# The $\theta$-formulation of the 2D elastica – Buckling and boundary layer
theory.

Gregory Kozyreff1, Emmanuel Siéfert2, Basile Radisson2 and Fabian Brau2
1Optique Nonlinéaire Théorique, Université libre de Bruxelles (U.L.B.), CP
231, Campus de la Plaine, 1050 Bruxelles, Belgium.
2Nonlinear Physical Chemistry Unit, Université libre de Bruxelles (U.L.B.), CP
231, Campus de la Plaine, 1050 Bruxelles, Belgium.

###### Abstract

The equations of a planar elastica under pressure can be rewritten in a useful
form by parametrising the variables in terms of the local orientation angle,
$\theta$, instead of the arc length. This “$\theta$-formulation” lends itself
to a particularly easy boundary layer analysis in the limit of weak bending
stiffness. Within this parameterization, boundary layers are located at
inflexion points, where $\theta$ is extremum, and they connect regions of low
and large curvature. A simple composite solution is derived without resorting
to elliptic functions and integrals. This approximation can be used as an
elementary building block to describe complex shapes. Applying this theory to
the study of an elastic ring under uniform pressure and subject to a set of
point forces, we discover a snapping instability. This instability is
confirmed by numerical simulations. Finally, we carry out experiments and find
good agreement of the theory with the experimental shape of the deformed
elastica.

##  I Introduction

The problem of finding the shape of an elastica under various loads and
constraints appears in many contexts of the applied sciences and at a wide
variety of scales, from biological cells to suspended bridges. After centuries
of analysis [1, 2], one may consider it to be solved, at least as far as two-
dimensional deformations is concerned. A general solution of the associated
differential equations is quoted in Love’s treatise [1] and further
developments abound in the literature on its basis [3, 4, 5, 6, 7, 8, 9, 10,
11]. It is thus often possible to explicitly write the coordinate of any point
of the elastic curve as a function of arc length in a wide range of
situations. A weakness of the known solutions, however, is that they involve
several elliptic functions, which can themselves appear as arguments of
inverse trigonometric functions. Furthermore, the parameters of these elliptic
functions are themselves functions of the roots of a fourth order polynomial
in which the mechanical parameters of the system appear. Hence, despite being
undeniably useful, the exact solution can be quite hard to analyse by
inspection. Neither does it simplify in the limit of a small bending stiffness
or long elastica. In response to this issue, the purpose of this paper is to
put forward another way of parameterising the elastic curve that may lead to
great simplification. Very simply, we propose to express the curvature and the
internal forces not as a function of arc length but as a function of the local
angle, $\theta$, made by the tangent of the elastic curve with a reference
direction. We thus call it the $\theta$-formulation of the elastica. It is no
more than the application of an old trick to reduce the order of a
differential system [12], which has been used before [13, 14]. What hasn’t
been appreciated so far, however, is its great advantage for asymptotic
analysis, i.e., in the treatment of boundary layers.

After deriving this $\theta$-formulation, we first show how it gives way to an
economical resolution of some classical problems. Next, we exploit the limit
of a long elastica or, equivalently, of weak bending stiffness. We assume that
a net pressure acts on one side of the elastica and allow that a set of point
forces be applied. In particular, we devise a general framework to treat
closed rings that are either compressed or inflated while being either locally
stretched or pinched. We also carry out experiments of such elastic rings and
find that the asymptotic approximation faithfully reproduce observations. In
the case of pinching with prescribed forces, we identify a snapping
instability, past which the ring collapses to self-contact. Finally, we
indicate how the same technique could be applied to more general problems.

##  II Derivation of the $\theta$-formulation

The differential equations governing the shape of an elastica in a plane can
be written as

$\displaystyle\partial_{s}n$ $\displaystyle=p-\kappa t,$ (1)
$\displaystyle\partial_{s}t$ $\displaystyle=\kappa n,$ (2) $\displaystyle
B\partial_{s}\kappa$ $\displaystyle=-n,$ (3)

where $\kappa$, $t$, and $n$ are the curvature, tension and shear force,
respectively, $s$ is the arc length, and $B$ is the bending stiffness. Given
the 1D nature of the elastica, the pressure $p$ appearing in Eq. (1) has
dimensions of Nm-1. Knowing $\kappa$, the local angle with respect to the
horizontal direction is deduced from

$\partial_{s}\theta=\kappa$ (4)

and the cartesian coordinates along the elastic curve are solution of
$\partial_{s}x=\cos\theta$, $\partial_{s}y=\sin\theta$. Multiplying Eq. (3) by
$\kappa$ and summing with (2), one easily finds that

$B\frac{\kappa^{2}}{2}+t=H,$ (5)

where $H$ is a constant (it is equivalent to the constant $\mu$ appearing in
various papers [15, 5, 6] and $C$ in [9]). This allows one to eliminate the
variable $t$ and reduce the order of the differential system (1)-(4) by one.
Note that if we write $H$ as $H=B\kappa_{0}^{2}/2$, then a state with
$\kappa=\kappa_{0}$ for all $s$ corresponds to $t=n=p=0$. In that case, $H$
measures the curvature of a stress-free elastic ring. On the other hand,
$\kappa=0$ everywhere corresponds to the state of a straight elastica with
$n=p=0$ and a uniform tension $t=H$. Let us also note that, if $t>0$ anywhere
in the elastica, then $H>0$. Stated differently, the elastica must be
everywhere under axial compression for $H$ to be negative.

Using (5) to eliminate $t$, Eq. (1) yields

$\partial_{s}n=p-\kappa\left(H-B\kappa^{2}/2\right),$ (6)

which, together with Eqs. (3) and (4), yields a third-order differential
system. A further reduction of order is achieved by expressing $n$ and
$\kappa$ as functions of $\theta$. Firstly, writing
$n=n\left(\theta(s)\right)$, we have $\partial_{s}n=\kappa\partial_{\theta}n$.
Hence, Eq. (6) becomes

$\partial_{\theta}n=\frac{p}{\kappa}-H+B\frac{\kappa^{2}}{2}.$ (7)

Secondly, if we write

$\frac{\kappa^{2}}{2}=q\left(\theta(s)\right),$ (8)

we obtain, by differentiating each side with respect to $s$,

$\partial_{s}\kappa=\partial_{\theta}q.$ (9)

Hence, Eq. (3) yields

$n=-B\partial_{\theta}q.$ (10)

Eventually, Eq. (7) becomes

B(∂^2_θq+q)+pκ= H. (11)

This equation, which we presently call the $\theta$-formulation, is not
particularly useful from a numerical point of view but can lead to significant
analytical progress, particularly in the limit of small and large curvature.
Here, “small” and “large” curvature respectively refer to situations where
$p/\kappa$ or $B(\partial^{2}_{\theta}q+q)$ dominate the left hand side,
respectively. Evidently, both limits considerably simplify the resolution of
Eq. (11). To make this statement more systematic, let us rescale $\kappa$ (and
therefore $q$) as

$\displaystyle\kappa$ $\displaystyle=\frac{p}{H}\,K,$ $\displaystyle q$
$\displaystyle=\left(\frac{p}{H}\right)^{2}Q,$ (12)

provided that $p$ is constant and not zero. With this scaling, we still have
$K^{2}/2=Q$ and Eq. (11) becomes

$\epsilon^{2}\left(\partial^{2}_{\theta}Q+Q\right)+\frac{1}{K}=1,$ (13)

where

$\epsilon^{2}=\frac{Bp^{2}}{H^{3}}.$ (14)

Viewing $x$ and $y$ as functions of $\theta$, we have
$\partial_{s}(x,y)=\kappa\partial_{\theta}(x,y)$. Hence, given
$\kappa(\theta)$, the shape of the elastica is obtained by integrating

$\displaystyle\partial_{\theta}x$
$\displaystyle=\frac{\cos\theta}{\kappa(\theta)},$
$\displaystyle\partial_{\theta}y$
$\displaystyle=\frac{\sin\theta}{\kappa(\theta)},$ (15)

or, in a more compact way,

$\displaystyle\partial_{\theta}z$
$\displaystyle=\frac{e^{i\theta}}{\kappa(\theta)},$ $\displaystyle z$
$\displaystyle=x+iy.$ (16)

The advantage of a complex representation is that translation by a vector
$(x_{0},y_{0})$ amount to adding of $x_{0}+iy_{0}$ to $z$, while rotation by
an angle $\alpha$ is achieved by a multiplication by $e^{i\alpha}$. This is
useful for constructing elastic curves as pieces of solutions of the above
differential equation joined together. Furthermore, from Eqs. (5) and (10), we
have

$\displaystyle t$ $\displaystyle=H\left(1-\epsilon^{2}Q\right),$
$\displaystyle n$ $\displaystyle=-\epsilon^{2}H\,\partial_{\theta}Q.$ (17)

So far, our analysis has been exact. In Secs. III and IV, we show how the
above formulation can be useful to treat some classical problems. Next, in
Sec. V, we derive asymptotic approximations of $K$ (hence, of $Q$) in the
small-$\epsilon$ limit using the method of matched asymptotic expansions [12].
More specifically, we derive simple formulas describing how the elastica
switches between a small positive curvature to a large negative one. As a
function of $\theta$, $K$ displays boundary layers near extremal value of
$\theta$, which are inflexion points. By convention here, the positivity of
the curvature is defined relative to the sign of the pressure: a positive
curvature is understood here as being of the same sign as $p/H$. With this
convention in mind, we may treat in one go elastic rings that are subjected to
an inflating pressure or to an external compression.

##  III Application 0: Euler’s elastica revisited

Before investigating boundary layers, we wish to highlight the usefulness of
writing Eq. (11) for the elastica problem posed and solved by the Bernoullis
and Euler [2]. In that case, $p=0$, so the rescaling (12) does not apply.
However, Eq. (11) for the curvature is now turned into a linear equation of
the simplest kind. Its solution is simply and exactly

$\displaystyle q(\theta)$
$\displaystyle=\frac{H}{B}\left(1+c\cos\theta\right),$
$\displaystyle\kappa(\theta)$
$\displaystyle=\pm\sqrt{\frac{H}{B}}\sqrt{2\left(1+c\cos\theta\right)}.$ (18)

where $c$ is an arbitrary constant and $\theta=0$ defines a direction of
reference. Specifying this direction of reference is equivalent to imposing
the second constant of integration of the equation for $q$. With $x$ and $y$
respectively defined as the cartesian coordinates along and perpendicular to
the reference direction, they satisfy Eqs. (15). They yield

$\displaystyle x(\theta)$
$\displaystyle=\sqrt{\frac{2B}{H}}\Bigg{\\{}x_{0}\pm\frac{\sqrt{1+c}}{c}\left[E\left(\frac{\theta}{2}\Big{|}\frac{2c}{1+c}\right)-\frac{1}{1+c}F\left(\frac{\theta}{2}\Big{|}\frac{2c}{1+c}\right)\right]\Bigg{\\}},$
(19) $\displaystyle y(\theta)$
$\displaystyle=\sqrt{\frac{2B}{H}}\left[y_{0}\mp\frac{1}{c}\left(\sqrt{1+c\cos\theta}-1\right)\right].$
(20)

Above, $x_{0}$ and $y_{0}$ are arbitrary constants, while the function
$F(\phi|m)$ and $E(\phi|m)$ are the incomplete elliptic integrals of the first
and second kinds, respectively [16]. Note that, up to a constant vertical
shift, the above expression makes it directly apparent that $y\propto\kappa$.

Eqs. (19) and (20) are an alternative parametric expression of the classical
elastica curves to what is found, e.g. in Refs. [1, 5]. It appears that the
$\theta$-formulation affords a somewhat more economical and direct route to
the solution, as well as a minimal use of elliptic functions. The shape of the
elastica is presently determined by the parameter $c$, while $H$ simply
contributes to a scale factor, see Fig. 1. It is required that $-1<c<\infty$,
with $c=0$ obviously corresponding to a perfect circle ($\kappa$ is constant
in that case). The value $c=1$ corresponds to a single loop and so does the
limiting case $c\to-1$. For $c>1$, $q(\theta)$ vanishes for some values of
$\theta$, indicating inflexion points. At these, different instances of Eqs.
(19) and (20) can be used with distinct values of $c_{x}$ and $c_{y}$ and
possibly a change of the sign of $\kappa$ to construct a smooth elastica curve
in a piecewise manner. As an illustration of this last point, we draw in Fig.
1 the elastic curve for $c=2.16279$. This is the limiting value for the
existence of self-crossing.

