# Metastability of a periodic network of threads:
what are the shapes of a knitted fabric ?

Jérôme Crassous  Univ Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR
6251, F-35000 Rennes, France<EMAIL_ADDRESS>Samuel Poincloux
Department of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe,
Sagamihara, Kanagawa 252-5258, Japan  Audrey Steinberger  Univ Lyon, ENS de
Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France

###### Abstract

Knitted fabrics are metamaterials with remarkable mechanical properties, such
as extreme deformability and multiple history-dependent rest shapes. This
letter shows that those properties may stem from a continuous set of
metastable states for a mechanically relaxed fabric, evidenced through
experiments, numerical simulations and analytical developments. Those states
arise from the frictional contact forces acting in the braid zone where the
threads interlace and follow a line in the configuration space accurately
described by a 2D-elastica model. The friction coefficient sets a terminal
point along this line, and the continuous set of relaxed states is obtained by
varying the braid inclination while contact forces remain on the friction
cone.

Assemblies of long, flexible, and intertwined fibers with frictional contacts
are materials involved in various phenomena, including surgical or shoe knots
johanns_strength_2023 ; daily_roles_2017 , nests and self-assembled natural
structures weiner_mechanics_2020 ; verhille.2017 , nonwoven fabrics with a
wealth of applications albrecht_nonwoven_2006 , or the degradation of ancient
manuscripts vibert_relationship_2024 . Despite being essential for most
applications, the mechanical response of fiber assemblies is intrinsically
non-linear, dissipative, and history-dependent, stemming from the fibers’
slenderness and the frictional contacts. Providing quantitative mechanical
predictions for the assembly from the properties of the fibers remains a
theoretical and numerical challenge, with recent advances made in simplified
situations with tight geometries poincloux.2021 ; seguin_twist_2022 ;
chopin.2024 . One particular class of ordered fiber assemblies, textiles, have
tremendous industrial importance in manufacturing long_design_2005  or geo-
engineering koerner_geotextiles_2016 . They also recently gained a renewed
interest as metamaterials with extensive programmability poincloux.2018b ;
singal_programming_2024  for emerging soft robotics and smart textile
applications chen_smart_2020 ; sanchez_textile_2021 . However, the prediction
of basic properties, like the rest shape of a knitted fabric given the length
by stitch of its constitutive yarn, is an old but still open question
munden_geometry_1959 ; lanarolle_geometry_2021  even though a reproducible
state can be achieved after repeated multidirectional stretching
allan_heap_prediction_1983 . One possible way of progress may emerge from
yarn-level simulation of knitted fabrics for which the computer graphics
community made enormous progress kaldor_simulating_2008 ;
sperl_estimation_2022 ; ding_unravelling_2023 , but the dynamics usually rely
on viscous dissipative forces at the contacts, ill-adapted to capture rest
shapes arrested by dry friction. Nonetheless, recent numerical advances allow
the combination of large fiber displacements with frictional interactions
cirio_yarn-level_2017 ; li_implicit_2018 ; liu_computational_2018 ;
crassous_discrete_2023  and open the way to explore the mechanics and
stability of frictional fiber assemblies quantitatively sano_randomly_2023 .
The complex geometry of the contact zones between fibers makes exact
theoretical modeling extremely complicated. In this letter, we show that a
simplified description of these zones can faithfully reproduce the mechanical
equilibrium properties of a knitted fabric. The postulate of a single form of
equilibrium must be abandoned. Even without applied stresses, the solid
friction between the threads stabilizes the materials in very different forms
depending on the system’s history.

Figure 1: (a) Photo of a Jersey stitch knit. (b) Experimental set-up. The
knitted fabric (light blue) is attached with metallic staples (orange) that
can freely slide over 4 cylindrical bars linked to a biaxial tensile machine.
(c) Geometry and numerical model. A rectangular cell of size
$\ell_{x}\times\ell_{y}$ is composed of 3 threads $(1),(2)\&(3)$. Segments of
cylinders composing the threads are colored. Dots $1a$, $1b$,… are the
endpoints of the different threads, which are constrained to be on the cell
boundaries with periodic boundary conditions.

