## Abstract

The postbuckling behavior of a clamped-clamped elastic fiber constrained inside a rigid cylinder is analyzed theoretically, numerically, and experimentally. We concentrate on characterizing the contact configuration between the fiber and the cylinder wall during initial postcontact stages of the fiber deformation, in which only a small segment of the fiber length maintains contact with the cylinder wall. This is apparently the first study of that phenomenon presenting an in-depth investigation of the fiber deformation and contact stages in experiments, along with a detailed examination of the effect of fiber geometrical imperfection. The main experimental challenge was to identify regions of contact between the fiber and the cylinder wall and to distinguish them from segments of the fiber that are very close to the cylinder wall but make no contact with it. To this end, we employed a novel experimental setup consisting of a transparent rigid cylinder filled with an opaque milky fluid, combined with image processing, and synchronized force measurements. The results of the experiments are supported by finite-element analysis and are also compared to available theoretical predictions based on the elastica model of an initially perfectly straight fiber. A small strain analytical solution reveals the role of minor initial geometrical imperfections in deriving the force–displacement relation during fiber bending. This study provides new understanding of the influence of key parameters on the behavior of such systems and may have practical implications in the fields of stent procedures, medical endoscopy, deep drilling, and more.

## 1 Introduction

The postbuckling behavior of a linearly elastic fiber subjected to lateral constraints is of practical importance in a variety of fields, ranging from medical procedures to various engineering applications. Examples of medical procedures include in vivo diagnosis and the threading of fiber for the purpose of medical imaging or for catheterization of the heart, urinary tract, and blood vessels. Understanding the nonlinear behavior of such systems, in particular, the mechanical interaction between the fiber (the guidewire) and the constraining walls (the artery) is imperative to ensure safe execution of the procedure [1]. In rare cases, the extensive deformations of the guidewire can result in fracture of the guidewire or induce damage to the artery during the procedure [2,3]. Other applications include the internal examination of pipe systems, the insertion of artificial fibers in industrial crimpers, drilling of wells from a platform to reach deep hydrocarbon or gas reservoirs [4], effects of delamination in composite materials [5,6], the insertion of paper into toner, the growth of plant roots [7], and the growth of filopodia in living cells [811].

Initially, engineering interest was mainly concerned with ways of avoiding critical deformations followed by buckling, and the scientific discussion centered on assessing critical forces [1215]. During the second half of the twentieth century, starting in the early 1950s, theoretical models of postbuckling behavior began to emerge. These early studies dealt with formulating and solving problems of laterally unconstrained, compressed columns, and curved beams supported by various types of boundary conditions [16,17]. In recent decades, the interest in postbuckling behavior of laterally constrained fibers has increased. In particular, theoretical and experimental studies have shown that a bilaterally constrained fiber undergoing plane deformations exhibits intriguing behavior. Ensuing studies have exposed a rather rich sequence of events under controlled axial end displacement [6,1821]. This sequence includes the formation of discrete (point contact) or continuous (line contact) regions of contact between the fiber and the constraining walls. A related topic of interest is the instantaneous transition from one equilibrium configuration to another due to the onset and the formation of new instability modes. Specific details of such deformation patterns and their dependence on process parameters such as fiber slenderness, the ratio between fiber radius of gyration and the width of the gap between the walls, loading rate, and friction are given in Ref. [22]. Theoretical studies have adopted various strategies and simplifying assumptions such as fixed constraints, frictionless walls, or small deformations [4,23]. Research objectives were to examine a range of possible equilibrium configurations and the evolution of contact between the fiber and the constraining walls [4]. In addition, numerical methods were employed to study the planar deformation of fibers subjected to more complex lateral constraints, such as nonparallel walls and noncontinuous and curved surfaces [2430]. Only a handful of studies consider the effects of friction [4,23], and an even smaller body of work has approached the realistic case of compliant (deformable) constraining walls [31,32].

The out-of-plane three-dimensional (3D) response of a fiber constrained inside a rigid cylinder is discussed in Ref. [33]. In addition to the formation of discrete and/or continuous contact regions, a transition between planar deformation and three-dimensional configurations occurs. Typically, initially, straight elastic fiber buckles into a planar wavy shape when subjected to edge thrust. As the edge thrust increases, the fiber contacts the cylinder wall, switches to a nonplanar deformation, and eventually twists and adopts a helix-like shape. In some applications, such as for drilling oil wells, an understanding of the details of this behavior is crucial. Once the fiber contacts the wall, the effectiveness of the drilling operation is considerably reduced. Moreover, locking might occur when the fiber turns into a helix-like shape with the extensive wall contact. A similar phenomenon also occurs in stent operations [2,3,22,34]. Studies of the 3D deformation of a laterally constrained fiber have been also conducted in the context of delamination frequently encountered in fiber-reinforced composites [35,36].

Theoretical studies investigating the 3D deformation of a fiber constrained inside a cylinder can roughly be divided into two main groups. The first group of studies assumes that the constraining cylinder is slender and that the deformation of the fiber is small. This group of studies employs the model of small strains and small rotations. Several different formulations for the critical loads and postcritical configurations were examined. A few papers considered the effects of friction [22], gravity [37,38], and the inclination angle of the constraining cylinder [12,39].

Nearly, all theoretical studies of finite deformations of a fiber constrained inside a cylinder have focused on the final stage of the deformation process where almost the entire length of the fiber contacts the cylinder wall, and the fiber adopts a helix-like configuration [811]. References [16,37], which are among the earliest in this direction, have utilized energy methods to extract the relation between edge thrust and pitch of the circular helix. To date, not much attention has been devoted to the initial postcontact stages of the fiber deformation that follow the first contact between the fiber and the cylinder wall. In this respect, Refs. [4042] provide valuable theoretical, numerical, and experimental information; the focus therein is on extremely slender cylinders (inner radius to length ratio of $∼10−4$) and on horizontal orientation, causing nearly 90% of the fiber length to be initially in contact with the cylinder, even before the external load is applied. In the recently published papers by Chen et al. [43,44], a rigorous elastica type model was developed to describe the postbuckling behavior of a perfectly straight fiber inside a rigid and frictionless cylinder. Before external force is applied, the fiber coincides with the centerline of the cylinder, making no contact with the cylinder wall. Numerical results for a relatively large inner radius to length ratio of ∼10−1 have demonstrated the many possible equilibrium configurations of an axially compressed fiber inside a constraining cylinder. However, there is a definite need for experimental studies that thoroughly investigate postcontact behavior in such processes over a range of parameters. In fact, little is known on process sensitivities to the main parameters that govern the mechanical interaction between the fiber and the cylinder.