Figure 1: Some examples of classic elastic curves ($p=0$) obtained for
different values of $c$ in Eqs. (20) and (19). With $c>1$, the curve contains
inflexion points, at which copies of Eqs. (20) and (19) with different
integration constants $c_{x,y}$ can be smoothly joined. An example is given
with $c=2.16279$, value above which such composite curves without self-
crossing can be constructed.

It will be useful in what follows to detail further the one-loop solution with
$c=-1$, also known as syntractix curve [17, p. 277]. Then
$q=2(H/B)\sin^{2}\theta/2$, $\kappa=\pm 2\sqrt{H/B}\sin\theta/2$ and the
equation for $x$ and $y$ can be integrated as

$\displaystyle x$
$\displaystyle=x_{0}\pm\sqrt{\frac{B}{H}}\left(2\cos\frac{\theta}{2}+\ln\left|4\tan\frac{\theta}{4}\right|\right),$
$\displaystyle y$ $\displaystyle=y_{0}\pm
2\sqrt{\frac{B}{H}}\sin\frac{\theta}{2}.$ (21)

In complex notation ($z=x+iy$), this is

$z=z_{0}\pm\sqrt{\frac{B}{H}}\left(2e^{i\theta/2}+\ln\left|4\tan\frac{\theta}{4}\right|\right).$
(22)

(The factor $4$ inside the logarithm above is introduced for convenience, in
view of studying this expression in the limit where the argument $\theta\to
0$.)

##  IV Application 1: buckling instability of a compressed elastic ring

To further demonstrate the interest of Eq. (11) or, equivalently, Eq. (13),
let us consider a circular ring subject to uniform inward pressure and compute
the buckling threshold at which it ceases to be circular. We thus look for a
solution of the form

$\displaystyle K$ $\displaystyle\sim K_{0}+\delta\cos(n\theta)+\dotsb,$
$\displaystyle Q$ $\displaystyle\sim K_{0}^{2}/2+\delta
K_{0}\cos\left(n\theta\right)+\dotsb,$ (23)

where $\delta\ll 1$ and $n$ is a positive integer. Above, $K_{0}$ is the
rescaled curvature prior to buckling. Substituting this ansatz in Eq. (13), we
get ($\epsilon^{2}$ is not necessarily small, here, and not necessarily
positive either)

$\epsilon^{2}\frac{K_{0}^{2}}{2}+\frac{1}{K_{0}}-1+\delta\left[\epsilon^{2}K_{0}\left(-n^{2}+1\right)-\frac{1}{K_{0}^{2}}\right]+O\left(\delta^{2}\right)=0.$
(24)

This yields two equations for the two unknowns $K_{0}$ and $\epsilon^{2}$:

$\displaystyle\epsilon^{2}\frac{K_{0}^{2}}{2}+\frac{1}{K_{0}}$
$\displaystyle=1,$
$\displaystyle\epsilon^{2}K_{0}\left(-n^{2}+1\right)-\frac{1}{K_{0}^{2}}$
$\displaystyle=0.$ (25)

Given that $n$ was defined as a positive integer, it directly follows from the
second equation above that $n\neq 1$ and, hence, $n\geq 2$. Given the ring
radius $R=1/\kappa$ prior to buckling, we have from (12) that $K_{0}=H/(pR)$.
Substituting this expression in the second of Eqs. (25) and using Eq. (14) to
eliminate $\epsilon^{2}$, one directly obtains the classical buckling pressure
derived by Halphen [18] (see also [15]):

$-p=\frac{B}{R^{3}}\left(n^{2}-1\right).$ (26)

Meanwhile, from the first of Eqs. (25), we obtain

$H=\frac{B}{2R^{2}}+pR=\frac{B}{R^{2}}\left(\frac{3}{2}-n^{2}\right),$ (27)

which completes the description. Note that $H<0$, and hence $\epsilon^{2}<0$
in the present situation. We thus see that the $\theta$-formulation provides a
particularly short route to the answer of this classical problem.
Incidentally, by Eq. (5), the compression along the elastica at buckling
threshold is $-t=(n^{2}-1)B/R^{2}$. In terms of its half-perimeter $L$, this
is $\pi^{2}(n^{2}-1)B/L^{2}$, to be compared with the critical buckling load
of compression $\pi^{2}n^{2}B/L^{2}$ for a hinged straight beam of the same
length [19]. Recently, the dynamical aspects of this buckling instability have
been examined both theoretically and experimentally [20, 21]. Interestingly,
they used the analogy with a straight beam under effective compression as a
toy model for their experiment and, in [21], started out their analysis by
introducing the dimensionless parameter $P=pR^{3}/B$. We will return to this
adimensional parameter in § VI.

##  V Application 2: small-$\epsilon$ hook shape

From now on, we assume a small bending stiffness, in the sense that
$0<\epsilon\ll 1$, and exploit this limit using boundary layer analysis.
Implicitly, we restrict our attention to $H>0$, which includes all situations
where $t>0$ at least at one point in the elastica. We first consider an
elementary curve made up of two segments, one with small curvature, the other
with a large negative curvature. The resulting hook-curve will be used in
latter sections as a building block to construct more complex shapes, such as
the experimental shape that will be produced in § VI, see Figs 3 and 5.

###  V.1 Region of small curvature

In Eq. (13), let us assume that $K=O(1)$. Then, letting $\epsilon\to 0$, we
obtain

$K\sim K_{I}=1+O(\epsilon^{2}).$ (28)

In view of Eq. (12), the above expression means $p\sim\kappa H$. Next, $H\sim
t$, see Eq. (17). Therefore, $K\sim 1$ simply expresses that

$p\sim\kappa t,$ (29)

which is nothing but the classical force balance equation of a membrane.
Furthermore, since the curvature is approximately constant, the elastica is
locally circular and we may infer without further calculation that,

$\begin{pmatrix}x\\\ y\end{pmatrix}\sim\begin{pmatrix}x_{I}\\\
y_{I}\end{pmatrix}\sim\begin{pmatrix}x_{0}\\\
y_{0}\end{pmatrix}+\frac{H}{p}\begin{pmatrix}\sin\theta\\\
-\cos\theta\end{pmatrix}+O(\epsilon^{2}),$ (30)

where $x_{0}$ and $y_{0}$ are constants of integration. In complex notation,
this is

$z\sim z_{I}\sim z_{0}-\frac{iH}{p}e^{i\theta}.$ (31)

###  V.2 Region of large absolute curvature

Where the curvature is large, let us assume that

$\displaystyle K$ $\displaystyle=\epsilon^{-1}K_{II},$ $\displaystyle Q$
$\displaystyle=\epsilon^{-2}Q_{II},$ (recall that $Q=K^{2}/2$) (32)

where $K_{II},Q_{II}=O(1)$. This leads to a different balance between terms in
Eq. (13), which becomes

$\partial^{2}_{\theta}Q_{II}+Q_{II}+\frac{\epsilon}{K_{II}}=1,$ (33)

Neglecting the $O(\epsilon)$ term above, we directly obtain

$\displaystyle Q_{II}$ $\displaystyle\sim
Q_{II}^{(0)}=1+c\cos\left(\theta-\alpha\right),$ $\displaystyle\to K_{II}$
$\displaystyle\sim
K_{II}^{(0)}=-\sqrt{2\left[1+c\cos\left(\theta-\alpha\right)\right]},$ (34)

where $c$ and $\alpha$ are constants to be determined. Since the pressure term
$\epsilon/K_{II}$ is negligible in first approximation, we recover, to leading
order, the classical Euler elastica of § III with the horizontal direction
rotated by an angle $\alpha$. The negative sign of $K_{II}$ will be justified
in the next section. In particular, we may directly infer in the case $c=-1$
that [see Eq. (22)]

$\displaystyle c$ $\displaystyle=-1:$ $\displaystyle z$ $\displaystyle\sim
z_{II}\sim\hat{z}_{0}\pm\sqrt{\frac{B}{H}}e^{i\alpha}\left(2e^{i(\theta-\alpha)/2}+\ln\left|4\tan\frac{\theta-\alpha}{4}\right|\right).$
(35)

In this last expression, $\sqrt{B/H}$ is in fact equal to $\epsilon H/p$,
which makes explicit that the region of space spanned by the large-curvature
region is a factor $\epsilon$ smaller than in low-curvature region. The factor
$e^{i\alpha}$ in (35) indicates that the elastica is rotated by an angle
$\alpha$ in the complex plane, and the argument $\theta-\alpha$ in the
trigonometric functions come from the fact that the direction of reference for
this functional dependence was also rotated by an angle $\alpha$ compared to §
III. Alternatively, one can note that
$\kappa(\theta)=\kappa_{\text{E}}(\theta-\alpha)$, where
$\kappa_{\text{E}}(\theta)$ is the curvature derived in § III. Then Eq. (16)
becomes

$\partial_{\theta}z=\frac{e^{i\theta}}{\kappa_{\text{E}}(\theta-\alpha)}=e^{i\alpha}\frac{e^{i(\theta-\alpha)}}{\kappa_{\text{E}}(\theta-\alpha)},$
(36)

and the link between (22) and (35) directly follows. Finally, the $\pm$ sign
in (35) is deduced as follows: in (34), $-\sqrt{2[1-\cos(\theta-\alpha)]}=\pm
2\sin[(\theta-\alpha)/2]$. Hence, if $\theta<\alpha$, a $+$ sign corresponds
to a negative curvature and vice versa.

###  V.3 Inflexion point

We now examine the vicinity of the inflexion point, where $\kappa$ vanishes,
i.e., where $\theta$ reaches an extremum. As a function of $\theta$, the
curvature undergoes rapid variations in order to achieve the connection
between the two limiting behaviours, $K_{I}$ and $\epsilon^{-1}K_{II}$.
Denoting the extremum value of $\theta$ by $\theta_{*}$, let us write

$\displaystyle K$ $\displaystyle=K_{III}(\xi),$ $\displaystyle Q$
$\displaystyle=Q_{III}(\xi),$ $\displaystyle\xi$
$\displaystyle=\frac{\theta-\theta_{*}}{\epsilon}.$ (37)

Written in this way, the functions $K_{III}$ and $Q_{III}$ vary rapidly in the
vicinity of $\theta_{*}$ in the small-$\epsilon$ limit. Eq. (13) then becomes

$\partial_{\xi}^{2}Q_{III}+\frac{1}{K_{III}}=1+O\left(\epsilon^{2}\right).$
(38)

Expressing this equation in terms of $K_{III}$ only yields

$\partial_{\xi}\left(K_{III}\partial_{\xi}K_{III}\right)+\frac{1}{K_{III}}\sim
1.$ (39)

Multiplying by $K_{III}\partial_{\xi}K_{III}$ and integrating, we obtain

$\frac{1}{2}\left(K_{III}\partial_{\xi}K_{III}\right)^{2}+K_{III}=\frac{1}{2}K_{III}^{2}+\mathcal{E}.$
(40)

where $\mathcal{E}$ is a constant. On the positive-curvature side of the
inflexion point, we have the matching condition

$\lim_{\xi\to\pm\infty}K_{III}\sim\lim_{\theta\to\theta_{*}}K_{I}=1.$ (41)

Above, the limit $\xi\to-\infty$ is implied if $\theta_{*}$ is a maximum and
$\xi\to\infty$ if $\theta_{*}$ is a minimum. In either case, it follows that
$\mathcal{E}=1/2$ in Eq. (40), which can now be rewritten in terms as

$\left(K_{III}\partial_{\xi}K_{III}\right)^{2}=\left(K_{III}-1\right)^{2},$
(42)

that is:

$K_{III}\partial_{\xi}K_{III}=\pm\left(K_{III}-1\right).$ (43)

This is easily integrated as

$e^{K_{III}-1}(K_{III}-1)=-e^{\pm\xi-1},$ (44)

where we used the fact $\xi=0$ is the inflexion point, i.e. that
$K_{III}(0)=0$. In order for the solution $K_{III}$ of the above equation to
be real, it is necessary that $\pm\xi<0$. If the inflexion angle $\theta_{*}$
is a local maximum, then $K_{III}$ is defined for negative values of $\xi$.
Conversely, if $\theta_{*}$ is a local minimum, then $K_{III}$ is defined for
positive values of $\xi$. Therefore, in the right hand side of Eq. (44), we
must have

$\displaystyle-e^{\xi-1}$ if $\theta_{*}$ is maximum,
$\displaystyle-e^{-\xi-1}$ if $\theta_{*}$ is minimum.

