# Longitudinal tension and mechanical stability of a pressurized straw tube

L. Glonti, T. Enik, V. Kekelidze, A. Kolesnikov, D. Madigozhin,
N. Molokanova, S. Movchan, Yu. Potrebenikov, S. Shkarovskiy
Joint Institute for Nuclear Research, 141980, Dubna, Russia Corresponding
author. Email address<EMAIL_ADDRESS>
###### Abstract

For the development of charged particle detectors based on straw tubes
operating in vacuum, a special measurement technique is required for the
evaluation of their mechanical properties. A summary of the known equations
that govern straw behavior under internal pressure is provided, and a new
experimental method of a strained pressurized straw tube study is presented in
this paper. The Poisson’s ratio of the straw wall, which defines the stability
conditions of a built-in tube, is measured for the NA62 spectrometer straw,
and its minimum pre-tension is estimated.

keywords: straw tracker, straw tube, vacuum, pressure

PACS[2010] 29.40.Cs 29.40.Gx

##  1 Introduction

During the past few decades, a series of experiments in high-energy physics
have been designed to investigate very rare decay modes of kaon (see, for
example, [1, 2]). In particular, the purpose of NA62 experiment at CERN SPS
[1] is to measure $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay branching ratio of the
order of $10^{-10}$ with an uncertainty of 10%. It will be a test of the
Standard Model and a probe to possible new physics. This is a challenging
task, requiring an unprecedented precision of $\pi^{+}$ momentum measurement
in order to reject the dominating background using the evaluation of event
missing mass. This require a precise measurement of the momentum and position
of the charged particle.

A minimal material budget requirement in such experiments stimulates the
development of gaseous particle detectors based on straw tubes containing gas
under atmospheric pressure and operating in vacuum. These tubes maintain an
internal overpressure of 1 atm, but higher overpressure may be desirable for a
slower drift of electrons in the gas.

From the mechanics of a column, it is known that, if a compressive axial load
is applied to a clamped thin-walled tube without overpressure, the load
critical value $F_{crit}$ for buckling is [3, 4]

$F_{crit}=4\pi^{2}EJ/L^{2}.$ (1)

where $R$ is the straw radius, $L$ is its length, and $h$ is the thickness of
the tube wall. Further, $E$ is the Young’s modulus and $J=\int\int
y^{2}dxdy=\pi R^{3}h$ is a second moment of the cross-section of the tube wall
. $EJ$ is known as the object flexural rigidity. For an NA62 straw tube
$F_{crit}\approx 0.5N$, which is a small value.

It should be considered that a large straw deviation may appear when this
limit is approached [4]. A shift of a few millimeters of the tube axis with
respect to the anode wire may cause an electric discharge and detector
malfunction. Only axial tension guarantees the buckling prevention for a long
straw tube, and it has been demonstrated earlier [4, 5] that an internal
overpressure changes this necessary minimum tension.

Moreover, pre-tension is required to maintain the curvature caused by gravity
to be small for a straw placed horizontally. For the given straw geometry and
overpressure, the only parameter for the curvature control is the preliminary
stretching force applied to the straw.

Therefore, estimation of the minimum pre-tension of the straw is necessary for
any design of a straw tracker operating in vacuum. However, the calculations
based on the published material properties of the straw wall are not
sufficiently precise, because these properties depend on the batch of
material. Therefore, an experimental procedure for the evaluation of material
properties of a straw under operating conditions may be very useful.

The main purpose of this article is to present a new experimental method to
estimate the minimum preliminary stretching force required to prevent the
pressurized straw buckling and to limit its gravitational curvature.

##  2 Mechanics of a pressurized straw tube

The pressurized tube problem is not new in the industry, but it is not well-
known in high-energy physics instrumentation. Therefore, we provide a summary
of the known equations driving the behavior of straw tubes under internal
pressure.

Only two stress directions are essential for the problem of a strained straw
under internal pressure: the axial direction along the straw axis
($\parallel$) and the circumferential direction ($\perp$) tangent to the
cylindrical surface of the straw in the plane normal to the axis. For a thin-
walled tube ($h\ll R$), it can be demonstrated that straw circumferential
(hoop) stress caused by the inner overpressure $P$ is always
$\sigma_{\perp}=\frac{PR}{h}$ [6]. The axial stress $\sigma_{\parallel}$
depends on the conditions at the straw ends.