The goal of the present article is to present further progress toward bridging this gap. We have systematically studied the initial deformation stages of a fiber constrained inside a rigid cylinder by means of novel experiments together with a finite-element analysis (FEA). Special effort has been made to develop an experimental method that enables the identification of contact characteristics. This undertaking proved to be a challenging task. Hence, even if a transparent cylinder is used, the curvature of the cylinder strongly affects the optics and makes it practically impossible to realistically identify zones of contact (or noncontact) between the fiber and the wall. The approach we have adopted is based on filling a transparent cylinder with an opaque white fluid and using a dark fiber along with postexperiment image processing. Synchronized force–displacement measurements have enabled accurate quantitative identification of the deformation pattern, including the important issue of contact behavior. A comparison of the results with the theoretical predictions of Ref. [44] supports the applicability of our findings over the range of parameters examined.

Furthermore, a standard postbuckling analysis, accounting for initial imperfections, within the framework of the Euler Bernoulli beam theory, has revealed good agreement with experimental data at least up to the second fiber/wall contact. We have assumed that the total imperfection is dominated by a linear combination of the (scaled) leading first symmetric and antisymmetric buckling modes. The scaling amplitudes of both modes have been determined by the comparison with measured force end shortening data, with the aid of a numerical best-fit procedure. Closed-form analytical solutions are derived for the fiber's initial postbuckling response, along with simple and useful asymptotic relations. Elegant approximations are given for axial force levels near wall contact, and fiber shape sensitivity to imperfections amplitudes is illustrated and discussed.

We turn now to a brief review of work by Chen et al. [4346], particularly the results presented in Ref. [44] that are relevant to the current research, and recapitulate the main theoretical findings therein. In earlier work, Chen [43] adopted the assumption of small deformations to study the postbuckling of a fiber constrained inside a rigid cylinder. The model considered a slender, isotropic, linear elastic, stress-free, and perfect fiber (no geometrical or material imperfections). The effects of gravity and friction were assumed to be negligible, and clamped-clamped boundary conditions were imposed. Thus, one end of the fiber is completely fixed (zero displacements and rotations) at the center of the cylinder cross section, while the other end can only move along the axis of the cylinder. The fiber bending response to edge thrust and eventual contact configurations were investigated. According to this model, the transition from one-point contact configuration to two-point contact configuration occurs at edge thrust, which corresponds to the critical (Euler) buckling load of a clamped-clamped column. Interestingly, this transition was found to involve a jump in the ends shortening upon additional loading. It has been argued that this peculiar jump phenomenon is due to the limitation of the small-deformation theory. To remedy this deficiency, a theory associated with the elastica model was developed in Ref. [44]. A similar approach was applied in Refs. [45,46] to study the deformation of a fiber subjected to end-twist rather than end-thrust. It was found that, contrary to the small-deformation theory, the planar one-point contact evolves to spatial (3D) one-point contact first and then gradually transforms to the two-point contact configuration. Moreover, seven distinct deformation shapes, each characterized by a different contact configuration, were identified [44]: (1) in no-contact, the fiber buckles into a curved shape as the force approaches Euler’s critical load; (2-1) contact forms between the fiber and the cylinder, leading to a planar (2D) one-point contact configuration, which in turn results in a sharp increase of the fiber response slope; (2-2) the fiber switches to a spatial (3D) one-point configuration, associated with a significant decrease of that slope; (3) the gradual evolution of a two-point contact configuration; (4) 3-point contact configuration; (5) point-line-point contact; (6) one-line contact; and (7) three-line contact.

The present article is organized as follows: In Sec. 2, we describe the methods and materials pertaining to the experimental system setup, the image-processing procedure, and the finite-element (FE) simulations used to characterize the contact configuration between the fiber and the cylinder wall. Then, in Sec. 3, we present a detailed small strain analysis of the bending response of an axially compressed, imperfect fiber, constrained inside a rigid cylinder. The experimental part of this research, which is presented and discussed in Sec. 4, is supplemented by analytical and numerical results. We provide a detailed comparison between experiments and theory over a range of parameters, including a sensitivity analysis. Finally, Sec. 5 summarizes the main observations drawn from this study and suggests several directions for future research.

## 2 Materials and Methods

In this section, we present the laboratory setup and experimental methodology for conducting the tests. This is followed by details of the image-processing equipment and technique employed in our research. Finally, we give a short account of the FE numerical calculations.

### 2.1 Experimental Setup.

Experiments were performed using an Instron 4483 machine, on which the designated experimental system was installed; see Fig. 1. The experimental system included five different setups with CSN EN 10270-1 steel wire fibers of length $L=530mm$, with three radii $(r=0.61mm,0.78mm,$$and0.88mm)$ constrained inside transparent Perspex cylinders (of radii $R=55mm$ and 100 mm). All three wires were tested in the $R=55mm$ cylinder. With the $R=100mm$ cylinder, we tested two wires with radii of $r=0.78mm$ and 0.88 mm. Thus, a total of five different specimen configurations were tested in the experiments. The cylinders were filled with an opaque white fluid (metalworking-cooling fluid, PVR-925S, mixed with water). Due to the inherent curvature of the cylinder, which strongly affects the optics, it is almost impossible to identify the onset and progress of contact between the fiber and the cylinder wall. Filling the transparent circular cylinder with the opaque white milky fluid enabled the progress of these contact regions to be identified and traced, as explained later. Special adapters were designed and installed to impose clamped boundary conditions at both ends of the fiber. The lower adapter was then fixed to the cylinder, while the upper adapter was attached to the moving arm of the Instron machine, so that the fiber axis coincided with the symmetry axis of the cylinder at the beginning of the experiment. During the experiment, the distance between the two ends of the fiber was slowly decreased upon loading by lowering the upper end of the Instron machine. This process resulted in the deformation of the fiber constrained by the cylinder. It should be noted that our method of shortening the distance between the two ends of the fiber, while the length of the fiber remains constant, differs from the method employed in Ref. [36]. There, the fiber was injected using two feeder rollers through a slave injector, thereby forming a slack loop. It was then pulled through a primary injector into the constraining glass cylinder. Besides the different conditions imposed on the fiber, the method employed in the present study is thought to introduce smaller friction enabling higher accuracy in measuring the fiber force. The experiments were carried out with two different cylinders of radii $R=55mm$ or 100 mm, corresponding to two values of the nondimensional ratio $ε=R/L$, namely, ɛ ≈ 0.1, 0.19. These values were chosen to enable quantitative comparison with the numerical results presented in Ref. [44], where values of ɛ = 0.1 and 0.2 were considered. End shortening, i.e., the decrease of the distance between the two clamps, was set by the displacement of the upper clamp as controlled by the Instron machine. Edge thrust (axial compressive force) applied to the fiber was measured by a static load cell, and together with the displacement adapter, both were synchronized with a digital camera (MAKO G-223 with CMOSIS/ams CMV2000 sensor, global shutter; 50 frames per second) that was used to record the experiment. To prevent plastic deformations, the maximum level of the end shortening was restricted.