The transcendental equation $we^{w}=z$ possesses two real solutions for $w$,
namely the Lambert functions $W_{0}(z)$ and $W_{-1}(z)$. Hence, two solutions
exist for $K_{III}$. The former is

$K_{III}=1+W_{0}(-e^{\pm\xi-1}),$ (45)

with far field behaviour

$\displaystyle K_{III}$ $\displaystyle\sim 1,$ $\displaystyle Q_{III}$
$\displaystyle\sim\frac{1}{2},$ $\displaystyle\text{as }\pm\xi\to-\infty.$
(46)

The latter is

$K_{III}=1+W_{-1}(-e^{\pm\xi-1}),$ (47)

with

$\displaystyle K_{III}$ $\displaystyle\sim\pm\xi,$ $\displaystyle Q_{III}$
$\displaystyle\sim\frac{\xi^{2}}{2}=\frac{\left(\theta-\theta_{*}\right)^{2}}{2\epsilon^{2}},$
$\displaystyle\text{as }\pm\xi\to-\infty.$ (48)

The two solutions coincide and vanish at the inflexion point $\xi=0$.
Moreover, it is immediate to check [compare (28) and (46)] that the solution
involving the function $W_{0}$ naturally matches with the low-curvature
solution $K_{I}$. Let us now consider the limiting behaviour of $Q$ in region
$II$ as $\theta\to\theta_{*}$. Expanding (34) to second order in
$\theta-\theta_{*}$ and recalling the scaling (32):

$\epsilon^{-2}Q_{II}\sim\epsilon^{-2}\left[1+c\cos\left(\theta_{*}-\alpha\right)-c\sin\left(\theta_{*}-\alpha\right)\left(\theta-\theta_{*}\right)-c\cos\left(\theta_{*}-\alpha\right)\frac{\left(\theta-\theta_{*}\right)^{2}}{2}\right].$
(49)

This matches (48) if $\alpha=\theta_{*}$ and $c=-1$. Furthermore, with
$W_{-1}$, the function $K_{III}$ is negative and, hence, so is the curvature,
in the high-curvature region. This justifies the negative sign of $K_{II}$ in
(34).

Finally, let us compute local approximations of the elastic curve in the
vicinity of the inflexion point:

$z\sim z_{III}\sim
z_{*}+\frac{H}{p}\int_{\theta_{*}}^{\theta}\frac{e^{i\theta^{\prime}}\mathrm{d}\theta^{\prime}}{K_{III}\left[\left(\theta^{\prime}-\theta_{*}\right)/\epsilon\right]}\sim
z_{*}+\frac{\epsilon
H}{p}e^{i\theta_{*}}\int_{0}^{\xi}\frac{\mathrm{d}\xi^{\prime}}{K_{III}\left(\xi^{\prime}\right)}\\\
=z_{*}\pm\frac{\epsilon
H}{p}e^{i\theta_{*}}\ln\left|1-K_{III}\left(\xi\right)\right|=z_{*}\pm\frac{\epsilon
H}{p}e^{i\theta_{*}}\ln\left|W_{k}(-e^{\pm\xi-1})\right|,$ (50)

where we used Eq. (43) to perform the integration. Above, $z_{*}$ is the
location of the inflection point and $k=0,-1$ depending on the sign of the
curvature.

On the positive-curvature side ($k=0$), we find the far-field behaviour

$\displaystyle K_{III}$ $\displaystyle>0:$ $\displaystyle z_{III}$
$\displaystyle\sim z_{*}+\frac{\epsilon H}{p}e^{i\theta_{*}}\xi,$
$\displaystyle\pm\xi\to-\infty.$ (51)

On the negative-curvature side ($k=-1$), we have

$\displaystyle K_{III}$ $\displaystyle<0:$ $\displaystyle z_{III}$
$\displaystyle\sim z_{*}\pm\frac{\epsilon
H}{p}e^{i\theta_{*}}\ln\left|\xi\right|,$ $\displaystyle\pm\xi\to-\infty.$
(52)

###  V.4 Composite approximations

Figure 2: Illustration of the composite approximation of the hook shape with
$\epsilon=0.1$ and $\theta_{*}=2.9$. (a) Curvature as a function of $\theta$.
The blue and orange curves are given by Eq. (54) and Eq. (57), respectively.
(b) Corresponding elastica shape in physical space, drawn using Eqs. (56) and
(60). The arrows indicates on which side of the elastica a net pressure $p$ is
applied.

Combining the previous approximations, composite, uniformly valid, asymptotic
approximations can be constructed (See, e.g., [12]). In the entire positive-
curvature region, we write the curvature as

$K\sim K_{I}+K_{III}-\text{common part},$ (53)

Where the “common part” is the function that $K_{I}$ and $K_{III}$ have in
common in the region where both approximations are simultaneously valid.
Presently, the common part is simply $1$. Hence, a uniformly valid
approximation in the positive-curvature region is

$K_{\text{low}}\sim 1+W_{0}(-e^{\pm\xi-1}).$ (54)

Let us further construct a uniform approximation for $z$. Comparing the
asymptotic behaviour of (31) in the limit $\theta\to\theta_{*}$ with (51), we
have the matching condition

$z_{0}-\frac{iH}{p}e^{i\theta_{*}}\left[1+i\left(\theta-\theta_{*}\right)\right]=z_{*}+\frac{H}{p}e^{i\theta_{*}}\left(\theta-\theta_{*}\right).$
(55)

This condition can be used to obtain the complex integration constant $z_{0}$,
given $z_{*}$. The two members of this equality are also the common part of
(31) and (50) in their overlapping region of validity. Hence, a uniformly
valid, composite expression of the elastica shape in the region of low,
positive curvature is obtained by adding the formulas for $z$ in (31) and (50)
and subtracting the left hand side of (55):

$z_{\text{low}}\sim\frac{H}{p}\left\\{-i\left(e^{i\theta}-e^{i\theta_{*}}\right)+e^{i\theta_{*}}\left[-\theta+\theta_{*}\pm\epsilon\ln\left|W_{0}\left(-e^{\pm\xi-1}\right)\right|\right]\right\\}.$
(56)

In this expression, we have omitted the constant $z_{*}$, which amounts to an
arbitrary translation in the complex plane.

Turning now to the negative-curvature side of the inflexion point, the common
part of $K_{III}$ and $\epsilon^{-1}K_{II}$ in the region of overlapping
validity is $\pm\xi$, see (48). Hence a composite approximation is

$K_{\text{high}}\sim
1+W_{-1}(-e^{\pm\xi-1})\pm\epsilon^{-1}\left[2\sin\left(\frac{\theta-\theta_{*}}{2}\right)-\theta+\theta_{*}\right].$
(57)

Next, comparing the far field limit $\theta\to\alpha$ of (35) with (52), we
obtain the matching condition

$\hat{z}_{0}\pm\frac{\epsilon
H}{p}e^{i\alpha}\left(2+\ln\left|\theta-\alpha\right|\right)=z_{*}\pm\frac{\epsilon
H}{p}e^{i\theta_{*}}\ln\left|\xi\right|.$ (58)

Noting that
$\ln\left|\xi\right|=\ln\epsilon^{-1}+\ln\left|\theta-\theta_{*}\right|$, we
obtain the matching conditions

$\displaystyle\alpha$ $\displaystyle=\theta_{*},$
$\displaystyle\hat{z}_{0}\pm\frac{2\epsilon H}{p}e^{i\alpha}$
$\displaystyle=z_{*}\pm\frac{\epsilon H}{p}e^{i\theta_{*}}\ln\epsilon^{-1}.$
(59)

The expression obtained for $\alpha$ is consistent with what we found earlier.
The common part of $z_{II}$ and $z_{III}$ in their overlapping region of
validity being given by both sides of Eq. (58). Adding $z_{II}$ and $z_{III}$
and subtracting the left side of Eq. (58), we obtain the following uniform
approximation in the high- negative-curvature region:

$z_{\text{\,high}}\sim\pm\frac{\epsilon
H}{p}e^{i\theta_{*}}\left[4ie^{i(\theta-\theta_{*})/4}\sin\frac{\theta-\theta_{*}}{4}+\ln\left|\frac{4}{\theta-\theta_{*}}\tan\frac{\theta-\theta_{*}}{4}\right|+\ln\left|W_{-1}\left(-e^{\pm\xi-1}\right)\right|\right],$
(60)

again, up to an arbitrary complex translation $z_{*}$.

Equations (54), (56), (57), and (60) are the main analytical result of this
paper and are illustrated for a given inflexion angle $\theta_{*}$ and reduced
bending stiffness $\epsilon$ in Fig. 2. They form the leading-order
description of the elastica in the small-$\epsilon$ limit and we will indeed
see that they can be used to explicitly construct solutions in a wide variety
of situations. In practice, Eqs. (56) and (60) yield curves that are
remarkably close to the actual solutions of the elastica solution even for
moderately small values of $\epsilon$. We note however that, away from the
inflexion point, the actual angle made by the tangent to curves drawn with
these formula differs from $\theta$ by an $O(\epsilon)$ quantity. In other
words, the parameter $\theta$ appearing in Eqs. (56) and (60) is not exactly
the angle made by the tangent of these curves with the horizontal.

In response to this issue, an alternative to Eqs. (56) and (60) is to
numerically integrate

$\displaystyle z$
$\displaystyle\sim\frac{H}{p}\int_{\theta_{*}}^{\theta}\frac{e^{i\theta^{\prime}}}{K(\theta^{\prime})}\mathrm{d}\theta^{\prime},$
$\displaystyle K$ $\displaystyle=K_{\text{low}},K_{\text{\,high}}.$ (61)

We will use this expression to compute the condition of self-contact in the
next section.

##  VI Application 3: pinching a ring into a peanut or a trefoil shape

###  VI.1 Experiments and construction of the shape

In order to assess the validity of this asymptotic theory, we carry
experiments and numerical simulations where an elastic ring of radius $R$ and
square cross section of side $h$ is loaded with an internal pressure $p$ and
perturbed with two diametrically opposed indenters.

We manufacture our elastic rings by casting a silicon elastomer (Elite Double
8 from Zhermack, Young Modulus $E=250\pm 15$ kPa) into 3D-printed moulds.
These elastic rings are then placed on the surface of a bath of distilled
water. For each experiment, we initially add a small quantity of dishsoap at
the surface of the bath inside the ring. This results in a sudden drop in the
surface tension inside the ring. The difference of surface tension on either
side of the elastica produces a net outward force $p$ per unit length on the
order of 0.04Nm-1 [22]. The elastica is additionally constrained by two
opposite indenters fixed at a given distance $d$ from each other, imposing
point forces on the structure. Given the side length $h$, $B=Eh^{4}/12$ and we
have the dimensionless bendability

$\epsilon_{0}=\sqrt{\frac{B}{pR^{3}}}.$ (62)

We will see shortly [Eq. (69)] that $\epsilon_{0}$ is closely related to
$\epsilon$. Compared to the latter, it has the advantage of being fixed for a
given experimental set-up, independently of deformation. By varying $h$ in the
range $[0.5,2]$ mm and $R$ in the range $[24,53]$ mm, we are able to
experimentally achieve $0.01<\epsilon_{0}<1.1$. Note that in their study of
dynamical buckling, Kodio et al. [21] used the equivalent dimensionless
parameter $P=1/\epsilon_{0}^{2}$. Their linear stability analysis in the
large-$P$ limit showed that the most unstable mode number scales as
$(P/2)^{1/2}$, suggesting the excitation of boundary layers by the
instability.

For small values of $\epsilon_{0}$, this setup leads to peanuts shapes with
two positive, nearly constant, low-curvature regions separated by two boundary
layers of large negative curvature near the indenters point where the two
forces are applied. As $\epsilon_{0}$ is increased, these boundary layers
expand [Fig. 3(a)].