###  2.1 Pressurized tube with free closed ends

We will begin with the simple formulae derived in the isotropy approximation,
when the material properties do not depend on the considered direction. If the
internal pressure acts not only on the cylindrical shell of the tube but also
on the free ends closed by airtight end plugs, a simple relation can be
derived between the hoop stress $\sigma_{\perp}$ and the free closed tube
axial stress $\sigma^{free}_{\parallel}=\frac{P\pi R^{2}}{2\pi
Rh}=\frac{PR}{2h}=\sigma_{\perp}/2$ (see [6] for cylindrical pressure vessel).

Hoop stress leads to the transverse strain of the material of the straw wall
defining the direct contribution to the relative radius increasing
$\epsilon^{d}_{\perp}=\sigma_{\perp}/E=\frac{PR}{Eh}$. For the typical straw
tube $\epsilon^{d}_{\perp}\ll 1$; thus, this value may be used as a small
parameter to determine the order of the considered contribution to a strain.
The direct contribution to the axial relative elongation
$\epsilon_{\parallel}=\frac{\Delta L}{L}$ for the free closed tube has the
same first order:
$\epsilon^{d}_{\parallel}=\sigma^{free}_{\parallel}/E=\frac{PR}{2Eh}$.

The Poisson effect leads to the shortening of the straw wall in the direction
perpendicular to the corresponding direct strain. The Poisson ratio $\mu$ is
defined as the ratio of the perpendicular relative shortening to the
corresponding direct relative elongation. Thus, we have the first
approximation for a free closed tube.

$\displaystyle\frac{\Delta
L}{L}=\epsilon_{\parallel}=\epsilon^{d}_{\parallel}-\mu\epsilon^{d}_{\perp}=\frac{PR}{2hE}(1-2\mu)$
(2) $\displaystyle\frac{\Delta
R}{R}=\epsilon_{\perp}=\epsilon^{d}_{\perp}-\mu\epsilon^{d}_{\parallel}=\frac{PR}{2hE}(2-\mu).$
(3)

In the paper [7], the equation (2) has been observed to be sufficiently
precise to the best of our knowledge of the material properties. The hoop
strain formula (3) has been derived for the pressure vessel in [8].

In an experimental set-up employing straws in vacuum, straw ends are usually
built into the rigid frame (built-in tube) [1]. For this case, the formulae
(2, 3) in general are incorrect, since the pressure force applied to the
built-in end is balanced by the rigid frame rather than by the axial tension
of the wall.

However, there is a special case wherein a built-in tube and a free closed
tube are equivalent. Further, it can be used to obtain the simplest estimation
of the minimum preliminary tension required to prevent straw buckling in
vacuum [9]. If a free closed pressurized straw is glued into a rigid frame
that exactly fits the straw length enlarged by the overpressure (2), the frame
does not apply any axial load to the built-in straw. If the frame requires a
longer straw, an overall axial tension is applied to the straw ends ensuring
the stability of the tube against buckling. Furthermore, no change occurs in
the state of the straw wall if we connect the inner straw volume with the gas
supply mounted in the frame.

Therefore, in order to prevent buckling, we require a minimum straw pre-
tension ensuring the elongation (2) prior to the vacuum creation around the
tube. The stress caused by straw preliminary tension $T_{0}^{min}$ should be
at least $\frac{T_{0}^{min}}{2\pi
Rh}=E\epsilon_{\parallel}=\frac{PR}{2h}(1-2\mu)$, and the minimum pre-tension
for buckling prevention is

$T_{0}^{min}=P\pi R^{2}(1-2\mu).$ (4)

###  2.2 Pressurized tube with built-in ends

We will assume the different material properties in the axial and
circumferential directions (orthotropic approximation), which is quite usual
for a straw wall material. For example, NA62 straws are made of Hostaphan®
polyethylene terephthalate (PET) film. The film manufacturer reports the
different values of transverse (transverse direction, TD) and longitudinal
(machine direction, MD) Young’s moduli [10] (see Table 1). Therefore, the
direct contribution to the relative radius change of straw becomes
$\epsilon^{d}_{\perp}=\sigma_{\perp}/E_{\perp}=PR/(E_{\perp}h)$.