Fig. 1
Fig. 1
Close modal

In each experiment, two essential characteristics of the fiber response were monitored and recorded, namely, the force end–displacement relation and details of the contact. To determine these variables, the axial force applied to the fiber was monitored along with the corresponding ends shortening. The analysis of the force versus end–displacement relation (loading path) provides the core information on the fiber loading path, revealing important aspects of the fiber behavior. The details of contact between the fiber and the cylinder were determined by analyzing the successive frames taken by the camera and complemented by image processing with matlab®, thus providing identification of the contact region between the fiber and the cylinder wall. Synchronization between the camera and the Instron machine enabled instructive qualitative and quantitative comparisons between the response observed in our experiments and the results predicted by FEA as well as the numerical results of Ref. [44].

### 2.2 Image Processing.

Each snapshot (image) underwent image processing with matlab® to identify the contact region between the fiber and the inner wall of the cylinder. To this end, the following procedure was applied: First, the image was converted to a digital array of scalar integers in the range of [0,255], where the array size is identical to the number of pixels in the image, and the scalar integer values represent the gray level of each pixel with values of 0 and 255 corresponding to black and white, respectively. Next, the image was corrected to produce a uniform background, that is, to ensure that all pixels of the white fluid have the same level of gray. The purpose of this step is to minimize the effects of nonuniform illumination and curvature of the cylinder wall. Without this correction, columns of the array (image) that are remote from the center are generally darker (have smaller gray level values). This correction involves multiplying each column by a different factor, so that the average values of the fluid pixels in all columns are identical. Finally, a threshold filter was applied to isolate pixels corresponding to contact between the fiber and the cylinder. The threshold level was calibrated by using the force–displacement to identify the first contact between the fiber and the cylinder wall. Since at this stage of deformation the contact configuration is necessarily of a point contact type, the threshold level was set as the gray level of that contact point. Further, the “size” of the contact region associated with the so-called (mathematically) point contact configuration was calibrated.

### 2.3 Finite-Element Analysis.

FEA was performed with the commercial FEA software abaqus®. A dynamic implicit analysis, designed to simulate the experimental system, included a 530 mm fiber that is clamped at both ends and laterally constrained by a rigid cylinder. The fiber model was meshed with hexahedral solid elements, type C3D8R (eight-node brick, accounting for geometrical nonlinearity), with over 50 elements in the fiber cross section and a total of 2700 elements in the entire fiber. The Young’s modulus of the fiber was set to 200 GPa, in accordance with prior tensile experiments that were performed with the Instron machine. Preliminary analyses with high-order brick elements and with a finer mesh (larger number of elements) led to similar results.

One end of the fiber was fixed to avoid all displacements and rotations. At the other end of the fiber, the only degree-of-freedom was the displacement in the x direction under a constraint that allowed for a predefined maximum displacement. This vertical displacement is equal to the end shortening, i.e., the decrease in distance between the two ends of the fiber. The edge thrust, equivalent to the force applied by the Instron machine in the experiments (see Fig. 1), was extracted from the simulation as the resultant force at that fiber end. The shortening rate was set to 10 mm/s, which is comparable with the rate at which the experiments were performed. The preliminary FEA showed that lower rates produced similar results, implying a quasi-static experimental response. To facilitate a fiber bending response from the outset and avoid a bifurcation analysis at the first buckling load, we introduced a realistic geometrical imperfection into the analysis, as discussed in Sec. 3. Thus, the stress-free configuration of the fiber was assumed to admit an initial imperfection y0(x), where x defines the fiber axis (see Fig. 2). Contact between the cylinder and the fiber was introduced by means of penalty stiffness in the normal direction of the contact surfaces (pressure-overclosure with hard contact and no penetration). In addition, tangential interaction, accounting for friction between the two bodies, was implemented in the model. A friction coefficient of μ = 0.05 was considered, representing the estimated friction coefficient between the metal fiber and the cylinder Perspex (including a greasy metalworking-cooling fluid as discussed earlier).

Fig. 2
Fig. 2
Close modal

## 3 Initial Imperfection Analysis

In this section, we present analytical derivations for the initial postbuckling response of the fiber, accounting for the presence of initial imperfections (see Fig. 2). Both symmetric and antisymmetric components are assumed, aiming at simple, if approximate, relations for the loading path. The theoretical analysis assumes that the thin elastic, and slightly imperfect fiber of length L with a circular cross section, behaves according to classical beam-bending theory. The fiber is uniform in its mechanical properties along its length; in the stress-free state, it is nearly straight and untwisted. The fiber deformation is constrained inside a straight hollow rigid circular cylinder with inner radius R, and the cylinder centerline coincides with the unstressed (perfect) fiber axis. Gravity and friction force are not considered. The diameter of the fiber cross section is negligible compared to that of the cylinder. It is also assumed that the fiber is completely clamped (no displacements and no rotations) at one end, while at the loaded end, the fiber is clamped laterally but is free to slide longitudinally. It is imperative that the solution method applied in the analysis should account for the expected deformation pattern at the outset. In that sense, one is instructively guided by the results of the theory presented in Ref. [44]. Introducing the parameter ɛ = R/L, it has been observed in Ref. [44] that, for a relatively slender cylinder, the first stages of the elastica deformation involve the sequence of one-point, two-point, three-point, and point-line-point contact configurations. It was also found that, even during the early stage of deformation, there are some differences between the predictions of small-deformation theory and the elastica model. According to the small-deformation theory [4,23], the one-point contact configuration is restricted to the planar form. Yet, the elastica model predicts also a 3D deformation pattern at the one-point contact stage. In addition, according to the small-deformation theory [4,23], the point-line-point contact configuration is the final stage of the deformation. As the radius of the constraining cylinder increases, the deformed patterns become less complex and the number of distinct configurations decreases before process termination when the fiber ends meet.

