{2})\boldsymbol{)}_{\text{E}}:\mathbb{R}^{2}\to\textbf{E}^{2}$
is a diffeomorphism (except at $x^{1}=x^{2}=0$) and Ricci tensor is
identically zero, i.e. $\boldsymbol{Ric}=\boldsymbol{0}$. Furthermore,
$\det(\partial_{1}x,\partial_{2}x;\partial_{1}y,\partial_{2}y)>0$, and thus,
the definition of the unit normal to the deformed surface is not violated
(again, except at $x^{1}=x^{2}=0$). Thus, half of change in the first
fundamental form tensor (i.e. planar strain) can be expressed as follows,

$\boldsymbol{a}(x^{1},x^{2})=\frac{1}{4}r^{2}\mathrm{diag}(1-\frac{8}{\pi^{2}},~{}\sin^{2}(x^{1})-\frac{8}{\pi^{2}}(x^{1})^{2})$

and the change in second fundamental form tensor (i.e. bending) can be
expressed as follows,

$\boldsymbol{b}(x^{1},x^{2})=-\frac{1}{2}\sqrt{2}r\mathrm{diag}(1,~{}\sin^{2}(x^{1}))~{}.$

Now, with this coordinate transform no such ‘stretch-free configuration’ can
exist for a plate with a radius $r$ that is being deformed into a hemisphere
with a $\frac{1}{2}\sqrt{2}r$, unless the radius of the plate is zero.

Above analysis shows that Efrati _et al._ [1] are not studying plates, but
they are studying nonlinear Koiter shells with an erroneous strain tensor. The
authors’ definition of the strain tensor leads to an incorrect change in
second fundamental form tensor, and thus, an overestimation of the bending
energy density of the shell per $h^{3}$ (see equation (3)). To attempt this
problem with mathematical precision, let
$\tilde{g}_{ij}(\boldsymbol{x})=\partial_{i}X_{\bar{k}}\partial_{j}X^{\bar{k}}$
be the metric of the reference configuration

$\boldsymbol{X}(\boldsymbol{x})=\boldsymbol{\sigma}(x^{1},x^{2})+x^{3}\frac{(\boldsymbol{\sigma}_{,1}\times\boldsymbol{\sigma}_{,2})}{||\boldsymbol{\sigma}_{,1}\times\boldsymbol{\sigma}_{,2}||}$

with respect to the curvilinear coordinate system
$\boldsymbol{x}=\boldsymbol{(}x^{1},x^{2},x^{3}\boldsymbol{)}$, where
$\boldsymbol{\sigma}:\mathbb{R}^{2}\to\textbf{E}^{3}$ is a sufficiently
differentiable immersion (Efrati _et al._ ’s reference metric [1] is derived
by $\bar{\boldsymbol{g}}=\tilde{\boldsymbol{g}}|_{x^{3}=0}$). Thus, in
nonlinear shell theory, one defines the strain tensor as
$\epsilon_{\alpha\beta}(\boldsymbol{u})=\frac{1}{2}(g_{\alpha\beta}(\boldsymbol{u})-\tilde{g}_{\alpha\beta})$,
where
$g_{\alpha\beta}(\boldsymbol{u})=\partial_{\alpha}r_{\bar{i}}(\boldsymbol{u})\partial_{\beta}r^{\bar{i}}(\boldsymbol{u})$,

$\boldsymbol{r}(\boldsymbol{u})=\boldsymbol{R}(\boldsymbol{u})+x^{3}\frac{(\partial_{1}\boldsymbol{R}(\boldsymbol{u})\times\partial_{2}\boldsymbol{R}(\boldsymbol{u}))}{||\partial_{1}\boldsymbol{R}(\boldsymbol{u})\times\partial_{2}\boldsymbol{R}(\boldsymbol{u})||}~{},$

$\boldsymbol{R}(\boldsymbol{u})=\boldsymbol{\sigma}+u^{\alpha}\partial_{\alpha}\boldsymbol{\sigma}+u^{3}\boldsymbol{N}$
and $\boldsymbol{u}(\boldsymbol{x})$ is the displacement field in curvilinear
coordinates. For more on nonlinear Koiter’s shells, please consult Ciarlet
[3], Koiter [8], and Libai and Simmonds [9].