Owing to the Poisson effect, the hoop strain causes an axial strain of the
opposite sign, and thus, the straw becomes shorter if the ends are not built-
in. This imaginary relative change of the length is
$\epsilon^{\mu}_{\parallel}=-\mu_{\perp}\epsilon^{d}_{\perp}$, where
$\mu_{\perp}$ is the transverse (circumferential for the tube) Poisson’s
ratio. However, the straw ends are fixed, which indicates an appearance of the
compensatory tension force $\Delta T$ and the corresponding axial stress
$\Delta T/(2\pi Rh)$, which returns the straw to its initial length:

$\frac{\Delta T}{2\pi
Rh}=-E_{\parallel}\epsilon^{\mu}_{\parallel}=\mu_{\perp}P\frac{R}{h}\frac{E_{\parallel}}{E_{\perp}}.$
(5)

From Maxwell’s reciprocity theorem, the known relation
$\mu_{\parallel}=\mu_{\perp}\frac{E_{\parallel}}{E_{\perp}}$ can be derived,
where $\mu_{\parallel}$ is the axial Poisson’s ratio [4]. Thus, the wall axial
tension is

$T=T_{0}+\Delta T=T_{0}+2\mu_{\parallel}P\pi R^{2}.$ (6)

The total force applied by a gas-filled straw in vacuum on the detector frame
can be considered as an “effective tension” $T_{P}=T-P\pi R^{2}$, if the
effect of external atmospheric pressure applied to the frame is calculated by
ignoring all holes made for the gas supply into the straws. For this effective
tension, we obtain

$T_{P}=T_{0}-(1-2\mu_{\parallel})P\pi R^{2}.$ (7)

Poisson’s ratio $\mu$ for plastics is usually less than 0.5. Thus, the vacuum
around a straw leading to its inner overpressure diminishes $T_{P}$ with
respect to the straw pre-tension $T_{0}$ in spite of the increase in true
straw tension $T$ (6).

###  2.3 Lateral effect of internal pressure

Figure 1: Forces applied to the curved tube element without flexural rigidity
subject to internal pressure and axial tension

We will evaluate the summary force applied by internal pressure to the wall of
a curved tube. Ignoring the flexural rigidity, consider a short element of a
slightly curved tube (Figure 1), limited by its two cross-sections normal to
the curved axis. The absolute longitudinal tension force
$|T_{AB}|=|T_{CD}|=|T|$ is a constant in this approximation, but the
directions of $T_{AB}$ and $T_{CD}$ forces are defined by different slopes at
the element ends.

The area of the arched side of the wall (upper side in Figure 1) is larger
than the area of the concave part. Consequently, a non-zero summary pressure
force $F_{W}$ applied to the wall appears, which is directed toward the arched
side of the wall. Thus, the internal overpressure always attempts to increase
the existing curvature of the tube.

The evaluation of summary pressure effect has been performed earlier in a few
ways: an imaginary ideal piston at the boundary of the pressurized tube
segment [11], the consideration of the forces applied to separated fluid
contained within a tube element [12], and a direct integration of the pressure
forces [4].

However, we can directly evaluate the pressure effect $F_{W}$ without
integration. For the given $P$, the summary force $F_{W}$ depends only on the
shape of the tube segment. Imagine an absolutely rigid shell consisting of a
curved tube wall with additional transverse plugs closing the cross sections
AB and CD tightly. A resultant force applied by inner overpressure to a closed
rigid shell is always zero. Therefore, the total pressure force applied to the
curved wall is equal, but with the opposite sign, to the vector sum of the
pressure forces $P\pi R^{2}$ applied to the end plugs, which are perpendicular
to the curved axis of the tube at the ends (see Figure 1). One of these forces
is directed against $T_{AB}$, and the other against $T_{CD}$, which
effectively diminishes the straw tension by the pressure-related force $P\pi
R^{2}$.

Thus, the lateral dynamics of a curved tube element are defined by the
effective tension $T_{P}=T-P\pi R^{2}=T_{0}-(1-2\mu_{\parallel})P\pi R^{2}$,
which replaces the true tension $T$ in all the formulae describing a straw
bending [12, 4, 5].

It is important to understand the physical difference between the effective
tension $T_{P}$ defining the straw bending phenomena and the tube wall tension
$T$ in the tensile strength formulae. Straw wall stress is normally increased
owing to the internal overpressure, whereas the pressurized straw as an
elastic body is effectively relaxed under the same conditions, and this
relaxation is described by the pressure-dependent effective tension $T_{P}$
behavior.