### 3.1 Loading Path up to First Contact.

Introducing a small initial imperfection provides a realistic modeling of the deviation from the perfectly straight shape and avoids numerical difficulties in treating the bifurcation point at the Euler critical load. Here, we present the small strain beam-bending solution for a clamped-clamped fiber in the presence of initial imperfection. This well-established model is presented here for completeness and follows Ref. [15], assuming y0(x) is the initial, stress-free, wavy shape of the fiber axis.

When a longitudinal compressive force P is applied at the end of the fiber, an additional bending deflection y1(x) develops, so the total shape of the deflection curve becomes y = y0 + y1. In the absence of a lateral load, the differential equation for the fiber bending response is expressed as follows:
$EId4y1dx4+P(d2y0dx2+d2y1dx2)=0$
(1)
where EI represents the usual flexural rigidity and x denotes the distance along the fiber with the origin located at the fiber center (see Fig. 2). When the load P increases beyond Pw (the compressive force at first contact between the fiber and the cylinder wall), an additional bending component y2(x) is activated as discussed below. To facilitate the analytical derivation, we assume the initial imperfection as a superposition of symmetric and antisymmetric modes, which represent the first two eigenmodes solution of the perfect beam-buckling equation, namely,
$y0=Af0(x)+Bg0(x),−L2≤x≤L2$
(2)
with
$f0(x)=12(1+cos2πxL),g0(x)=2xL−sin2γxLsinγ$
Here, A and B are the scaling amplitudes (as yet unknown) of the symmetric and antisymmetric imperfection modes, respectively, and γ ≈ 4.493 is the first antisymmetric eigenvalue obtained from the equation tan (γ) − γ = 0. By substituting (2) back in (1) and implementing the clamped boundary condition $(x=±L2:y1=0,dy1/dx=0)$, the additional bending displacement y1(x) is obtained from Eq. (1) in the form of
$y1=A(P¯1−P¯)f0(x)+B(P¯(γ/π)2−P¯)g0(x)$
(3)
with
$P¯=PPcr,Pcr=4π2EIL2$
(3)
where Pcr denotes the critical (symmetric) buckling force of a clamped-clamped fiber.
The total bending displacement of the fiber y(x) therefore becomes
$y=y1+y0=(A1−P¯)f0(x)+(B1−(π/γ)2P¯)g0(x)$
(4)

As expected, Eq. (4) implies that the fiber bending shape is highly amplified near the two critical loads, $P¯=1$ and $P¯=(γ/π)2≈2.046$. The values of the imperfection scaling amplitudes (A, B) can be assessed from experimental data, as discussed in the following section.

Relation (4) is valid up to first wall contact force, which can be obtained by a simple first approximation. We assume that A > 0 and |B| ≪ A (to be verified by experimental data) and accordingly neglect the antisymmetric branch at the onset of the first wall contact (y = y1 + y0 = R, at x = 0) when $P¯=P¯w$. Using (4), the symmetric imperfection amplitude A can be assessed from the simple approximate relation
$A=(1−P¯w)R$
(5)
using the measured value of first contact force. As a result of bending, the fiber end displacement is given by the standard geometrical relation
$Δ=12∫−L/2L/2[(dy0dx+dy1dx)2−(dy0dx)2]dx=12∫−L/2L/2[(dy1dx)2+2dy0dxdy1dx]dx$
(6)
with further substitution of Eqs. (2)(4), this relation provides the nondimensional end shortening
$Δ¯=(π2A¯)2[(1−P¯)−2−1]+γ2B¯2[(1−(πγ)2P¯)−2−1]whereΔ¯=ΔL,A¯=AL,B¯=BL$
(7)

Coefficients $A¯$ and $B¯$ can now be evaluated from $Δ¯−P¯$ measurements along the loading path, using relation (5) as first guess along with (7), or more accurately by a best-fit procedure (least-squares method) based on Eq. (7) for fitting the measurements of the end shortening as a function of the applied load, up to first contact. Notice that, by (4), as $P¯→1$ the leading asymptotic term of (7) is given by the asymptotic approximation $P¯≈1−πA¯/2Δ¯$. Similarly, when $P¯→(γ/π)2$, the leading asymptotic term is $P¯≈(γ/π)2(1−γB¯/Δ¯)$. The accuracy of these approximations depends on the magnitude of the initial imperfection and on the ratio ɛ = R/L. In fact, (5) can be written as $P¯w=1−A¯/ε$, implying that the load needed for first wall contact increases with increasing ɛ. These simple algebraic approximations are useful for fitting the imperfection amplitudes of Eq. (7) with experimental data.

To simplify the analysis, we shift now to the nondimensional coordinate ξ = x/L and rewrite the total deflection curve (4) as follows:
$y¯(ξ)=(A¯1−P¯)f0(ξ)+(B¯1−(π/γ)2P¯)g0(ξ),−12≤ξ≤12$
(8)
with, by Eq. (2),
$f0(ξ)=12(1+cos(2πξ)),g0(ξ)=2ξ−sin(2γξ)sinγ$
(9)
The exact axial location of the first wall contact ξw can be evaluated from Eqs. (8) and (9) by solving the two equations:
$y¯=εdy¯dξ=0atξ=ξw,P¯=P¯w$
(10)
for the two unknowns $(ξw,P¯w)$. Assuming that the first contact is near the fiber midpoint, with |ξw| ≪ 1, we find from (10) the first-order approximations:
$P¯w=1−A¯ε,ξw=(1−γsinγ)B¯π2ε[1−(π/γ)2P¯w]$
(11)

Notice that the contact force in (11) is identical with the simplified relation (5). Tracing the loading path (7), along with approximations (11), and comparing with the experimental data, we can determine the imperfection amplitudes $(A¯,B¯)$ to fit best with test data, as described in Sec. 4.

