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\dateposted{2025-05-12}
\begin{document}


\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Projective continuum mechanics}

\alttitle{M\'{e}canique projective des milieux continus}

\author{\firstname{Tam\'as} \lastname{Baranyai}\CDRorcid{0000-0002-0512-7161}}
\address{Department of Morphology and Geometric Modelling, Budapest
University of Technology and Economics, M\H{u}egyetem rkp. 1-3, 1111
Budapest, Hungary}
\email{baranyai.tamas@epk.bme.hu}

\begin{abstract}
The description of Cauchy stress and infinitesimal strain tensors is
given, such that it is compatible with the homogeneous coordinate
description of projective geometry. It is shown that neither material
isotropy nor global material orthotropy are projective invariants thus
the transformation of known solutions is useful only for statically
determinate problems. As membrane shells are often statically
determinate, they are identified as a potential area of application.
The transformation of the graph of the Airy stress function is given in
a point-wise projective three dimensional way, which preserves its
correspondence with the shape function, as established by Pucher's
equation of a unidirectionally loaded membrane shell.
\end{abstract}


\begin{altabstract}
La description des tenseurs de contrainte de Cauchy et de
d\'{e}formation infinit\'{e}simale est donn\'{e}e, de mani\`{e}re \`{a}
\^{e}tre compatible avec la description en coordonn\'{e}es
homog\`{e}nes de la g\'{e}om\'{e}trie projective. Il est
d\'{e}montr\'{e} que ni l'isotropie ni l'orthotropie globale des
mat\'{e}riaux ne sont des invariants projectifs ; la transformation de
solutions connues n'est donc utile que pour les probl\`{e}mes
statiquement d\'{e}termin\'{e}s. Les coques membranaires \'{e}tant
souvent statiquement d\'{e}termin\'{e}es, elles constituent un domaine
d'application potentiel. La transformation du graphe de la fonction de
contrainte d'Airy est donn\'{e}e de mani\`{e}re tridimensionnelle
projective point par point, pr\'{e}servant ainsi sa correspondance avec
la fonction de forme, telle qu'\'{e}tablie par l'\'{e}quation de Pucher
d'une coque membranaire charg\'{e}e unidirectionnellement.
\end{altabstract}

\keywords{\kwd{Cauchy stress tensor}\kwd{Infinitesimal strain
tensor}\kwd{Collineations}\kwd{Membrane shells}\kwd{Airy stress
function}\kwd{Pucher's equation}}

\altkeywords{\kwd{Tenseur de contrainte de Cauchy}
\kwd{Tenseur de d\'{e}formation infinit\'{e}simale}
\kwd{Collin\'{e}ations}\kwd{Contraintes membraires}
\kwd{Fonction de contrainte d'Airy}
\kwd{\'{E}quation de Pucher}}

\thanks{Ministry of Innovation and Technology of Hungary (PD
142720)} 

\maketitle

\end{noXML}    

\newcommand{\scal}[2]{\langle #1,#2\rangle}
\def\eqref#1{(\ref{#1})}

\section{Introduction}

When giving the homogeneous coordinate description of projective
lines~\cite{plucker1868neue}, Pl\"ucker was heavily infuenced by how he
viewed mechanics~\cite{plucker_mechanics}.  This lead to a view of
mechanics that had a lasting influence on the description of rigid body
dynamics~\cite{davidson2004robots,gallardo2016kinematic} or elastic
beams~\cite{SORGONA2024112847}. Here we show how to embed problems of
continuum mechanics into this framework, which currently seems to be
missing from the theory of mechanics. As finding analytical solutions
to continuum problems is in general not easy, the possibility to solve
problems by transformation~\cite{rankine1857transformations} has been
thoroughly investigated and is worked out for the case of affine
transformations~\cite{affinetransformations_2006_Ostrosablin}. This
paper extends this possibility to include collineations (more general
projective transformations). As global material orthotropy and thus
isotropy will turn out not to be projective invariants, the use of
collineations is mechanically feasible if the stress distribution is
not influenced by the material properties (the structure is statically
determinate). Membrane shells are often made statically determinate and
as such the paper concludes with the description of how the graphs of
the membrane surface and the Airy stress function need to be
manipulated if one wishes to rely on Pucher's
equation~\cite{pucher1934spannungszustand},  an often used tool in
engineering literature.

\section{Preliminaries}\label{sec:2}

We will use the known coordinate
description~\cite{pottmann2001computational} of real $n$ dimensional
projective space (here $\mathcal{P}^n$), which we briefly recall here
for the engineering audience. The coordinate description identifies $m$
dimensional projective subspaces of $\mathcal{P}^n$ with $m+1$
dimensional linear subspaces of $\mathbb{R}^{n+1}$. Formally one can
introduce the equivalence relation $\mathbi{x} \sim \mathbi{y} \iff 
\mathbi{x}=\lambda\mathbi{y}$ ($\mathbi{x},\mathbi{y}\in \mathbb{R}^{n+1},
\lambda\in \mathbb{R}$) and have points of $\mathcal{P}^n$ as
equivalence classes of $\mathbb{R}^{n+1} \setminus \{ 0 \}$ with
respect to $\sim$. These equivalence classes in themselves capture
incidence properties, to be able to handle the metric nature of
elasticity we have to fix preferred (appropriately chosen)
representants. Let us denote $\mathbb{R}^{n+1}$ with $\mathbb{V}$  for
simplicity, chose an orthonormal base $\{\mathbi{e}_i\}$ in it ($i \in
\{0,1,\ldots,n \}$) and denote the usual inner product in it with
$\scal{ \ }{ \ }$. Our preferred choice of representants will be such
that a finite (not at infinity) point $p_f$ should be represented with
vector $\mathbi{p}_f$ satisfying $\scal{\mathbi{p}_f}{\mathbi{e}_0}=1$ while
ideal (at infinity) point $p_i$ with vector $\mathbi{p}_i$ satisfying
$\scal{\mathbi{p}_i}{\mathbi{e}_0}=0$. We may represent an $m-1$ dimensional
projective subspace spanned by points $q_i$ ($i \in \{1,\ldots,m\}$)
through the exterior product as $ \mathbi{q}_1 \wedge,\ldots,\wedge
\mathbi{q}_m \in \bigwedge^m\mathbb{V}$. The space $\bigwedge^m\mathbb{V}$
is a vector space, it has an orthonormal base determined by
$\{\mathbi{e}_i\}$ that is spanned by $\{\mathbi{e}_{i_1}\wedge \cdots \wedge
\mathbi{e}_{i_m} \ \vert \ 0\leq i_1 &lt;\cdots &lt; i_m\leq n \}$. To simplify
notation we will omit the wedges and the multiple $\mathbi{e}$-s, for
instance $\mathbi{e}_{i}\wedge\mathbi{e}_{j} \wedge \mathbi{e}_{k}$ will be
written as $\mathbi{e}_{ijk}$ ($i&lt;j&lt;k$). In some cases it will be shorter
to write the terms missing, such that $\mathbi{e}_{\widehat{i_1\ldots
i_j}}$ will contain all the base vectors in lexicographic order, whose
indices are missing from the list $i_i \ldots i_j$.