For $T_{P}<0$, a summary transverse force pushes a curved straw element toward
its arched side, and without flexural rigidity, the curvature increases until
the tension (increased owing to the curved tube elongation) becomes equal to
the pressure force everywhere along the tube: $T=P\pi R^{2}$.

From $T_{P}>0$ and (7), we obtain the minimum longitudinal straw pre-tension
$T_{0}$ to prevent buckling [4]

$T_{0}^{min}=(1-2\mu_{\parallel})P\pi R^{2}.$ (8)

The same limit is defined by (4) derived in a different way.

###  2.4 Equilibrium of a horizontal straw

For a horizontally placed straw, we should ensure that the maximum vertical
deviation (sagitta $Sag$) caused by gravitation is small. Consider a
horizontal straw with built-in ends subject to vertically distributed
gravitational load $q$ and an internal overpressure $P$. We choose the origin
of the axial coordinate $x$ at the center of the straw. The axis of a straw
lateral deviation $y$ is directed downwards, thus $y(0)=Sag$. The linear
gravitational load is $q=g\rho$, where $\rho$ is the linear density of the
straw and $g$ is the gravitational acceleration.

The straw equilibrium equation for this case was derived in [4, 5]:

$\frac{dF}{dx}=-EJ\frac{d^{4}y}{dx^{4}}+T_{P}\frac{d^{2}y}{dx^{2}}+q=0,$ (9)

where $\frac{dF}{dx}$ is a resulting force per unit of length, which is zero
for the case of static equilibrium.

Straw ends in the NA62 spectrometer are glued into the frame (clamped).
Therefore, the boundary conditions are $\frac{dy}{dx}(\pm\frac{L}{2})=0$ and
$y(\pm\frac{L}{2})=0$. The symmetric solution ($y(x)=y(-x)$) for this static
case is

$y_{s}(x)=\frac{q}{2T_{P}}(\frac{L^{2}}{4}-x^{2}+\frac{L}{k}\cdot\frac{cosh(kx)-cosh(kL/2)}{sinh(kL/2)}),$
(10)

where $k=\sqrt{\frac{T_{P}}{EJ}}$ (see [5]). Sagitta $y_{s}(0)$ is increased
by the internal overpressure owing to the decrease in $T_{P}$ (7).

###  2.5 Low-frequency oscillations

The interesting consequence of the pressure lateral effect is a straw
vibration pressure dependence caused by the behavior of $T_{P}$ used instead
of string tension for straw transverse dynamics. For the case of oscillations,
we should set $\frac{dF}{dx}=\rho\frac{d^{2}y}{dt^{2}}$ instead of zero in
(9). We will determine the solution in the form of
$y(x,t)=y_{s}(x)+y_{f}(x,t)$, where $y_{s}(x)$ is a solution of the static
equation (9). It leads to the wave equation

$\rho\frac{d^{2}y_{f}}{dt^{2}}=-EJ\frac{d^{4}y_{f}}{dx^{4}}+T_{P}\frac{d^{2}y_{f}}{dx^{2}}$
(11)

set up by Lord Rayleigh [13].

We will determine the symmetric solution with fixed ends
($y_{f}(\pm\frac{L}{2})=0$) and with “pinned” boundary conditions
($\frac{d^{2}y_{f}}{dx^{2}}(\pm\frac{L}{2})=0$). Such a simple solution is
sufficient for the qualitative understanding of straw oscillations. These
boundary conditions are satisfied for $y_{f}(x,t)=cos((1+2n)\pi x/L)e^{i2\pi
ft}$ with any integer $n$. For $n=0$, the equation (11) results in $(2\pi
f)^{2}\rho=(\pi/L)^{2}(EJ(\pi/L)^{2}+T_{P})$, and thus the lowest frequency of
the straw vibration is [14]

$f=\frac{1}{2L}\sqrt{\frac{EJ(\pi/L)^{2}+T_{P}}{\rho}}.$ (12)

Notably, the flexural rigidity of the straw results in an addition of
$EJ(\pi/L)^{2}$ to the effective tension in (12). The frequency is decreased
by the overpressure owing to the behavior of $T_{P}$ (7).