### 3.2 Post-First Contact Range.

With a further increase of the applied load, beyond first contact with the cylinder wall $(P¯>P¯w)$, solution (8) does not apply, as two separate zones of the fiber need to be considered. To this end, we denote by $y¯2(ξ)$ the additional fiber deflection for $P¯>P¯w$, so the total deflection in the post-first-contact range is expressed as follows:
$y¯(ξ)=y¯2(ξ)+y¯w(ξ)$
(12)
with
$y¯w(ξ)=A¯1−P¯wf0(ξ)+B¯1−(π/γ)2P¯wg0(ξ)$
(13)
denoting, by (8), the fiber shape at first contact $(P¯=P¯w)$. The bending curve is now governed by the following equation:
$d4dξ4(y¯2+y¯w−y¯0)+(kL)2d2dξ2(y¯2+y¯w)=0$
(14)
with
$(kL)2=4π2P¯$
(15)
However, at first contact, $y¯2=0$, and (14) is reduced to
$d4dξ4(y¯w−y¯0)+(kwL)2d2dξ2y¯w=0$
(16)
with
$(kwL)2=4π2P¯w$
(17)
Subtracting (16) from (14), we arrive at the differential equation, for the additional postcontact deflection,
$d4y¯2dξ4+(kL)2d2y¯2dξ2=−[(kL)2−(kwL)2]d2y¯wdξ2$
(18)
with $y¯w$ given by (13). The solution of (18) is now separated into two zones, under the simplifying assumption that the first point of contact (ξ = ξw = 0) remains almost at the center of the fiber. Accordingly, we write the two branches of solution, on the left $(y¯2L)$ and right $(y¯2R)$ to the center ξ = 0, as follows:
$−12≤ξ≤0:y¯2L(ξ)=C1Lsin(kLξ)+C2Lcos(kLξ)+C3Lξ+C4L+Ff0(ξ)+Gg0(ξ)$
(19)
$0≤ξ≤12:y¯2R(ξ)=C1Rsin(kLξ)+C2Rcos(kLξ)+C3Rξ+C4R+Ff0(ξ)+Gg0(ξ)$
where (CiL, CiR, i = 1…4) are integration constants, and (F, G) are two process parameters (scaling the nonhomogeneous solution of (18)) defined by
$F=(P¯−P¯w)A¯(1−P¯w)(1−P¯)andG=(P¯−P¯w)B¯[1−(π/γ)2P¯w][P¯−(γ/π)2]$
(20)
The eight integration constants in (19) are determined from eight boundary conditions, four for each branch. In our simplified model, the point of first contact (ξ = 0) remains constant for loads beyond $P¯w$, and apart from a jump of fiber cross-sectional shear resultant (due to the wall reaction), clamped conditions apply at ends of fiber branches. Thus,
$ξ=−12,0:y¯2L=0,dy¯2Ldξ=0$
(21)
$ξ=0,12:y¯2R=0,dy¯2Rdξ=0$
Compliance with that boundary data results in the integration constants
$C1L=kLFU1+βGU2,C2L=kLFU3+βGU4,C3L=−(kL)2FU1+kLβGU3,C4L=kLFU2−βGU4$
(22)
and
$C1R=−kLFU1+βGU2,C2R=kLFU3−βGU4C3R=(kL)2FU1+kLβGU3,C4R=kLFU2+βGU4$
with
$β=2(1−γsinγ)=11.207$
(23)
The four load-dependent functions Ui(i = 1…4) are specified in the  Appendix. The solutions (19) for the fiber bending shape beyond initial contact $(P¯>P¯w)$ are valid as long as $y¯2+y¯w<ε$. This analysis is of course restricted by the assumptions of classical beam theory. Nevertheless, along with (8), the present small strain comparison model provides insight regarding the nature of fiber behavior, particularly the role of initial imperfections as manifested by the fiber response. Regarding end shortening in the post-first-contact range, we have the overall shortening:
$Δ¯=π(rL)2P¯+12∫−1212[(dy¯2dξ+dy¯wdξ)2−(dy¯0dξ)2]dξ$
(24)
where, for completeness, we have added (in the first term of (24)) the fiber contraction due to axial compression. In evaluating (24), we use the expression for end shortening at first contact:
$Δ¯w=π(rL)2P¯w+12∫−1212[(dy¯wdξ)2−(dy¯0dξ)2]dξ$
(25)
and rewrite (24) as follows:
$Δ¯=π(rL)2(P¯−P¯w)+Δ¯w+12∫−1212[(dy¯2dξ)2+2dy¯wdξdy¯2dξ]dξ$
(26)
for $P¯≥P¯w$. Integration of (26) is performed over both branches of (19), with $y¯w(ξ)$ given by (13), and $Δ¯w$ follows from (25) as a minor generalization of (7):
$Δ¯w=π(rL)2P¯w+(π2A¯)2[(1−P¯w)−2−1]+γ2B¯2[(1−(πγ)2P¯w)−2−1]$
(27)

The simplified beam-bending solution given above provides a useful reference for the fiber contact up to the second contact and, after the curve fitting procedure, predicts quite accurately the loading path history. Also, response sensitivity to the level of initial imperfection is clearly revealed.

## 4 Experimental Results and Discussion

### 4.1 Estimating Imperfection Scaling Amplitudes via Curve Fitting.

Figure 3 displays experimental data for the dependence of the axial compressive force on end shortening, for a fiber of r = 0.61 mm and cylinder of R = 55 mm, along with curves of theoretical predictions. Similar plots for the other four experiments are provided in the  Appendix, see Fig. 12. The values of imperfection scaling amplitudes have been determined according to test results up to first contact point, as detailed in the previous chapter, and are summarized in Table 1 Relevant test data have been considered as force–displacement measurements in the range of sufficiently developed fiber bending shapes to filter out minor imperfection components. This is compatible with basic imperfection theory, where the symmetric and antisymmetric modes become dominant as the axial force approaches the first and second critical loads, respectively.