In this study, we use a Jersey stitch knit, which is both simple and widely
used. It consists of a single yarn forming interlocked loops as in fig. 1(a).
Experimentally, we make a knitted fabric of $70\times 70$ stitches from a
polyamide (Nylon) thread (Madeira Monofil n°40, $E=1.05~{}GPa$, $\mu=0.50$) of
diameter $d=0.155~{}mm$. The length of thread per stitch is $\ell=9.7~{}mm$.
The central $N\times N$ stitches ($N=50$) are attached to a biaxial tensile
machine (fig. 1.b), where the spacing $\Delta x$ along the courses and $\Delta
y$ along the wales can be varied by stepper motors. The forces per row
$f_{x}\mathrel{\mathop{\mathchar 58\relax}}=F_{x}/N$ and columns
$f_{y}\mathrel{\mathop{\mathchar 58\relax}}=F_{y}/N$ are measured using strain
gauge force sensors. The network’s periodicity $(\ell_{x},\ell_{y})$ is
obtained from images recorded with a camera. The network is also studied using
numerical simulations based on Discrete Elastic Rods (DER) coupled with dry
frictional contacts crassous_discrete_2023 . Threads are decomposed into
segments of circular cylinders connected by springs, which account for the
elastic forces of traction, flexion, and torsion. A mesh comprises 3 rods as
shown in fig 1.c. The endpoints of these 3 wires are constrained to the
$\pm\ell_{x}/2$ or $\pm\ell_{y}/2$ planes. Periodic boundary conditions in
terms of positions and forces are applied at the junctions between the
strands: for example, the strands $1$ and $2$ satisfy ${\bf r}_{2a}-{\bf
r}_{1a}=\ell_{y}{\bf e}_{y}$, $\alpha_{2a}=\alpha_{1a}$, ${\bf
r}^{{}^{\prime}}_{2a}={\bf r}^{{}^{\prime}}_{1a}$ and ${\bf
r}^{{}^{\prime\prime}}_{2a}={\bf r}^{{}^{\prime\prime}}_{12}$. $\alpha$ is the
torsion angle, ${}^{{}^{\prime}}$ and ${}^{{}^{\prime\prime}}$ are the 1st and
2nd derivatives with respect to the curvilinear abscissa $s$. Conditions on
the derivatives ensure the continuity of the forces and moments.

Figure 2: Relaxed configurations of a knitted fabrics ($\ell/d=62$,
$\mu=0.5$). (open symbols): Experimental configurations during relaxation,
color-coded by the value of $\log(f/f_{min})$. The dotted lines are
guidelines. (black diamonds): Relaxed, experimental configurations; (red
squares): Configurations obtained from DER simulations. There are no relaxed
configurations above or to the left of the terminal points $T$, noted
$T_{exp}$ (resp. $T_{num}$) for experimental (resp. numerical) data. (black
plain line): Possible configurations expected from a 2D-elastica model with no
applied stress.

The relaxed states are obtained as follows. Experimentally, the knitted fabric
is stretched to an initial state $(\Delta x_{0},\Delta y_{0})$, then $\Delta
x$ and $\Delta y$ are varied to reduce $f\mathrel{\mathop{\mathchar
58\relax}}=(f_{x}^{2}+f_{y}^{2})^{1/2}$ until it becomes lower than
$f_{min}=8$ mN. Numerically, the knitted fabric is stretched, the forces are
calculated. The stitch length is varied by $\delta\ell_{i}=-\lambda f_{i}$,
with $i=x,y$ and $\lambda$ a numerical constant, until the force is smaller to
a given threshold. Fig.2 shows the mesh sizes obtained following this
experimental and numerical protocol. Firstly, the shape of the knitted fabric
after the relaxation of the applied forces varies strongly with the initial
state considered and is not uniquely defined. Secondly, these states are
located on a curve which acts as an attractor in the space of
$(\ell_{x},\ell_{y})$ configurations. This curve ends at a terminal point
$T\mathrel{\mathop{\mathchar 58\relax}}=(\ell_{x_{T}},\ell_{y_{T}})$. A knit
verifying $\ell_{y}>\ell_{y_{T}}$ or $\ell_{x}<\ell_{x_{T}}$ is impossible
without external forces. Finally, the experimental and numerical data
agreement is very satisfactory: assimilating a polyamide thread to an elastic
fiber whose cross-sectional deformations are neglected is a reasonable
hypothesis. The differences can be understood by noting that plastic
deformations occur during knitting, meaning the stress-free yarn is no longer
ideally rectilinear.