###  2.6 Radius correction

$T_{P}$ depends on the straw radius $R$ considered so far as a constant value.
However, $R$ depends on $P$, whereas the radius of the built-in end plug
remains unchanged. It leads to the formation of a radius transition zone on
the tube wall near the straw end. However, the axial component of the pressure
force applied to the transition zone is subtracted from the axial wall tension
applied to the end plug. Therefore, for $T_{P}$ calculation, we can consider
the radius of the end plug to be equal to the pressure-dependent straw radius
$R$.

The direct contribution of overpressure to the relative change of radius is
$\frac{\sigma_{\perp}}{E_{\perp}}=\frac{PR}{E_{\perp}h}$, but if we consider
the additional tension applied to the built-in ends, this term becomes
$(1-\mu_{\parallel}\mu_{\perp})\frac{PR}{E_{\perp}h}$. Moreover, the applied
pre-tension $T_{0}$ diminishes the straw radius owing to the Poisson’s effect.
Therefore, we have

$R=R_{0}(1+(1-\mu_{\parallel}\mu_{\perp})\frac{PR_{0}}{E_{\perp}h}-\frac{\mu_{\parallel}T_{0}}{E_{\parallel}2\pi
R_{0}h}),$ (13)

where $R_{0}$ is the initial radius of the straw. For $T_{P}$ calculations, if
$T_{0}\approx 1$ kgf, the radius change (13) leads to the next-order
correction in terms of the small parameter $\frac{PR}{Eh}$.

##  3 Poisson’s ratio measurement

It can be observed from (8) that the straw buckling limit for a given
overpressure is defined by the tube radius $R$ and the Poisson’s ratio
$\mu_{\parallel}$. The Poisson’s ratio is typically not provided in the PET
film specifications. The published independent measurements of $\mu$ are not
related to the specific batch of material used in the straw tubes production
for the given detector. Moreover, the straw production process may change some
foil properties. Therefore, the $\mu_{\parallel}$ measurement procedure
applicable to a welded straw is required in order to evaluate the minimum
straw pre-tension for the specific detector design.

###  3.1 Straw specimens

Two straw specimens have been tested (see Table 1). The tubes are made of PET
Hostaphan® foils using longitudinal welding [1]. The parameters of the foils
can be found in the manufacturer specifications [10].

Table 1: Properties of the tested straws Property | 9.8-mm straw | 18-mm straw
---|---|---
Diameter, mm | 9.8(1) | 18.0(1)
Length, m | 2.10(1) | 1.90(1)
Material density, g/cm3 | 1.4(1) | 1.4(1)
Wall thickness, $\mu$m | 36(1) | 54.4(8)
Tube linear density, g/m | 1.55(12) | 4.31(6)
$E_{\parallel}$, N/mm2 | 4500(500) | 4000(500)
$E_{\perp}$, N/mm2 | 5000(500) | 5500(500)

The first specimen (9.8-mm straw) is obtained from a party of approximately
7000 straws produced at the Joint Institute for Nuclear Research (Dubna) for
NA62 spectrometer [1]. This 9.8-mm straw is coated inside the tube with two
thin metal layers ($0.05\mu m$ of Cu and $0.02\mu m$ of Au) in order to
provide electrical conductivity on the cathode and to improve the
impermeability of the straw tube. The wall material volume density [10] is
used to estimate the linear density of the 9.8-mm straw. The contributions of
metal layers and the air mass inside the tube to linear density are
negligible.

The second specimen (18-mm straw) is made of a thicker Hostaphan® film. In
this case, the linear density of the tube has been measured by weighing, since
the wall thickness is not strictly defined in the manufacturer specifications.

###  3.2 Test bench for studies of a strained straw under pressure

A special test bench (see Figure 2) has been manufactured in order to study
the properties of a built-in straw with an initial pre-tension and an inner
overpressure applied subsequently. The longitudinal force applied to the straw
end $T_{P}=T-P\pi R^{2}$ is measured using a tensometer $Tm$ based on a
single-point aluminum load cell (Tedea-Huntleigh, model 1022) [15].

Figure 2: The test bench scheme: B – rigid basement; Tm – tensometer; e1 –
closed end plug; e2 – end plug with a gas supply channel; C – end cap with a
seal; P – pressure supply; S – straw; O – optical coupler; A - amplifier.

