
==== Front
Open Res Eur
Open Res Eur
Open Research Europe
2732-5121
F1000 Research Limited London, UK

10.12688/openreseurope.17329.1
Research Article
Articles
On elastic deformations of cylindrical bodies under the influence of the gravitational field
[version 1; peer review: 1 approved, 2 approved with reservations]

Barzegar Hamed Conceptualization Formal Analysis Validation Visualization Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0002-6472-3115
1
Chruściel Piotr T. Conceptualization Funding Acquisition Methodology Project Administration Resources Supervision Validation Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0001-8362-7340
2
Steininger Florian Conceptualization Data Curation Formal Analysis Investigation Methodology Software Visualization Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0003-0092-3043
a2
1 Centre de Recherche Astrophysique de Lyon, Universite Claude Bernard Lyon 1, Villeurbanne, Auvergne-Rhône-Alpes, 69622, France
2 Faculty of Physics, University of Vienna, Vienna, Vienna, 1090, Austria
a f.steininger@univie.ac.at
*Preprint UWThPh-2024-3

No competing interests were disclosed.

10 5 2024
2024
4 9810 4 2024
Copyright: © 2024 Barzegar H et al.
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Background

Large elastic deformations of gravitating cylindrical bodies play a significant role in everyday life. On the other hand, tiny such deformations are relevant for state-of-the art experiments, as they affect the physical properties of materials under consideration, impacting wave propagation. This is of key importance for a recently planned experiment to explore the influence of the gravitational field on entangled photons propagating in waveguides. The purpose of this work is to determine this influence.

Methods

We use the methods of linear elasticity, including thermoelasticity, to determine the stresses and strains of the medium. For this, the symmetry of the cylinder allows us to solve the problem by using Mitchell’s solutions of the equations satisfied by the Airy functions. The boundary conditions are implemented by an approximation of the Hertz contact method.

Results

We calculate the displacements, the stresses and strains for several classes of boundary conditions, and give explicit solutions for a number of physically motivated configurations. The influence of the resulting deformations on the planned GRAVITES experiment is determined.

Conclusions

The results are relevant for fiber interferometry experiments sensitive to the effects of the gravitational field on photon propagation. Our calculations give stringent bounds on the environmental variables, which need to be controlled in such experiments.

linear elasticity
GRAVITES
waveguides
Airy stress
Michell solution
Horizon 2020 Framework Programme740021 Horizon Europe Framework Programme101071779 Austrian Science FundP34274 This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 740021), and under the Horizon Europe Framework Programme [101071779] and the Austrian Science Fund [P34274]. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
==== Body
pmc1 Introduction

An experiment is currently being built [ 1] with the aim to measure the effect of the gravitational field on entangled states of photons propagating in an optical fiber. The experiment requires displacing vertically an optical fiber, with circular cross-sections, in a spooled configuration, see Figure 1.1. Such a displacement in the gravitational field of the Earth leads to a minute phaseshift, which is expected to be measurable with the current state-of-the art photonic technology. The displacement is associated with a change of the ambient gravitational acceleration, of temperature, and of atmospheric pressure, leading to an elastic deformation of the fiber. The deformation affects the shape, the length, and the propagation properties of light in the fiber, leading to an additional phaseshift which needs to be determined for a correct interpretation of the results of the experiment. The objective of this work is to develop a framework which can be used to determine this elastic deformation. The resulting effects on the propagation properties of light in the fiber will be determined elsewhere.

Figure 1.1. The GRAVITES experiment.

The upper arm of the interferometer is moved vertically. The change of the gravitational field due to the change of height affects the propagation of light, resulting in a height-dependent phase shift. Here and unless otherwise indicated, the y-axis is aligned vertically, so that the gravitational force acts anti-colinearly along the y-axis. © C. Hilweg, reproduced with kind permission of the author.

The planned configuration is that of a spool with its axis of symmetry aligned vertically. Since the radius of the spool is very large compared to the radius of the fiber, we ignore the spooling and consider a very long elastic cylinder. We then consider several models for the problem at hand:

As a first model we start with a cylindrical waveguide resting on a horizontal contact line (see Figure 1.2a and Section 4.1.1). The next model is a configuration where the waveguide is squeezed between two contact lines, to take into account the pressure arising from the layers pressing from above (see Figure 1.2b and Section 4.1.2). The results obtained in both cases are unacceptable, with an infinite deformation at each contact line; this is of course a well known problem of such models (cf. [ 2, Chapter 8.4.7]). However, the model appears useful for its simplicity, we will return to this shortly.

Figure 1.2. The four main models.

The y-axis is again along the vertical in all four figures. Figures ( a) and ( b) show the cross-section of an infinite cylinder resting on a rigid support, with a supplementary pressure from the top in Figure ( b). Figures ( c) and ( d) show the spool of Figure 1.1 as cut by a vertical plane; we assume that the radius of the spool is very large compared to the radius of the waveguide. The circles represent the consecutive returns of the waveguide, exercising pressure on the neighboring strands. The dashed region represents the rigid spool.

To avoid the above problems we pass to a Hertz-contact-type calculation, where we first analyse a configuration with the waveguide resting on a rigid support, followed by one where the waveguides are stacked upon each other; in the last case the influence of the upper layer on a lower one is modelled by contact with a rigid plane. There arises a parameter describing the pressure from the waveguides stacked above the section of the fiber under consideration. We leave this parameter free in our calculations, its value can be determined by the number of layers and windings of the spool whenever a specific configuration is considered.

The above takes into account the vertical neighbours of any given section of the fiber, but ignores the side neighbours. To address this we consider two further configurations, where each fiber has four neighbours as in Figure 1.2c (see Section 4.2.3), or each fiber has six neighbours as in Figure 1.2d and Section 4.2.4. The last model seems to us to provide the best approximation to the problem at hand. In such models new parameters arise, associated with presence of new neighbours. We leave these parameters free again, and show in Appendix A how one of these parameters can be determined for each layer in the quadrilateral-contact case.

Our results show that in all models the relative deformations are similar, for the practical purposes of our interest, close enough to the center of waveguide, where the guiding core resides. Therefore we expect that the first, simplest model, will be sufficient for the applications we have in mind. We plan to return to this question in a future full treatment of the influence of small elastic deformations on the dispersion relation in optical fibers.

We include a constant ambient pressure term, as well as temperature effects in all configurations, in order to model possible variations of the environment.

2 Waveguide deformation in earth’s gravity

The unperturbed waveguide is modelled as a very long, homogeneous cylinder with radius a, length L and density ρ, supported by a rigid plane with gravity acting as a body force via a constant gravitational acceleration g. We take U = {( r, θ, z) : 0 < r ≤ a, –π < θ ≤ π, 0 ≤ z ≤ L} to be the interior of the waveguide and write ∂U for the surface r = a. Note that a ⪡ L.

It is known that in isotropic and homogeneous materials the generalized Hooke’s law in three dimensions yields the following stress-strain relation (cf., e.g., [ 2, Eq. (4.2.7)] and [ 3, Eq. (4.6)]),

σij=λϵkkδij+2μϵij,(2.1)

where λ and µ are Lamé’s first and second parameters of the material, respectively, with µ being called the shear modulus which is sometimes denoted by G in the literature; repeated indices are summed over unless explicitly indicated otherwise. Here, and elsewhere, all tensors are expressed in orthonormal frames. With σ kk = (3 λ + 2 µ) ϵ kk we find

ϵij=1+νEσij−νEσkkδij,(2.2)

where E := µ(3 λ + 2 µ) /( λ + µ) is Young’s modulus and ν := λ/[2( λ + µ)] is Poisson’s ratio. Note that E = 2 µ(1 + ν).

When the changes of temperature are not negligible, the above needs to be revised as follows: According to [ 2, Section 4.4], in a thermally-isotropic and thermally-linear medium, Equation ( 2.2) should be replaced by

ϵij=1+νEσij−νEσkkδij+α(T−T0)δij,(2.3)

where α is the coefficient of linear thermal expansion. This is a consequence of the assumptions of linear thermo-elasticity, for which the strain decomposes into independent thermal and elastic components. Inverting this relation, the stress tensor is given by

σij=λϵkkδij+2μϵij−(3λ+2μ)α(T−T0)δij.(2.4)

We allow the waveguide to stretch in the z direction linearly in z (compare Figure 2.1 (a)–(c)):

Figure 2.1. Four models for a very long cylinder with boundary conditions covered by our calculations.