A dual description of projective subspaces is through linear
functionals ${\bpsi}\in \mathbb{V}^\vee$, such that ${\bpsi}$
represents a projective hyperplane, containing all the points $p$
having representants satisfying ${\bpsi}(\mathbi{p})=0$. Lower
dimensional subspaces may also be represented through the exterior
algebra of $\mathbb{V}^\vee$: given hyperplanes represented by
${\bpsi}_i$, their intersection is represented by $ {\bpsi}_1
\wedge,\ldots,\wedge {\bpsi}_m \in \bigwedge^m\mathbb{V}^\vee$, which
is an $n-m$ dimensional subspace. The space
$\bigwedge^m\mathbb{V}^\vee$ is also a vector space. Denoting the dual
base of $\mathbb{V}^\vee$ with $\{{\balpha}_i\}$ such that
${\balpha}_i(\mathbi{e}_j)=\delta_{ij}$ (the Kronecker-delta), the
corresponding orthonormal base in $\bigwedge^m\mathbb{V}^\vee$ is
$\{{\balpha}_{i_1 \ldots i_m} \ \vert \ 0\leq i_1 &lt;\cdots &lt; i_m\leq n
\}$.

To transition between the primal and dual representations of forces and
displacements we combine the Hodge-star operator with the isomorphism
${\balpha}_i \leftrightarrow \mathbi{e}_i$ giving the map
{\begin{equation}
\dag : \bigwedge_{i\in \mathcal{I}} \mathbi{e}_i \mapsto (-1)^t
\bigwedge_{j\in \mathcal{J}} {\balpha}_j \label{eq:transition}
\end{equation}}\unskip
where $\{0, \ldots, n \}$ is the disjoint union of $\mathcal{I}$ and
$\mathcal{J}$ and $t$ is the number of transpositions required to bring
$\{\mathcal{I},\mathcal{J}\}$ to $(1 \ldots n)$. With some possible
abuse of notation we will use $\dag$ both in the $\bigwedge^m
\mathbb{V} \to \bigwedge^{n+1-m} \mathbb{V}^\vee$ and
$\bigwedge^m\mathbb{V}^\vee \to \bigwedge^{n+1-m} \mathbb{V}$ sense.

The inner product of $\mathbi{x},\mathbi{y} \in \bigwedge^m\mathbb{V}$ is
defined through the Hodge-star as
{\begin{equation}
\scal{\mathbi{x}}{\mathbi{y}}:=\star(\mathbi{x} \wedge \star
\mathbi{y})=\star(\star\mathbi{x} \wedge  \mathbi{y}).
\end{equation}}\unskip
This can be used to express the evaluation rule of ${\bchi} \in
\bigwedge^m \mathbb{V}^{\vee}$ as
{\begin{equation}
{\bchi}(\mathbi{y})=\scal{\mathbi{x}}{\mathbi{y}} \quad \text{where } \mathbi{x}=\sum
x_{i_1 \ldots i_m} \mathbi{e}_{i_1 \ldots i_m} \text{ and } {\bchi}=\sum
x_{i_1 \ldots i_m} {\balpha}_{i_1 \ldots i_m}
\end{equation}}\unskip
holds.   


The primal description lends itself to naturally represent forces: what
would be force $\underline{\mathbi{F}}$ acting at point
$\underline{\mathbi{x}}$ ($\underline{\mathbi{x}},\underline{\mathbi{F}}\in
\mathbb{R}^n$), gets expressed as 
$\mathbi{f}=(1,\underline{\mathbi{x}})\wedge(0,\underline{\mathbi{F}})$ doubling
as projective line coordinates for the line of action of the force.
Pure moments can be thought of as having lines of action at infinity.
Infinitesimal motions are dual vectors to forces, given small
displacement ${\bphi} \in \bigwedge^2\mathbb{V}^\vee $ the work of
$\mathbi{f}$ on ${\bphi}$ is ${\bphi}(\mathbi{f})$. If ${\bphi}$ is
a pure rotation it can be written as a product ${\bphi}={\bchi}_1
\wedge{\bchi}_2$ if it is a pure translation as
${\bphi}={\balpha}_0 \wedge{\bchi}$ having the axis
(point-wise fix subspace) of rotation lying in the ideal hyperplane
$\alpha_0$. 