Figure 3: (Color): Curves of constant elastic energy $E$ for cells made of
frictionless threads ($\mu=0$, $l/d=62$), color-coded by the values of
$\log(E-0.01)$. (Dotted lines) Relaxation of fabrics with frictional thread
($\mu=0.5$) between an initial configuration (open square) and a relaxed
configuration (filled square). $T_{num}$ is the terminal point. (Dashed
curve): Relaxation of a fabric, initially at $T_{num}$, when $\mu$ is
gradually decreased.

To describe the set of relaxed states, we first consider the elastic energy
$E(\ell_{x},\ell_{y})$ of a frictionless knitted fabric. We obtain this
quantity from DER simulation for each cell size at mechanical equilibrium. The
iso-energy curves are shown fig.3. The traction zones (large $\ell_{x}$ or
$\ell_{y}$) are separated from the compressive zones (small $\ell_{x}$ or
$\ell_{y}$) by a valley that descends towards small $\ell_{y}$ and large
$\ell_{x}$: the knitted fabric relaxes its bending energy by tending to align
its yarns parallel to ${\bf e}_{x}$. The milder slope in this energy landscape
acts as an attractor for knitted fabrics of finite friction. A knitted fabric
initially placed on one side or the other of this line relaxes towards it and
stops in its vicinity. Friction stops the deformation and allows
metastability. The following numerical experiment can demonstrate this. We
prepare a fabric with $\mu=0.5$ at its terminal point $T_{num}$. The friction
coefficient is then decreased of $\delta\mu=-0.05$: the knitted fabric
descends along the valley and stabilizes in a new position. By gradually
reducing the friction, the line of milder slope (dashed line of fig.3) is
followed toward the bottom of the valley. This line is the locus of terminal
points obtained for different $\mu$.

Therefore, describing the set of metastable configurations requires
i)understanding the shape of the valley, which a priori is independent of
friction, and ii)knowing how the friction $\mu$ controls the position of the
terminal point $T$ on the line of milder slope.

Figure 4: (a) Contact forces ${\bf p}(s)$ acting in the braid zone (blue
rectangle) between two fibers. (b) Equilibrium of forces on the thread (1).
The points $(1a)$ and $(2a)$ are on the vertical line of middle point
$O^{\prime}$, and ${\bf{\overline{t}}}$ is the braid direction. (c) (Red
line): Projection of the vector ${\bf r}_{2}-{\bf r}_{1}$, joining the
centerlines, on the planes perpendicular to ${\bf{\overline{t}}}$. $v$ is the
coordinate along an axis of direction
$-{\bf{\overline{n}}}\mathrel{\mathop{\mathchar
58\relax}}={\bf{\overline{t}}}\times{\bf e}_{z}$. The circle is of unit
radius. Contact occurs in two points $\alpha$ and $\beta$ (blue dots) that
makes angles $\varphi_{\alpha}$ and $\varphi_{\beta}=-\varphi_{\alpha}$ with
the $z=0$ plane. (d) Orientations of the contact force at a contact point
$\xi$. The plane $\Pi$ containing ${\bf t}_{1}$ and ${\bf t}_{2}$ is normal to
${\bf N}_{\xi}$. ${\bf T}_{\xi}$ is the direction of the tangential contact
force, which makes an angle $\Psi_{\xi}$ with ${\bf{\overline{t}}}$.

Interactions between the yarns take place in the area where they become
entangled, creating both normal and tangential forces to the threads. This
zone, schematically depicted in fig. 4.a, needs to be described. We introduce
the curvilinear abscissa $s$ along the centerline of thread (1), and ${\bf
p}(s)$ as the linear density of contact force exerted by (2) on (1). Because
friction is present, ${\bf p}(s)$ is not necessarily aligned to the normal
vector ${\bf n}(s)$. We define