A straw specimen is closed on the tensometer side by the end plug $e1$ with
the end cap $C$ using glue for rigid sealing. The end cap $C$ is connected to
the tensometer $Tm$ using two flexible joints and a rigid rod. The other straw
end contains a plug $e2$ with a gas supply channel. This straw end is glued
into the solid support, which may be moved along the rigid basement $B$ and
fixed at a specific place in order to create a preliminary straw tension prior
to the test. Subsequently, the pressure supply is opened, and the changing
pressure values $P$ are recorded together with the corresponding results of
the tensometer measurement $T_{P}$ (see points in Figure 3 and Figure 4A).

Straw oscillations are studied using the optical coupler O [1], which emits
constant intensity infrared radiation and registers the radiation reflected
from the straw wall. When the straw oscillations are mechanically excited, the
registered radiation intensity is modulated by the changing distance to the
straw wall. The obtained signal is amplified and sent to the oscilloscope with
a fast Fourier transform function. The lowest frequency peak position is
registered as the lowest frequency of the straw (points in Figure 4B).

The end cap $C$ is slightly shifted down owing to its weight of $10-20$ gf.
However, for the horizontal force $T_{P}>300$ gf (assuming the cap weight of
$25$ gf), the vertical shift of the cup is less than $4\ mm$. It leads to the
relative elongation of 2-m straw $\approx 10^{-4}$, which is much less than
the minimum straw elongation caused by the tested preliminary tension
($10^{-3}$). Thus, the straw ends may be considered to be fixed during the
test.

###  3.3 Effective tension and Poisson’s ratio measurement

It is known that any material becomes nonlinear for a large stress, whereas
the linear properties of the material are defined for zero-stress limit.
However, on the built test bench, precision measurement becomes problematic
for a low effective tension $T_{P}$. Therefore, we must consider the possible
nonlinearity in such a way that the resulting $\mu_{\parallel}$ could be
easily extracted for $T\to 0$.

Accordingly, we postulate a weak linear dependence of Poisson’s ratio as a
function of axial stress. For the fits of our experimental results, we use the
tension-dependent value

$\nu=\mu_{\parallel}-\frac{kT}{2\pi Rh}$ (14)

instead of only $\mu_{\parallel}$. Apart from the material nonlinearity, the
coefficient $k$ also absorbs the effect of the set-up deformation under
tension and the next-order effects ignored in the formulae. Expression (14)
provides a physically motivated interpolation and extrapolation of the
measurement results, whereas in order to compare the resulting Poisson’s ratio
with the other measurements, we can consider the measured $\mu_{\parallel}$
and ignore the stress-dependent term.

$T$ and $R$ variables in (14) depend on the Poisson’s ratio value. Therefore,
we implement an iterative procedure starting with a tension $T=T_{0}$, nominal
straw radius $R=R_{0}$, and the starting Poisson’s ratio value of
$\nu=\mu_{\parallel}$. In each iteration, new $T,\nu,R$ values were
calculated, and three iterations were sufficient for the precise calculation.

Two free parameters ($\mu_{\parallel}$ and $k$) describe the measured tensions
and pressures satisfactorily. Figure 3 and Figure 4A show the measured
effective tensions for the NA62 straw along with the result of their fit with
the formula

$T_{P}=T_{0}-(1-2\nu)P\pi R^{2}.$ (15)

The MINUIT [16] package and ROOT [17] interface were used to obtain the
resulting parameter values and their fit errors.

$P(at)$$T_{P}(gf)$

Figure 3: 9.8-mm straw effective tension $T_{P}$ versus the overpressure $P$ for different initial tensions $T_{P}(0)$. Circles – measurements; solid lines – fit by the formula (15). Table 2: Poisson’s ratio measurement results | 9.8 mm straw | 18 mm straw
---|---|---
|  $k$, $\frac{\mu m^{2}}{gf}$ | $\mu_{\parallel}$ | $\mu_{\perp}$ |  $k$, $\frac{\mu m^{2}}{gf}$ | $\mu_{\parallel}$ | $\mu_{\perp}$
Central value | 3.98 | 0.3055 | 0.3394 | -2.89 | 0.2960 | 0.4070
Radius correction | 0.43 | 0.0005 | 0.0006 | 0.92 | 0.0005 | 0.0007
Radius value | 0.14 | 0.0083 | 0.0092 | 0.59 | 0.0039 | 0.0054
$T_{P}$ scale | 0.06 | 0.0099 | 0.0110 | 1.24 | 0.0088 | 0.0121
$P$ scale | 0.27 | 0.0094 | 0.0104 | 1.03 | 0.0084 | 0.0116
$\delta\frac{E_{\perp}}{E_{\parallel}}$ | 0 | 0 | 0.0306 | 0 | 0 | 0.0296
Systematic error | 0.53 | 0.0160 | 0.0353 | 1.95 | 0.0128 | 0.0344
Statistical error | 0.18 | 0.0005 | 0.0006 | 14.35 | 0.0150 | 0.0206
Total error | 0.56 | 0.0160 | 0.0353 | 14.48 | 0.0197 | 0.0401