Under projective transformations of $\mathcal{P}^n$ we will mean
collineations. They are incidence preserving maps, that is to say given
line $l$, point $p$ and their images $l'$ and $p'$; $p'$ lies on $l'$
exactly if $p$ lies on $l$. They correspond to equivalence classes of
invertible $(n+1) \times (n+1)$ matrices under equivalence relation
$\sim$. In what follows we choose a representant matrix $\mathbi{T}$ from
the appropriate equivalence class and treat it as given. We treat every
vector as a column vector and multiply accordingly. Point representants
$\mathbi{p}$ and $\mathbi{p}'$ satisfy
{\begin{equation}
\mathbi{p}' \sim \mathbi{T}\mathbi{p}.
\end{equation}}\unskip
The effect of $\mathbi{T}$ on $\bigwedge^m \mathbb{V}$ is given by the
$m$th cofactor matrix~\cite{prasolov1994problems} $C_m\mathbi{T}$ as
{\begin{equation}
(\mathbi{T} \mathbi{p}_1 \wedge \cdots \wedge \mathbi{T}
\mathbi{p}_m)=C_m\mathbi{T}(\mathbi{p}_1 \wedge \cdots \wedge
\mathbi{p}_m )
\end{equation}}\unskip
implying 
{\begin{equation}
(\mathbi{p}_1 \wedge \cdots \wedge \mathbi{p}_m )' \sim
C_m\mathbi{T}(\mathbi{p}_1 \wedge \cdots \wedge \mathbi{p}_m ).
\end{equation}}\unskip
On functionals ${\bchi} \in \bigwedge^m \mathbb{V}^{\vee}$ we will
use the transpose of the $m$th adjugate~\cite{aitken2017determinants}
$\text{adj}_m\mathbi{T}^T$  to describe transformations which we will
write as
{\begin{equation}
{\bchi}' \sim \text{adj}_m\mathbi{T}^T {\bchi}
\end{equation}}\unskip
for short. This preserves all incidences due to
{\begin{equation}
{\bchi}'(\mathbi{y}')=\scal{\mathbi{x}'}{\mathbi{y}'}=\scal{\text{adj}_m
\mathbi{T}^T\mathbi{x}}{C_m\mathbi{T}\mathbi{y}}=\scal{\mathbi{x}}{\text{adj}_m\mathbi{T}
C_m\mathbi{T}\mathbi{y}}=|\mathbi{T}|\scal{\mathbi{x}}{\mathbi{y}}\label{eq:adj_incidnce}
\end{equation}}\unskip
which can only be $0$ if $\scal{\mathbi{x}}{\mathbi{y}}=0$ ($|\mathbi{T}|$
denotes the determinant of $\mathbi{T}$). When transforming forces and
infinitesimal rotations we have to be stricter than equivalence
relations in order to preserve static equilibrium and kinematic
compatibility. We have:
{\begin{eqnarray}
&amp;\displaystyle \mathbi{f}\mapsto  \lambda_f C_2\mathbi{T} \mathbi{f} \label{eq:ftrafo} \Seqnsplit
&amp;\displaystyle {\bphi}\mapsto  \lambda_{\phi} \text{adj}_2\mathbi{T}^{T} {\bphi} 
\label{eq:phitrafo}
\end{eqnarray}}\unskip
where the scalars $\lambda_f$ and $\lambda_{\phi}$ account for
$\mathbi{T}$ being an arbitrary (fixed) representant. We stress that these
transformations are known to preserve static equilibrium and kinematic
compatibility. The work of $\mathbi{f}$ on ${\bphi}$ transforms as
{\begin{equation}
{\bphi}(\mathbi{f})\mapsto \lambda_f \lambda_{\phi} |\mathbi{T}|
{\bphi}(\mathbi{f}).\label{eq:worktrafo}
\end{equation}}\unskip

In order for the transformations to make mechanical sense we will
restrict our investigation to the case where both the domain of the
elasticity problem and its image is in the Euclidean part of the
projective plane; that is no part of the domain is mapped to a point at
infinity.

\section{Algebraic description of stress and strain tensors}

The theory of elasticity~\cite{sadd_elasticity} establishes the stress
and (infinitesimal) strain tensors as linear maps. We will take these
linear maps as granted and incorporate them into the multilinear
algebraic description outlined above. 



\subsection{Stress tensor}

Let us denote the matrix representation of the symmetric Cauchy stress
tensor with $[\bar{\sigma}_{i,j}]$ with indices $i,j \in \{ 1,\ldots,
n\}$. In the traditional engineering approach this represents a map
$(\mathbb{R}^n)^{\vee} \to \mathbb{R}^n$. The fact that we
want to describe an $n$-dimensional problem in $n+1$ coordinates means
we will need a larger matrix ${\bsigma}=[\sigma_{i,j}]$ with indices
$i,j \in \{ 0,\ldots, n\}$ such that
$[\sigma_{i,j}]=[\bar{\sigma}_{i,j}]$ if $i \neq 0 \neq j$ and 
$[\sigma_{i,j}]=0$ otherwise. We interpret this matrix as it was an
$n+1$ dimensional stress matrix for a thin piece of $n+1$ dimensional
material between planes ${\balpha}_0(\mathbi{v})=1$ and
${\balpha}_0(\mathbi{v})=1+\mathrm{d}t$.

We may describe stresses at point $p$ in the projective problem by
cutting the material in half, considering a small patch of the cut
surface at $p$ and the resultant of the force system acting on that
patch. What appears in the projective setting as an $(n-1)$ simplicial
patch with vertices $p$ and $q_i$ ($i \in \{1,2,\ldots,n-1\}$) is in
the affine $n+1$ dimensional setting an $n$ dimensional polytope, with
vertices
$\mathbi{p},\mathbi{q}_i,(1+\mathrm{d}t)\mathbi{p},(1+\mathrm{d}t)\mathbi{q}_i$ (the
projective planar case is drawn in Figure~\ref{fig:d}). As in the
projective problem $q_i \to p$ such that $\mathrm{d}t$ and all angles
of the (simplex) at $p$ stay constant, the $n$ dimensional polytope
tends to a prism having the simplex as a base. Observing that
{\begin{equation}
{\balpha}_0(\mathbi{q}_i-\mathbi{p})=0 \implies  {\balpha}_0 \wedge
{\balpha}_{i_2} \wedge \cdots \wedge {\balpha}_{i_{n-1}}
((\mathbi{q}_1-\mathbi{p})\wedge,\ldots,\wedge(\mathbi{q}_{n-1}-\mathbi{p}))=0
\end{equation}}\unskip
holds for all functionals starting with ${\balpha}_{\mathbf{0}}$ we conclude
that the first $n$ coordinates of the multivector
{\begin{equation}
\frac{1}{(n-1)!}
\mathrm{d}t\mathbi{p}\wedge((\mathbi{q}_1-\mathbi{p})\wedge,\ldots,\wedge(\mathbi{q}_{n-1}-\mathbi{p}))= 
\frac{1}{(n-1)!}\mathrm{d}t\mathbi{p}\wedge\mathbi{q}_1\wedge,\ldots,\wedge\mathbi{q}_{n-1}
\end{equation}}\unskip
contain the $n$-volumes of the faces of the prism that stand on the
faces of the simplex. As such by introducing the dual vector
{\begin{equation}
{\bnu}:=\frac{1}{(n-1)!}\dag(\mathrm{d}t\mathbi{p}\wedge\mathbi{q}_1\wedge,\ldots,\wedge
\mathbi{q}_{n-1}) \label{eq:nudef}
\end{equation}}\unskip
one can approximate the contact force on the patch to arbitrary
precision with $\mathrm{d}\mathbi{F}\approx{\bsigma}{\bnu}$ as $q_i \to
p$. Although $\mathrm{d}\mathbi{F}$ acts at the centroid of the contact
patch and not in $p$, when writing up the moment with respect to the
origin the effect of the difference in point of action will be
quadratic in the sides of the parallelepiped and thus negligible
compared to the term $\mathbi{p}\wedge \mathrm{d} \mathbi{F}$. As such
for the moment of the stresses we keep only the term $\mathbi{p}\wedge
\mathrm{d} \mathbi{F}$ which we calculate through
matrix-multiplication. There is a unique matrix
$[\mathbi{p}\boldsymbol{\wedge}]$, such that $\mathbi{p}
\boldsymbol{\wedge}
\mathbi{x}=[\mathbi{p}\boldsymbol{\wedge}]\mathbi{x}$ for all 
$\mathbi{x} \in \bigwedge^m\mathbb{V}$ (for a fixed $m$). We will also
use the transpose of this matrix acting on dual coordinates as
$[\mathbi{p}\boldsymbol{\wedge}]^T: \ \bigwedge^{2}\mathbb{V}^\vee \to
\bigwedge\mathbb{V}^\vee $. With it have  