Such shapes can be asymptotically described by combining solutions of the
forms (56) and (60) derived in the previous section. A point force at a given
arc length $s_{0}$ is modelled by the addition of a delta function in the
right hand side of Eq. (1): $p\to p-f\delta\left(s-s_{0}\right)$. This
produces the discontinuities

$\left[B\partial_{s}\kappa\right]^{+}_{-}=-\left[n\right]^{+}_{-}=f,$ (63)

In the $\theta$-formulation, the jump is

$\displaystyle\left[\partial_{\theta}q\right]^{+}_{-}$ $\displaystyle=f/B,$
$\displaystyle\iff$ $\displaystyle\left[\partial_{\theta}Q\right]^{+}_{-}$
$\displaystyle=\frac{f}{H\epsilon^{2}}.$ (64)

Bearing this discontinuity in mind, it is simple to stitch the solutions (54)
and (57) into a piecewise function that approximates the curvature along the
deformed ring. This is illustrated in Fig. 3. In the case of a peanut shape,
there are 4 inflection points with angles $\theta_{1}$ to $\theta_{4}$ as
defined in the Fig. 3(c). By symmetry, only $\theta_{1}$ needs to be
specified, since

$\displaystyle\theta_{2}$ $\displaystyle=\pi-\theta_{1},$
$\displaystyle\theta_{3}$ $\displaystyle=\pi+\theta_{1},$
$\displaystyle\theta_{4}$ $\displaystyle=2\pi-\theta_{1}.$ (65)

The solution can be constructed by substituting $\theta_{*}=\theta_{1}$,
$\theta_{2}$, $\theta_{3}$, and $\theta_{4}$ and adding appropriate complex
translations in the formulas of § V (See Mathematica files as Supplementary
Material for examples of implementation). Given $\epsilon$, it appears from
these formulas that the shape is entirely determined, up to a scale factor
$H/p$, by $\theta_{1}$. Fig. 3(b) and (c) displays the result obtained by
evaluating Eqs. (54) and (57) and Eqs. (56), and (60), respectively, for
$\epsilon=0.1$ and $\theta_{1}=2.9$. It is visually clear that the existence
of an inflexion point requires $\theta_{1}>\pi/2$, while selfcontact
necessarily arises as $\theta_{1}\to\pi$ in the $\epsilon\to 0$ limit.

The slope discontinuities in Fig. 3(b) at $\theta=\pi/2$ and $3\pi/2$ are due
to the point forces. Let us now evaluate them. On either side of
$\theta=\pi/2$, in the vicinity of the applied force, we have
$Q_{-}\sim\left[1-\cos\left(\theta-\theta_{1}\right)\right]/\epsilon^{2}$ and
$Q_{+}\sim\left[1-\cos\left(\theta-\theta_{2}\right)\right]/\epsilon^{2}$,
respectively. Evaluating expression (64) at $\theta=\pi/2$, we thus find

$f=-2H\cos\theta_{1}.$ (66)

Meanwhile, the large-absolute curvature regions of the ring only make up an
$O(\epsilon)$ fraction of its length. Hence, the length of the deformed ring
is, to leading order, given by $4\theta_{1}/\kappa_{0}$, where $\kappa_{0}$ is
the nearly constant curvature away from the pinch. If the ring is
inextensible, this length must be equal to $2\pi R$:

$\frac{4\theta_{1}}{\kappa_{0}}\sim 2\pi R+O\left(\epsilon\right).$ (67)

Now $\kappa_{0}$ corresponds to $K\sim 1$, so $\kappa_{0}\sim p/H$. Hence,
$H\sim\pi pR/(2\theta_{1})$. Substituting this expression into Eq. (66), we
eventually obtain

peanut: $\displaystyle f$ $\displaystyle\sim-p\pi
R\frac{\cos\theta_{1}}{\theta_{1}}+O\left(\epsilon\right),$
$\displaystyle\pi/2\leq\theta_{1}<\pi.$ (68)

This relates the applied point force to the resulting inflexion angle
$\theta_{1}$, i.e., to the deformation of the ring. Using the expression of
$H$ that we just derived, we further have

$\displaystyle\epsilon^{2}$
$\displaystyle=\frac{B}{pR^{3}}\left(\frac{2\theta_{1}}{\pi}\right)^{3}\equiv\epsilon_{0}^{2}\left(\frac{2\theta_{1}}{\pi}\right)^{3},$
$\displaystyle\epsilon_{0}$ $\displaystyle=\sqrt{\frac{B}{pR^{3}}}.$ (69)

where the bendability parameter $\epsilon_{0}$, introduced earlier in this
section, is the value of $\epsilon$ obtained in the limit
$\theta_{1}\to\pi/2$, i.e., $f\to 0$. It is important to realise that the
small parameter $\epsilon$ on which the asymptotic development is based
depends on $H$ and, hence, varies in the course of deformation, see Eq. (69).
While $\epsilon$ is more natural and convenient from an analytical point of
view, the alternative small parameter $\epsilon_{0}$ is more sensible from an
experimental point of view. Indeed, its value is fixed once and for all given
$B$, $p$ and $R$ and independently of the point force $f$.

Taking the value of the inflection angle $\theta_{1}$ given by numerical
simulations (detailed below), we may thus reconstruct the shape of the peanuts
without any fitting parameter [dashed lines in Fig. 3(a)]. As expected, we
obtain a very good agreement between experiments and theory in the small
$\epsilon_{0}$ limit up to $\epsilon_{0}\approx 0.1$, past which the accuracy
of the asymptotic theory gradually decreases.

Finally, evaluating the length $\ell$ of the peanut shape on the basis of the
outer solution, we easily derive, in the limit $\epsilon\to 0$,

$\ell(\theta_{1})=\frac{2H}{p}\left(1-\cos\theta_{1}\right)=\pi
R\frac{1-\cos\theta_{1}}{\theta_{1}}.$ (70)

The surprising feature of this formula is that $\ell(\theta_{1})$ passes by a
maximum value of $\ell(2.33)\approx 2.28R$ in the course of deformation, with
$\ell(\pi)=\ell(\pi/2)=2R$. This follows entirely from the inextensibility
constraint.

Figure 3: (a) Comparison between experimental and analytical shapes of the
peanut for increasing values of $\epsilon_{0}$ ($h=0.4,0.8,1,1.4$ mm ,
$R=47.2,53.4,51.5,47.1$ mm, corresponding to
$\epsilon_{0}=0.011,0.037,0.061,0.138$ respectively; $d=19.2$ mm). The value
of the inflexion angle $\theta_{1}$ required to compute the shape is taken
from the numerical simulation with the same parameters. (b) $\kappa$ vs
$\theta$ in an elastic circular ring subjected to outward pressure and two
opposite inward point forces. $\epsilon=0.1$. (c) the resulting peanut shape,
constructed piecewise using Eqs. (54), (56), (57), and (60). The circled
numbers indicate correspondance between the two graphs. (d) Inflexion angle
$\theta_{1}$ as a function of the rescaled point force $f/p\pi R$. The solid
line corresponds to the theory, see Eq. (68), the dashed line to the numerical
simulations with $\epsilon_{0}=5.2\times 10^{-3}$ . Beyond the rightmost
point, the elastica snaps to contact. Animations showing deformation and
snapping are given in Supplementary Information.

###  VI.2 Snap-through buckling

Equation (68) implies, unexpectedly, that there is a maximum value of $f$
beyond which there is no solution, see Fig. 3(d). This limit point (LP)
corresponds to a value of $\theta_{1}$ given by $\theta_{1,LP}\approx 2.80$,
where $f=f_{LP}\approx 0.34p\pi R$. To explain this observation, note that, at
the inflexion point, $H$ is exactly given by $t$. Therefore, with the aid of
Fig. 3, it is immediate to recognize in (66) the horizontal force balance

$f\sim-2t\cos\theta_{1}$ (71)

over the entire negative-curvature region. Indeed, recalling that its extent
is only of $O\left(\epsilon\right)$, the pressure contribution is negligible
to leading order in that equilibrium condition. The absence of a static
solution for $f>f_{LP}$ can thus be simply understood by the fact that the
tension $t$ inside the ring is not sufficient to balance the point force $f$.
Henceforth, we conclude that the ring must snap into self-contact. Controlling
the force rather than the displacement in the experiments is challenging, and
we therefore turn to numerical simulations to check this prediction.

We simulate the ring by discrete Cosserat rod equations, which we integrate
using the method of Ref. [23]. In this scheme, rotation matrices are output
for every discrete point of the elastica and the angle $\theta(s)$ can
directly be retrieved from it. We study separately displacement-controlled and
force-controlled deformations.

In the first scenario, we impose the distance $d$ between two diametrically
opposed points of the ring. We use values of $B$, $R$ and $p$ directly
inferred from the experimental parameters, which fixes the value of
$\epsilon_{0}$. Starting from the unstrained state $d=2R$, we quasistatically
decrease $d$ and monitor the inflexion point $\theta_{1}$ as a function of
$d$. This value of $\theta_{1}$ is then substituted in the analytical formulas
to compare analytical and experimental curves in Fig. 3(a).

In the second scenario, the value of the force $f$ is imposed and $\theta_{1}$
and $d$ are monitored as a function of $f$. For very small values of
$\epsilon_{0}$, self-contact arises, as predicted, by a snap-through at the
limit point value $f_{LP}$ and very good agreement is found with the
analytical formula, Eq. (68), as shown in Fig. 3(d). As $\epsilon_{0}$ is
increased, the influence of the boundary layer becomes more important. For
$\epsilon_{0}>0.13$, the deformation in the boundary layers is sufficiently
large that self-contact happens for $f<f_{LP}$. In that case, no snap-through
buckling is observed, and the two scenarii (displacement-controlled and force-
controlled) become equivalent.

Figure 4: (a) Pictures of experimental peanuts shape when selfcontact is
imposed for increasing values of $\epsilon_{0}$ ($\epsilon_{0}=0.01$, 0.04 and
1.1 respectively). (b) Critical angle $\theta_{1,\text{SC}}$ at the transition
to selfcontact as a function of $\epsilon_{0}$ for the theory, numerical
simulations and experiments.

.

###  VI.3 Transition to selfcontact

In order to quantitatively assess the regime of validity of the asymptotic
theory, we enquire under which condition self-contact arises in both scenarios
(force- or displacement-controlled).

As mentioned before, when the control parameter is the applied force $f$, the
leading-order theory as $\epsilon_{0}\to 0$ predicts a critical inflexion
angle $\theta_{1,LP}\approx 2.80$ at which the elastica snaps to contact. As
$\epsilon_{0}$ is increased from zero, however, selfcontact can occur before
this critical angle is reached. In such a case, controlling the displacement
or the force are thus equivalent.

The condition of self-contact is simply expressed by saying that
$x(\theta=\pi/2)$ in the negative-curvature region is equal to
$x(\theta=\pi/2)$ in the positive curvature region. Using Eq. (61), the
condition is

$\int_{0}^{\theta_{1}}\frac{\cos{\theta}}{K_{\text{low}}(\theta,\epsilon)}\mathrm{d}\theta=\int_{\pi/2}^{\theta_{1}}\frac{\cos{\theta}}{K_{\text{\,high}}(\theta,\epsilon)}\mathrm{d}\theta.$
(72)

This equation yields the value of $\theta_{1,\text{SC}}$ at which self-contact
happens as a function of $\epsilon$. Solving Eq. (72) as an equation for
$\theta_{1}$ with different values of $\epsilon$, we eventually obtain

self-contact:
$\displaystyle\theta_{1,\text{SC}}\approx\pi\times\frac{1+189\epsilon+474\epsilon^{2}}{1+192\epsilon+722\epsilon^{2}}\approx\pi\times\frac{1+103\epsilon_{0}+522\epsilon_{0}^{2}}{1+109\epsilon_{0}+778\epsilon_{0}^{2}},$
(73)

where we recall that $\epsilon_{0}=\epsilon/(2\theta_{1}/\pi)^{3/2}$. For
instance, with $\epsilon_{0}=0.1$, we find $\theta_{1}\approx 2.64$ at self-
contact and estimate through Eq. (68) that this happens for an applied force
$f\approx 0.332\times p\pi R$.

In Fig. 4, we compare the above analytical estimate with the experimental
values for displacement-controlled deformation and numerical simulations for
both displacement- and force-controlled deformation. The value
$\epsilon_{0}^{*}\approx 0.13$ marks the separation between numerical outcomes
when either $d$ or $f$ is imposed in the numerical code. For
$\epsilon_{0}<0.13$, snap-through is numerically observed and the inflexion
angle corresponding to this instability is seen to tend to the analytically
predicted value as $\epsilon_{0}$ is further decreased. An example of snap-
through dynamic observed in our numerical simulations for a fixed force $f$
slightly larger than the critical value is shown in the supplementary movie
for $\epsilon_{0}=9\times 10^{-3}$.