Even if Efrati _et al._ [1] obtain the correct form of the strain tensor for
shells, they are still unjustified in using the shell strain tensor to model
plates. To explain this matter with mathematical rigour, let $\mathscr{S}$ be
a two-dimensional plane and let $\mathscr{S}^{\prime}$ be a two-dimensional
surface. What the authors fail to grasp is that an arbitrary mapping from
$\mathscr{S}$ to $\mathscr{S}^{\prime}$ (i.e.
$\boldsymbol{\sigma}:\mathscr{S}\subset\mathbb{R}^{2}\to\mathscr{S}^{\prime}\subset\textbf{E}^{3}$)
is not the same as deforming the plane $\mathscr{S}$ into the surface
$\mathscr{S}^{\prime}$ (i.e.
$\\{\mathscr{S}\cup\\{\boldsymbol{u}\in\textbf{E}^{3}\\}\subset\textbf{E}^{3}\\}=\\{\mathscr{S}^{\prime}\subset\textbf{E}^{3}\\}$).
The former is a simple coordinate transform (which may or may not be related
to deforming the body), while the latter is a unique vector displacement
(unique up to a rigid motion). To understand the distinction between a
coordinate transform and a vector displacement, please consult section 1 and
section 2 of Morassi and Paroni [10].

##  3 Conclusions

In conclusion, Efrati _et al._ ’s [1] publication is not on plate theory: it
is on shell theory with an incorrect strain tensor. We showed that the authors
made the fundamental error of assuming that an arbitrary mapping from a plane
to a curved surface is the same as deforming a plane in to a curved surface,
when deriving their model. Thus, the authors numerical results imply that a
thin object can be stretched substantially with very little force, which is
physically unrealistic and mathematically disprovable. All the theoretical
work of the authors, i.e. nonlinear plate equations in curvilinear
coordinates, can easily be rectified with the inclusion of both a sufficiently
differentiable $\mathbb{R}^{2}\to\textbf{E}^{2}$ diffeomorphism and a set of
external loadings, such as an external strain field.

## References

  * Efrati et al. [2009] E. Efrati, E. Sharon, R. Kupferman,  Elastic theory of unconstrained non-euclidean plates,  Journal of the Mechanics and Physics of Solids 57 (2009) 762–775.
  * Ball [1976] J. M. Ball,  Convexity conditions and existence theorems in nonlinear elasticity,  Archive for rational mechanics and Analysis 63 (1976) 337–403.
  * Ciarlet [2000] P. G. Ciarlet, Theory of Shells, Mathematical Elasticity, Elsevier Science, 2000\.
  * Willmore and Willmore [1996] T. J. Willmore, T. J. Willmore, Riemannian geometry, Oxford University Press, 1996.
  * Evans [2010] L. C. Evans, Partial Differential Equations, Graduate studies in mathematics, American Mathematical Society, 2010.
  * Ciarlet [1997] P. G. Ciarlet, Theory of Plates, Mathematical Elasticity, Elsevier Science, 1997\.
  * Reddy [2006] J. N. Reddy, Theory and analysis of elastic plates and shells, CRC press, 2006.
  * Koiter [1966] W. T. Koiter,  On the nonlinear theory of thin elastic shells. i- introductory sections. ii- basic shell equations. iii- simplified shell equations(nonlinear theory of thin elastic shells, discussing surface geometry and deformation, equations of equilibrium and boundary conditions and stress functions),  Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series B 69 (1966) 1–54.
  * Libai and Simmonds [2005] A. Libai, J. G. Simmonds, The nonlinear theory of elastic shells, Cambridge university press, 2005.
  * Morassi and Paroni [2009] A. Morassi, R. Paroni, Classical and Advanced Theories of Thin Structures: Mechanical and Mathematical Aspects, CISM International Centre for Mechanical Sciences, Springer Vienna, 2009\.