Vertical dashed regions represent fixed ends in the z-direction, whereas the dotted bars represent rigid supports. The remaining boundaries in the figures can move freely. Note that the model ( d) requires the vanishing of the elongation coefficient κ of ( 2.5).

uz=κz.(2.5)

All remaining fields are assumed to be z-independent. Equation ( 2.5) leads to a non-vanishing z-component of the strain tensor

ϵij=12(∂iuj+∂jui),(2.6)

namely

ϵzz=κ,butϵxz=ϵyz=0,(2.7)

giving an additional contribution to the usual plane-strain relations (cf. [ 4] for related considerations).

Having reduced the dimensionality of the problem, the equilibrium equations (cf., e.g., [ 2, Eq. (3.6.4)] or [ 5, Eq. (2.34)]),

∂jσij+Fi=0,(2.8)

with σ ij the stress tensor and F i = −∂ iV the body force due to gravity, where V = g ρy, simplify to

∂xσxx+∂yσxy−∂xV=0,(2.9)

∂xσxy+∂yσyy−∂yV=0.(2.10)

Note that the stress in the z direction does not necessarily vanish. Indeed, the zz-component of ( 2.3) gives

κ=σzzE−νE(σrr+σθθ)+α(T−T0),(2.11)

so that

σzz=ν(σxx+σyy)+Eκ−Eα(T−T0).(2.12)

Adapting our coordinate system to the symmetry of the setup by choosing cylindrical coordinates, from now on tensor components will refer to the following orthonormal frame, which in Cartesian coordinates is given by

er:=(cos(θ),sin(θ),0),eθ:=(−sin(θ),cos(θ),0),ez:=(0,0,1).(2.13)

In this frame the stress tensor can be expressed in terms of derivatives of the Airy stress function ϕ as (cf. [ 6] and, e.g., [ 7, Eqs. (8.12)–(8.13)])

σrr=1r∂rϕ+1r2∂θ2ϕ+V,(2.14)

σθθ=∂r2ϕ+V,(2.15)

σrθ=−∂r(1r∂θϕ),(2.16)

which has to satisfy the compatibility condition (see, e.g., [ 2, Eq. (7.5.5) or [ 5, Eq. (7.17b)], together with [ 2, Eq. (12.3.7)]),

Δδ2ϕ=−1−2ν1−νΔδV−Eα1−νΔδT,(2.17)

where Δ δ is the Laplace operator of the Euclidean metric on ℝ 2. (Note that V is only defined up to a constant, which can be absorbed by a redefinition of ϕ.) In this work we consider a steady-state configuration, which requires Δ δT = 0 (cf. [ 2, Section 12.1]). Further, since V = g ρy we have Δ δV = 0 as well, implying that the compatibility conditions reduce to the homogeneous biharmonic equation.

Since the frame { e r, e θ, e z } is orthonormal, the strain is related to the stress via ( 2.3):

ϵrr=12μ[(1−ν)σrr−νσθθ]−νκ+(1+ν)α(T−T0),(2.18)

ϵθθ=12μ[(1−ν)σθθ−νσrr]−νκ+(1+ν)α(T−T0),(2.19)

ϵrθ=12μσrθ.(2.20)

One can now determine the displacement vector u i from the usual equations, where u r and u θ are frame components of u (cf., e.g., [ 2, Eq. (7.6.1)])

ϵrr=∂rur,(2.21)

ϵθθ=1r(∂θuθ+ur),(2.22)

ϵrθ=12(1r∂θur+∂ruθ−1ruθ).(2.23)

Having compiled all relevant equations from linear elasticity, we set up an adapted coordinate system as in Figure 2.2, with

r∈(0,a]andθ∈(−π,π].(2.24)

Figure 2.2. Convention for polar coordinates.

There exists a general solution for ( 2.17) in polar coordinates, denoted by 1

ϕ(r,θ)=A0lnr+B0+C0r2lnr+D0r2+(a0lnr+b0+c0r2lnr+d0r2)θ+(A1rlnr+B1r+C1r3+D1r+E1rθ+F1rθlnr)cos(θ)+(a1rlnr+b1r+c1r3+d1r+e1rθ+f1rθlnr)sin(θ)+∑n≥2[(Anrn+Bnr−n+Cnrn+2+Dnr2−n)cos(nθ)+(anrn+bnr−n+cnrn+2+dnr2−n)sin(nθ)],(2.25)

known as the ‘Michell solution’ [ 8].

In practical terms, we are mostly concerned with determining the coefficients in ( 2.25) from the boundary conditions we impose. An immediate simplification can be achieved by only considering solutions which have the mirror symmetry x ↦ −x (equivalently θ ↦ −θ), since the forces considered are invariant under this symmetry; see Appendix B for a treatment without imposing mirror symmetry at the outset. Additionally, we require regularity at r = 0 and 2 π-periodicity in θ. Restricting to mirror-symmetric boundary conditions finally leads to a requirement of mirror-symmetry in σ rr and σ θθ and antisymmetry for σ rθ . Lastly, note that the parameters B 0 and D 1 do not contribute to the stress as by Equations ( 2.14)–( 2.16), which is straightforwardly verified. They may be considered degeneracies of the solution space and set to zero without loss of generality (cf. [ 7]). The remaining terms in ( 2.25) are

ϕ(r,θ)=D0r2+C1r3cos(θ)+∑n≥2(Anrn+Cnrn+2)cos(nθ),(2.26)

which is the form considered for all applications below.

The displacement can be calculated via ( 2.21)–( 2.23) 2 . We have for the remaining terms in ϕ,

ur(r,θ)=12μ{2(1−2ν)D0r+(1−4ν)C1r2cos(θ)−12gρ(1−2ν)r2cos(θ)+∑n≥2[−nAnrn−1+(2−4ν−n)Cnrn+1]cos(nθ)}−Ξcos(θ)−νκr+(1+ν)α(T−T0)r,(2.27)

uθ(r,θ)=12μ{(5−4ν)C1r2sin(θ)−12gρ(1−2ν)r2sin(θ)+∑n≥2[nAnrn−1+(4−4ν+n)Cnrn+1]sin(nθ)}+Ξsin(θ)+c*r,(2.28)

where c ∗, Ξ are integration constants. These are fixed by the boundary conditions imposed at the contact point u r ( a, 0) = 0 = u θ ( a, 0), which imply c ∗ = 0 and

Ξ:=12μ{2(1−2ν)D0a+(1−4ν)C1a2−12gρ(1−2ν)a2+∑n≥2[−nAnan−1+(2−4ν−n)Cnan+1]}−νκa+(1+ν)α(T−T0)a,(2.29)

provided that the sums converge.

3 Boundary conditions

We now have general expressions for the stresses ( 2.14)–( 2.16), the strains ( 2.18)–( 2.20), and the displacements ( 2.27)–( 2.28), in terms of the coefficients { D 0, C 1, A n , C n } for n ≥ 2. These coefficients can be determined from the boundary conditions algebraically.

At the boundary of the cylinder we consider an angle dependent pressure f( θ) acting in the radial direction and a shear force per unit area g( θ). The boundary conditions in linear elasticity require the stresses at the boundary to react to the external forces as

σrθ|∂U=g(θ),(3.1)

σrr|∂U=f(θ).(3.2)

Our assumption of mirror-symmetry implies that the boundary conditions can be Fourier decomposed into sine and cosine series respectively,

σrθ|∂U=∑n≥0gnsin(nθ),(3.3)

σrr|∂U=12f0+∑n≥1fncos(nθ),(3.4)

with the coefficients given explicitly by

gn=2π∫0πdθg(θ)sin(nθ),(3.5)

fn=2π∫0πdθf(θ)cos(nθ).(3.6)

On the other hand, we know that the solution can be written using ( 2.14)–( 2.16) with ( 2.26) for the Airy stress function; explicitly

σrθ|∂U=2C1asin(θ)+∑n≥2[(n−1)An+a2(n+1)Cn]nan−2sin(nθ),(3.7)

σrr|∂U=2D0+(2C1−gρ)acos⁡(θ)−∑n≥2[(n−1)nAn+a2(n2−n−2)Cn]an−2cos(nθ).(3.8)

Comparing term by term determines the coefficients. We are generally interested in the case with vanishing shear forces, i.e. g( θ) = 0, which corresponds to only normal forces or the frictionless limit. Then,

C1=0andCn=1−na2(1+n)An,(3.9)

and for the radial stress

σrr|∂U=2D0−agρcos⁡(θ)−2∑n≥2An(n−1)an−2cos(nθ).(3.10)

Comparing coefficients with ( 3.4), we find

D0=14f0,(3.11)

An=−12(n−1)a2−nfn,(3.12)

for n ≥ 2, with

f1=−agρ.(3.13)

We see that the Fourier coefficient f 1 in the function f in ( 3.2) is not arbitrary, and is determined by the body force.