${\bf f}_{c}=\int_{s_{in}}^{s_{out}}{\bf p}(s)ds$ (1)

the resultant of contact forces with $s_{in}$ the entrance of the contact
zone: ${\bf p}(s_{in})\neq 0$ and ${\bf p}(s)=0$ for $s<s_{in}$, with
reciprocal definition for $s_{out}$. For any relaxed state, we must have ${\bf
f}_{c}\cdot{\bf e}_{y}=0$. Indeed (see fig.4.b), in the relaxed state, the
total stress on each cell edge is $0$, so the external force on $(1b)$ is
${\bf f}_{1b}=0$. Since the force exerted by thread (2) on thread (1) is ${\bf
f}_{c}$, we have ${\bf f}_{1a}=-{\bf f}_{c}$. No applied stress condition also
implies ${\bf f}_{1a}+{\bf f}_{3a}=0$. Symmetry with respect to the $y$ axis
at point (2m) implies ${\bf f}_{1a}\cdot{\bf e}_{y}={\bf f}_{3a}\cdot{\bf
e}_{y}$, which leads to ${\bf f}_{c}\cdot{\bf e}_{y}=0$. It has been checked
that the results of the simulations fulfilled all those requirements.

Let $s_{c}$ be the arbitrary abscissa at which ${\bf f}_{c}$ is applied. In
the limit $d/\ell\ll 1$, we consider the bidimensional problem of finding the
value of $f_{c}\mathrel{\mathop{\mathchar 58\relax}}=\|{\bf f}_{c}\|$ such
that the 2D elastic curve ${\bf r}(s)$ representing the strand (1) is at
equilibrium. The curve must satisfy the applied external force ${\bf f}_{c}$
at $s=s_{c}$ and $-{\bf f}_{c}$ at $s=s_{1a}=0$, and verifying $dy/ds=0$ at
$s=s_{1b}=\ell/4$ and $\Bigl{(}{\bf r}(s_{c})+(d/2){\bf n}(s_{c})-{\bf
r}(0)\Bigr{)}\cdot{\bf e}_{x}=0$. The last condition imposes contacts between
the two threads in $O^{\prime}$. As shown on fig.4.c, the threads actually
touch in two points but not in the middle point $O^{\prime}$. However, the
distance between the centerlines in $O^{\prime}$ is typically $1.03~{}d\simeq
d$. The solution of this 2D-elastica problem may be expressed with equations
involving elliptic integrals. Since there is no explicit solutions, we solved
this problem numerically. For each arbitrary value of $s_{c}$ we obtain
$f_{c}$, ${\bf r}(s_{c})$ and ${\bf n}(s_{c})$. Using symmetries of the cell,
we obtain:

$\displaystyle\mathcal{L}_{x}(s_{c})$ $\displaystyle=\Bigl{(}4\bigl{[}{\bf
r}(s_{c})-{\bf r}(\ell/4)\bigr{]}+2d~{}{\bf n}(s_{c})\Bigr{)}\cdot{\bf e}_{x}$
(2a) $\displaystyle\mathcal{L}_{y}(s_{c})$
$\displaystyle=\Bigl{(}2\bigl{[}{\bf r}(s_{c})-{\bf r}(0)\bigr{]}+d~{}{\bf
n}(s_{c})\Bigr{)}\cdot{\bf e}_{y}$ (2b)

$(\mathcal{L}_{x}(s_{c}),\mathcal{L}_{x}(s_{c}))$ is the parametric curve in
the plane $(\ell_{x},\ell_{y})$ shown on fig.2. This simple 2D approximation
adequately captures the set of metastable points obtained from experiments and
DER simulations.

Figure 5: (a):
$\mu~{}cos(\psi)/\bigl{[}\cos(\varphi)+\mu\sin(\varphi)\sin(\psi)\bigr{]}$ vs
$\tan\theta$. Symbols are DER simulation results, and line is eq.(6). Insets
are the threads centerlines for $\tan(\theta)\simeq 0$, and
$\tan(\theta)\simeq\tan(\theta_{M})$. The dashed rectangles are the meshes of
sizes $\ell_{x}\times\ell_{y}$. (b): Evolution of $\psi$ with $\tan(\theta)$.
The maximal inclination $\tan(\theta_{M})$ (dashed line) is obtained for
$\psi\simeq\pi$. Data of (a) and (b) are obtained for $\mu=0.5$. (c):
Variation of $\tan(\theta_{M})$ with $\mu$. Dotted line is eq.(7) with
$\varphi=129~{}deg$