\begin{Lemma}\label{lem:1}
$\dag (\mathbi{p}\wedge\mathbi{q}_1\wedge,\ldots,\wedge\mathbi{q}_{n-1})=(-1)^{n+1}
[\mathbi{p}\wedge]^T
\dag(\mathbi{q}_1\wedge,\ldots,\wedge\mathbi{q}_{n-1})$.
\end{Lemma}
The proof of which is omitted due to being technical and straightforward.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:d}For the projective planar case we consider a 3 dimensional
solid with thickness $\mathrm{d}t\ll 1$. What appears as a cut along a line
segment in the projective problem is a trapezoid cut surface in the
higher dimensional affine problem.}
\end{figure}

We define the projective stress tensor (in matrix form, corresponding to point $p$) as
{\begin{equation}
{\bSigma} := 
\mathrm{d}t(-1)^{n+1}[\mathbi{p}{\boldsymbol{\wedge}}]
{\bsigma}[\mathbi{p}{\boldsymbol{\wedge}}]^T
\end{equation}}\unskip
where $\mathrm{d}t=1$ may be used for practical purposes. The stress matrix has
a structure and can be thought of as a block matrix, namely
{\begin{equation}
{\bSigma} =
\begin{bmatrix}
[\overline{\sigma}_{i,j}] &amp; {\bmu}^T \\
{\bmu} &amp;  {\bzeta}
\end{bmatrix}.
\end{equation}}\unskip
To see this, recall that ${\bSigma}={\bSigma}^T$ and it represents a
map acting on dual line coordinates returning primal line coordinates.
Thus for any pair of base vectors of the form
${\mathbi{e}}_0\wedge\mathbi{e}_i$ and
$\mathbi{e}_0\wedge{\mathbi{e}}_j$ the expression  
$\scal{\mathbi{e}_0\wedge\mathbi{e}_i}{{\bSigma} \dag
(\mathbi{e}_0\wedge\mathbi{e}_j})$ is the
$\mathbi{e}_{\mathbi{i}}$ directional component of
$\mathrm{d}\mathbi{F}$ if the cut is orthogonal to $\mathbi{e}_j$ and
has unit $n-1$ volume, explaining the $[\overline{\sigma}_{i,j}]$
block. The block ${\bmu}$ represents the moments of the stresses and
its rows are linear combinations of the rows of
$[\overline{\sigma}_{i,j}]$ according to how moments are defined. Due
to the incidence condition $\mathbi{p} \wedge ( \mathbi{p} \wedge
\mathrm{d}\mathbi{F} )=\mathbf{0}$ if $[\overline{\sigma}_{i,j}]$ is
non-degenerate the rank of ${\bSigma}$ has to be $n$, thus the rows of
${\bzeta}$ are linear combinations of the rows of ${\bmu}^T$ with the
same coefficients as in the case of $[\overline{\sigma}_{i,j}]$ and
${\bmu}$.


In order to derive how the stress matrix transforms under collineations
we must keep in mind that
{\begin{equation}
\mathbi{q}_i \mapsto \mathbi{q}'_i=\frac{1}{{\balpha}_0(\mathbi{T}\mathbi{q}_i)} \mathbi{T}\mathbi{q}_i
\end{equation}}\unskip 
is a non-linear map due to the term in the denominator. The fact that
no point of the mechanics problem is mapped into infinity ensures that
${\balpha}_0 (\mathbi{T}\mathbi{q}_i ) \neq 0 \neq
{\balpha}_0 (\mathbi{T}\mathbi{p} )$ holds. Introducing
notation $\lambda_{q_i}={\balpha}_0(\mathbi{T}\mathbi{q}_i)$ one may
arrive at 
{\begin{equation}
{\bSigma}'=\frac{\lambda_f \Pi_{i=1}^{n-1}
\lambda_{q_i}}{|\mathbi{T}|}C_2\mathbi{T}{\bSigma}C_2\mathbi{T}^T 
\end{equation}}\unskip
by relying on the usual commutativity requirement of tensor
transformations and identities of the cofactor and adjugate matrices.
This equation cannot hold in this form, since it would make the
transformation rule dependant on the simplex. However, since we are
interested in the limiting case $q_i \to p$ only, we may prescribe
{\begin{equation}
{\bSigma}'=\frac{\lambda_f \lambda_p^{(n-1)}}{|\mathbi{T}|}C_2\mathbi{T}
{\bSigma}C_2\mathbi{T}^T \label{eq:fesztrafo}
\end{equation}}\unskip
as the transformation rule for the stress matrix. Since this is a
continuous map (on the domain of the problem) and we merely embedded
the usual stress matrix in a larger matrix, static equilibrium is
preserved and the transformed projective stress matrix is a good
description of the stresses of the transformed problem. 