# Attentive Fusion: A Transformer-based Approach to Multimodal Hate Speech
Detection

Atanu Mandal1† Gargi Roy2‡ The work was carried out when the author was at
Jadavpur University. Amit Barman3†
Indranil Dutta4† Sudip Kumar Naskar5†
†Jadavpur University, Kolkata, INDIA
‡Optum Global Solutions Private Limited, Bengaluru, INDIA
{1atanumandal0491, 2roygargi1997, 3amitbarman811<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

With the recent surge and exponential growth of social media usage,
scrutinizing social media content for the presence of any hateful content is
of utmost importance. Researchers have been diligently working since the past
decade on distinguishing between content that promotes hatred and content that
does not. Traditionally, the main focus has been on analyzing textual content.
However, recent research attempts have also commenced into the identification
of audio-based content. Nevertheless, studies have shown that relying solely
on audio or text-based content may be ineffective, as recent upsurge indicates
that individuals often employ sarcasm in their speech and writing. To overcome
these challenges, we present an approach to identify whether a speech promotes
hate or not utilizing both audio and textual representations. Our methodology
is based on the Transformer framework that incorporates both audio and text
sampling, accompanied by our very own layer called “Attentive Fusion”. The
results of our study surpassed previous state-of-the-art techniques, achieving
an impressive macro F1 score of 0.927 on the Test Set.

##  1 Introduction

In recent years, the explosive growth of digital communication platforms has
facilitated unprecedented levels of information exchange, enabling individuals
from diverse backgrounds to interact and share ideas. However, this surge in
online interactions has also led to the emergence of a concerning issue: the
increase of hate speech Davidson et al. (2017). Hate speech, characterized by
offensive, discriminatory, or derogatory language targeting individuals or
groups based on race, ethnicity, religion, gender, or sexual orientation,
poses significant challenges to maintaining a safe and inclusive online
environment Schmidt and Wiegand (2017).

Traditional methods of hate speech detection primarily focused on analyzing
text-based content, leveraging natural language processing (NLP) techniques to
identify offensive language patterns. While these approaches have yielded some
success, they often struggle to capture the nuanced nature of speech, as the
exact text might be interpreted differently when considering context, tone,
and intent Fortuna and Nunes (2018). To address these limitations, researchers
are turning to a more holistic approach combining both text and speech
modalities to enhance the accuracy and robustness of hate speech detection
systems Rana and Jha (2022).

Figure 1: Identification of “Hate” or “Not Hate” using multimodality approach

This multidimensional approach referred to as multimodal hate speech
detection, leverages not only the textual content of messages but also the
acoustic cues and prosodic features present in speech. By simultaneously
analyzing both text and speech-based characteristics, this approach aims to
capture a more comprehensive representation of communication, considering not
only the words used, but also the emotional nuances conveyed through speech
intonation, pitch, and rhythm. Figure 1 illustrates two examples each for
“Hate” and “Not Hate” using the Multimodality. In the two cues shown (figure
1),  represents the speech cue and  represents the text cue.

In this paper, we investigate multimodal hate speech detection exploring the
synergies between text and speech for identifying hate speech instances. We
examine the challenges posed by hate speech in the digital age, the
limitations of traditional text-based detection methods, and the potential
advantages of integrating speech data into the detection process. By
leveraging insights from various disciplines such as NLP, audio signal
processing, and machine learning, multimodal approaches hold promise in
achieving higher detection accuracy and reducing false positives, ultimately
fostering safer and more inclusive online environments.

Our methodology sets itself apart from other state-of-the-art (SOTA)
methodologies in the subsequent manner:

  * •

Our system consists of a sequence of interconnected systems enclosing the
Transformer framework111Code is publicly available in GitHub..

  * •

We have introduced a layer termed “Attentive Fusion” that augments the
results.