The fit to all the measured $T_{P}$ values for the given straw is performed
with a common set of free parameters and the same assumed measurement error.
The error ($\pm 3.6$ gf for the 9.8-mm straw and $\pm 20.47$ gf for the 18-mm
straw) is defined in such a way that the resulting $\chi^{2}/ndf=1$, in order
to estimate the parameter statistical uncertainties.

The fit results and systematic uncertainty contributions are shown in Table 2.
The coefficients of correlation between $\mu_{\parallel}$ and $k$ are 0.991
for the 9.8-mm straw and 1.000 for the 18-mm straw. Thus, for the second
specimen, the presence of the non-zero term $k$ is not confirmed.

The radius correction effect is considered, but it is also completely included
in the systematic errors as a “Radius correction” contribution. The Gaussian
width of the NA62 straw diameter distribution is approximately 0.03 mm, and
the systematic shift of its central value from the nominal number has the same
size [1]. Thus, considering a conservative value of the possible diameter
systematic uncertainty of 0.1 mm for the tested specimens, we have obtained
the “Radius value” contribution to the systematic error shown in Table 2.
Moreover, the effects of a systematic scale shift of 5% on both the measured
effective tension and measured pressure are considered as independent
contributions (“$T_{P}$ scale” and “$P$ scale”) to the systematic errors.

Considering the manufacturer’s information about Young’s moduli (see Table 1),
the $\mu_{\perp}$ central values shown in Table 2 have been extracted as
$\mu_{\perp}=\mu_{\parallel}\frac{E_{\perp}}{E_{\parallel}}$. Their
uncertainties depend on the error of the modules ratio, which may be
approximately estimated from the significant digits of the provided values as
$\delta\frac{E_{\perp}}{E_{\parallel}}\approx 0.1$.

Typically, the reported Poisson’s ratio values for the oriented PET films are
0.37-0.44 [18], and thus, the obtained $\mu_{\perp}$ values are consistent
with them. However, the resulting $\mu_{\parallel}$ is approximately $0.3$,
which may be a specific property of Hostaphan® foil or the consequence of the
production of straw tube using longitudinal welding [1].

##  4 Oscillation frequency measurements

The oscillation frequency measurement was the last test of the straw tensions
performed on the assembled NA62 spectrometer modules [1]. The same test has
been repeated on the present test bench for a qualitative verification of the
pressure effect.

The length of the tube on the test bench is defined by the position of the
movable support and the end cup $C$ position. Unfortunately, when a straw
tension is applied to the end cup fixed on the short rod with flexible joints,
an effective oscillator is formed with a frequency close to the measured
frequency of straw. Therefore, using this test bench, only a qualitative
understanding of straw oscillations can be obtained using the simple frequency
formula (12).

The results of the lowest frequency measurements are shown in Figure 4B along
with the corresponding calculation results based on (12). The error bars show
the frequency uncertainty of 1 Hz defined by the width of the observed
spectrum peaks.

The prediction (12) for the 9.8-mm straw satisfactorily describes the
measurement results in the vicinity of NA62 design parameters ($T_{0}=1500$
gf, $P=1$ at). However, the overall discrepancy reaches 2 Hz, which is more
than the measurement precision. Nevertheless, the obtained qualitative
description of the straw vibration confirms that the lowest frequency of the
straw is significantly diminished by the internal overpressure in spite of the
increase in straw wall stress according to (6).

$P(at)$$T_{P}(gf)$$f(Hz)$AB

Figure 4: A: effective tensions $T_{P}$ versus $P$ for different initial
tensions $T_{P}(0)$. B: lowest frequency for the corresponding $P$ and
$T_{P}$. Filled circles – 9.8-mm straw, open circles – 18-mm straw. Curves:
formula (15) for the plot A and (12) for B; solid lines – 9.8-mm straw, dashed
lines – 18-mm straw.