The boundary conditions in the z-direction are given by the models shown in Figure 2.1. We focus on subfigure (c), with the special case without tension force F T corresponding to subfigure (b). By definition, the displacement boundary conditions are given by ( 2.5), since we allow for elongation of the open end. The boundary condition for the stress is given by

FT=∫U(z=L)σzzrdrdθ,(3.14)

i.e. the forces on the end face of the cylinder have to match the stresses on the end face. Using ( 2.12), this can be rewritten as

FT=∫U(z=L)ν(σxx+σyy)rdrdθ+πa2E[κ−α(T−T0)].(3.15)

The integral can be calculated either explicitly or numerically for the solutions in Section 4, providing thus a relation between the forces acting on the system, κ, and the remaining parameters that appear in the problem.

This can be restated as an equation for the elongation κ as a function of tension F T and the plane stresses along the fiber:

κ=−1πa2E∫U(z=L)ν(σxx+σyy)rdrdθ+α(T−T0)+FTπa2E.(3.16)

For a waveguide of total length L, this implies a change in length

L↦L+κL=L[1−1πa2E∫U(z=L)ν(σxx+σyy)rdrdθ+α(T−T0)+FTπa2E].(3.17)

4 Contact models

Having formally solved the problem for arbitrary boundary conditions in the preceding section, we now implement the boundary conditions for the configurations shown in Figures 1.2a– 1.2d.

As already pointed out in the Introduction, the solutions for the configurations of Figures 1.2a– 1.2b, that we are about to derive, with the boundary conditions corresponding to contact lines, describe unphysical displacement fields, diverging at the contact interfaces. We show that this can be cured by deriving a solution involving extended contact regions, using the Hertz contact deformations formalism. We show that even though the displacements differ between these approaches, the stresses near the center of the waveguide are in reasonable agreement.

4.1 Line contacts

4.1.1 Resting on a contact line. The simplest physical model is given by a cylindrical waveguide lying on an infinite plane, contacting as a first approximation only on a line. This is reminiscent of the famous Flamant solution with circular cross-section (see in particular [ 7, Problem 3 of Chapter 12]).

The boundary conditions are given by

σrθ|∂U=0,(4.1)

σrr|∂U=δ(θ)−p,(4.2)

where is the pressure with which the contact line is resisting the weight of the waveguide and the ambient pressure, taken to be constant. To put this into physical terms, we apply neither tangential nor radial forces, except for the contact line with the “contact wire”.

This model is particularly convenient, since the δ-distribution has a Fourier series expansion as

δ(θ)=12π[1+2∑1≤ncos(nθ)].(4.3)

We find for the coefficients in the Michell solution

D0=4π−p2,(4.4)

C1=0,(4.5)

An=−2πan−2(n−1),(4.6)

Cn=2πan(n+1),(4.7)

with

=−πagρ.(4.8)

This last constraint implements the physicality of the calculations, since in the absence of other forces there can only be an equilibrium configuration if the integrated body force of the waveguide

∫(−Fy)rdrdθ=∫gρrdrdθ=πa2gρ

matches the reactive force exterted by the plane, given by

∫δ(θ)rdθ|∂U=−πa2gρ.

The sum in ( 2.26) converges for coefficients ( 4.4)–( 4.7), yielding

ϕ(r,θ)=−12r2p+14rgρ[r(a+rcos(θ))−4a2arctan⁡(rsin(θ)a−rcos(θ))sin(θ)],(4.9)

and further for the stresses

σrr=A[r(6a2+r2)cos(θ)−a(a2+3r2+2(a2+r2)cos(2θ)−arcos(3θ))]−p,(4.10)

σrθ=A[4a(a2+r2)cos(θ)−r(5a2+r2+2a2cos(2θ))]sin(θ),(4.11)

σθθ=A[−r(2a2+r2)cos(θ)+a(r2−a2+2(a2+r2)cos(2θ)−arcos(3θ))]−p,(4.12)

with

A:=gρ(a2−r2)2[a2+r2−2arcos(θ)]2.

A representative plot of the stresses can be seen in Figure 4.1.

Figure 4.1. Typical plot of internal stresses for the case of a waveguide resting on an infinitely thin line, with unrealistic parameters arbitrarily chosen for illustration purposes listed in Table 4.1.

Darker colors encode larger stresses, with blue color indicating compressive (negative) stresses.

Table 4.1. Set of parameters chosen in our visualizations.

For conciseness, we express lengths in multiples of a, and pressure in multiples of the shear modulus μ, leading to dimensionless quantities for the remaining parameters used in numerical calculations.

ρg	ν	, κ, α	P˜	P^ = P∨	
0.01	0.17	0	1	0.3	

Equations ( 2.27)–( 2.29) do not make sense, as the sum in Ξ does not converge. But one can find explicit expressions for u r and u θ by integration; the resulting formulae are lengthy and not very enlightening, therefore we did not include them here. Not unexpectedly, and similar to the Flamant solution [ 2, Chapter 8.4.7] , the singularity in the function A at r = a and cos θ = 1 leads to an infinite displacement there. Hence the boundary condition u( r = a, θ = 0) = 0 cannot be imposed. However, the displacements obtained by direct integration are finite away from the contact point, in particular near the center of the waveguide (cf. [ 9]). One can also truncate the series ( 2.27)–( 2.28) which leads to finite solutions, illustrated in Figure 4.2.

Figure 4.2. Illustrative deformation for the case of a waveguide resting on an infinitely thin line, after truncating the sums in ( 2.27)–( 2.29) to n = 20, drawn in red.

The undeformed reference is drawn in black. Here and in similar figures below, the inset shows a magnified section around the contact point.

4.1.2 Squeezed between two lines. A second straightforward case is given by the waveguide being squeezed between two lines, with an arbitrary pressure P˜ pushing from the top. See also [ 2, Example 8–10] and [ 7, Problem 1 of Chapter 12], where a similar problem is modelled by superposition of three particular stress fields, including two Flamant solutions together with a uniform radial tension loading. The boundary conditions are given by

σrθ|∂U=0,(4.13)

σrr|∂U=δ(θ)+P˜δ(θ−π)−p.(4.14)

The calculation is completely analogous to the above, with the boundary condition enforcing

D0=+P˜4π−p2,(4.15)

C1=0,(4.16)

An=−+(−1)nP˜2πan−2(n−1),(4.17)

Cn=+(−1)nP˜2πan(n+1),(4.18)

with

=P˜−πagρ.(4.19)

Again, the physical interpretation is that now the force from above necessitates a reactive force from below larger than in ( 4.8) to achieve equilibrium.

Denoting by σijI the right-hand sides of ( 4.10)–( 4.12), the stress components read

σrr=σrrI−B(a2−r2)[a4−2a2r2−r4+2a4cos(2θ)],(4.20)

σrθ=σrθI+2Ba2(a4−r4)sin(2θ),(4.21)

σθθ=σθθI−B[a6+5a4r2+a2r4+r6−2a2(a4+a2r2+2r4)cos(2θ)],(4.22)

with B:=P˜π(a2−r2)[a4+r4−2a2r2cos(2θ)]−2 . Again, visualizing the result gives a clear picture. See Figure 4.3.

Figure 4.3. Typical plot of internal stresses for the case of a waveguide squeezed between two lines; same parameters and color coding as in Figure 4.1.

As before, the deformation diverges at the “contact wires”. A truncated sum is shown in Figure 4.4.

Figure 4.4. Illustrative deformation for the case of a waveguide squeezed between two lines, after truncating the sums in ( 2.27)-( 2.29) to n = 20, drawn in red.

The undeformed reference is drawn in black.