Overall, a good agreement is found between the three approaches for small
enough $\epsilon_{0}$. For large $\epsilon_{0}$, the quantitative predictions
of the boundary layer theory become inaccurate. Both numerical simulations and
experiments indicate a plateau in Fig. 4(b). For such large values of
$\epsilon_{0}$ and, hence, $\epsilon$, the pressure term in Eq. (11) becomes
irrelevant and we recover the classical problem of § III. In that frame, we
note that selfcontact arises at $c=2.16279$ (Fig. 1). Using this value in Eq.
(18), we find an inflexion angle $\arccos(-1/2.16279)\approx 2.05$ (dashed
black line to the right in Fig. 4b). Such a situation also corresponds to the
racket shape obtained experimentally by Py et al. in Ref. [24].

Figure 5: (a) An elastic circular ring deformed into a trefoil by three
opposite point forces. Left: $\epsilon=0.1$ and $\theta_{1}=2\pi/3$. Right:
$\epsilon=0.1$ and $\theta_{1}=3.34$. The latter configuration is mechanically
unstable if the control parameter is $f$ (see text). (b) Comparison between
experimental and theoretical trefoil shapes for increasing values of
$\epsilon_{0}$ ($\epsilon_{0}=0.01$, 0.04, 0.09).

.

###  VI.4 Trefoil and $N$-foil shape

Thanks to the simplicity brought about by the preceding asymptotic argument,
we can easily generalize our discussion to the case of $N>2$ point forces
applied symmetrically to an elastic ring. The relation (66) continues to hold
on each side of the applied point force but this time, the inextensibility
condition is

$\frac{2N\theta_{1}}{\kappa_{0}}\sim 2\pi R+O\left(\epsilon\right).$ (74)

Hence, for a symmetric $N$-foil shape, we have

$N$-foil : $\displaystyle f_{N}$ $\displaystyle\sim-p\pi
R\frac{2\cos\theta_{1}}{N\theta_{1}}+O\left(\epsilon\right),$
$\displaystyle\epsilon$
$\displaystyle=\epsilon_{0}\left(\frac{N\theta_{1}}{\pi}\right)^{3/2},$ (75)

where, as before, $\theta_{1}=\pi/2$ correspond to a vanishing compressing
force applied horizontally and $\epsilon_{0}=\sqrt{B/pR^{3}}$. The maximum
force for which a solution exists decreases as $1/N$ but the maximum inflexion
angle remains unchanged. The symmetric trefoil is illustrated for two values
of $\epsilon$ and $\theta_{1}$ in Figs. 5. (Mathematica file to generate the
figure given in Supplementary Information.) One of the situation depicted
correspond to $\theta_{1}>\theta_{1,LP}$ and is therefore mechanically
unstable in the case of a prescribed value of $f$. Again, if the displacement
at the pinching point is prescribed, rather than $f$, no instability occurs.
Furthermore, we note that, contrary to the peanut, a nonzero value of
$\epsilon$ allows one to attain values of $\theta_{1}$ larger than $\pi$. The
existence of a limit point again implies a snapping instability in the force-
driven deformation and this is again confirmed by numerical simulations. It is
remarkable that the LP remains associated to the same value of $\theta_{1}$,
independently of $N$ and $p$.

#### Remark

So far, we have confined our attention to symmetrically applied forces, as
this automatically ensures static equilibrium. More general sets of forces can
obviously be applied, provided that they are again in static equilibrium,
i.e., their vector sum and total moment both vanish. One technical difference
applies in that case: the tension $t$ also becomes discontinuous at the points
where loads are applied. Indeed the applied forces now generally contain a
non-vanishing component along the tangent of the elastica. Calling this
component of force $f_{t}$, we have the discontinuous jump

$\left[t\right]_{-}^{+}=-f_{t}.$ (76)

This results in a discontinuity

$\left[H\right]_{-}^{+}=-f_{t}$ (77)

in the constant of integration to be used on either side of the point force.
Bearing this modification in mind, the same piecewise construction as before,
based on Eqs. (54) and (57) can in principle be used.

##  VII Application 4: the stretched ring

Figure 6: Elastic circular ring subjected to inward pressure and two opposite
outward point forces with $\epsilon=0.15$ and $\theta_{1}=0.32$. (a) Top:
$\kappa$ vs $\theta$. (b) Bottom: piecewise analytically computed profile. (c)
Comparison with the experiment: dishsoap is added to the bath on the outside
of the elastic ring.

Let the pressure now be applied externally on the ring, while two opposite
point forces are applied outwards. This is the mechanical inverse of the
situation leading to the peanut shape, so we might call it the ‘anti-peanut’.
The formalism derived in previous sections directly applies here too and is
depicted in Fig. 6. Interestingly, the piecewise construction of the function
$\kappa(\theta)$ yields a qualitatively different picture than with the
peanut: branches of solutions intersect in the large-curvature side in Fig. 6
instead of the low-curvature side in Fig. 3(b). This is due to the fact that
the range of values of $\theta$ where $Q\gg 1$ makes up the majority of the
domain of $\kappa(\theta)$ in the anti-peanut configuration.

As before, we may capitalize on previous calculation to directly infer that
$f\sim 2t\cos\theta_{1}$ and that $\kappa t\sim p$ in the low-curvature
region, with the same inextensibility constraint as before. We then
immediately obtain

$\displaystyle f$ $\displaystyle\sim p\pi
R\frac{\cos\theta_{1}}{\theta_{1}}+O\left(\epsilon\right),$ $\displaystyle
0<\theta_{1}.$ (78)

This formula for $f$ is consistent with intuition: flatter shapes correspond
to smaller $\theta_{1}$ and hence to larger stretching forces. Above, there is
a maximum allowed value of $\theta_{1}$ to prevent self-crossing of the two
segments of low curvature. The maximum allowed value of $\theta_{1}$ increases
with $\epsilon$ as follows. In the low-curvature region ($K\sim 1$), the
curvature is nearly constant, so $\theta(s)\sim\kappa_{0}s$. At the inflexion
point $\theta_{1}\sim\kappa_{0}s_{1}$. Meanwhile, the vertical difference
between the center of the stretched elastica and the inflexion point is
approximately $\kappa_{0}s_{1}^{2}/2=\theta_{1}^{2}/(2\kappa_{0})$. To prevent
self-crossing this vertical elevation must be met by an equivalent opposite
vertical displacement in the region of large curvature (see Fig. 6), the
latter being on the order of $\epsilon/\kappa_{0}$. Hence, the maximum allowed
value of $\theta_{1}$ is $O\left(\epsilon^{1/2}\right)$.

At this stage, it is clear that the above configuration can easily be
generalized to a larger number of stretching forces, just as in the previous
section.

##  VIII Generalization with body forces

The success of Eq. (11) to address complex elastica shape was due the
reduction of the original problem by two differential orders: one thanks to
the existence of the first integral $H$, the other by switching the curve
parameterization to $\theta$. In the presence of body forces, $H$ is not
constant anymore but an asymptotic treatment along the previous lines can
still be envisaged. Eqs. (1) to (3) become

$\displaystyle\partial_{s}n$ $\displaystyle=p-\kappa t-\mathcal{F}_{n},$
$\displaystyle\partial_{s}t$ $\displaystyle=\kappa n-\mathcal{F}_{t},$
$\displaystyle B\partial_{s}\kappa$ $\displaystyle=-n.$ (79)

where $\mathcal{F}_{t}$ and $\mathcal{F}_{n}$ are external axial and shear
force per unit length, respectively. Recently, for instance, an analytically
solvable model for elastic wrinkling on a nonlinear foundation was introduced,
in which $\mathcal{F}_{t}=0$ while $\mathcal{F}_{n}$ scales as some power of
the local displacement of the elastica [25, 26, 27] (See also Refs. [28, 29,
30, 31, 32] for more experimental and theoretical results on the subject.)

We may continue to define $H$ as $B\kappa^{2}/2+t$. It now satisfies

$\partial_{s}H=-\mathcal{F}_{t}.$ (80)

In the $\theta$-formulation, we have

$\displaystyle
B\left(\partial^{2}_{\theta}q+q\right)+\frac{p-\mathcal{F}_{n}}{\kappa}$
$\displaystyle=H,$ $\displaystyle\partial_{\theta}H$
$\displaystyle=\frac{-\mathcal{F}_{t}}{\kappa}.$ (81)

In regions of large curvature, we continue to benefit from the simple leading
order problem

$\displaystyle B\left(\partial^{2}_{\theta}q+q\right)$ $\displaystyle\sim H,$
$\displaystyle H$ $\displaystyle\sim\text{const.},$ (82)

and we recover the approximation of previous sections. This time, however, the
value of $H$ generally varies between boundary layers. On the other hand, if
$\kappa$ is ‘small’, we have $H\sim t$ and we may neglect terms proportional
to $B$ in Eqs. (81). We thus have

$\displaystyle\kappa$ $\displaystyle\sim\frac{p-\mathcal{F}_{n}}{H},$
$\displaystyle\frac{1}{H}\partial_{\theta}H$
$\displaystyle\sim\frac{-\mathcal{F}_{t}}{p-\mathcal{F}_{n}}.$ (83)

Suppose for instance that gravity acts in the vertical direction, so that
$\mathcal{F}_{n}=-\rho g\cos\theta$, $\mathcal{F}_{t}=-\rho g\sin\theta$,
where $\rho$ is the mass per unit length. The equation for $H$ above is easily
solved in that case. We have

$H\sim\frac{\rho gh}{a+\cos\theta},$ (84)

for some constant $h$, where we define

$a=\frac{p}{\rho g}.$ (85)

Meanwhile $\kappa\sim\left(a+\cos\theta\right)^{2}/h$. Then, integration of
Eqs. (15) yields, up to an arbitrary translation,

$\displaystyle x$
$\displaystyle\sim\frac{2h}{\left(1-a^{2}\right)^{3/2}}\operatorname{arctanh}\left(\frac{1-a}{\sqrt{1-a^{2}}}\tan\frac{\theta}{2}\right)-\frac{ah\sin\theta}{\left(1-a^{2}\right)\left(a+\cos\theta\right)},$
$\displaystyle y$ $\displaystyle\sim\frac{h}{a+\cos\theta}.$ (86)

Finally near an inflexion point, the same local analysis as in Sec. V.3
applies to leading order.

##  IX Conclusions and perspectives

The present boundary layer theory leads to a composite solution that can be
used as a building-block to describe a wide variety of complex shapes. With
$\theta$ as the curve parameter, the solution takes an elementary form. The
only concession to the use of special functions is in the vicinity of
inflexion points, where the Lambert functions $W_{0}$ and $W_{1}$ arise.
Thanks to this simplified picture, simple local expressions of force balance
can be derived, allowing intuitive mechanical analysis. This gives confidence,
in particular, in the surprising prediction of snapping instability of pinched
peanuts and $N$-foil shapes past a certain applied force. It would be
interesting to see if such a scenario could find application in engineering or
in microbiology. Furthermore, a good agreement was obtained both with direct
numerical simulations and experiments.

Not all possible analyses afforded by the simple $\theta$-formulation have
been exposed here. First, it is in principle not difficult to compute the
first corrections to the leading order expressions that we have derived to
obtain more accurate formula and enlarge the range of values of the small
parameter $\epsilon$ where they are reliable. Secondly, not all possible
shapes have been envisaged within the present framework. We have already
mentioned how contact points with friction, i.e., with a non vanishing tangent
component, can be handled. Another situation that we haven’t explored is the
possibility to have segments of large, opposite curvature in succession,
rather than the low-high curvature detailed in Sec. V. Let us briefly explain
how this could in principle be handled. One would need to match two regions of
large curvatures with opposite signs, each described asymptotically by
expressions of the form (34). The intermediate region is now still described
as in Sec. V.3, only this time the constant of integration $\mathcal{E}$
appearing in (40) differs from $1/2$. This equation can still be integrated
implicitly, yielding $\xi$ as a function of $K_{III}$. Setting
$\mathcal{E}=1/2+b^{2}$, one can derive

$\sqrt{b^{2}+\left(K_{III}-1\right)^{2}}+\operatorname{arcsinh}\left(\frac{K_{III}-1}{b}\right)=\pm\xi+c$
(87)

where $c$ is a constant, but the inversion of the resulting formula is not
straightforward. In the limit $\xi=\pm\infty$, one obtains the desired
behaviour $K_{III}\sim\pm\xi$. One limit in which the above expression can be
inverted asymptotically is $b\to\infty$. We do not pursue the analysis here
but note that this limit could be relevant in situations where two point
forces of opposite signs are applied at close locations of the elastica rather
than far appart.