\subsection{Strain tensor}

Strains of the material correspond to the relative translations of the
two endpoints $p$ and $q$ of an infinitesimally short chord of the
material. The usual projective treatment of translations is that they
are rotations around $n-2$ dimensional subspaces at infinity (lying in
the ideal hyperplane). However, if we are only interested in the
displacement of a single point, multiple infinitesimal rotations can
cause the same displacement of $q$ relative to $p$. We will have to use
this property, otherwise we would only be able to describe affine
transformations leaving the ideal hyperplane fixed. In general the
strain tensor corresponding to point $p$ will give rotations the axis
of which is lying in a specific hyperplane $\psi_{\mathrm{ref}}$ that does not
contain point $p$. We will start by introducing the strain tensor in
the traditional setting where ${\bpsi}_{\mathrm{ref}}={\balpha}_0$ then
give a method to change the reference hyperplane.


Let $[\bar{\epsilon}_{i,j}]$ denote the $n$ dimensional engineering
strain tensor at point $p$ and $[\epsilon_{i,j}]$ the $n+1$ dimensional
one including extra zeroes, similar to the case of stresses. If the two
endpoints of an infinitesimally short chord of the material are $p$ and
$q$ (with properly chosen representants $\mathbi{p}$ and $\mathbi{q}$), we
may approximate their relative displacement
$(0,\underline{\bDelta})$ to arbitrary precision with
$[\epsilon_{i,j}](\mathbi{q}-\mathbi{p})$. If the reference-hyperplane is the
hyperplane at infinity, the translation may be expressed as
$\mathrm{d}{\bphi}={\balpha}_0 \wedge [\epsilon_{i,j}](\mathbi{q}-\mathbi{p})$.
Motivated by this we define the projective strain matrix as
{\begin{equation}
\mathbi{E}^{{\balpha}_{\mathbf{0}}}:=[{\balpha}_{\mathbf{0}}
\boldsymbol{\wedge}] [\epsilon_{i,j}][{\balpha}_{\mathbf{0}}\boldsymbol{\wedge}]^T
\end{equation}}\unskip
that will act on primal line coordinates in the form of $\mathbi{p}\wedge
\mathbi{q}$ and the upper index denotes the reference hyperplane. To
justify this we may observe that the only non-zero elements of
$[{\balpha}_{\mathbf{0}}\boldsymbol{\wedge}]^T$ are those with row index $j$ and column
index $0j$ where $j\in\{1,\ldots,n\}$ and all these entries have value
1. Thus we have $(\mathbi{q}-\mathbi{p})=[{\balpha}_{\mathbf{0}}\boldsymbol{\wedge}]^T
(\mathbi{p}\wedge \mathbi{q})$. If the reference hyperplane is the ideal
hyperplane the strain matrix has the block structure
{\begin{equation}
\mathbi{E}^{{\balpha}_{\mathbf{0}}}=
\begin{bmatrix}
[\overline{\epsilon}_{i,j}] &amp; \mathbf{0}_{\mathbi{n} \times \binom{\mathbi{n}}{\mathbf{2}}} \\
\mathbf{0}_{\binom{\mathbi{n}}{\mathbf{2}} \times \mathbi{n}} &amp;  
\mathbf{0}_{\binom{\mathbi{n}}{\mathbf{2}} \times \binom{\mathbi{n}}{\mathbf{2}}}
\end{bmatrix}
\end{equation}}\unskip 
which can be seen through a similar reasoning as in the case of
${\bSigma}$. To describe the change of reference-hyperplane we will
need the following observation:


\begin{Lemma}\label{lem:2}
If $\psi_{\mathrm{ref}}$ is not the ideal hyperplane there is a linear map $L: \
\mathbb{V}^\vee \rightarrow \mathbb{V}^\vee$ describing the change of
reference-hyperplane, corresponding to a central--axial collineation of
$\mathcal{P}^n$ with center $p$.
\end{Lemma}


\begin{proof}
Since $p$ does not lie on $\psi_{\mathrm{ref}}$ we may chose a representant
satisfying ${\bpsi}_{\mathrm{ref}}(\mathbi{p})=1$ and a generator-system $\{
{\balpha}_0, {\bpsi}_i \}$ with $i\in \{1,\ldots,n\}$ such that
${\bpsi}_i(\mathbi{p})=0$ holds. The map is determined by  equations
{\begin{equation}
\mathbi{L}{\balpha}_0  =   {\bpsi}_{\mathrm{ref}} \quad \text{ and  } \quad
\mathbi{L}{\bpsi}_i = {\bpsi}_i
\end{equation}}\unskip
where $\mathbi{L}$ represents a central axial collineation since all
hyperplanes incident with point $p$ are fixed. Now consider point $p$
(with representant  $\mathbi{p}=(1,\underline{\mathbi{p}})$) undergoing
translation $\underline{\bDelta}\in \mathbb{R}^n$. Taking the
hyperplane at infinity as reference hyperplane this motion can be
represented as ${\balpha}_0\wedge {\bDelta} $ where
${\bDelta}=(-\scal{\underline{{\bDelta}}}{\underline{\mathbi{p}}},
\underline{\bDelta})$ represents a hyperplane incident with $p$. 
This can be seen by choosing unit vector $\underline{\mathbi{u}}\in
\mathbb{R}^n$ and the corresponding ideal point
$\mathbi{u}=(0,\underline{\mathbi{u}})$ and calculating the $\mathbi{u}$
directional component of the translation as 
{\begin{equation}
({\balpha}_0\wedge {\bDelta})(\mathbi{p}\wedge \mathbi{u})=
\begin{vmatrix}
{\balpha}_0(\mathbi{p}) &amp; {\balpha}_0(\mathbi{u}) \\
{\bDelta}(\mathbi{p}) &amp; {\bDelta}(\mathbi{u})
\end{vmatrix}
={\balpha}_0(\mathbi{p}) {\bDelta}(\mathbi{u})=\scal{\underline{{\bDelta}}}
{\underline{\mathbi{u}}}
\end{equation}}\unskip
(recall that ${\bDelta}(\mathbi{p})=0$). The image of this motion under
$L$ is the rotation ${\bpsi}_{\mathrm{ref}} \wedge {\bDelta} $
which causes the same translational motion at $p$ since 
{\begin{equation}
({\bpsi}_{\mathrm{ref}}\wedge {\bDelta})(\mathbi{p}\wedge \mathbi{u})=
\begin{vmatrix}
{\bpsi}_{\mathrm{ref}}(\mathbi{p}) &amp; {\bpsi}_{\mathrm{ref}}(\mathbi{u}) \\
{\bDelta}(\mathbi{p}) &amp; {\bDelta}(\mathbi{u})
\end{vmatrix}
={\bpsi}_{\mathrm{ref}}(\mathbi{p}) {\bDelta}(\mathbi{u})=
\scal{\underline{{\bDelta}}}{\underline{\mathbi{u}}}
\end{equation}}\unskip
(recall that ${\bpsi}_{\mathrm{ref}}(\mathbi{p})=1$). Thus given any
$\Psi_{\mathrm{ref}}$ and any displacement having axis incident with it
the inverse of $L$ maps the displacement to the ideal hyperplane while
preserving its effect of point $p$, which completes the proof.
\end{proof}