Fig. 3
Fig. 3
Close modal
Table 1

Best-fit imperfection amplitudes and experimental/analytical data for contact points, for each of the five fibers

R (mm)r (mm)$A¯$$B¯$ξw (11)$Δ¯w(exp)$$P¯w(exp)$$P¯w(28)$$Δ¯ww(exp)$$P¯ww(exp)$$P¯ww(30)$Eeff/E (32)
55 ɛ ≈ 0.1040.613.47 × 10−22.91 × 10−30.02310.02010.6410.6660.02321.8331.9650.00504
0.781.12 × 0−21.83 × 10−30.01730.01880.8690.8920.02111.9981.9950.01053
0.884.3 × 10−38.28 × 10−40.00850.02060.9620.9580.02332.0212.0220.01071
100 ɛ ≈ 0.1890.783.17 × 10−21.54 × 10−30.00780.07950.8380.8320.08911.8832.0220.00233
0.881.27 × 10−21.13 × 10−30.00630.08090.9470.9320.03421.9432.0280.00824
R (mm)r (mm)$A¯$$B¯$ξw (11)$Δ¯w(exp)$$P¯w(exp)$$P¯w(28)$$Δ¯ww(exp)$$P¯ww(exp)$$P¯ww(30)$Eeff/E (32)
55 ɛ ≈ 0.1040.613.47 × 10−22.91 × 10−30.02310.02010.6410.6660.02321.8331.9650.00504
0.781.12 × 0−21.83 × 10−30.01730.01880.8690.8920.02111.9981.9950.01053
0.884.3 × 10−38.28 × 10−40.00850.02060.9620.9580.02332.0212.0220.01071
100 ɛ ≈ 0.1890.783.17 × 10−21.54 × 10−30.00780.07950.8380.8320.08911.8832.0220.00233
0.881.27 × 10−21.13 × 10−30.00630.08090.9470.9320.03421.9432.0280.00824

It is worth noting that, in accordance with practical experience, both amplitudes (symmetric and antisymmetric) take smaller values for larger fiber thickness (diameter). The amplitude of the antisymmetric imperfection is about an order of magnitude smaller than that of the symmetric component, implying by (11) that the axial location of first contact is practically at the fiber center point; see Table 1. It is noted that although the curve fitting procedure is based on the test data limited to the range ahead of first contact, the analytical predictions remain in close agreement with test data up to the second contact, and even beyond in the 3D range. A possible explanation for that wide range of applicability, of the initial imperfection analysis, is that wall reaction at the second contact point is much smaller than that at the first contact point. Also, the 3D range appears to be highly unstable with several competing equilibrium states at nearly identical load levels.

A close inspection of the fiber deformation response near both points of contact is displayed in Fig. 4 for the fiber of r = 0.61 mm and cylinder R = 55 mm (similar plots for the other four fibers are provided in the Appendix, see Fig. 13). Evidently, the uniform pattern that emerges for all five fibers shows that a symmetric bending shape dominates the deformation up to the first contact. With increasing load, the antisymmetric branch becomes more noticeable up to the second contact. This observation is supported by the imperfection sensitivity chart displayed in Fig. 5, which illustrates the predicted $P¯−Δ¯$ relations for various values of $A¯$ and $B¯$. These include a nearly perfect fiber (“NP”), the actual fiber with the best-fit imperfections that appear in Table 1 (“BF”), and four additional cases. In fact, the role of imperfections can be illustrated in a simple thought experiment. Consider first a hypothetical fiber with the symmetric imperfection only with $(B¯=0)$, and so from (11), the onset of first contact is given by ξ = 0, with
$P¯w=1−A¯ε$
(28)
Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal
The values of $P¯w$ obtained by (28), with best fit $A¯$, are shown in Table 1, along with the experimental results. The prediction of (28) remains close to the test data, as expected from the best-fit procedure. Likewise, for a hypothetical fiber with only an antisymmetric imperfection $(A¯=0)$, we find from (8) and (9) that the maximum value of the antisymmetric branch at contact follows from the relation:
$ε=1.398B¯1−(πγ)2P¯ww$
(29)
where $P¯ww$ denotes the compressive force at second contact. Thus,
$P¯ww=(γπ)2(1−1.398B¯ε)$
(30)

As shown in Table 1, the predictions of (30) are in good agreement with the experimental measurement of compressive force at the second contact. This simple analysis suggests that the antisymmetric imperfection mode is activated as the second point of contacts approached. The complete analytical solution displayed in Fig. 4 (see also Fig. 13), complemented by the sensitivity to imperfection shown in Fig. 5, reflects this observation for all five fibers. Thus, the difference between fiber shapes at first and second points of contact (Fig. 4) is nearly purely antisymmetric. Also, while sensitivity to symmetric imperfections is pronounced ahead of first contact, sensitivity to the antisymmetric imperfection is clearly observed in the post second contact range (Fig. 5).

Finally, it is apparent from Fig. 3 that the fiber response shows hardening in the transition zone between the two points of contact. An approximate measure of this hardening is provided by the relation:
$η=P¯ww−P¯wΔ¯ww−Δ¯w$
(31)
where both force and shortening data are experimental. Thus, relative to axial stiffness of a straight fiber under compression, EA (A is the fiber cross-sectional area), we find the fiber-effective stiffness between the two points of contact:
$EeffE=η(πε)2(rR)2$
(32)
for the circular cross section. The ratio Eeff /E, obtained from (32) is shown in Table 1, revealing values of the order one percent.

### 4.2 Experimental Results.

In this section, we present the results of the experiments (details of the setup and procedures are provided in Sec. 2.1), focusing on the measured $P¯−Δ¯$ relation and observed contact characteristics. These results are compared with the results in Ref. [44] obtained for a perfect fiber based on the elastica model, and with the results of our finite-element analysis, which accounts for initial imperfection. The imperfection introduced in the FE model was deduced from the analytical model presented in the previous section, where the imperfection magnitudes, $A¯$ and $B¯$, for each fiber are summarized in Table 1.

Figure 6 displays the dependence of the compressive force $P¯$ on fiber end displacement, $Δ¯$, over the entire experimental range, for each of the three fibers constrained inside the R = 55 mm cylinder. The experimental results are compared with the theoretical elastica predictions of Ref. [44] (dashed curve). The fiber slenderness parameter, ɛ = R/L, in these experiments is similar to the value of ɛ = 0.1 considered in Ref. [44], allowing for a quantitative comparison. According to the theoretical model of Ref. [44], the $P¯−Δ¯$ curve is divided into distinct stages of deformation, which are indicated in Fig. 6 for convenience. These are (1) no-contact; (2-1) planar, 2D, one-point contact configuration; (2-2) spatial, 3D, one-point configuration; (3) two-point contact; (4) three-point contact; and (5) point-line-point contact.