###### Acknowledgements.

G.K. is a Research Associate of the Fonds de la Recherche Scientifique - FNRS
(Belgium.) This project has received funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant
agreement n∘ 101027862, from the F.R.S.-FNRS under the research grant n∘
J.0017.21 (CDR “FASTER”) and from the Federation Wallonia-Brussels (Concerted
Research Actions “Capture”).

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# A modification of the mixed joint universality theorem
for a class of zeta-functions

Benjaminas Togobickij, Roma Kačinskaitė

###### Abstract

The property of zeta-functions on mixed joint universality in the Voronin’s
sense states that any two holomorphic functions can be approximated
simultaneously with accuracy $\varepsilon>0$ by suitable vertical shifts of
the pair consisting from the Riemann zeta- and Hurwitz zeta-functions. In [1],
it was shown rather general result, i.e., an approximating pair was composed
of the Matsumoto zeta-functions’ class and the periodic Hurwitz zeta-function.
In this paper, we prove that this set of shifts has a strict positive density
for all but at most countably many $\varepsilon>0$. Also, we give the
concluding remarks on certain more general mixed tuple of zeta-functions.

Keywords: mixed joint universality, joint value distribution, periodic Hurwitz
zeta-function, Matsumoto zeta-function, simultaneous approximation.

AMS classification: 11M06, 11M41, 11M36.

##  1 Introduction

Denote by ${\mathbb{N}}$, ${\mathbb{N}}_{0}$, ${\mathbb{P}}$, ${\mathbb{Q}}$,
${\mathbb{R}}$, and ${\mathbb{C}}$ the sets of positive integers, non-negative
integers, prime numbers, rational numbers, real numbers, and complex numbers,
respectively, and by $s=\sigma+it$ a complex variable.

As it is well-known, the Riemann zeta-function $\zeta(s)$ is defined by the
Dirichlet series and has a representation by the Euler product over primes,
i.e.,

$\zeta(s)=\sum_{m=1}^{\infty}\frac{1}{m^{s}}=\prod_{p\in{\mathbb{P}}}\bigg{(}1-\frac{1}{p^{s}}\bigg{)},\quad\sigma>1.$

The Hurwitz zeta-function $\zeta(s,\alpha)$ with the real parameter $\alpha$,
$0<\alpha\leq 1$, is given by the series

$\zeta(s,\alpha)=\sum_{m=0}^{\infty}\frac{1}{(m+\alpha)^{s}},\quad\sigma>1.$

Both of the functions $\zeta(s)$ and $\zeta(s,\alpha)$ have an analytic
continuation to the whole complex plane, except for a simple pole at $s=1$
with residue 1. Note that the function $\zeta(s,\alpha)$ has an Euler product
over primes only for the cases $\alpha=1$ and $\alpha=\frac{1}{2}$, when
$\zeta(s,1)=\zeta(s)$ and $\zeta(s,\frac{1}{2})=(2^{s}-1)\zeta(s)$.

In the first decade of XXI century, the new type of universality was
discovered by H. Mishou [2] and independently by J. Steuding and J. Sander
[3]. They have opened so-called mixed joint universality in Voronin’s sense
for the Riemann zeta- and the Hurwitz zeta-functions. More precisely, they
proved that a pair of analytic functions is simultaneously approximated by
shifts of a pair $\big{(}\zeta(s+i\tau),\zeta(s+i\tau,\alpha)\big{)}$ with
transcendental $\alpha$.

For the brevity, throughout this paper we use following notations and
definitions. Let $D(a,b)=\\{s\in{\mathbb{C}}:a<\sigma<b\\}$ for every real
numbers $a<b$. Denote by $\mathrm{meas}\\{A\\}$ the Lebesgue measure of a
measurable set $A\subset{\mathbb{R}}$, by $\mathcal{B}(S)$ the set of all
Borel subsets of a topological space $S$, and by $H(D)$ the set of all
holomorphic functions on $D$. For any compact set $K\subset{\mathbb{C}}$,
denote by $H^{c}(K)$ the set of all complex-valued continuous functions
defined on $K$ and holomorphic in the interior of $K$, while by $H^{c}_{0}(K)$
denote the subset of $H^{c}(K)$, consisting of all elements which are non-
vanishing on $K$.

######  Theorem 1 (see [2, Theorem 2]).

Suppose that $\alpha$ is a transcendental number. Let $K_{1}$ and $K_{2}$ be
compact subsets of $D(\frac{1}{2},1)$ with connected complements. Then, for
any $f_{1}(s)\in H^{c}_{0}(K_{1})$, $f_{2}(s)\in H^{c}(K_{2})$ and every
$\varepsilon>0$, it holds that

$\liminf_{T\to\infty}\frac{1}{T}\mathrm{meas}\bigg{\\{}\tau\in[0,T]:\sup_{s\in
K_{1}}\left|\zeta(s+i\tau)-f_{1}(s)\right|<\varepsilon,\sup_{s\in
K_{2}}\left|\zeta(s+i\tau,\alpha)-f_{2}(s)\right|<\varepsilon\bigg{\\}}>0.$
(1)

In 2013, a statement of universality theorems in terms of density was proposed
by J.-L. Mauclaire [4] and independently by A. Laurinčikas and L. Meška [5].
Such a statement of Theorem 1 was given in [6]. To be precise, let

$L(\alpha,{\mathbb{P}}):=\big{\\{}(\log
p:p\in{\mathbb{P}}),(\log(m+\alpha):m\in{\mathbb{N}}_{0})\big{\\}}.$

Then the following modification of Theorem 1 was obtained.

######  Theorem 2 (see [6, Theorem 2]).

Suppose that the elements of the set $L(\alpha,{\mathbb{P}})$ are linearly
independent over the field of rational numbers ${\mathbb{Q}}$. Let $K_{1}$,
$K_{2}$, $f_{1}(s)$, $f_{2}(s)$ be as in Theorem 1. Then the limit

$\lim_{T\to\infty}\frac{1}{T}\mathrm{meas}\left\\{\tau\in[0,T]:\sup_{s\in
K_{1}}\left|\zeta(s+i\tau)-f_{1}(s)\right|<\varepsilon,\sup_{s\in
K_{2}}\left|\zeta(s+i\tau,\alpha)-f_{2}(s)\right|<\varepsilon\right\\}>0$

exists for all but at most countably many $\varepsilon>0$.

More general results of the same type can be found, for example, in [7].

The aim of this paper is to show an analogous result as of Theorem 2 for
rather general classes of zeta-functions.

##  2 Statement of new result

A generalization of the Hurwitz zeta-function $\zeta(s,\alpha)$ was introduced
by A. Laurinčikas [8]. For the periodic sequence
$\mathfrak{B}=\\{b_{m}\in{\mathbb{C}}:m\in{\mathbb{N}}_{0}\\}$ with a minimal
period $k\in{\mathbb{N}}_{0}$ and a fixed real $\alpha$, $0<\alpha\leq 1$, the
periodic Hurwitz zeta-function $\zeta(s,\alpha;\mathfrak{B})$ is defined by
the Dirichlet series

$\zeta(s,\alpha;\mathfrak{B})=\sum^{\infty}_{m=0}\frac{b_{m}}{(m+\alpha)^{s}},\quad\sigma>1.$

Since the sequence $\mathfrak{B}$ is periodic sequence, then

$\zeta(s,\alpha;\mathfrak{B})=\frac{1}{k^{s}}\sum_{m=0}^{k-1}b_{m}\zeta\bigg{(}s,\frac{m+\alpha}{k}\bigg{)}.$

From this we deduce that the function $\zeta(s,\alpha;\mathfrak{B})$ has an
analytic continuation to the whole complex plane except for a simple pole at
the point $s=1$ with residue $b=\frac{1}{k}\sum_{m=0}^{k-1}b_{m}$.

A class of the Matsumoto zeta-functions is a second class under our interest,
particularly, since it covers a wide class of classical zeta-functions having
the Euler product representation over primes. It was introduced by K.
Matsumoto in [9]. For every $m\in{\mathbb{N}}$, let $g(m)\in{\mathbb{N}}$,
and, for $j\in{\mathbb{N}}$ with $1\leq j\leq g(m)$, $f(j,m)\in{\mathbb{N}}$.
Denote by $p_{m}$ the $m$th prime number, and $a^{(j)}_{m}\in{\mathbb{C}}$.
Assume that

$g(m)\leq
C_{1}p_{m}^{\alpha_{0}}\quad\text{and}\quad\left|a_{m}^{(j)}\right|\leq
p_{m}^{\beta_{0}}$

with a positive constant $C_{1}$ and non-negative constants $\alpha_{0}$ and
$\beta_{0}$. Define polynomials

$A_{m}(X)=\prod_{j=1}^{g(m)}\left(1-a_{m}^{(j)}X^{f(j,m)}\right)$

of degree $f(1,m)+\dots+f(g(m),m)$. The function

$\widetilde{\varphi}(s)=\prod_{m=1}^{\infty}\left(A_{m}(p_{m}^{-s})\right)^{-1}$
(2)

is called the Matsumoto zeta-function. The product on right-hand-side of the
equality (2) converges absolutely for $\sigma>\alpha_{0}+\beta_{0}+1$. In this
region, the function $\widetilde{\varphi}(s)$ has a Dirichlet series expansion
as

$\widetilde{\varphi}(s)=\sum_{k=1}^{\infty}\frac{\widetilde{c}_{k}}{k^{s}}$

with the coefficients satisfying an estimate
$\widetilde{c}_{k}=O(k^{\alpha_{0}+\beta_{0}+\varepsilon})$ for every
$\varepsilon>0$ if all prime factors of $k$ are large (for the details, see
[10]). For brevity, we define its shifted version by

$\varphi(s):=\widetilde{\varphi}(s+\alpha_{0}+\beta_{0})=\sum_{k=1}^{\infty}\frac{{c}_{k}}{k^{s}},$

where $c_{k}=\widetilde{c}_{k}k^{-\alpha_{0}-\beta_{0}}$. Then it is easy to
see that $\varphi(s)$ is absolutely convergent for $\sigma>1$.

Also we assume that the function $\varphi(s)$ satisfies the following
conditions:

  1. (i)

$\varphi(s)$ can be meromorphically continued to the region
$\sigma\geq\sigma_{0}$, $\frac{1}{2}\leq\sigma_{0}<1$, and all poles in this
region belong to a compact set which has no intersections with the line
$\sigma=\sigma_{0}$;

  2. (ii)

$\varphi(\sigma+it)=O\left(\left|t\right|^{C_{2}}\right)$ for
$\sigma\geq\sigma_{0}$ and a positive constant $C_{2}$;

  3. (iii)

it holds the mean-value estimate

$\int_{0}^{T}\left|\varphi(\sigma_{0}+it)\right|^{2}dt=O(T).$

All of functions satisfying above mentioned conditions construct the class of
Matsumoto zeta-functions, and we denote the set of all such $\varphi(s)$ as
$\mathcal{M}$.

In 2015, R. Kačinskaitė and K. Matsumoto proved [1] a mixed joint universality
theorem for wide class of Matsumoto zeta-functions and for the periodic
Hurwitz zeta-function with transcendental parameter.

For the proof of mixed joint universality, the Bagchi method [11] can be used,
but, in the case of the whole class $\mathcal{M}$ it is difficult to prove the
denseness lemma. Therefore, we use one more restriction class, namely, the
Steuding class ${\widetilde{S}}$ (see [12]).