##  2 Dataset Description

Dataset | Number of Samples
---|---
Train | Dev | Test
CMU-MOSEI |  |  |
Bagher Zadeh et al. (2018) | 597 | 133 | 130
CMU-MOSI |  |  |
Zadeh et al. (2016) | 181 | 40 | 39
Common Voice |  |  |
Ardila et al. (2020) | 8,050 | 1,768 | 1,733
LJ Speech |  |  |
Ito and Johnson (2017) | 102 | 23 | 23
MELD |  |  |
Poria et al. (2019) | 393 | 87 | 85
Social-IQ |  |  |
Zadeh et al. (2019) | 325 | 74 | 69
VCTK |  |  |
Yamagishi et al. (2019) | 138 | 31 | 30
| 9,786 | 2,156 | 2,109
Table 1: Statistics of the dataset used for Identification of Hatred.

For our experiments, we used fragments of the DeToxy dataset Ghosh et al.
(2022)222Ghosh et al. (2022) used 20,271 data consisting of CMU-MOSEI, CMU-
MOSI, Common Voice, IEMOCAP, LJ Speech, MELD, MSP-Improv, MSP-Podcast, Social-
IQ, Switchboard, and VCTK of which IEMOCAP, MSP-Improv, MSP-Podcast,
Switchboard are not open-sourced therefore we were unable to use the dataset.,
a dataset for detecting Hatred within spoken English speech. This dataset is
derived from diverse open-source datasets. The specifics regarding the number
of samples utilized from various datasets are precisely outlined in Table 1.

(a) Training Data

(b) Development Data

(c) Test Data

Figure 2: Pictorial representation of the contribution of datasets | Hate | Not Hate
---|---|---
Dataset | Train | Dev | Test | Train | Dev | Test
CMU-MOSEI | 149 | 33 | 35 | 448 | 100 | 95
CMU-MOSI | 47 | 10 | 10 | 134 | 30 | 29
Common Voice | 2,013 | 442 | 433 | 6,037 | 1,326 | 1,300
LJ Speech | 28 | 6 | 6 | 74 | 17 | 17
MELD | 99 | 22 | 21 | 294 | 65 | 64
Social-IQ | 83 | 18 | 19 | 242 | 56 | 50
VCTK | 34 | 8 | 8 | 104 | 23 | 22
| 2,453 | 539 | 532 | 7,333 | 1,617 | 1,577
Table 2: Data Statistics of “Hate” and “Not Hate”  Figure 3: Sample count for
“Hate” and “Not Hate”

Our experiments were carried out on a comprehensive dataset that encompassed
all seven datasets combined. Each dataset contained entries that fell into
either the "Hate" or "Not-Hate" category, along with a transcription for each
audio. To facilitate understanding, we have depicted the distribution of each
dataset’s contribution to our framework through a pie chart, as showcased in
Figure 2. Figures 2(a), 2(b), and 2(c) accordingly illustrate the respective
contributions of the training data, development data, and testing data. There
exists a significant disparity in the number of samples across the various
datasets but, the proportional representation of the training, development,
and test datasets remains consistent. Notably, Common Voice comprises the
majority of the data, while LJ Speech is the least represented. The
statistical analysis of the “Hate” and “Not Hate” classes is presented in
Table 2. Meanwhile, the bar plot showcasing the sample count for both classes
can be seen in Figure 3. A comprehensive description of datasets is described
in Section 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, and 2.7.

(a) CMU-MOSEI

(b) CMU-MOSI

(c) Common Voice

(d) LJ Speech

(e) MELD

(f) Social-IQ

(g) VCTK

Figure 4: Sample count for “Hate” and “Not Hate”

(a) CMU-MOSEI

(b) CMU-MOSI

(c) Common Voice

(d) LJ Speech

(e) MELD

(f) Social-IQ

(g) VCTK

Figure 5: Scatter representation of Datasets according to audio length

###  2.1 CMU-MOSEI

Carnegie Mellon University - Multimodal Opinion Sentiment and Emotion
Intensity (CMU-MOSEI) Bagher Zadeh et al. (2018) is considered the largest and
most extensive dataset for emotion recognition tasks and multimodal sentiment
analysis. Figure 4(a) provides the information on the number of samples for
“Hate” and “Not Hate” and Figure 5(a) provides the pictorial information of
the number of samples to audio duration.