##  5 Minimum straw pre-tension evaluation

We can use the obtained Poisson’s ratio to evaluate the minimum straw pre-
tension $T_{0}$ in the straw-based NA62 spectrometer operating in vacuum [1].

There are two requirements for $T_{0}$. The buckling limit $T_{P}>0$ (where
$T_{P}$ is calculated from (15) for $P=1$ at) is defined by the Poisson’s
ratio and the straw radius. The flexural rigidity contribution (1) to the
buckling prevention is small for the long NA62 straw and may be included into
the safety margin. Moreover, the experimental design requirements define the
gravitational deviation (sagitta) limit for a horizontal straw. Sagitta
$Sag=y_{s}(0)$ calculated from (10) depends on the effective tension, flexural
rigidity, and straw length.

The sagitta requirement $Sag<100$ $\mu$m of NA62 [1] cannot be satisfied using
a reasonable pre-tension for the complete 2.1-m straw. Hence, special
supporting spacers dividing each straw into three equal parts are implemented
in the NA62 spectrometer [1]. These spacers fix a straw only in one of the
lateral directions, and thus, they do not change the small buckling critical
load value (1) ignored in this study.

Straw symmetry near each spacer ensures that the boundary conditions of (10)
are satisfied. Hence, we use this solution with $L=0.7$ m in order to
calculate the sagitta.

AB$T_{0}$, gf$T_{0}$, gf$Sag$, $\mu$m$Sag$, $\mu$m

Figure 5: Sagitta in vacuum $Sag$ versus pre-tension $T_{0}$ of 70-cm-long
straw segment for 9.8-mm straw (A) and 18-mm straw (B). Vertical lines mark
the buckling limit $T_{P}=0$. Solid line is calculated for the central values
for all the parameters. Dashed lines show the worst case (see text).

The results of the sagitta calculation and buckling limit estimations are
shown in Figure 5A for the 9.8-mm straw in the NA62 spectrometer. The
corresponding results for the 18-mm straw, assuming the same straw length of
70 cm, are shown in Figure 5B for comparison.

Apart from the most probable sagitta values and buckling limits, the worst
cases are also shown in Figure 5. The worst case for each straw corresponds to
$\nu$ (14) central value diminished by the tripled total uncertainty evaluated
for each $T_{0}$ from the measured $\mu_{\parallel}$ and $k$ considering their
correlation. All other parameters for this worst-case scenario are considered
at their uncertainty limits leading to the largest sagitta and the easiest
buckling.

It can be observed from Figure 5 that, for the NA62 spectrometer, a straw pre-
tension value above 900 gf guarantees $Sag<100$ $\mu$m (NA62 requirement),
whereas the worst case of the buckling limit is below 400 gf. Thus, both the
NA62 nominal straw pre-tension of $1.5$ kgf and the factual minimum pre-
tension ($\approx 1.2$ kgf) [1] have a good safety margin.

However, if the 18-mm straw is used in the same detector design, the minimum
$T_{0}$ would be defined by the increased buckling limit above $1.15$ kgf with
a small safety margin, whereas the gravitational sagitta will be always below
100 $\mu$m for the pre-tension above the buckling limit.

## Conclusions

A new technique for the study of mechanical properties of straw tubes
subjected to inner pressure and longitudinal tension has been tested using a
specially built test bench. It includes the measurement of Poisson’s ratio of
a straw wall, which defines the buckling limit of a straw with a given radius
under a definite inner overpressure.

The axial Poisson’s ratio $\mu_{\parallel}$ for Hostaphan® foil is measured
for two specimens under the conditions close to those of a detector operating
in vacuum. The lateral Poisson’s ratio $\mu_{\perp}$ is evaluated using the
elasticity moduli in two directions provided by the foil producer. The spectra
measurements of straw oscillations qualitatively confirm the effective tension
predictions based on the measured $\mu_{\parallel}$.

The minimum pre-tension requirement for the NA62 spectrometer is re-evaluated
based on the measurement results. The obtained limit confirms the detector
design pre-tension with a safety margin of approximately 600 gf. The tested
technique can be used for the development of future straw trackers.

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