We say that $\varphi(s)$ belongs to the class $\widetilde{S}$ if the following
assumptions are fulfilled:

  1. (a)

$\varphi(s)$ has a Dirichlet series expansion
$\varphi(s)=\sum_{m=1}^{\infty}a(m)m^{-s}$ with $a_{m}=O(m^{\varepsilon})$ for
every $\varepsilon>0$;

  2. (b)

there exists $\sigma_{\varphi}<1$ such that $\varphi(s)$ can be
meromorphically continued to the region $\sigma>\sigma_{\varphi}$, and is
holomorphic there, except for a pole at $s=1$;

  3. (c)

for any fixed $\sigma>\sigma_{\varphi}$ and any $\varepsilon>0$, there exists
a constant $C_{3}\geq 0$ such that
$\varphi(\sigma+it)=O(\left|t\right|^{C_{3}+\epsilon})$ ;

  4. (d)

there exists the Euler product expansion

$\varphi(s)=\prod_{p\in{\mathbb{P}}}\prod_{j=1}^{l}\left(1-\frac{a_{j}(p)}{p^{s}}\right)^{-1};$

  5. (e)

there exists a constant $\kappa>0$ such that

$\lim_{x\to\infty}\frac{1}{\pi(x)}\sum_{p\leq x}\left|a(p)\right|^{2}=\kappa,$

where $\pi(x)$ denotes the number of primes $p$ up to $x$.

Let $\varphi(s)\in{\widetilde{S}}$, and suppose that $\sigma^{*}$ is an
infimum of all $\sigma_{1}$ for which

$\frac{1}{2T}\int_{-T}^{T}|\varphi(\sigma+it)|^{2}dt\sim\sum_{m=1}^{\infty}\frac{|a(m)|^{2}}{m^{2\sigma}}$

holds for any $\sigma\geq\sigma_{1}$. Then $\frac{1}{2}\leq\sigma^{*}<1$, and
we see that ${\widetilde{S}}\subset{\mathcal{M}}$.

In 2015, the first result on the mixed joint universality for the tuple
$\big{(}\varphi(s),\zeta(s,\alpha;\mathfrak{B})\big{)}$ was proved by R.
Kačinskaitė and K. Matsumoto (see [1]). Later it was proved in a more general
situation extending collection of periodic Hurwitz zeta-functions (see [10]).

######  Theorem 3 ([1, Theorem 2.2]).

Suppose that $\varphi(s)\in\widetilde{S}$, and $\alpha$ is a transcendental
number. Let $K_{1}$ be a compact subset of $D(\sigma^{*},1)$, $K_{2}$ be a
compact subset of $D\big{(}\frac{1}{2},1\big{)}$ and both with connected
complements. Then, for any $f_{1}(s)\in H^{c}_{0}(K_{1})$, $f_{2}(s)\in
H^{c}(K_{2})$ and $\varepsilon>0$, it holds that

$\liminf_{T\to\infty}\frac{1}{T}\mathrm{meas}\big{\\{}\tau\in[0,T]:\sup_{s\in
K_{1}}\left|\varphi(s+i\tau)-f_{1}(s)\right|<\varepsilon,\sup_{s\in
K_{2}}\left|\zeta(s+i\tau,\alpha;\mathfrak{B})-f_{2}(s)\right|<\varepsilon\big{\\}}>0.$

The aim of the present paper is to prove the modification of Theorem 3 in
terms of density and to give a further certain generalizations.

Now we state the main result of the present paper.

######  Theorem 4.

Suppose that $\varphi(s)\in{\widetilde{S}}$ and $\alpha$ is a transcendental
number. Let $K_{1}$, $K_{2}$, $f_{1}(s)$, $f_{2}(s)$ be as in Theorem 3. Then,
for all but at most countably many $\varepsilon>0$, it holds that

$\lim_{T\to\infty}\frac{1}{T}\mathrm{meas}\big{\\{}\tau\in[0,T]:\sup_{s\in
K_{1}}\left|\varphi(s+i\tau)-f_{1}(s)\right|<\varepsilon,\sup_{s\in
K_{2}}\left|\zeta(s+i\tau,\alpha;\mathfrak{B})-f_{2}(s)\right|<\varepsilon\big{\\}}>0.$

######  Remark.

The transcendence of $\alpha$ can be replaced by the assumption that the
elements of the set $L(\alpha,{\mathbb{P}})$ are linearly independent over
${\mathbb{Q}}$ as it is done in Theorem 2.

##  3 Two probabilistic results

For the proof of Theorem 4, the probabilistic approach is used. In this
section, we present joint mixed limit theorems on weakly convergent
probability measures in the space of analytic functions and a proposition on
the support of probability measure.

Since now, we are interested in the proof of a joint limit theorem for the
tuple $\big{(}\varphi(s),$ $\zeta(s,\alpha;\mathfrak{B})\big{)}$, we deal with
more specified regions than $D(\sigma^{*},1)$ and $D(\frac{1}{2},1)$ (for the
arguments, we refer to [1] or [13]). As it is know, the function $\varphi(s)$
has finitely many poles by condition (i) (denote them by
$s_{1}(\varphi),\dots,s_{l}(\varphi)$), then put

$D_{\varphi}:=\\{s\in{\mathbb{C}}:\sigma>\sigma_{0},\ \sigma\neq\Re
s_{j}(\varphi),\ 1\leq j\leq l\\}.$

Since the function $\zeta(s,\alpha;\mathfrak{B})$ can be written as a linear
combination of the Hurwitz zeta-functions $\zeta(s,\alpha)$, it is entire or
has at most a simple pole at $s=1$. Let

$D_{\zeta}:=\begin{cases}\\{s\in{\mathbb{C}}:\sigma>\frac{1}{2}\\},&\text{if}\
\zeta(s,\alpha;\mathfrak{B})\ \text{is entire},\\\
\\{s\in{\mathbb{C}}:\sigma>\frac{1}{2},\ \sigma\neq 1\\},&\text{if}\ s=1\
\text{is a pole of}\ \zeta(s,\alpha;\mathfrak{B}),\end{cases}$

and $D_{1}$ and $D_{2}$ be two open regions of $D_{\varphi}$ and $D_{\zeta}$,
respectively. By $H^{2}(D)$ we mean the Cartesian product of the spaces
$H(D_{1})$ and $H(D_{2})$. Let $T>0$, and for
$A\in\mathcal{B}\left(H^{2}(D)\right)$, define

$P_{T}(A)=\frac{1}{T}\mathrm{meas}\big{\\{}\tau\in[0,T]:Z(\underline{s}+i\tau)\in
A\big{\\}}$

with $\underline{s}+i\tau=(s_{1}+i\tau,s_{2}+i\tau)$, $s_{1}\in D_{1}$,
$s_{2}\in D_{2}$, and

$Z(\underline{s})=\big{(}\varphi(s_{1}),\zeta(s_{2},\alpha;\mathfrak{B})\big{)}.$

For the definition of limit measure, we need a certain probability space. Let
${\widehat{\gamma}}=\\{s\in{\mathbb{C}}:|s|=1\\}$. Define two tori

$\Omega_{1}=\prod_{p\in{\mathbb{P}}}\gamma_{p}\quad\text{and}\quad\Omega_{2}=\prod_{m=0}^{\infty}\gamma_{m},$

where $\gamma_{p}={\widehat{\gamma}}$ for all $p\in{\mathbb{P}}$ and
$\gamma_{m}={\widehat{\gamma}}$ for all $m\in{\mathbb{N}}_{0}$, respectively.
With the product topology and pointwise multiplication, both tori $\Omega_{1}$
and $\Omega_{2}$ become compact topological Abelian groups. Therefore, on
$(\Omega_{1},\mathcal{B}\left(\Omega_{1}\right))$ and
$(\Omega_{2},\mathcal{B}\left(\Omega_{2}\right))$, there exist the probability
Haar measures $m_{1H}$ and $m_{2H}$ respectively. Thus, we get the probability
spaces $(\Omega_{1},\mathcal{B}\left(\Omega_{1}\right),m_{1H})$ and
$(\Omega_{2},\mathcal{B}\left(\Omega_{2}\right),m_{2H})$. Denote by
$\omega_{1}(p)$ the projection of $\omega_{1}\in\Omega_{1}$ to the coordinate
space $\gamma_{p}$, $p\in{\mathbb{P}}$, while, for $m\in{\mathbb{N}}_{0}$, let
$\omega_{1}(m):=\omega_{1}^{\alpha_{p_{1}}}(p_{1})\cdots\omega_{1}^{\alpha_{p_{r}}}(p_{r})$
according to the factorization of $m$ into prime divisors
$m=p_{1}^{\alpha_{1}}\cdots p_{r}^{\alpha_{r}}$, and $\omega_{2}(m)$,
$m\in{\mathbb{N}}_{0}$, the projection to the coordinate space $\gamma_{m}$.

Now let $\Omega=\Omega_{1}\times\Omega_{2}$, and denote the elements of
$\Omega$ by $\omega=(\omega_{1},\omega_{2})$. Since $\Omega$ is a compact
topological Abelian group, we can define the probability Haar measure
$m_{H}:=m_{1H}\times m_{2H}$ on $(\Omega,\mathcal{B}\left(\Omega\right))$.
This leads to a probability space
$(\Omega,\mathcal{B}\left(\Omega\right),m_{H})$.

On $(\Omega,\mathcal{B}\left(\Omega\right),m_{H})$, define $H^{2}(D)$-valued
random element $Z(\underline{s},\omega)$ by the formula

$Z(\underline{s},\omega)=\big{(}\varphi(s_{1},\omega_{1}),\zeta(s_{2},\alpha,\omega_{2};\mathfrak{B})\big{)}.$

Here $\underline{s}=(s_{1},s_{2})\in D_{1}\times D_{2}$,

$\displaystyle\varphi(s_{1},\omega_{1})=\sum_{m=1}^{\infty}\frac{c_{m}\omega_{1}(m)}{m^{s_{1}}}\quad\text{and}\quad\zeta(s_{2},\alpha,\omega_{2};\mathfrak{B})=\sum_{m=0}^{\infty}\frac{b_{m}\omega_{2}(m)}{(m+\alpha)^{s_{2}}}$

are $H(D_{1})$-valued and $H(D_{2})$-valued random elements defined on
$(\Omega_{1},\mathcal{B}\left(\Omega_{1}\right),m_{1H})$ and $(\Omega_{2},$
$\mathcal{B}\left(\Omega_{2}\right),m_{2H})$, respectively. Denote by $P_{Z}$
the distribution of the random element $Z(\underline{s},\omega)$, i.e.,

$P_{Z}(A)=m_{H}\big{\\{}\omega\in\Omega:Z(\underline{s},\omega)\in
A\big{\\}},\quad A\in\mathcal{B}\left(H^{2}(D)\right).$

Now we are in position to state a mixed joint limit theorem for the tuple of
the class of zeta-functions.

######  Theorem 5.

Suppose that $\varphi(s)\in{\mathcal{M}}$, and $\alpha$ is a transcendental
number. Then the measure $P_{T}(A)$ converges weakly to $P_{Z}(A)$ as
$T\to\infty$.

###### Proof.

The proof of this theorem is given in [1, Section 3, Theorem 2.1]. We only
note that the transcendence of $\alpha$ plays an essential role in the proof.
∎

The second probabilistic result used in the proof of Theorem 4 is that we need
to construct an explicit form of the support of the measure $P_{Z}$. For an
obtaining of the mentioned result, we use the positive density method.
Therefore, it is necessary to assume that the function $\varphi(s)$ belongs to
the Steuding class ${\widetilde{S}}$, particularly, the condition (e) must to
be satisfied (for the details, see [1, Section 4, Remark 4.4]).

Let $\varphi(s)$, $K_{1}$, $K_{2}$, $f_{1}(s)$ and $f_{2}(s)$ be as in Theorem
3. Then there exists a real number $\sigma_{0}$, $\sigma^{*}<\sigma_{0}<1$ and
a sufficiently large positive number $M$ such that $K_{1}$ belongs to

$D_{M}=\\{s\in{\mathbb{C}}:\sigma_{0}<\sigma<1,\ |t|<M\\}.$

Since $\varphi(s)\in{\widetilde{S}}$, it has only one pole at $s=1$, then put
$D_{\varphi}=\\{s\in{\mathbb{C}}:\sigma>\sigma_{0},\ \sigma\not=1\\}$.
Therefore, $D_{M}\subset D_{\varphi}$. Analogously we can find a sufficiently
large positive number $N$ such that $K_{2}$ belongs to

$D_{N}=\bigg{\\{}s\in{\mathbb{C}}:\frac{1}{2}<\sigma<1,\ |t|<N\bigg{\\}}.$

Now, if in Theorem 5 take $D_{1}=D_{M}$ and $D_{2}=D_{N}$, we get an explicit
form of the $P_{Z}$’s support.