###  2.2 CMU-MOSI

Carnegie Mellon University - Multimodal Corpus of Sentiment Intensity (CMU-
MOSI) Zadeh et al. (2016) is another dataset by Carnegie Mellon University,
which consists of 2199 video clips of different opinions, annotated with
sentiment. It is annotated in the range $[-3,3]$, using various parameters for
sentiment intensity, subjectivity, and per-millisecond annotations of audio
features. It contains 97% non-toxic and nearly 3% toxic utterances. Figure
4(b) provides the information on the number of samples for “Hate” and “Not
Hate” and Figure 5(b) provides the pictorial information of the number of
samples to audio duration.

###  2.3 Common Voice

This dataset Ardila et al. (2020) by Mozilla Developer Network is an open-
source, dataset of voices of multiple languages for the use of training
speech-enabled systems, with 20,217 hours of recorded audio and 14,973 hours
of validated speech audios. Figure 4(c) provides the information on the number
of samples for “Hate” and “Not Hate” and Figure 5(c) provides the pictorial
information of the number of samples to audio duration.

###  2.4 LJ Speech

This is another open-source dataset Ito and Johnson (2017) which has 13,100
clips of short audio segments, with one speaker reading texts from a
collection of seven books of non-fiction. Every clip is transcribed and has a
varying length of 1 to 10 seconds. Figure 4(d) provides the information on the
number of samples for “Hate” and “Not Hate” and Figure 5(d) provides the
pictorial information of the number of samples to audio duration.

###  2.5 MELD

Multimodal Emotion Lines Dataset (MELD) Poria et al. (2019) has over 1,400
dialogues and 13,000 dialogues from the television show “Friends”. Each
utterance in dialogue has been labelled by one of the emotions – Anger,
Disgust, Sadness, Joy, Neutral, Surprise, and Fear. MELD also has annotations
for sentiments – positive, negative, and neutral. Figure 4(e) provides the
information on the number of samples for “Hate” and “Not Hate” and Figure 5(e)
provides the pictorial information of the number of samples to audio duration.

###  2.6 Social-IQ

Another dataset Zadeh et al. (2019) by Carnegie Mellon University has videos
that are thoroughly validated and annotated, along with questions, answers,
and annotations for the level of complexity of the said questions and answers.
Figure 4(f) provides the information on the number of samples for “Hate” and
“Not Hate” and Figure 5(f) provides the pictorial information of the number of
samples to audio duration.

###  2.7 VCTK

The VCTK corpus Yamagishi et al. (2019) contains 110 speakers’ speech data
spoken in English, having various accents. Every single speaker reads a
passage, selected from newspapers, archives, and so on. Figure 4(g) provides
the information on the number of samples for “Hate” and “Not Hate” and Figure
5(g) provides the pictorial information of the number of samples to audio
duration.

##  3 Experiments

This section demonstrates our innovative techniques for detecting Hatred
within a speech. The section is divided into numerous subsections for
understanding our approach with ease. Section 3.1 presents the methods we used
to prepare the dataset for our suggested framework. Section 3.2 describes our
suggested framework. Section 3.3 discusses the parameters used for our
proposed framework and Section 4 discusses the results of our approach with
other benchmark frameworks.