Relying on this one can use the
{\begin{equation}
\mathbi{E}^{\mathbi{L}{\balpha}_{\mathbf{0}}}=C_2 \mathbi{L}
\mathbi{E}^{{\balpha}_{\mathbf{0}}} C_2 \mathbi{L}^T
\end{equation}}\unskip
transformation rule to describe the change of reference-hyperplane. 

The transformation rule of the strain tensor under collineations may be
derived similarly to the case of the stress tensor to be
{\begin{equation}
\mathbi{E}'=\frac{\lambda_{\phi}
\lambda_p^{2}}{|\mathbi{T}|}\text{adj}_2\mathbi{T}^T\mathbi{E}\text{adj}_2\mathbi{T}. 
\label{eq:epstrafo}
\end{equation}}\unskip
Again, as we merely embedded the strain matrix in a larger matrix and
the map is continuous on the domain of the continuum problem the
transformed matrix is a good description of the transformed problem and
kinematic compatibility is preserved.

{\vspace*{-2pt}}

\subsection{Work}

{\vspace*{-2pt}}

The work of stresses on strains may be expressed from the matrices
above through the Hilbert--Schmidt inner product as
{\begin{equation}
W=\text{trace}(\mathbi{E}^T {\bSigma}) \label{eq:workdef}
\end{equation}}\unskip
which does not depend on the choice of $\psi_{\mathrm{ref}}$. To see
the independence we recall that $\mathbi{L}$ was created not to alter the
effect of infinitesimal rotation $\mathrm{d}{\bphi}$ on lines of the shape
$\mathbi{p}\wedge \mathbi{q}$. That is for any $\mathrm{d}{\bphi}$:
{\begin{equation}
\scal{\mathrm{d}{\bphi}}{\mathbi{p}\wedge \mathbi{q}}=
\scal{C_2\mathbi{T}\mathrm{d}{\bphi}}{\mathbi{p}\wedge \mathbi{q}}=
\scal{\mathrm{d}{\bphi}}{C_2 \mathbi{L}^T(\mathbi{p}\wedge \mathbi{q})}
\end{equation}}\unskip  
holds, implying $C_2 \mathbi{L}^T(\mathbi{p}\wedge \mathbi{q})=\mathbi{p}\wedge
\mathbi{q}$. As the columns of ${\bSigma}$ are exactly of shape
$\mathbi{p}\wedge \mathbi{q}$ we have $C_2 \mathbi{L}^T {\bSigma}
={\bSigma}$. Similarly, as each entry of $\mathbi{E}^T {\bSigma}$ is
a inner product of shape $\scal{\mathrm{d}{\bphi}}{\mathbi{p}\wedge \mathbi{q}}$ we
have  $C_2\mathbi{L}{\mathbi{E}^{{\balpha}_{\mathbf{0}}}}^T
{\bSigma}={\mathbi{E}^{{\balpha}_{\mathbf{0}}}}^T {\bSigma}$. Thus
$C_2\mathbi{L}{\mathbi{E}^{{\balpha}_{\mathbf{0}}}}^T C_2\mathbi{L}^T
{\bSigma}={\mathbi{E}^{{\balpha}_{\mathbf{0}}}}^T  {\bSigma}$ and the
expression of the work is indeed reference-hyperplane independent. The
correctness of Equation~(\ref{eq:workdef}) may be seen from the block
structure of the matrices, considering the form where the
reference-hyperplane is the hyperplane at infinity. 


Under projective transformations the work of the transformed stresses
on the transformed strains may be expressed from the original as
{\begin{equation}
\begin{array}{l}
\displaystyle W'=\frac{\lambda_f \lambda_p^{(n-1)}}{|T|} 
\frac{\lambda_{\phi} \lambda_p^{2}}{|T|}\text{trace}
((C_2\mathbi{T}{\bSigma}C_2\mathbi{T}^T )^T 
\text{adj}_2\mathbi{T}^T\mathbi{E}\text{adj}_2\mathbi{T})\\[0.7pc]
\displaystyle W'=\frac{\lambda_f \lambda_{\phi}\lambda_p^{(n+1)}}{|T|^2}|T|^2
\text{trace}(C_2\mathbi{T}{\bSigma}^T\mathbi{E}C_2\mathbi{T}^{{-}1})\\[0.7pc]
\displaystyle W'=\lambda_f \lambda_{\phi}\lambda_p^{(n+1)}W.
\end{array}
\end{equation}}\unskip

In the special case of affine transformations it is possible to chose
$\mathbi{T}$ such that $\lambda_p=1$ for all $p$ and if
$\lambda_{\phi}\lambda_{f}=1$ then the (point-wise) work is invariant,
returning the previous result of
Ostrosablin~\cite{affinetransformations_2006_Ostrosablin}.