######  Theorem 6.

The support of the measure $P_{Z}$ is the set $S:=S_{\varphi}\times H(D_{N})$,
where $S_{\varphi}:=\\{f_{1}(s)\in H(D_{M}):f_{1}(s)\neq 0\ \text{for all}\
s\in D_{M},\ \text{or}\ f_{1}(s)\equiv 0\\}$.

###### Proof.

The proof of the theorem can be found in [1, Lemma 4.3]. ∎

##  4 Proof of Theorem 4

First we remind two propositions used in the proof of the main result of the
paper.

Recall that a set $A\in\mathcal{B}\left(S\right)$ is said to be a continuity
set of the probability measure $P$ if $P(\partial A)=0$, where $\partial A$ is
the boundary of $A$. Note that the set $\partial A$ is closed, therefore, it
belongs to the class $\mathcal{B}\left(S\right)$. We are interested on the
property of probability measures defined in terms of continuity sets, which is
equivalent to weak convergence. Therefore we use a following fact.

######  Theorem 7.

Let $P_{n}$ and $P$ be probability measures on
$\left(S,\mathcal{B}\left(S\right)\right)$. Then the following assertions are
equivalent:

  1. 1)

$P_{n}$ converges weakly to $P$ as $n\to\infty$,

  2. 2)

$\lim\limits_{n\to\infty}P_{n}(A)=P(A)$ for all continuity sets $A$ of $P$.

###### Proof.

For the proof, see [14, Theorem 2.1]. ∎

We also recall the Mergelyan theorem on the approximation of analytic
functions by polynomials.

######  Theorem 8.

Let $K\subset{\mathbb{C}}$ be a compact subset with connected complement, and
let $f(s)$ be a continuous function on $K$ analytic inside $K$. Then, for any
$\varepsilon>0$, there exists a polynomial $p(s)$ such that

$\sup_{s\in K}|f(s)-p(s)|<\varepsilon.$

###### Proof.

The proof of the theorem can be found in [15]. ∎

###### Proof of Theorem 4.

Since $f_{1}(s)\not=0$ on $K_{1}$, by the Mergelyan theorem, there exist
polynomials $p_{1}(s)$ and $p_{2}(s)$ such that, for every $\varepsilon>0$,

$\displaystyle\sup_{s\in
K_{1}}\big{|}f_{1}(s)-\exp(p_{1}(s))\big{|}<\frac{\varepsilon}{2}\quad\text{and}\quad\sup_{s\in
K_{2}}\big{|}f_{2}(s)-p_{2}(s)\big{|}<\frac{\varepsilon}{2}.$ (3)

In view of Theorem 6, an element $\big{(}\exp(p_{1}(s)),p_{2}(s)\big{)}$
belongs to the set $S$, i.e., to the support of the measure $P_{Z}$.

Consider the set

$G=\bigg{\\{}(g_{1},g_{2})\in H^{2}(D):\sup_{s\in
K_{1}}\left|g_{1}(s)-\exp(p_{1}(s))\right|<\frac{\varepsilon}{2},\ \sup_{s\in
K_{2}}\left|g_{2}(s)-p_{2}(s)\right|<\frac{\varepsilon}{2}\bigg{\\}}.$

This set is an open subset in $H^{2}(D)$ and, by Theorem 6, an open
neighbourhood of an element $\big{(}\exp({p_{1}(s)}),p_{2}(s)\big{)}$.
Therefore, by Theorems 5 and 7, the inequality $P_{Z}(G)>0$ holds.

Now, for $f_{1}(s)$ and $f_{2}(s)$ fulfilling the conditions of Theorem 4,
define the set $G_{\varepsilon}$ by

$G_{{\varepsilon}}=\bigg{\\{}(g_{1},g_{2})\in H^{2}(D):\sup_{s\in
K_{1}}|g_{1}(s)-f_{1}(s)|<\frac{\varepsilon}{2},\ \sup_{s\in
K_{1}}|g_{2}(s)-f_{2}(s)|<\frac{\varepsilon}{2}\bigg{\\}}$

with the boundary

$\displaystyle\partial G_{\epsilon}=$ $\displaystyle\big{\\{}(g_{1},g_{2})\in
H^{2}(D):\sup_{s\in K_{1}}|g_{1}(s)-f_{1}(s)|<\varepsilon,\ \sup_{s\in
K_{2}}|g_{2}(s)-f_{2}(s)|=\varepsilon\big{\\}}$
$\displaystyle\cup\big{\\{}(g_{1},g_{2})\in H^{2}(D):\sup_{s\in
K_{1}}|g_{1}(s)-f_{1}(s)|=\varepsilon,\ \sup_{s\in
K_{2}}|g_{2}(s)-f_{2}(s)|<\varepsilon\big{\\}}$
$\displaystyle\cup\big{\\{}(g_{1},g_{2})\in H^{2}(D):\sup_{s\in
K_{1}}|g_{1}(s)-f_{1}(s)|=\varepsilon,\ \sup_{s\in
K_{2}}|g_{2}(s)-f_{2}(s)|=\varepsilon\big{\\}}.$

Easy to see that with different $\varepsilon_{1}>0$ and $\varepsilon_{2}>0$
the boundaries $\partial G_{\varepsilon_{1}}$ and $\partial
G_{\varepsilon_{2}}$ are disjoint. Therefore, only countable many sets
$\partial G_{\varepsilon}$ can have positive measure $P_{Z}$. Hence,
$P_{Z}(\partial G_{\varepsilon})=0$ for at most countable set of values
$\varepsilon>0$, i.e., $G_{\varepsilon}$ is a continuity set of $P_{Z}$ for
all but at most countable many $\varepsilon>0$. Moreover, in view of (3),
$G\subset G_{\varepsilon}$. Therefore, by Theorem 5, we have

$\lim_{T\to\infty}P_{T}({G_{\varepsilon}})=P_{Z}(G_{\varepsilon})>0$

for all but at most countable many $\varepsilon>0$. This together with the
definitions of $P_{T}$ and $G_{\varepsilon}$ prove the theorem. ∎

##  5 Concluding remarks

Theorem 4 can be generalised in a following direction.

Suppose that $\alpha_{j}$ is a real number such that $0<\alpha_{j}<1$, and
$l(j)$ is a positive integer, $j=1,\dots,r$. Let $\lambda=l(1)+\dots+l(r)$.
For each $j$ and $l$, $1\leq j\leq r$, $1\leq l\leq l(j)$, let
$\mathfrak{B}_{jl}=\\{b_{mjl}:m\in{\mathbb{N}}_{0}\\}$ be a periodic sequence
of complex numbers $b_{mjl}$ with minimal period $k_{jl}$, and let
$\zeta(s,\alpha_{j};\mathfrak{B}_{jl})$ be the corresponding periodic Hurwitz
zeta-function. Denote by $k_{j}$ the least common multiple of periods
$k_{j1},\dots,k_{jl(j)}$. Let $B_{j}$ be a matrix consisting of elements
$b_{mjl}$ from the periodic sequences $\mathfrak{B}_{jl}$, $j=1,\dots,r$,
$l=1,\dots,l(j)$, i.e.,

$B_{j}:=\begin{pmatrix}b_{1j1}&\cdots&b_{1jl(j)}\\\ \vdots&\ddots&\vdots\\\
b_{k_{j}j1}&\cdots&b_{k_{j}jl(j)}\end{pmatrix},\quad j=1,\dots,r.$

######  Theorem 9.

Suppose that $\alpha_{1},\dots,\alpha_{r}$ are algebraically independent over
${\mathbb{Q}}$, $\operatorname{rank\\!}\left(B_{j}\right)=l(j)$, $1\leq j\leq
r$, and $\varphi(s)$ belongs to the class $\widetilde{S}$. Let $K_{1}$ be a
compact subset of $D(\sigma^{*},1)$ and $K_{2jl}$ be a compact subset of
$D(\frac{1}{2},1)$, all of them with connected complements. Suppose that
$f_{1}(s)\in H^{c}_{0}(K_{1})$ and $f_{2jl}(s)\in H^{c}(K_{2jl})$. Then, for
all but at most countably many $\varepsilon>0$, it holds that

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\mathrm{meas}\bigg{\\{}\tau\in[0,T]:$
$\displaystyle\sup_{s\in K_{1}}|\varphi(s+i\tau)-f_{1}(s)|<\varepsilon,$ (4)
$\displaystyle\sup_{1\leq j\leq r}\sup_{1\leq l\leq l(j)}\sup_{s\in
K_{2jl}}|\zeta(s+i\tau,\alpha_{j};\mathfrak{B}_{jl})-f_{2jl}(s)|<\varepsilon\bigg{\\}}>0.$
(5)

###### Proof.

In [10], the joint mixed universality it was proved under the same conditions
as in the theorem instead $\lim$ studying $\liminf$ for every $\varepsilon>0$.
Therefore, arguing in similar way as in the proof of Theorem 4, we can show
the universality inequality (4).

However, we give some highlights. Let
$H^{\lambda+1}(D):=H(D_{1})\times\underbrace{H(D_{2})\times
H(D_{2})}_{\lambda}$, and let $p_{2jl}(s)$ be a polynomials satisfying the
second inequality of (3) for each $j=1,\dots,r$, $l=1,...,l(j)$.

Instead of the set $G$ in the proof of Theorem 4, we consider the set

$\displaystyle{\underline{G}}=\bigg{\\{}$
$\displaystyle(g_{1},g_{211},\dots,g_{21l(1)},\dots,g_{2r1},\dots,g_{2rl(r)})\in
H^{\lambda+1}(D):$ $\displaystyle\quad\sup_{s\in
K_{1}}|g_{1}(s)-\exp(p_{1}(s))|<\frac{\varepsilon}{2},\ \sup_{1\leq j\leq
r}\sup_{1\leq l\leq l(j)}\sup_{s\in
K_{2jl}}|g_{2jl}(s)-p_{2jl}(s)|<\frac{\varepsilon}{2}\bigg{\\}},$

and show that $P_{\underline{Z}}(\underline{G})>0$, where $P_{\underline{Z}}$
is a distribution of the $H^{\lambda+1}(D)$-random element constructed for the
collections of zeta-functions in the theorem. For details, we refer to [10].

Next we define the set

$\displaystyle{\underline{G}_{\varepsilon}}=\bigg{\\{}$
$\displaystyle(g_{1},g_{211},\dots,g_{21l(1)},\dots,g_{2r1},\dots,g_{2rl(r)})\in
H^{\lambda+1}(D):$ $\displaystyle\quad\sup_{s\in
K_{1}}|g_{1}(s)-f_{1}(s)|<{\varepsilon},\ \sup_{1\leq j\leq r}\sup_{1\leq
l\leq l(j)}\sup_{s\in K_{2jl}}|g_{2jl}(s)-f_{2jl}(s)|<{\varepsilon}\bigg{\\}}$

and obtain that it is a continuity set of the measure $P_{\underline{Z}}$ for
all but at most countably many $\varepsilon>0$. Again, arguing as for $G$ and
$G_{\varepsilon}$, we get that
${\underline{G}}\subset{\underline{G}}_{\varepsilon}$. Therefore, for all but
at most countably many $\varepsilon>0$,
$\lim\limits_{T\to\infty}P^{*}_{T}({\underline{G}}_{\varepsilon})=P_{\underline{Z}}({\underline{G}}_{\varepsilon})>0$.
In view of the similarity of $P^{*}_{T}$’s construction to $P_{T}$ extending a
collection of the periodic Hurwitz zeta-functions (for exact definition of
$P^{*}_{T}$, see p. 195 in [10]), this and the definition of
${\underline{G}}_{\varepsilon}$ complete the proof. ∎

Finally, we would like to mention that Theorem 9 can be shown under different
conditions than the algebraic independence over ${\mathbb{Q}}$ of the
parameters the parameters $\alpha_{1},\dots,\alpha_{r}$. Particularly, we can
prove that the universality inequality (4) holds if the elements of the set
$\big{\\{}(\log
p:p\in{\mathbb{P}}),(\log(m+\alpha_{j}):m\in{\mathbb{N}}_{0},j=1,...,r)\big{\\}}$
are linearly independent over ${\mathbb{Q}}$.

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Address of the auhtors:
Department of Mathematics and Statistics,
Faculty of Informatics,
Vytautas Magnus University,
Vileikos 8, Kaunas LT-44404, Lithuania

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