###  3.1 Dataset Pre-processing

In the task of pre-processing, we carefully selected the data that possessed
comparable lengths of audio. We disregarded instances with excessively long or
short duration. Our inclination to overlook excessively long audio duration
stemmed from the understanding that it would necessitate extensive
computational resources. Conversely, audio with extremely short duration
lacked the richness of audio features.

| Values
---|---
Sample Rate | 16,000 Hz
Number of FFT | 400
Number of MELs Channel | 80
Hop Length | 160
Chunk Length | 30
Number of Samples | 4,80,000
Number of Frames | 3,000
Number of Samples per Token | 320
Frames per Second | 10 ms
Tokens per Second | 25 ms
Table 3: Audio feature extraction parameters

We conducted a series of experiments with our framework, exploring various
methods of extracting features such as Mel-frequency cepstral coefficients
(MFCCs) and filter banks. However, we discovered that the “log mel
spectrogram” yields superior accuracy in comparison to other alternatives, as
it captures auditory information in a manner akin to human perception. To
extract these features, we established the optimal parameters empirically,
which are detailed in Table 3.

For feature extraction from text, we used the pre-trained Albert Tokenizer Lan
et al. (2020) from IndicBART Dabre et al. (2022) developed by AI4Bharat. To
tokenize each sentence, we introduced the symbols “$<s>$” and “$</s>$” at the
start and end, respectively, signifying the commencement and conclusion of the
sentences.

###  3.2 Framework

We have used the Transformer Vaswani et al. (2017) framework which has gained
widespread recognition and is considered SOTA in the domains of Speech
Recognition and Machine Translation (MT) due to its exceptional ability to
handle the complexities of these complex tasks. To provide a clear overview of
our methodology, Figure 6 presents an overview of our framework.

Figure 6: Overview of our approach

The Speech Feature is extracted by the “log mel spectrogram” technique, which
has been discussed in section 3.1. This technique involves the computation of
a spectrogram that represents the frequency content of an audio signal over
time, using a logarithmic scale for the frequency axis. The resulting
spectrogram has a dimension of “$(80\times time\\_step)$” and is then passed
to the Speech Sampling Block. The Speech Sampling Block is responsible for
selecting a subset of the input spectrogram, based on certain criteria
(described in Section 3.2.1). On the other hand, the tokenized Text, which is
obtained through a process described in section 3.1, has a dimension of
“$(max\\_length\times 1)$” and is passed to the Text Sampling Block (discussed
in Section 3.2.2). The Text Sampling Block performs a similar function as the
Speech Sampling Block but on the tokenized Text instead of the spectrogram.

The resulting subset of Speech Sampling is fed to the Encoder of the first
Transformer module and the Decoder of the second Transformer module.
Similarly, the resulting subset of Text Sampling is fed to the Decoder of the
first Transformer module and the Encoder of the second Transformer module. The
motivation behind this approach is to investigate whether the text in the
Decoder can learn from the audio in the Encoder, and vice versa. This is
inspired by the idea of MT, where the target text in the Decoder learns from
the source text in the Encoder. By applying this concept to the audio and text
domains, we aim to explore the potential for cross-modal learning and the
transfer of knowledge between different modalities.

To further process the outputs of the two Transformer modules, we introduce a
Long short-term memory (LSTM) block that consists of a single LSTM layer. This
LSTM block is responsible for sequentially learning the knowledge from each
step of the output. After going through this process, we obtain two outputs:
one from the first LSTM and another from the second LSTM. The combination of
the first Transformer with the first LSTM is represented as “Pipeline 1” and
the combination of the second Transformer with the second LSTM is represented
as “Pipeline 2”. These two outputs are then passed to the proposed “Attentive
Fusion” Layer (described in Section 3.2.3). The Attentive Fusion Layer is
designed to learn the knowledge from both outputs in a joint manner, combining
the information from the two pipelines. The output of the Attentive Fusion
Layer is then fed to a Linear Layer with Softmax activation, where it
undergoes further processing and classification according to the specific
classes.

A comprehensive exploration of the Audio Sampling Module, Text Sampling
Module, and our proposed Attentive Fusion layer are presented in Section
3.2.1, 3.2.2, and 3.2.3, respectively. All hyperparameter configurations can
be found in Section 3.3.