All relaxed configurations $(\mathcal{L}_{x}(s_{c}),\mathcal{L}_{y}(s_{c}))$
are obtained using the condition ${\bf f}_{c}=f_{c}{\bf e}_{x}$ necessary to
cancel the applied stress on one cell. We now discuss to which condition this
constraint may be fulfilled. We consider the details of the contact force
distribution ${\bf p}(s)$. The braid comprises two twisted fibers in a
geometry similar to the one occurring in knots where the threads are twisted
of $n$ turns. The cases $n=1$ ($3_{1}$ knot) audoly_2007 ; clauvelin_2009 ,
$n=2$ ($5_{1}$ knot) clauvelin_2009  and $n\gg 1$ jawed_2015  have been
considered previously. Threads in the braid are approximately a helix
contacting in a few points. It seems that our braid with $n=1/2$ has not been
previously considered, but it exhibits a very similar behavior. The threads
have nearly helical shape twisted around a common line of direction
${\bf{\overline{t}}}=\bigl{[}{\bf t}_{1}(s_{1})+{\bf
t}_{2}(s_{2})\bigr{]}/\|{\bf t}_{1}(s_{1})+{\bf t}_{2}(s_{2})\|$, where ${\bf
t}_{1}$ and ${\bf t}_{2}$ are the tangent vectors of centerlines of threads
(1) and (2). The figure 4.c represents the relative position ${\bf
r}_{2}(s_{2})-{\bf r}_{1}(s_{1})$ of the threads, where ${\bf r}_{1}(s_{1})$
and ${\bf r}_{2}(s_{2})$ are the projections of the centerlines on the plane
perpendicular to ${\bf{\overline{t}}}$. The fibers are in contacts in two
points $\alpha$ and $\beta$, and then:

${\bf f}_{c}=\int{\bf p}(s)ds=\sum_{\xi=\alpha,\beta}\bigl{[}f_{\xi}^{(n)}{\bf
N}_{\xi}+f_{\xi}^{(t)}{\bf T}_{\xi}\bigr{]}$ (3)

with ${\bf N}_{\xi}$ the normal vector to the contact point, ${\bf T}_{\xi}$
the orientation of the tangential contact forces (see fig.4(d)), and
$f_{\xi}^{(n)}$ and $f_{\xi}^{(t)}$ the normal and tangential forces. For all
equilibrium configurations, we found
$f_{\alpha}^{(n)}=f_{\beta}^{(n)}\mathrel{\mathop{\mathchar
58\relax}}=f^{(n)}$ and $({\bf N}_{\alpha}+{\bf N}_{\beta})\cdot{\bf
e}_{z}=0$. Noting $\varphi_{\xi}$ the angle between ${\bf N}_{\xi}$ and $z=0$
(see fig.4.c), we have
$\varphi_{\alpha}=-\varphi_{\beta}\mathrel{\mathop{\mathchar
58\relax}}=\varphi$, with $\varphi\simeq 3\pi/4$ for each relaxed
configuration.

When the meshes are steadily deformed, the threads in contact make the braid
slide. The friction is then fully mobilized, and $f_{\xi}^{(t)}=\mu
f_{\xi}^{(n)}$. Since ${\bf N}_{\xi}\cdot{\bf t}_{i}(s_{\xi})=0$ for the two
threads $i=1,2$, ${\bf N}_{\xi}$ is perpendicular to the plane containing
${\bf t}_{1}(s_{\xi})$ and ${\bf t}_{2}(s_{\xi})$. ${\bf T}_{\xi}$ belongs to
this plane, and we may write:

${\bf
T}_{\xi}=\cos(\psi_{\xi})~{}{\bf{\overline{t}}}+\sin(\psi_{\xi})~{}\bigl{[}{\bf{\overline{t}}}\times{\bf
N}_{\xi}\bigr{]}$ (4)

where $\psi_{\xi}$ is the angle between ${\bf T}_{\xi}$ and
${\bf{\overline{t}}}$ (see fig.4.d). Using eq.(3) and eq.(4), we may calculate
${\bf f}_{c}$. With $\varphi_{\alpha}=-\varphi_{\beta}$, the equilibrium
conditions ${\bf f}_{c}\cdot{\bf e}_{z}=0$ leads to
$\psi_{\alpha}=-\psi_{\beta}\mathrel{\mathop{\mathchar 58\relax}}=\psi$, and
finally:

${\bf
f}_{c}=2f^{(n)}\times\Bigl{(}\mu\cos(\psi){\bf{\overline{t}}}+\bigl{[}\cos(\varphi)+\mu\sin(\varphi)\sin(\psi)\bigr{]}{\bf{\overline{n}}}\Bigr{)}$
(5)

The condition of mechanical equilibrium of a relaxed configuration ${\bf
f}_{c}\cdot{\bf e}_{y}=0$ may then be written as:

$\tan(\theta)=\frac{\mu~{}cos(\psi)}{\cos(\varphi)+\mu\sin(\varphi)\sin(\psi)}$
(6)

In this equation, $\theta=\arccos({\bf{\overline{t}}}\cdot{\bf e}_{y})$ is the
angle between the braid and y-axis. Fig.5.a shows the variations in the right
and left members of eq.(6) for a fixed coefficient of friction $\mu=0.5$. The
values of $\theta$, $\psi$, and $\varphi$ are measured at the various stopping
points in fig.3, and the relationship eq.(6) is checked. The different
$\theta$ values correspond to different aspect ratios of the meshes. For
examples, insets of fig.5 shows configuration elongated in $x$ for
$\theta\simeq 0$ and a configuration close to the terminal point $T$ for
$\theta\simeq 40~{}deg$.

Given the expression of eq.(6), at a given friction coefficient, and $\varphi$
being roughly constant for all configurations, the only way to vary $\theta$
is to vary $\psi$, as can be seen clearly in fig.5.b. The tangential forces at
the two contact points $\alpha$ and $\beta$ rotate on either side of the braid
axis ${\bf{\overline{t}}}$. The effect of those rotations is to vary the
amplitude of the total friction force along the ${\bf{\overline{t}}}$ axis,
even if individual friction forces always evolve at the Coulomb threshold.
Another consequence of eq.(6) is that $\tan(\theta)$ has a maximum value. With
$\varphi\simeq 3\pi/4$ constant, $\tan(\theta)$ is maximal for a rotation
$\psi=\pi-\arcsin(\mu)\simeq\pi$ for $\mu\ll 1$ (note that $\cos(\varphi)$ and
$\cos(\psi)$ are negative), as shown in fig.5.b. This maximal value of
$\theta$ is obtained when the frictional forces are roughly aligned with the
common tangent of the centerline. Approximating $\psi=\pi$ in eq.(6), we
finally obtain for small $\mu$:

$\tan\theta\leq\tan(\theta_{M})\mathrel{\mathop{\mathchar
58\relax}}=\frac{\mu}{|\cos(\varphi)|}$ (7)

where $\theta_{M}$ is the maximal value of $\theta$. This relation may be
checked by measuring $\tan(\theta_{M})$ for different values of $\mu$. Fig.5.c
shows that $\tan(\theta_{M})$ indeed follows eq.(7) with a constant
$\varphi=129~{}deg$. For a given $\mu$, the maximal value of $\ell_{y}$ and
minimal value of $\ell_{x}$ (corresponding to the terminal point $T$ of fig.2
and fig.3) is attained for $\theta=\theta_{M}$. When $\mu$ is varied, the line
of terminal points (dashed line of fig.3) is traveled. On this line,
$\theta=\theta_{M}(\mu)$, and $\psi\simeq\pi$, whatever is $\mu$: the friction
forces are aligned and opposite to the relative displacements of the threads.