4.2 Hertz contact deformations

The proper solution for the contact problem in linear elasticity is given by the Hertz contact deformation [ 10], with a modern derivation given by [ 3, Chapter 9]. In the special case of two cylinders contacting length-wise the two-dimensional contact region extends to a contact strip, which can be described as a contact line in terms of the cross-section (cf. [ 3, Problem 2 of Chapter 9]). The total force per unit length pushing two bodies into each other is distributed over a region { x : −s ≤ x ≤ s}, where

s=4Fπ(1−ν2E+1−ν′2E′)RR′R+R′,(4.23)

with {ν, E, R} the material properties and radius of curvature of the body in question and { ν′, E′, R′} for the body in contact. The pressure distribution over the contact region, which we denote by P y , is taken to be

Py=2Fπs1−x2s2.(4.24)

This can be restated in the form of boundary conditions

σrθ|∂U=0,(4.25)

σrr|∂U={2Fπatan(Θ)1−tan2(θ)tan2(Θ)−Θ≤θ≤Θ0otherwise,(4.26)

where

Θ=arctan(sa).(4.27)

For the case of an infinite cylinder resting on a rigid plane, we simply take R′, E′ → ∞ and ν′ → 0.

Finding the cosine expansion of ( 4.26) is not obvious. Instead, one can approximate the Hertz profile with a step-function as in Figure 4.5, with straightforward cosine expansion, allowing us to find exact expressions for the Airy functions and stresses in the settings discussed below. The half-width of the step function can be determined, based on the Hertz solution, to

sHertz=4aFπ1−ν2E,(4.28)

ΘHertz=arctan(sHertza),(4.29)

Figure 4.5. Comparison of the pressure distribution in the contact region for the exact Hertz solution ( 4.26) and an approximation using constant pressure in the contact region.

The shaded and dashed regions cause overshoot and undershoot in the displacement figures below.

via ( 4.23) and ( 4.27), ensuring a physically motivated boundary pressure distribution in response to external forces.

This approach yields exact solutions closely matching the full Hertz problem, avoiding the undesirable behaviour at the contact point exhibited by the solutions above. We expect that this approach is sufficient for weak forces and away from the contact region.

4.2.1 Resting on a rigid plane. Approximating the Hertz contact solution of Section 4.2 by a step-function of constant pressure P for a single contact region from below leads to the boundary conditions

σrθ|∂U=0,(4.30)

σrr|∂U=Pχ[−Θ,Θ](θ)−p(1−χ[−Θ,Θ](θ))=:P′χ[−Θ,Θ](θ)−p,(4.31)

where

χA(x):={1x∈A0else,(4.32)

is the indicator function and Θ ≈ Θ Hertz.

A Fourier cosine series for the boundary-values of the Airy function leads to the following coefficients in ( 2.26):

D0=P′Θ2π−p2,(4.33)

C1=0,(4.34)

An=−P′sin(nΘ)πan−2n(n−1),(4.35)

Cn=P′sin(nΘ)πann(n+1),(4.36)

with

P′=agρπ2sin(Θ).(4.37)

The summed expression for the Airy function is

ϕsingle=14gρ[(2a2+r2)rcos(θ)−ar2Θsin(Θ)+asin(Θ)(ζ(θ−Θ)−ζ(θ+Θ))]−12r2p,(4.38)

where the subscript “single” refers to a single contact region, with

ζ(x):=[a2+r2−2arcos(x)]arctan(rsin(x)a−rcos(x)).(4.39)

All the sums converge, leading to an admissible displacement field. The stresses and the displacements are shown in Figure 4.6 and Figure 4.7.

Figure 4.6. Typical internal stresses for a waveguide resting on an extended contact zone.

Here, and in the following figures, we use the same parameters and the same color coding as in Figure 4.1.

Figure 4.7. Deformation for the case of a waveguide resting on a rigid plane, drawn in red.

The undeformed reference is drawn in black. Here, and in similar figures that follow, the (barely visible) indentation is an artefact of the approximation illustrated in Figure 4.5.

4.2.2 Squeezed between two rigid planes. The next simplest description of a spooled waveguide is one with extended contact regions both above and below. We model this by squeezing the fiber between two rigid planes in the spirit of Section 4.2.1, i.e. boundary conditions

σrθ|∂U=0,(4.40)

σrr|∂U=Pχ[−Θ,Θ](θ)+P˜χ[π−Θ˜,π+Θ˜](θ)−p[1−χ[−Θ,Θ](θ)−χ[π−Θ˜,π+Θ˜](θ)]=:P′χ[−Θ,Θ](θ)+P˜′χ[π−Θ˜,π+Θ˜](θ)−p,(4.41)

with Θ now an angle which approximates half of the contact region from below and Θ˜ the same from above.

The Fourier coefficients for the Airy function are found to be

D0=P′Θ+P˜′Θ˜2π−p2,(4.42)

C1=0,(4.43)

An=−P′sin(nΘ)+(−1)nP˜′sin(nΘ˜)πan−2n(n−1),(4.44)

Cn=P′sin(nΘ)+(−1)nP˜′sin(nΘ˜)πann(n+1),(4.45)

with

2sin(Θ)P′=2P˜′sin(Θ˜)−agρπ.(4.46)

The re-summed expression for the Airy function is

ϕdouble=ϕsingle+P˜′2π[r2(Θ˜+Θsin(Θ˜)sin(Θ))+ξ(θ−Θ˜)−ξ(θ+Θ˜)−sin(Θ˜)sin(Θ)(ζ(θ−Θ)−ζ(θ+Θ))],(4.47)

where

ξ(x):=[a2+r2+2arcos(x)]arctan(rsin(x)a+rcos(x)).(4.48)

All the sums converge, leading to a finite displacement field. The stresses and the displacements can be seen in Figure 4.8 and Figure 4.9 respectively. Here, and in the plots that follow, the parameters are related to those of the two contact-lines case via the correspondence

P˜contactline=P˜extendedcontactregion×2Θ˜,(4.49)

Figure 4.8. Internal stresses for the case of a waveguide squeezed between two rigid planes with extended contact region.

Figure 4.9. Deformation for the case of a waveguide squeezed between rigid planes of finite width, drawn in red.

The undeformed reference is drawn in black.

Pcontact line = Pextended contact region × 2Θ , (4.49)

which guarantees an identical integrated reaction force from the support. It is of interest to compare the line-contact cases to the corresponding extended region cases, focusing on the central region containing the guiding core of the waveguide. In Figure 4.10 and Figure 4.11 we show the differences for the same set of parameters as used in the plots. We can quantify the differences away from the contact points by comparing the maximal values over the regions

U1/2={(r,θ):0<r≤a2,−π<θ≤π}

Figure 4.10. The difference ∆ σ between the line contact Figure 4.1 and extended contact region Figure 4.6.

Figure 4.11. Difference ∆ σ between the line contact of Figure 4.3 and the extended contact region of Figure 4.8.

and

U1/10={(r,θ):0<r≤a10,−π<θ≤π},

and setting

ΔU(σ1,σ2)=2maxU|σ1−σ2|maxU|σ1|+maxU|σ2|.(4.50)

The results, for the parameters of Table 4.1 (used in all our figures), are shown in Table 4.2 and Table 4.3.

Table 4.2. Relative differences for the single contact region models.

	σ rr	σ rθ	σ θθ	σ xx	σ xy	σ yy	
∆ U 1/2	2 .9%	4 .4%	5 .5%	11 .7%	6 .5%	2 .9%	
∆ U 1/10	0 .6%	1 .1%	0 .9%	2 .5%	2 .3%	0 .6%	

A reasonable approximation is obtained in the central region for the values of parameters considered if a 11% error is less than the measurement errors at hand.

Table 4.3. Relative differences for the double-contact-region models.