{\vspace*{-2pt}}

\section{Invariant mechanical properties}

{\vspace*{-2pt}}

As projective transformations are defined to preserve incidences we
investigate what mechanical properties are related to incidence. The
stress tensor maps vectors representing $n-2$ dimensional projective
subspaces to vectors representing lines; while the strain tensor maps
vectors representing lines to vectors representing $n-2$ dimensional
projective subspaces. If the projective subspaces represented by the
starting and the image vectors intersect, the corresponding mechanical
phenomenon is that there is no normal stress on the cut determined by
the $n-1$ dimensional subspace and there is no normal strain in the
direction determined by the line. We may also note that the kernels of
the tensors correspond to projective subspaces and thus we have

\begin{corollary}\label{thm:cor0}
At point $p$ the absence of (normal) stresses/strains  along a
hyperplane/line containing $p$ is a projective invariant.
\end{corollary}

As for the material properties, we have a negative result. Practical
problems are often isotropic or at least globally orthotropic. (Under
global we mean that the material is orthotropic such that its
eigen-directions are the same in all of its points.)  

\begin{proposition}
Global orthotropy and thus isometry are not projective invariants. 
\end{proposition}

\begin{proof}
A planar construction (shown in Figure~\ref{fig:geogebra}) is given
showing the proposition, which implies the higher dimensional cases as
well. Consider a planar central--axial collineation with axis $a$,
center $c$, point $p$ and its image $p'$. If global orthotropy was a
projective planar invariant we would have, for any two orthogonal lines
$d_1$ and $d_2$ passing through $p$ the following: If $p$ is in
uniaxial stress state such that the stress is acting in the direction
$d_1$, then the relative motion of the endpoints of a $d_1$ directional
chord at $p$ would be a rotation around a point on line $d_2$ and the
relative motion of the endpoints of a $d_2$ directional chord would be
a rotation around a point lying on $d_1$; furthermore the same would
hold for the transformed lines $d'_1$ and $ d'_2$. If $p$ is in general
position lines $d_i$ and $d'_i$ ($i \in \{1,2\}$) intersect at points
$t_i$ on $a$. Due to the orthogonality conditions both $p$ and $p'$
must lie on a Thales circle having diameter $t_1 t_2$. Since $a$, $p$
and $p'$ uniquely determine the circle the orthotropy of the material
is preserved for exactly one pair of lines $d_1,d_2$ and crucially for
the proposition the directions of these lines depend on $p$ and are not
the same on the whole projective plane.
\end{proof}

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:geogebra}Construction finding the directions in which
ortothropy is preserved. Transforming an isotropic material by a
central--axial collineation determined by axis $a$, center $c$, point
$p$ and its image $p'$, there is a single pair of orthogonal directions
$d_1$ and $d_2$ that are mapped to eigen-directions of the material of
the transformed problem.}
\end{figure}


Due to this negative result the explicit transformation rule of the
material properties is not pursued. Instead the paper focuses on a type
of problem where the material properties do not influence the stress
distribution, the problem of statically determinate membrane shells. 


\section{Membrane shells}

Membrane theory describes thin shells based on the assumption that the
shell is thin enough to neglect its resistance against bending and
torsion and the effect of stresses in the shell may be approximated by
an internal force system in the plane of the middle-surface of the
shell~\cite{csonka}. These problems often admit a description where the
middle surface is given as the graph of the height function $H(p)$
above a horizontal plane. Let us embed our problem into $\mathcal{P}^3$
such that the horizontal reference plane is the projective plane
${\balpha}_3(\mathbi{p})=0$ and the graph of $H(p)$ constitutes of
points of the form
{\begin{equation}
(1,{\balpha}_1(\mathbi{p}),{\balpha}_2(\mathbi{p}),H(p) )^T.
\end{equation}}\unskip
Often the internal forces acting in the middle surface are also
projected to the ${\balpha}_3(\mathbi{p})=0$ plane where they become
stresses of a planar elasticity problem. If the geometry of the domain
is simply connected the stresses of this planar problem may be uniquely
represented through the Airy stress function~\cite{airy} $\Psi(p)$. 
Often the dominant load of the shell is $e_3$ directional and its
intensity is described with a function of the form $g(p)$. As
everything is parametrized through points in the
${\balpha}_3(\mathbi{p})=0$ plane, we will investigate transformations
leaving this plane invariant, as well as leaving the $e_3$ directional
lines $e_3$ directional. These maps have the form of
{\begin{equation}
\mathbi{T}=
\begin{bmatrix}
\mathbi{B} &amp; \mathbf{0}_{3 \times 1} \\
\mathbf{0}_{1 \times 3} &amp; \lambda_3
\end{bmatrix} \label{eq:inducedmap}
\end{equation}}\unskip
where $\lambda_3 \in \mathbb{R} \setminus \{0\}$ and the block
$\mathbi{B}$ is invertible. These maps will commute with projection in the
$e_3$ direction which we omit to show since it is technical and
straightforward. The difficulty in the use of these transformations in
the analysis of membrane shells is expected to come from the fact how
the load function transforms. Assuming that the load corresponds to a
stress state where the only non-zero component is the $e_3$ directional
normal stress with the intensity of the load, the transformation rule
may be expressed from Equation~(\ref{eq:fesztrafo}) as
{\begin{equation}
g'(p')=\frac{\lambda_3 \lambda_f}{|\mathbi{B}|}\lambda_p ^4 g(p)
\end{equation}}\unskip      
(We used the Symbolic Toolbox of MatLab). To some extent this
prescription on the load may be achieved by altering the thickness of
the shell, as noted by Dischinger~\cite{statischen_Massenausgleich}.

Puchers' equation~\cite{pucher1934spannungszustand} gives a
correspondence between functions $H(p)$, $\Psi(p)$ and the load, which
in this case simplifies into
{\begin{equation}
\frac{\partial^2 H}{\partial x_1^2}\frac{\partial^2 \Psi}{\partial
x_2^2}+\frac{\partial^2 H }{\partial x_2^2}\frac{\partial^2 \Psi
}{\partial x_1^2}-2\frac{\partial^2 H }{\partial x_1 \partial
x_2}\frac{\partial^2 \Psi }{\partial x_1 \partial x_2} +g=0
\end{equation}}\unskip
showing that the role of the shape and stress functions may be
interchanged. This leads to the question if it is possible to have
transformations acting point-wise on the graphs of $H(p)$ and $\Psi(p)$
preserving this correspondence? The answer is yes, but we should be
careful with the role of the multiplier $\lambda_f$. If we are given a
planar transformation with the corresponding $\mathbi{B}$ and $\lambda_f$
we can create an \emph{induced} map $\mathbi{B}^I$ by setting
$\lambda_3=|\mathbi{B}|/\lambda_f$ in \eqref{eq:inducedmap}. When applying
this, now projective 3 dimensional transformation we should remember
not to apply $\lambda_f$ again, and the load in this case will
transform as $g'(p')=\lambda_p ^4 g(p)$. This way we can rely on the
following result:
\begin{proposition}
The induced map maps the graph of the Airy stress function of a planar
elasticity problem to the graph of the Airy stress function of the
transformed elasticity problem.
\end{proposition}