Our study rationalizes the relaxed states of a periodic yarn assembly. The
relaxed state is not unique, but forms a continuous subset of the space of
possible periodic configurations $(\ell_{x},\ell_{y})$ of a knitted fabrics. A
terminal point (named T) bounds this subset. Those findings have many
implications. Applying successive stretching cycles along $y$, the stitch size
will converge to the terminal point $T$. The configuration corresponding to T
would be the reproducible shape of a relaxed knitted fabric, even if
metastability make other relaxed shapes possible. The existence of a continuum
of relaxed states has important consequences for the macroscopic mechanical
properties. The restoring forces are, therefore, weak over a wide range of the
configuration space. Knitted fabrics are soft objects for deformations that
remain in this zone but are relatively rigid when we move away from it.
Finally, variations in aspect ratios mean that the area per stitch
$\ell_{x}\times\ell_{y}$ can be varied at zero external force. A flat knitted
fabric can thus be stretched to cover a surface with non-zero Gaussian
curvature without any external forces being applied. This study can also serve
as a ground basis for exploring further the mechanics of knitted fabrics or,
more generally, of periodic structures made of threads with out-of-plane
deformations or three-dimensional greer_2023 . The numerical and theoretical
models can be adapted to different knitting topology singal_programming_2024 ,
but also for fibers closer to applications by modifying the elastic properties
of the rod or smaller aspect ratio $l/d$. The methods introduced here are not
limited to relaxed states but can also be adapted to explore the role of
friction in the force vs. strain responses of textiles, including hysteresis
poincloux.2018b  and slip-induced fluctuations poincloux.2018 .

###### Acknowledgements.

A.S. thanks D. Le Tourneau and P. Metz for building the biaxial tensile
machine. J.C. acknowledges CNRS-Physique for hosting in delegation. S.P.
acknowledges financial support from the Japanese Society for the Promotion of
Science as a JSPS International Research Fellow. This work has been supported
by Agence Nationale de la Recherche Grant ANR-23-CE30-0015.