	σ rr	σ rθ	σ θθ	σ xx	σ xy	σ yy	
∆ U 1/2	2.4%	9.9%	7.7%	11.0%	17.2%	3.9%	
∆ U 1/10	1.4%	2.0%	1.7%	5.1%	10.4%	1.5%	

4.2.3 Four contact regions. We consider now a configuration as in Figure 1.2c, with four contact regions for the waveguide. Using step-function boundary-stresses for the contact region gives

σrθ|∂U=0,(4.51)

σrr|∂U=Pχ[−Θ,Θ](θ)+P˜χ[π−Θ˜,π+Θ˜](θ)+P¯[χ[π2−Θ^,π2+Θ^](θ)+χ[3π2−Θ^,3π2+Θ^](θ)]−p[1−χ[−Θ,Θ](θ)−χ[π−Θ˜,π+Θ˜](θ)−χ[π2−Θ^,π2+Θ^](θ)−χ[3π2−Θ^,3π2+Θ^](θ)]=:P′χ[−Θ,Θ](θ)+P˜′χ[π−Θ˜,π+Θ˜](θ)+P¯′[χ[π2−Θ^,π2+Θ^](θ)+χ[3π2−Θ^,3π2+Θ^](θ)]−p,(4.52)

with Θ now an angle which approximates half of the contact region from below and Θ˜ the same from above and Θ^ from the sides.

The resulting Fourier coefficients in the Airy function are

D0=P′Θ+P˜′Θ˜+2P˜′Θ^2π−p2,(4.53)

C1=0,(4.54)

An=−P′sin(nΘ)+(−1)nP˜′sin(nΘ˜)+2P¯′cos(nπ2)sin(nΘ^)πan−2n(n−1),(4.55)

Cn=P′sin(nΘ)+(−1)nP˜′sin(nΘ˜)+2P¯′cos(nπ2)sin(nΘ^)πann(n+1),(4.56)

with

P′=P˜′sin(Θ˜)sin(Θ)−agρπ2sin(Θ).(4.57)

All the sums converge, leading to an admissible displacement field. The stresses and the displacement can again be illustrated graphically – see Figure 4.12 and Figure 4.13.

Figure 4.12. Internal stresses for the case of a waveguide squeezed between four rigid planes from below, above and the sides as depicted in Figure 1.2c, with extended contact region.

Figure 4.13. Deformation for the case of a waveguide squeezed between rigid planes of finite width as in Figure 1.2c, drawn in red.

The undeformed reference is drawn in black.

4.2.4 Squeezed between six rigid planes. Our final model for spooled waveguides is given by Figure 1.2d, which we model by imposing the boundary conditions

σrθ|∂U=0,(4.58)

σrr|∂U=Pχ[−Θ,Θ](θ)+P˜χ[π−Θ˜,π+Θ˜](θ)+P^[χ[π3−Θ^,π3+Θ^](θ)+χ[5π3−Θ^,5π3+Θ^](θ)]+P∨[χ[2π3−Θ∨,2π3+Θ∨](θ)+χ[4π3−Θ∨,4π3+Θ∨](θ)]−p[1−χ[−Θ,Θ](θ)−χ[π−Θ˜,π+Θ˜](θ)−χ[π3−Θ^,π3+Θ^](θ)−χ[5π3−Θ^,5π3+Θ^](θ)−χ[2π3−Θ∨,2π3+Θ∨](θ)+χ[4π3−Θ∨,4π3+Θ∨](θ)]=P′χ[−Θ,Θ](θ)+P˜′χ[π−Θ˜,π+Θ˜](θ)+P^′[χ[π3−Θ^,π3+Θ^](θ)+χ[5π3−Θ^,5π3+Θ^](θ)]+P′∨[χ[2π3−Θ∨,2π3+Θ∨](θ)+χ[4π3−Θ∨,4π3+Θ∨](θ)]−p,(4.59)

with Θ now an angle which approximates half of the contact region from below, and Θ˜ the same from above, and Θ^ and Θ∨ from the sides.

The associated Fourier cosine series for the Airy functions has coefficients

D0=P′Θ+P˜′Θ˜+2P^′Θ^+2P′∨Θ∨2π−p2,(4.60)

C1=0,(4.61)

An=−1πan−2n(n−1)[P′sin(nΘ)+(−1)nP˜′sin(nΘ˜)+2P^′cos(nπ3)sin(nΘ^)+2P′∨cos(2nπ3)sin(nΘ∨)],(4.62)

Cn=1πann(n+1)[P′sin(nΘ)+(−1)nP˜′sin(nΘ˜)+2P^′cos(nπ3)sin(nΘ^)+2P′∨cos(2nπ3)sin(nΘ∨)],(4.63)

with

P′=P˜′sin(Θ˜)sin(Θ)−P^′sin(Θ^)sin(Θ)+P′∨sin(Θ∨)sin(Θ)−agρπ2sin(Θ).(4.64)

All the sums converge, leading to a finite displacement field. The stresses and the displacement are seen in Figure 4.14 and Figure 4.15.

Figure 4.14. Internal stresses for the case of a waveguide squeezed between six rigid planes with extended contact regions, as in Figure 1.2d.

Figure 4.15. Deformation for the case of a waveguide squeezed between rigid planes of finite width as in Figure 1.2d, drawn in red.

The undeformed reference is drawn in black.

5 Estimates for GRAVITES

The planned GRAVITES experiment [ 1] will use single-mode optical fibers made of silica glass, which we assume to be homogeneous, with material properties and experimental parameters given in Tables 5.1– 5.2. The numbers used below, such as the change of pressure δ and the change of temperature T− T 0, and their outcome for the experiment, correspond to a change in height of 1 m at sea level for an unshielded experiment.

Table 5.1. Properties of silica oxide glass from [ 11].

ν	E	ρ	α	
0.17	73.1 GPa	2.2 g/cm 3	1.8 · 10 −7 K −1	

Table 5.2. Parameters for GRAVITES from [ 1].

a	δg	δp	T − T 0	L	β	
62.5 µm	3 µm/s 2	10 Pa	10 –2 K	10 5 m	6 · 10 6 m –1	

It should be kept in mind that the real experiment will be in a vacuum chamber. There are various outcomes possible, depending upon the details of the experimental implementation of the problem:

1. The effects scale in an obvious way with the residual pressure, in which case our calculations provide bounds on the quality of the vacuum needed for measurability of the gravitational effect.

2. The gradients of the temperature of the residual gas affect the temperature of the waveguide during the timescale of the experiment, in which case our calculations provide bounds on the residual temperature gradients.

The above can of course be circumvented by ensuring that there are neither pressure gradients nor temperature gradients, of the kind considered here, inside the vacuum chamber at the timescale of the experiment, so that the effects determined here become irrelevant. Our calculations thus give guidance concerning the experimental setup.

In what follows we will also need to estimate the spooling force. We choose

FT=1N,

which appears to be standard for commercially available spooling machines. In a first very rough estimate the lateral pressures P^ and P∨ , for the hexagonal case, are taken to be the same as P¯ from the quadrilateral case, as given in ( A.2).

Finally, the maximal pressure P˜ for fibers at the bottom can be estimated by the weight of the spooled fiber ~ 10 kg distributed over the area of a horizontal cross-section of the spool, resulting in

P˜max=1.3kPa.

The actual value of P˜ decreases with height along the spool, vanishing for the top layer.

For the following numeric estimates we use this maximal value.

The GRAVITES estimates for the averaged differences of stresses are denoted by,

δσ¯rr:=1πa2∫U|σrr(r,θ)|@1m−σrr(r,θ)|@0m|rdrdθ,(5.1)

the averaged difference of displacements at the boundary are written as

δu¯=12π∫∂U‖u|@1m−u|@0m‖dθ,(5.2)

where u = u r e r + u θ e θ . We estimate the change in fiber length as

δL=L[−νE(δσ¯rr+δσ¯θθ)+α(T−T0)],(5.3)

and we use

δφ=βδL(5.4)

to denote the change of phase of light due to the change of length of the waveguide. The results are summarised in Table 5.3 for the single and double plane cases covered in Sections 4.2.1 and 4.2.2. The configurations with four and six contact planes only give small corrections compared to the two plane case, since the additional pressures are independent of the height. This should be compared with the gravitational effect

Table 5.3. Estimates for the GRAVITES experiment described in [ 1].

The numbers correspond to a difference of height between the arms of 1 m. The columns isolate the effects with respect to their sources. We write δg single for the effect calculated in the configuration of Section 4.2.1 and δg double for that in Section 4.2.2; δ σ¯ rr is defined in ( 5.1); δ u¯ is the average transverse deformation of the waveguide as defined in ( 5.2); δφ is the change of phase of light due to the elastic elongation of the waveguide.