####  3.2.1 Speech Sampling

Figure 7: Overview of Speech Sampling

Our module for “Speech Sampling” was influenced by the work of Radford et al.
(2022) with minor modifications. This module comprises a pair of Convolutional
layers, with each layer being accompanied by a Gaussian Error Linear Unit
(GELU) activation function. The outcome of the Convolutional layer was passed
through a Positional Encoder and an LSTM layer separately. The results from
the Positional Encoder and LSTM were combined. The Speech Sampling framework
is shown in Figure 7.

####  3.2.2 Text Sampling

Our “Text Sampling” module comprises a simplistic approach containing Word
Embedding and a Positional Encoder. The raw text was tokenized by appending
with “$<s>$” and “$</s>$” at the commencement and conclusion of the sentences
(refer to Section 3.1) and passed on to Word Embedding. The subsequent output
is then directed to the Positional Encoder. Subsequently, the output of the
Word Embedding and the Positional Encoder are combined and conveyed to the
subsequent hierarchical module. The representation of the “Text Sampling”
framework can be seen in Figure 8.

Figure 8: Overview of Text Sampling

####  3.2.3 Attentive Fusion Layer

The layer we have named the “Attentive Fusion” layer is a layer that we have
devised for the purpose of detecting hatred within a speech. In our
methodology (as illustrated in equation 1), we have seamlessly integrated the
outcomes from Pipeline 1 and Pipeline 2, allowing them to flow into their
respective Linear layers individually, thereby ensuring the preservation of
their unique characteristics.

$\displaystyle L_{1}=Linear(x_{1})$ (1) $\displaystyle L_{2}=Linear(x_{2})$

The result, $L_{1}$ and $L_{2}$ underwent a process of cross multiplication,
after which a hyperbolic tangent function ($tanh$) was used. To enhance the
disparity of each tensor value ($w_{i}$), an exponential ($e$) function was
applied, as demonstrated in equation 2.

$w_{i}=e^{(tanh(L_{1}\times L_{2}))}$ (2)

The outcome of equation 2 $w_{i}$ underwent division by the summation of every
element of $w_{i}$. To prevent division by zero, we introduced an epsilon
($\epsilon$). The entire outcome was then subjected to multiplication with
$w_{i}$ itself. Equation 3 illustrates our approach.

$w^{{}^{\prime}}_{i}=\frac{w_{i}}{\sum_{i}w_{i}+\epsilon}\times w_{i}$ (3)

The value, $w^{{}^{\prime}}_{i}$ obtained from equation 3 was introduced into
the subsequent module that incorporates a Linear Layer to differentiate
between different classes.

###  3.3 Hyperparameters

####  3.3.1 Speech Sampling

For the two Convolutional layers, we used filter sizes of “4096” and “1024”,
respectively and kernel size of “3” for both. Strides of “1” for $1^{st}$ and
“2” for $2^{nd}$ Convolutional layer was used. For LSTM layer units of “512”
with activation function “tanh” and recurrent activation of “sigmoid” was
used. For the Positional Encoder vocab size of “64,014”, the hidden dimension
of “512” was passed.

####  3.3.2 Text Sampling

For the Word Embedding, we used a vocab size of “64,014”, and a hidden
dimension of “512” with True mask zero. For the Positional Encoder vocab size
of “64,014”, the hidden dimension of “512” was passed.

####  3.3.3 Transformer

For the transformer framework, the number of heads and the number of layers
for the Encoder and Decoder were kept “4”. The hidden dimensions were kept
“512”, and the dropout Srivastava et al. (2014) rate of “0.3”.

####  3.3.4 LSTM

For the LSTM layer, units of “512” with activation function “tanh” and
recurrent activation of “sigmoid” were used, with a dropout rate of “0.3”. Use
of bias and Forget bias with return sequence kept True.

####  3.3.5 Learning Rate

We use “AdamW” optimizer Loshchilov and Hutter (2019) with $\beta_{1}$ of
“0.9”, $\beta_{2}$ of “0.98”, $\epsilon$ of “$1\times 10^{-6}$” and decay of
“0.1” with adaptive learning rate.