\begin{proof}
The Airy stress function is known~\cite{phillips} to correspond to
moments as follows: Consider a planar curve $p(s)$ running in the
material. Denoting the resultant collected from $s=0$ acting on the
material on the left side of the curve (according to the direction
given by the parametrization $s$) with $\mathbi{f}(s)$, the value of the
Airy stress function is the moment of the resultant with respect to the
point on the curve (its work on a unit rotation orthogonal to the
plane):
{\begin{equation}
\Psi(p(s))=\dag(\mathbi{e}_{\mathbf{3}}\wedge \mathbi{p}) ( \mathbi{f}(s))=
\star(\mathbi{e}_{\mathbf{3}}\wedge \mathbi{p} \wedge \mathbi{f}(s))
\end{equation}}\unskip    
for all $s$. As the stress function is defined on the whole plane in
what follows we omit the arclength parametrization of the curve from
the notation. Relying on this mechanical interpretation and identities
of linear algebra we may arrive at $\Psi'(p')=\lambda_p^{-1} \lambda_f
|\mathbi{T}| \Psi(p)$. The graph of the stress function consists of points
having representants
$(1,{\balpha}_1(\mathbi{p}),{\balpha}_2(\mathbi{p}),\Psi(p))^T$ which transform into
{\begin{equation}
(\lambda_p,{\balpha}_1(\mathbi{B}^I\mathbi{p}),{\balpha}_2(\mathbi{B}^I\mathbi{p}),
|\mathbi{B}|\lambda_f\Psi(p))^T \sim (1,{\balpha}_1(\mathbi{p}'),
{\balpha}_2(\mathbi{p}'),\Psi'(p'))^T
\end{equation}}\unskip
completing the proof.
\end{proof}


\subsection{Example}

As an example consider functions $H=\Psi=- x_1^2 - x_2^2 + 1$ which is
a valid shape and stress function pair with load $g(x_1,x_2)=-8$ and
the stress tensor of the projected planar problem is of shape
{\begin{equation}
{\bSigma}=
\begin{bmatrix}
-2 &amp; 0 &amp; 2 x_2 \\
0 &amp; -2 &amp; -2 x_1\\
2 x_2 &amp; -2 x_1 &amp; - 2 x_1^2 - 2 x_2^2
\end{bmatrix}.
\end{equation}}\unskip

Consider map
{\begin{equation}
\mathbi{B}=
\begin{bmatrix}
2 &amp; 0 &amp; -1 \\
0 &amp; 2 &amp; 0\\
-1  &amp; 0 &amp; 2
\end{bmatrix}
\end{equation}}\unskip
with $\lambda_f =1$. The planar transformation maps the unit circle
into the ellipse with equation $3x_1^2  + 4x_2^2  = 4$ while the
induced map surface $H$ to $H'=-(9 x_1^2 + 12 x_2^2 - 12)/(2 x_2 + 4)$.
The induced map may be imagined as transforming the paraboloidal shell
over the unit circle to a skewed hyperboloidal shell over the ellipse
as shown in Figure \ref{fig:memb}. The transformed stress tensor of
the planar problem will be 
{\begin{equation} 
{\bSigma}'=
\begin{bmatrix}
-(9 x_1^{2} + 36)/(x_2 + 2)^3 &amp;  -(9 x_1)/(x_2 + 2)^2 &amp;  ({-}18 x_1^{2} + 
36 x_2)/(x_2 + 2)^3 \\
-(9 x_1)/(x_2 + 2)^2 &amp; {-}9/(x_2 + 2) &amp; -(18 x_1)/(x_2 + 2)^2 \\
(- 18 x_1^{2} + 36 x_2)/(x_2 + 2)^3 &amp; -(18 x_1)/(x_2 + 2)^2 &amp; 
-(36 x_1^{2} + 36 x_2^{2})/(x_2 + 2)^3
\end{bmatrix}
\end{equation}}\unskip
expressed in global coordinates, such that
$\bSigma' \dag (1,x_1,x_2)^T=\mathbf{0}$. The load to which solution
$\bSigma'$ corresponds is 
{\begin{equation}
g'=-648/(x_2 + 2)^4. \label{eq:g'}
\end{equation}}\unskip


\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:memb}Transformation of a membrane shell induced by
a planar projective transformation. Left: starting shape $x_3=- x_1^2 -
x_2^2 + 1$ drawn over the unit circle. Right: image shape $x_3=-(9
x_1^2 + 12 x_2^2 - 12)/(2 x_2 + 4)$ drawn over the ellipse $3x_1^2  +
4x_2^2  =4$.}
\end{figure}


\section{Conclusions}

The Cauchy stress and infinitesimal strain tensors have been embedded
in the coordinate description of projective geometry, such that they
may be transformed in a way that preserves static equilibrium and
kinematic compatibility. It was shown that in general these projective
transformations do not keep the isotropy or global orthotropy of the
material and thus if one wishes to keep the material law to be the same
in every point of the material, one may transform either the stresses
or strains and calculate the other. This makes this approach mostly
useful for statically determinate problems. As such a method has been
outlined to extend previously catalogued analytic solutions of
statically determinate membrane problems through collineations. As the
currently catalogued analytic solutions usually correspond to shapes
where the shape of the shell has a vertical axis of symmetry which
these induced maps do not keep vertical, this allows for the design of
membrane shells having a wider array of shapes than previously
catalogued analytic solutions.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Funding}

This work was supported by the Ministry of Innovation and Technology of
Hungary from the National Research, Development and Innovation Fund
under the PD 142720 funding scheme.

\section*{Acknowledgment}

The author is very grateful to Dr.\ Istv\'an Sajtos for the discussions
on the applicability of the theory.

\CDRGrant[ITM]{PD 142720}

\back{}

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\refinput{crmeca20250307-reference.tex}

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