## References

  * (1) Paul Johanns, Changyeob Baek, Paul Grandgeorge, Samia Guerid, Shawn A Chester, and Pedro M Reis.  The strength of surgical knots involves a critical interplay between friction and elastoplasticity.  Sci. Adv., 9(23):eadg8861, 2023.
  * (2) Christopher A Daily-Diamond, Christine E Gregg, and Oliver M O’Reilly.  The roles of impact and inertia in the failure of a shoelace knot.  Proc. R. Soc. A, 473(2200):20160770, 2017.
  * (3) Nicholas Weiner, Yashraj Bhosale, Mattia Gazzola, and Hunter King.  Mechanics of randomly packed filaments—the “bird nest” as meta-material.  J. Appl. Phys., 127(5), 2020.
  * (4) Gautier Verhille, Sébastien Moulinet, Nicolas Vandenberghe, Mokhtar Adda-Bedia, and Patrice Le Gal.  Structure and mechanics of aegagropilae fiber network.  Proc. Natl. Acad. Sci. U.S.A., 114(18):4607–4612, 2017.
  * (5) Wilhelm Albrecht, Hilmar Fuchs, and Walter Kittelmann.  Nonwoven fabrics: raw materials, manufacture, applications, characteristics, testing processes.  John Wiley & Sons, 2006.
  * (6) Caroline Vibert, Anne-Laurence Dupont, Justin Dirrenberger, Raphaël Passas, Denise Ricard, and Bruno Fayolle.  Relationship between chemical and mechanical degradation of aged paper: fibre versus fibre–fibre bonds.  Cellulose, pages 1–19, 2024.
  * (7) Samuel Poincloux, Tian Chen, Basile Audoly, and Pedro M. Reis.  Bending response of a book with internal friction.  Phys. Rev. Lett., 126:218004, May 2021.
  * (8) Antoine Seguin and Jérôme Crassous.  Twist-controlled force amplification and spinning tension transition in yarn.  Phys. Rev. Lett., 128(7):078002, 2022.
  * (9) Julien Chopin, Animesh Biswas, and Arshad Kudrolli.  Energetics of twisted elastic filament pairs.  Phys. Rev. E, 109:025003, Feb 2024.
  * (10) Andrew Craig Long.  Design and manufacture of textile composites.  Elsevier, 2005.
  * (11) Robert Koerner.  Geotextiles: from design to applications.  Woodhead Publishing, 2016.
  * (12) Samuel Poincloux, Mokhtar Adda-Bedia, and Frédéric Lechenault.  Geometry and elasticity of a knitted fabric.  Phys. Rev. X, 8:021075, Jun 2018.
  * (13) Krishma Singal, Michael S. Dimitriyev, Sarah E. Gonzalez, A. Patrick Cachine, Sam Quinn, and Elisabetta A. Matsumoto.  Programming mechanics in knitted materials, stitch by stitch.  Nat. Commun., 15(1):2622, March 2024.
  * (14) Guorui Chen, Yongzhong Li, Michael Bick, and Jun Chen.  Smart textiles for electricity generation.  Chem. Rev., 120(8):3668–3720, 2020.
  * (15) Vanessa Sanchez, Conor J. Walsh, and Robert J. Wood.  Textile Technology for Soft Robotic and Autonomous Garments.  Adv. Funct. Mater., 31(6):2008278, February 2021.
  * (16) D. L. Munden.  THE GEOMETRY AND DIMENSIONAL PROPERTIES OF PLAIN-KNIT FABRICS.  Journal of the Textile Institute Transactions, 50(7):T448–T471, July 1959.
  * (17) Gamini Lanarolle.  Geometry of compact plain knitted structures.  Res. J. Text. Appar., 25(4):330–345, November 2021.
  * (18) S. Allan Heap, Peter F. Greenwood, Robert D. Leah, James T. Eaton, Jill C. Stevens, and Pauline Keher.  Prediction of Finished Weight and Shrinkage of Cotton Knits— The Starfish Project: Part I: Introduction and General Overview.  Text. Res. J., 53(2):109–119, February 1983.
  * (19) Jonathan M. Kaldor, Doug L. James, and Steve Marschner.  Simulating knitted cloth at the yarn level.  ACM Trans. Graph., 27(3):1–9, August 2008.
  * (20) Georg Sperl, Rosa M. Sánchez-Banderas, Manwen Li, Chris Wojtan, and Miguel A. Otaduy.  Estimation of yarn-level simulation models for production fabrics.  ACM Trans. Graph., 41(4):1–15, July 2022.
  * (21) Xiaoxiao Ding, Vanessa Sanchez, Katia Bertoldi, and Chris H. Rycroft.  Unravelling the Mechanics of Knitted Fabrics Through Hierarchical Geometric Representation, July 2023.  arXiv:2307.12360 [cond-mat].
  * (22) Gabriel Cirio, Jorge Lopez-Moreno, and Miguel A. Otaduy.  Yarn-Level Cloth Simulation with Sliding Persistent Contacts.  IEEE Trans. Vis. Comput. Graph., 23(2):1152–1162, February 2017.
  * (23) Jie Li, Gilles Daviet, Rahul Narain, Florence Bertails-Descoubes, Matthew Overby, George E Brown, and Laurence Boissieux.  An implicit frictional contact solver for adaptive cloth simulation.  ACM Trans. Graph., 37(4):1–15, 2018.
  * (24) Dani Liu, Bahareh Shakibajahromi, Genevieve Dion, David Breen, and Antonios Kontsos.  A Computational Approach to Model Interfacial Effects on the Mechanical Behavior of Knitted Textiles.  J. Appl. Mech., 85(4):041007, April 2018.
  * (25) Jérôme Crassous.  Discrete-element-method model for frictional fibers.  Phys. Rev. E., 107(2):025003, 2023.
  * (26) Tomohiko G Sano, Emile Hohnadel, Toshiyuki Kawata, Thibaut Métivet, and Florence Bertails-Descoubes.  Randomly stacked open cylindrical shells as functional mechanical energy absorber.  Commun. Mater., 4(1):59, 2023.
  * (27) B. Audoly, N. Clauvelin, and S. Neukirch.  Elastic knots.  Phys. Rev. Lett., 99:164301, Oct 2007.
  * (28) N. Clauvelin, B. Audoly, and S. Neukirch.  Matched asymptotic expansions for twisted elastic knots: A self-contact problem with non-trivial contact topology.  J. Mech. Phys. Solids, 57(9):1623–1656, 2009.
  * (29) M. K. Jawed, P. Dieleman, B. Audoly, and P. M. Reis.  Untangling the mechanics and topology in the frictional response of long overhand elastic knots.  Phys. Rev. Lett., 115:118302, Sep 2015.
  * (30) Widianto P. Moestopo, Sammy Shaker, Weiting Deng, and Julia R. Greer.  Knots are not for naught: Design, properties, and topology of hierarchical intertwined microarchitected materials.  Sci. Adv., 9(10):eade6725, 2023.
  * (31) Samuel Poincloux, Mokhtar Adda-Bedia, and Frédéric Lechenault.  Crackling dynamics in the mechanical response of knitted fabrics.  Phys. Rev. Lett., 121:058002, Jul 2018.