$\displaystyle arg_{1}=\sqrt{cs}$ (4) $\displaystyle arg_{2}=cs\times
ws^{-1.5}$ $\displaystyle lr=\sqrt{d_{model}}\times\min(arg_{1},arg_{2})$
$\displaystyle lrate=\min(lr,0.0004)$

In equation 4, $cs$ is current step, warmup steps ($ws$) were set to “2048”
and $d_{model}$ of “512”.

##  4 Results

Table 4 provides the benchmark macro F1 scores of the frameworks proposed by
Ghosh et al. (2022). The proposed frameworks were trained only with the audio
sequences. Our study suggests that only the audio sequences cannot provide a
better understanding of hatred in speech. Current trends show that persons use
hateful words in spoken sentences but the tones, frequency and amplitudes are
kept normal, which can also be remarked as “Sarcasmic Behaviour”. To overcome
the situation we used multimodality where an audio specimen along with its
transcripts are used. Using multimodality has an increase in macro F1 score
compared with the F-Bank framework proposed by Ghosh et al. (2022).

In cases of unfrozen wav2vec-2.0. the differences are very nominal as
wav2vec-2.0 provides an embedding knowledge of each token of the audio
specimen. In contrast, we didn’t use any embedding knowledge of speech tokens,
which will be experimented with in our upcoming works. The researcher has
shown two wav2vec-2.0 among which is one wav2vec-2.0 (9 layer). In this
system, the researcher took the representation token from the $9^{th}$ layer.

System | Category | Dev | Test
---|---|---|---
F-Bank | - | 0.610 | 0.620
wav2vec-2.0 | Freezed | 0.448 | 0.457
Unfreezed | 0.877 | 0.869
| wav2vec-2.0
---
(9 layer)
Unfreezed | 0.897 | 0.877
| Proposed
---
Framework
- | 0.931 | 0.927
Table 4: Evaluation and Result of Different Systems Proposed by Ghosh et al.
(2022). Unfrozen wav2vec-2.0 setup with representations taken from the
$9^{th}$ layer as reported by Ghosh et al. (2022)

##  5 Ablation Studies

###  5.1 “Attentive Fusion layer” vs “Concatenate layer”

Instead of using our proposed Attentive Fusion Layer, we used the Concatenate
Layer to study the effectiveness of the Attentive Fusion layer and found it
even outperformed the Concatenate layer. The differences in results are shown
in Table 5.

| Dev | Test
---|---|---
Concatenate Layer | 0.908 | 0.909
Attentive Fusion Layer | 0.931 | 0.927
Table 5: Macro F1 Score Result on replacing “Attentive Fusion layer” with
“Concatenate layer”

###  5.2 Using Pipeline 1 and Pipeline 2 separately

We also checked whether alone text learning from audio representation or audio
learning from textual representation outperformed our baseline result or not.
For the evaluation, we used Pipeline 1 and Pipeline 2 separately followed by
Linear Layer, and found that it is unable to score at par result to our
baseline. The differences in results are shown in Table 6.

| Dev | Test
---|---|---
Pipeline 1 | 0.910 | 0.909
Pipeline 2 | 0.910 | 0.899
Our Baseline | 0.931 | 0.927
Table 6: Macro F1 Score Result on Pipeline 1 and Pipeline 2 separately and
compared with our baseline.

##  6 Conclusion

In this work, we proposed a framework that can classify whether a speech
promotes Hatred or not. For the speech feature extraction, we used a log mel
spectrogram feature extraction technique. Our framework consists of Speech
Sampling and Text sampling followed by two separate transformer frameworks
that serve different efforts. Each Transformer framework is followed by an
LSTM layer, the output of which is fed to our proposed layer, and further sent
to Linear Layer for Classification. The whole framework was able to outperform
the existing benchmark macro F1 score. The only limitation of our approach is
it is limited to the English language. In future work, we would like to test
its robustness for other languages.

## Acknowledgements

This research was supported by the TPU Research Cloud (TRC) program, a Google
Research initiative.

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