Fig. 6
Fig. 6
Close modal

It is evident that the force–displacement relationship for the fiber with r = 0.88 mm (dark curve) is consistent with the theoretical results in Ref. [44]. As expected, deviations of the test results from the theoretical perfect fiber predictions increase with the level of imperfection, especially in the initial stages of the deformation. This is pronounced for the thinner fibers, with r = 0.78 mm (curve) and r = 0.61 mm (azure curve). Another expected finding is that the effect of geometrical imperfections decreases after contact forms between the fiber and the constraining cylinder. In fact, the curves of the r = 0.78 mm and r = 0.88 mm fibers are practically similar (and consistent with the perfect fiber) once contact forms. On the other hand, for the thinner fiber with r = 0.61 mm, the presence of imperfections is noticeable throughout the entire range of end shortening. For all three fibers, the first contact occurs at almost the same fiber–tip displacement $Δ¯≈0.02$, as expected, which is also similar to that of the perfect fiber. The fluctuations of the measured force are presumably due to friction between the fiber and cylinder, causing stick-slip-like behavior. Contrary to the effect of geometrical imperfection, these fluctuations increase as deformation progresses due to higher normal contact force and contact area that develop during advanced stages of fiber deformation.

Fiber wall contact was identified based on the image-processing procedure described in Sec. 2.2. The results are presented in Fig. 7 for each of the three fibers, with the top and bottom rows showing side-view photographs at different fiber–tip displacements before and after image process (see Sec. 2.2). These results suggest that, for the fibers with r = 0.78 mm and 0.88 mm, the deformation stages and contact development are consistent with the theoretical findings in Ref. [44]. This is also reflected in the results presented in Fig. 6. It is our view that the concept of point contact [44] is a theoretical idealization of the real nature of fiber wall contact mechanics. In practice, a small segment of contact zone should be considered as a manifestation equivalent to the theoretical point contact. Thus, all images in Fig. 7 up to stage e, reflect a single-point contact configuration. After stage e, the images clearly show the emergence of two distinct regions of contact that appear to move further apart with increasing fiber–tip displacement, as predicted by the perfect fiber elastica model for stages f–h. The image-processing procedure also reveals three separate regions of contact at stage i (r = 0.78 mm), which is again consistent with the results presented in Ref. [44]. The qualitative agreement of contact characteristics between our experimental results and the theoretical results of Ref. [44] is consistent with the quantitative agreement of the force–displacement relation. In contrast, for the fiber with r = 0.61 mm, the contact stages observed in the experiment are not consistent with the theoretical predictions of Ref. [44]. This is also reflected in the force–displacement curve shown in Fig. 6, which deviates significantly from the results in Ref. [44] due to geometrical imperfection.

Fig. 7
Fig. 7
Close modal

Additional experiments investigated the behavior for ɛ ≈ 0.19, see Fig. 8, using a cylinder with an inner radius of R = 100 mm and fibers with r = 0.78 mm and 0.88 mm. The corresponding theoretical results in Ref. [44] are also shown in Fig. 8 for comparison and suggest that the deformation patterns become less complex for the larger radius of the constraining cylinder. For ɛ = 0.2, the theory predicts that only four contact configurations should occur (see also the brief review in Sec. 1), namely, configurations (1), (2-1), (2-2), and (3). Additional contact configurations, (4) through (7), are expected to take place for ɛ = 0.1, but not for ɛ = 0.2. More important is the prediction that once spatial 3D deformation is activated for the case of ɛ = 0.2, just ahead of second buckling load (at force $P¯≈2$), the force will no longer increase but will slowly decrease. In contrast, in the case of ɛ = 0.1, the force increases up to a level of nearly $P¯≈3$, with the deformation progressing from configuration (2-2) to configurations 3–5. Thus, the theoretical findings of Ref. [44] are consistent with the test results shown in Fig. 8. Along similar lines, we find that predictions of the theoretical model are consistent with the development of contact between fiber and cylinder wall as shown in Fig. 9.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal

Figure 10 displays force–displacement experimental curves for all five fibers, compared to our FEA results. The FEA predictions are in good agreement with the test results for all fibers, thus confirming the FE analysis combined with the method of characterizing initial imperfection from the experiments. Bending induced by the presence of imperfections begins at the initial stages of loading with considerable weakening as the first point of contact is approached. This is followed by a stage of strengthening up to the second point of contact, beyond which the fiber response is getting progressively less planar. Figure 11 shows the deformed shape of the fiber along with contact locations and force, extracted from the FEA results, for r = 0.88 mm and R = 55 mm. The first row shows a side view of the deformed fiber, where a lighter color indicates interaction with the wall (in these images, the schematic cylinder is shown for clarity/orientation, but the images are not on an identical scale to allow focusing on the contact region). The second row shows top view, where contact points are marked by small filled circles. All images in the second row are of the identical scale. These results agree well with the development of contact as observed in the experiment. They also demonstrate that after contact develops, especially for relatively large $Δ¯$, segments of the fiber may be in very close proximity to the cylinder wall, yet not really touching it. This fact combined with the influence of the cylinder curved wall on the optics makes experimental validation extremely difficult. This experimental challenge has been the main motivation for this work and led to the development of the experimental setup described earlier, which has proved to be successful in characterizing the contact configuration even for the aforementioned challenging conditions.

Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

## 5 Summary and Conclusions

We have investigated experimentally, numerically, and analytically the response of an elastic clamped-clamped fiber constrained inside a rigid cylinder, including the postcontact range. This problem serves as a prototype of similar processes encountered in the engineering practice of several fields. A novel experimental setup was designed to enable the identification of contact regions between the fiber and the cylinder. Experimental findings, for five different geometries, are well supported by finite-element results and by small strain postbuckling analysis. The fiber response predicted by the standard small strain postbuckling approximation, accounting for minor initial imperfections, remains in close agreement with computational results obtained from the large strain FEA up to, and even beyond, the two-point contact configuration. Both methods provide loading histories close to the test data, where the FEA enabling very good predictions, consistent with the experiments, for the entire investigated range of end shortening. Still, the advantage of the small strain analysis is that it provides closed-form analytical expressions and insights, e.g., for the first and second contact forces. Importantly, the closed-form solution also enables extracting from the experiments the actual initial fiber imperfection in a simple and straight-forward way. The resolved imperfection can then be implemented in the FE model to allow for more accurate predictions.