	δg single	δg double	δ = 10 Pa	δT = 10 –2 K	
δ σ¯ rr [Pa]	2.5 · 10 –7	3.4 · 10 –7	10	0	
δ u¯ [µm]	8.3 · 10 –15	9.6 · 10 –15	8.4 · 10 –9	1.7 · 10–7	
δL[µm]	1.2 · 10 –7	1.6 · 10 –7	4.7	180	
δφ[rad]	7.0 · 10 –7	9.4 · 10 –7	28	1080	

(5.5)

in the expected GRAVITES signal.

It has been proposed to place the interferometer in a centrifuge, with horizontal plane of rotation, with the arms of the interferometer placed at different distances from the center of the centrifuge, achieving an acceleration gradient δg ≈ 10 m/s 2 between the arms. To estimate the difference of phase between the arms we invoke the equivalence principle, modelling this problem by a radial gravitational field directed away from the axis of rotation of the centrifuge, with strength depending upon the distance from the axis, and ignoring the Earth gravitational field which acts with constant strength normal to the plane of rotation. (Clearly a more precise model would be needed for the real experiment.) For future reference, the corresponding values are given in Table 5.4; we do not include effects related to the change of temperature or pressure there, as such effects would depend strongly upon the experimental setup.

Table 5.4. Estimates for a centrifuge experiment with δg = 10 m/s 2 at one meter separation.

Notation as in Table 5.3.

	δg single	δg double	
δ σ¯ [Pa]	0.8	1.8	
δ u¯ [µm]	2.3 · 10 −8	3.8 · 10 −8	
δL[µm]	0.4	0.8	
δφ[rad]	2.3	4.8	

A   Relation between P¯ and κ

The pressure terms P¯ in the quadrilateral configuration, as well as P¯ and P∨ in the hexagonal configuration are linked to the spooling tension. We make this relation explicit for the case of a waveguide pressing against a rigid cylinder of radius R.

We assume a static configuration, and we ignore boundary effects at the point where the contact between the waveguide and the spool is lost. We consider a waveguide, consisting of an optical fiber, which is wound N times in one layer around a rigid cylinder of radius R. We suppose that the contact interface of the waveguide and the cylinder has constant width w, and that the pressure, which we denote by P¯ , exerted on the cylinder by the waveguide is constant across the whole area of contact. We assume that the axis of the cylinder is vertical, and we are interested in the pressure felt by the waveguide at the contact interface between the waveguide and the cylinder.

In a real-life situation there would be several layers of the waveguide, with various radii, with only the lowest one in contact with the cylinder, and the further ones in contact with neighbouring layers. Here we only consider the first layer, which is in contact with the cylinder. Similar considerations can be applied to describe successive layers; one would then also need to take into account the fact that the neighbouring layers are elastic and not rigid, as well as deformations of the neighbouring layers arising from the Poisson ratio. Our analysis in the main body of the paper sets the ground for further such investigations, which are left to future work.

We relate the pressure P¯ of Figure 1.2c to the tension force F T by considering the free-body diagram of a half of a single turn of a waveguide, i.e., we consider the intersection of the spool together with the waveguide and a plane on which the axis of the spool lies. The left Figure A.2 shows the intersection when viewed from above the spool of Figure A.1.

Figure A.1. A waveguide wound around a rigid cylinder with radius R.

We show a single layer of waveguide wrapped around the spool. Equal tension forces are applied at both the starting and end points so that the spool is in equilibrium. The spacing between the neighboring sections of the fiber has been increased for visual clarity.

Figure A.2. Free-body diagram of half of a single turn of waveguide around the spool (this corresponds to a view from above in Figure 1.2c or Figure A.1, so that the vertical is orthogonal to the plane of the picture).

The rectangle on the right-hand side shows the contact area with the width w as viewed from the left. The arrows illustrate the direction of the (uniform) reaction force.

Let us imagine that we have a constant pressure P¯ acting from the side as in Figure 1.2c, generating a force orthogonal to the contact region with the spool. This pressure generates a force ℱ which should equate the tension forces, i.e., ℱ = 2 F T , where

ℱ:=∫−π/2π/2P¯cosφAdφ=2Rw∫−π/2π/2P¯cosφdφ(A.1)

is the normal force due to the pressure P¯ acting on the contact area and φ is the angle shown in Figure A.2 cover 180 degrees of a semicircle. Hence, ℱ is the effective force caused by the pressure P¯ .

We assume that the tension force is constant along the waveguide. Therefore, we find ℱ = 4 Rw P¯ , which after equating the forces yields

P¯=FT2Rw.(A.2)

Recall that the constant appearing in the Hertz constant problem is defined as the total force per unit length pushing the bodies into each other, which in our context translates to P¯ = / w. Comparing with ( A.2) we find

F=FT2R.(A.3)

(Note that this is independent of the number of windings in the spool, and that a virtual work calculation gives the same result.) This value of can now be used in Section 4.2 to calculate w by solving the contact problem.

This formula is valid for the first layer, as counted starting from the core of the spool. One can generalize the result for the case where we have more layers that exert forces upon each other by writing

P¯n−P¯n+1=FT2[R+(n−1)a]wn,(A.4)

where w n is the contact width at the n’th layer, while P¯ n and P¯ n+1 are the contact pressures there.

B   Extending to non-symmetric solutions

Although the mirror symmetry assumption made in Section 2 is well-motivated for our purposes, we nevertheless give the full solution for completeness, demanding only regularity at r = 0 and 2 π-periodicity in θ.

Discarding only terms incompatible with regularity and 2 π-periodicity in ( 2.25), as well as B 0, D 1 and d 1 which do not contribute to the stresses, we have

ϕ(r,θ)=D0r2+C1r3cos(θ)+c1r3sin(θ)+∑n≥2[(Anrn+Cnrn+2)cos(nθ)+(anrn+cnrn+2)sin(nθ)].(B.1)

Again, the displacement can be derived by integrating ( 2.21)–( 2.23), yielding

ur(r,θ)=12μ{2(1−2ν)D0r−12gρ(1−2ν)r2cos(θ)+(1−4ν)r2(C1cos(θ)+c1sin(θ))+∑n≥2([−nAnrn−1+(2−4ν−n)Cnrn+1]cos(nθ)+[−nanrn−1+(2−4ν−n)cnrn+1]sin(nθ))}−Ξcos(θ)+Ξ2sin(θ)−νκr+(1+ν)α(T−T0)r,(B.2)

uθ(r,θ)=12μ{(5−4ν)r2(C1sin(θ)+c1cos(θ))−12gρ(1−2ν)r2sin(θ)+∑n≥2([nAnrn−1+(4−4ν+n)Cnrn+1]sin(nθ)+[−nanrn−1−(4−4ν+n)cnrn+1]cos(nθ))}+Ξsin(θ)+Ξ2cos(θ)+c∗r,(B.3)

with the integration constants determined by the boundary conditions u r ( a, 0) = 0 = u θ ( a, 0) as above. Thus,

Ξ=12μ{2(1−2ν)D0a+(1−4ν)C1a2−12gρ(1−2ν)a2+∑n≥2[−nAnan−1+(2−4ν−n)Cnan+1]}−νκa+(1+ν)α(T−T0)a,(B.4)

and further

Ξ2=12μ{(5−4ν)a2c1+∑n≥2[nanan−1+(4−4ν+n)cnan+1]}−c∗a,(B.5)

provided that the sums converge.

Continuing to the boundary conditions, we immediately restrict to the frictionless case, taking

σrθ|∂U=0.(B.6)

The boundary condition for σ rr can now be a general Fourier series which we write in the suggestive form

σrr|∂U=f(θ)=f0+∑n≥1[(f−n+fn)cos(nθ)+i(f−n−fn)sin(nθ)],(B.7)

with

fn=12π∫−ππdθf(θ)einθ.(B.8)

These boundary conditions determine the coefficients in ( B.1) algebraically by comparing to the corresponding stresses ( 2.14)–( 2.16). We find

C1=c1=0,Cn=1−na2(1+n)Anandcn=1−na2(1+n)an,(B.9)

via ( B.6), leading to

σrr|∂U=2D0−agρcos(θ)+∑n≥22(1−n)an−2[sin(nθ)an+cos(nθ)An],(B.10)

and comparing with ( B.7) for the remaining coefficients

D0=12f0,(B.11)

An=−12(n−1)a2−n(f−n+fn),(B.12)

an=−i2(n−1)a2−n(f−n−fn),(B.13)

for n ≥ 2, as well as

f−1+f1=−agρ,(B.14)

f−1−f1=0.(B.15)

Note that, similarly to the symmetric case the space of possible boundary conditions is constrained by the body force. Without any body forces oriented along the x-direction ( B.15) requires that the sin( θ) term in ( B.7) vanishes.

In the simplest case, considering line forces from the left and right and a line support from the bottom, i.e. boundary conditions

σrθ|∂U=0,(B.16)

σrr|∂U=δ(θ)+P^δ(θ−π/2)+P∨δ(θ+π/2),(B.17)

Equation ( B.15) implies P^ = P∨ , thus forcing mirror symmetry.

Acknowledgements

Useful discussions with Thomas Mieling and Philip Kornreich are acknowledged. We are very grateful to Arash Yavari for advice and comments about a previous version of the manuscript. Research supported in part by the Austrian Science Fund (FWF), Project P34274 and by the European Union (ERC, GRAVITES, project no 101071779). H.B. was supported in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement ERC advanced grant 740021–ARTHUS, PI: Thomas Buchert). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

A previous version of this submission is available as a preprint here: https://arxiv.org/pdf/2401.16949.pdf.

Data availability

No data are associated with this article.

10.21956/openreseurope.18729.r41793
Reviewer response for version 1
Brun Michele 1Referee https://orcid.org/0000-0002-4760-9062

1 University of Cagliari, Cagliari, Italy
23 8 2024 Copyright: © 2024 Brun M
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Version 1recommendationapprove-with-reservations
The paper presents some semi-analytical results for an elongated cylinder subjected to a transverse gravitational load and various contact forces.

The study is motivated by the need to model a planned experiment that aims to explore the influence of the gravitational field on entangled photons propagating in waveguides.

The approach used is classical and well-executed, determining the solutions in terms of Airy potentials. However, the advantage of this solution is currently limited by the availability of numerical techniques, particularly in the linear regime. Nonetheless, the work has its validity. The proposed results can be considered as an application of existing techniques to a specific problem.

Overall, I believe the work deserves acceptance after the following minor revisions have been addressed: The authors refer to large elastic deformations in the Background section, but the entire model has been developed in the small deformation (geometrically linear) regime.

Figure 1.1 is not sufficiently explanatory for a non-specialist, and additional details should be included. It is strongly recommended to visualize the reference system in most of the figures.

Page 4, Section 2: expression (2.1) is not specific to an isotropic homogeneous material, but rather to a linear isotropic material. Note that inhomogeneity only implies that material constants are functions of position.

The approximation indicated in Figure 4.5 is not validated. Would considering a piecewise constant approximation with additional subintervals for the pressure distribution improve the precision of the solution? In this regard, it should also be noted that the difference formula (4.50) is based on solutions that are relatively far from the load application neighborhoods.

I think that formula (4.64) is not correct since it appears has a linear momentum balance along y-direction and the components of the radial stresses resultants should be considered.

In eqns. (5.1) and (5.2) the notation @ is not explained.

In Appendix a the second \bar\mathcal{P} should be \hat\mathcal{P}.

I do not understand the factor 2 on the right handside of eqn. (A.1). Following classical Mariotte formula the 2 should be absent.

In the figure where contour plots of stress fields are shown legends should be inserted. These are of particular importance in Fig. 4.10.

In conclusion I think that the work can be indexed after a revision.

Is the study design appropriate and does the work have academic merit?

Partly

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Solid Mechanics

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

10.21956/openreseurope.18729.r41521
Reviewer response for version 1
Maitra Matthew 1Referee
1 ETH Zurich, Zürich, Zurich, Switzerland
27 7 2024 Copyright: © 2024 Maitra M
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Version 1recommendationapprove-with-reservations
SUMMARY

This paper investigates how the thermo-elastic deformation of an optical fibre will impact the phase-shift of photons passing through it. The paper supports the GRAVITES experiment -- a project that seeks to determine how Earth's gravity affects entangled photons -- by deriving some `extra' effects that will need to be corrected for when analysing the experimental data. 

The GRAVITES experiment will essentially be an interferometer (placed in a vacuum chamber) whose two arms are optical fibres spaced a vertical metre apart. Entangled photons will pass through those arms, and the phase shift between the two arms will be measured. However, the two arms are expected to undergo different thermo-elastic deformations due to the differences in gravity, pressure and temperature between the arms. Crucially, this could lead to a change in the length of the arms, which would in turn lead to an extra phase-shift between the photons that would need to be corrected for. Ultimately, this change in length is the crucial quantity that the present paper seeks to derive.

This paper therefore studies the thermo-elastic deformation of a long, homogenous, isotropic cylinder; the cylinder is clamped at one end and a force F_{T} is applied to the other, with the other external forces acting on the cylinder being ambient pressure and its own weight. The cylinder represents each of the arms of the interferometer. In reality the fibres will be spooled, but the authors argue (quite reasonably) that the radius of the spool will be so much larger than the the radius of the fibre that one may reasonably analyse instead the deformation of a long, straight cylinder subject to a `spooling tension force' (F_{T}).

In Section 2 the authors state and solve the equations of static (thermo-)elastic deformation, first reducing the dimensionality of the problem by taking the cylinder's along-axis deformation to be precisely linear in z (eq. 2.5), then using the Airy stress-function to write down a general solution. In Section 3 they state the boundary conditions, as well as the crucial result (eq. 3.17) concerning the relationship between external parameters and the fibre's overall elongation. They match their solution to the boundary conditions for a variety of different representative geometries in Section 4, before discussing the application of their results to GRAVITES in Section 5.  

The paper's language is clear and its analysis seems sound -- I appreciated Section 4's figures and adaptation of Hertz' method --  but I found the overall structure and aims quite hard to grasp. Multiple readings were required before I actually understood what Table 5.3 is saying (assuming I have now come to a correct understanding!). In what follows I will therefore mainly focus on structure and notation.

DISCUSSION

1

Please lead the reader more clearly through the analysis, stating clearly why certain equations are being derived and certain assumptions made:

1a

As discussed above (and hopefully my interpretation is correct) the crucial relationship derived in the paper is eq.(3.17). But the authors do not flag the importance of this relationship explicitly, nor is eq.(3.17) even *referenced* in Section 5 when the GRAVITES estimates are made! Please make sure that this equation's importance is emphasised, and that its centrality to equations (5.1),(5.2),(5.3) -- and thus to the whole paper -- is stated.

1b

In that connection, please make EXPLICIT that Table 5.3 refers to DIFFERENCES BETWEEN THE RESPECTIVE INTERFEROMETER ARMS. It might be obvious after a few readings of the paper -- and I have little doubt that it is trivial to people who have immersed themselves in this topic for months -- but I don't think that the present manuscript makes clear what is being compared to what. I would appreciate something like the following to be added to the end of Section 3, after eq.(3.17):

``Given a spooling force and upon specifying ambient gravity, pressure and temperature, eq.(3.17) allows the computation of the elongation of an optical fibre, that is of each arm of the interferometer. Now, due to the arms' one-metre vertical spacing, the ambient variables could be different for each arm; the spooling force could be different too because two different spools will be used. We should therefore expect each arm of the interferometer to be of different length. This difference in length will lead to an extra phase-shift that must be corrected for and that is derived directly from eq.(3.17). In Section 4 we proceed to derive expressions for \sigma_{xx} and \sigma_{yy} in terms of ambient pressure and gravity, so that eq.(3.17) can indeed be seen to express a relationship between each arm's actual length and: p,T,g and F_{T}.''

If words to this effect had been included in the manuscript (perhaps even in the Introduction) I think I would have had a *much* easier time understanding where things were going.

1c

Please introduce and describe the spooling force F_{T} earlier in the main text, rather than just mentioning it in passing in the caption of Figure 2.1c. Preferably, state at the beginning of Section 2 that the model of the waveguide that this paper analyses in detail is the version from Figure 2.1c, and that we therefore have displacement-free boundary conditions on one end while needing to match the stress at the other end to the externally imposed force F_{T}. Incidentally, this would give a much clearer motivation for imposing eq.(2.5) than what the manuscript offers at present. 

1d 

Following on directly from 1c, please motivate eq.(2.5) more clearly (I didn't find the statement "compare Figure 2.1 (a)–(c)" to help much in that regard). Once the spooling force has been foregrounded, this should be easy. In addition, I would appreciate a note after eq.(2.10) stating that the z-component of the force-balance equation is automatically satisfied.

1e (miscellaneous)

Eq.(2.1): 

What are \sigma and \epsilon?! A reader without much background in elasticity would be stuck here; please define all terms used in the paper. In addition, eq.(2.6) should logically precede eq.(2.1): how can you define strain without any concept of deformation? Also, the material need not be homogeneous for eq.(2.1) to hold.

Eq.(2.11): 

\sigma_{rr} + \sigma_{\theta\theta} should probably read \sigma_{xx} + \sigma_{yy} because the cylindrical coordinate system has not yet been introduced.

Eq.(4.32):

Please define both `A' and the concept of `indicator function'. (Presumably A here is different from the A in equations (4.10), (4.11) and (4.12)?!)

Eq.(5.3): 

\delta\overline{\sigma}_{\theta\theta} has not yet been formally defined. Eq.(5.1) should be adjusted to include this definition.

2

Temperature and thermo-elasticity.

A temperature T_{0} is introduced in eq.(2.3) as a `baseline', with T being the actual (constant) temperature of the optical fibre. Assuming that \delta{T} in Table 5.3 represents the difference between the *actual* temperatures of the interferometer's respective arms, then eq.(5.3) should have (T-T_{0}) replaced by \delta{T}. (This caused me a lot of confusion at first!)

3

I would like to see more uncertainty estimates, please.

3a

What is the rough uncertainty on the spooling force. How different will it therefore be between the two arms?

3b

What is the rough uncertainty on phase-shift due to gravity (eq. 5.5)? Without that, it is tough to get a feel for the importance of the numbers in Table 5.3.  

3c (tangential but related)

Please could there be a little discussion on the practical feasibility of controlling \delta{T} and \delta{p} such that one can measure \delta\phi_{GRAVITES} with sensible uncertainty?

3d (another tangent)

I would also like to see a discussion of the practical feasibility of controlling vertical vs horizontal gradients of T and p. For the purposes of Section 2, 3 and 4's analysis, T is taken as constant. Yet in Section 5 it is allowed to vary vertically so as to be different for each arm. But if T is constant along the fibre, then in real life it must also be constant along the spool -- and that would seem to rule out vertical gradients. Essentially we seem to require that the temperature not vary vertically along the length scale of the spool, but that it potentially vary vertically between the interferometer's arms. Is this a contradiction? In this connection, it would be nice to know roughly how tall the spools will be.

4

Miscellaneous

4a

I do not agree that ``[l]arge elastic deformations of gravitating cylindrical bodies play a significant role in everyday life". Spherical bodies, yes. But cylindrical ones... surely not?! Is it simply a misprint, with the authors actually wishing to state the opposite? That would make better sense to me, given that the following sentence starts by drawing a contrast (``On the other hand [...]"). Whatever the case, I really think that this first sentence must be changed or removed. 

4b

It is not (necessarily) the problem of these authors, but does Fig. 1.1 really need that background that seems to be representing curved spacetime? It makes the cartoon rather hard to read. How are the two spools distributed in space with respect to each other? A background that helps to show that distribution would be much more helpful, at least to me.

4c

In fact, Figure 1.1 is not particularly helpful on its own. It is appealing as a cartoon visualising GRAVITES, but I would also appreciate a proper diagram that shows the effective setup analysed within the paper. That is, I would like to see two parallel lines whose spacing is clearly indicated, each of which has its own environmental parameters (p, T, g) associated with it, with coordinate systems fully marked etc. 

4d

Very last paragraph:

The authors state that the centrifuge setup would require a more precise model. Perhaps they could outline what aspects of the present paper's analysis would need to be changed in that case? In particular, would it still be reasonable to assume a certain along-fibre deformation and then take all other fields to depend only on coordinates tangential to the fibre?

CONCLUSION

I will be glad to see this interesting paper (and its promised follow-up!) accepted for official approval subject to dealing with the points discussed above.

Is the study design appropriate and does the work have academic merit?

Yes

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Theoretical continuum mechanics; theoretical seismology

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

10.21956/openreseurope.18729.r41517
Reviewer response for version 1
Beig Robert 1Referee
1 Gravitational Physics/Physics Faculty, Universitat Wien, Vienna, Austria
19 7 2024 Copyright: © 2024 Beig R
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Version 1recommendationapprove
An experiment is currently developed aiming at measuring the influence of gravity on entangled states of photons in an optical fiber. Gravity will also deform the shape of the waveguide, which potentially changes the phase shift to be measured in the experiment. The paper by Barzegar .,et al calculates these elastic deformations, using linearized (thermo-) elasticity for isotropic, homogenous materials - which should be a valid approximation under present circumstances. The main issue in their calculations comes from boundary conditions at the rigid plane where the cylindrical wave guide is placed. In a first step the authors assume this contact to occur across a line. The results in that case, as the authors point out, and as is known from the literature they cite, is 'unphysical' - by which they mean that the obtained deformation diverges at the contact interfaces. In a second part the authors improve these solutions by employing Hertz-type boundary conditions at these interfaces. Finally, in section 5, the authors, estimate the influence on elastic deformations on the planned GRAVITES experiment. They expect this influence on the measured phase shift to be 2 orders of magnitude smaller than that due to electromagnetism. More details are promised from electromagnetic calculations under way.

I find the topic of this paper timely. The paper is well written, Without having checked details, I expect the calculations to be correct.

I would like to raise one point concerning the presentation. Due to symmetry the work in this paper takes place in 2d. Here, if the external force field. Comes from a potential which is harmonic (as to good approximation is the case for terrestrial gravity) and temperature is constant, the field equations boil down to the biharmonic equation for a scalar called an Airy function in the literature plus suitable boundary conditions. (One can then use an explicit expression for the general solution to the latter equation, Fourier expanded with respect to angle, known as the Michell solution in the literature.) It would greatly benefit a general theoretical physics audience if the authors cared to spend a few lines explaining the process leading from the standard field equations to the biharmonic equation rather than only referring to the classical, engineering-type literature. The basic observation (apparently due to Airy) is that a divergence-free symmetric tensor in 2d can be written as. 

σ ij = ⸹ ij ΔΦ - ∂ i∂ JΦ

The authors might also mention that the 'compatibility condition' leading from there to the biharmonic equation is actually a purely geometric condition on a symmetric tensor in order to be the linearized strain tensor of a deformation.

Is the study design appropriate and does the work have academic merit?

Yes

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

General Relativity, Relativistic Elasticity

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

1 The redefinition θ → θ − π2 can be absorbed by the constants.

2 A convenient form for the full parametrized solution can be found in [ 7, Tab. (9.1)].

Competing interests: No competing interests were disclosed.

Competing interests: No competing interests were disclosed.

Competing interests: No competing interests were disclosed.
==== Refs
1 Hilweg C Massa F Martynov D : Gravitationally induced phase shift on a single photon. New J Phys. 2017;19 : 033028. 10.1088/1367-2630/aa638f
2 Sadd MH : Elasticity: theory, applications and numerics. second edition ed., Aca-demic Press,2009.
3 Landau LD Lifshitz EM : Theory of elasticity. Course of theoretical physics. Elsevier Butterworth-Heinemann,1986;7.
4 Zhenye W Shiping L : The generalized plane strain problem and its application in three-dimensional stress measurement. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts. 1990;27 (1 ):43–49. 10.1016/0148-9062(90)90007-O
5 Gould PL : Introduction to linear elasticity. 3rd ed. 2013. ed., Springer New York,2013. 10.1007/978-1-4614-4833-4
6 Airy GB : On the strains in the interior of beams. Philos Trans R Soc London. 1863;153 :49–79. 10.1098/rstl.1863.0004
7 Barber JR : Elasticity. Springer Netherlands,2010. 10.1007/978-90-481-3809-8
8 Michell JH : On the direct determination of stress in an elastic solid, with application to the Theory of Plates. Proc Lond Math Soc. 1899;s1–31 (1 ):100–124. 10.1112/plms/s1-31.1.100
9 Timoshenko S Goodier JN : Theory of Elasticity. second ed., McGraw-Hill,1951.
10 Hertz H : On the contact of rigid elastic solids and on hardness. Ch 6: Assorted Papers,1882. Reference Source
11 Crystran Ltd: Silica glass (sio2).Data Sheet published online,2012.