The main conclusion, from the uniform pattern that emerges from this study, is that beyond onset of first wall contact, the fiber response (for given cylinder geometry) is dominated by the restraining rigid cylinder wall. Fiber slenderness and the presence of initial imperfections are apparent mainly in the range before first point of contact. Although the influence of imperfection is smaller in postcontact stages, it may still be significant if the initial imperfection is large.

The range of process parameter examined (fibers geometry and cylinder dimensions) extend the available experimental database, thus contributing to understanding of this complex nonlinear problem. For example, previous experiments were limited to extremely slender constraining cylinders, namely, ɛ < 10−4, or for cases where almost the entire fiber was in contact with the cylinder.

The FE simulations employed in the present work provide an efficient platform for investigating features of fiber behavior and may provide information that cannot be studied experimentally. For example, postprocessing of FE simulations can be used to analyze the deformation of the fiber or the stresses that develop. A similar approach, where FE simulations have been used to complement experiments of a fiber constrained inside a deformable cylindrical tube, has been adopted in Ref. [47].

Characterizing the behavior of the constrained fiber along with postcontact local response and friction included remains a major challenge for future studies. We believe that the experimental method introduced in our work can be developed further. The experimental difficulties due to uncertainties involved in fiber wall interactions at points of contact, along with perturbations induced by out-of-plane fiber imperfections, call for future studies in these directions.

## Acknowledgment

This work was partially supported by the Israel Science Foundation Grant ISF 581/17 and ISF 1598/21 to S.G. The financial support to Y.D. by RAFAEL is gratefully acknowledged.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

• k =

$PEI$

•
• r =

fiber radius

•
• x =

distance along the fiber

•
• y =

fiber deflection

•
• A =

scaling amplitude of symmetric imperfection

•
• B =

scaling amplitude of antisymmetric imperfection

•
• E =

Young’s modulus

•
• F =

$(P¯−P¯w)A¯(1−P¯w)(1−P¯)$

•
• G =

$(P¯−P¯w)B¯[1−(π/γ)2P¯w][P¯−(γ/π)2]$

•
• I =

inertia moment of fiber cross section

•
• L =

fiber length

•
• P =

compressive force

•
• R =

inner radius of the cylinder

•
• $P¯$ =

$PPcr$

•
• f0 =

$12(1+cos2πxL)$

•
• g0 =

$2xL−sin2γxLsinγ$

•
• y0 =

initial imperfection

•
• y1 =

additional deflection up to first contact

•
• y2L =

additional deflection at the left side, for compressive force larger than Pw

•
• y2R =

additional deflection at the right side, for compressive force larger than Pw

•
• CiL =

left branch of solution integration constant

•
• CiR =

right branch of solution integration constant

•
• Eeff =

$ηPcrA$

•
• $P¯cr$ =

critical (Euler) buckling force

•
• Pw =

force applied to the fiber at first wall contact

•
• $P¯w$ =

$PwPcr$

•
• Pww =

force applied to the fiber at second wall contact

•
• $P¯ww$ =

$PwwPcr$

•
• U1−4 =

four load-dependent functions (specified in the  Appendix)

### Greek Symbols

• β =

$2(1−γsinγ)=11.207$

•
• γ =

first antisymmetric eigenvalue from tan (γ) − γ = 0

•
• Δ =

end shortening (the decrease in distance between fiber ends)

•
• $Δ¯$ =

nondimensional end shortening

•
• $Δ¯w$ =

nondimensional end shortening at first contact between fiber and cylinder

•
• ɛ =

$RL$

•
• ξ =

$xL$

•
• η =

$P¯ww−P¯wΔ¯ww−Δ¯w$

### Appendix: Additional Calculation Details and Figures

The purpose of this appendix is to provide additional relations regarding the derivation that appears in Sec. 3.2. The four load dependent functions of (22) are defined by
$(U1,U2)=(2sinkL2),(−kLsinkL2−2coskL2+2)kL(kLsinkL2+4coskL2−4)(U3,U4)=(2−2coskL2),(kLcoskL2−2sinkL2)kL(kLsinkL2+4coskL2−4)$
(33)
The post-first-contact end shortening (26) is given by
$Δ¯=π(rL)2(P¯−P¯w)+Δ¯w+12∫−1212[(y¯2dξ)2+2dy¯wdξy¯2dξ]dξ=π(rL)2(P¯−P¯w)+Δ¯w+[(kL)2(k2L2U12F2+β2U22G2)I1+(kL)2(k2L2U1U3F2+β2U2U4G2)I2−2(kL)2(k2L2U12F2−β2U2U3G2)I3−2(kL)2F2(U1I4+U3I8)+2kLβG2(U2I5+U4I9)+(kL)2(k2L2U32F2+β2U42G2)I6−2(kL)2(k2L2U1U3F2−β2U3U4G2)I7+(kL)2(k2L2U12F2+β2U32G2)I10−2(kL)2F2U1+0.25F2π2+γ2G2]+[−2(kL)2A~F(U1I4+U3I8+U1)+2kLB~βG(U2I5+U4I9)+0.5A~Fπ2+2B~Gγ2]$
(34)
where the integrals I1 through I14 are expressed as follows:
$I1=cos2(kLξ),I2=12sin(2kLξ),I3=cos(kLξ),I4=−πsin(2πξ)cos(kLξ),I5=2cos(kLξ)[1−γcos(2γξ)sin(γ)],I6=sin2(kLξ),I7=sin(kLξ),I8=−πsin(2πξ)sin(kLξ),I9=2sin(kLξ)[1−γcos(2γξ)sin(γ)],I10=1,I11=−πsin(2πξ),I12=2[1−γcos(2γξ)sin(γ)],I13=π2sin2(2πξ),I14=4[1−γcos(2γξ)sin(γ)]2$
(35)

Complementing the results shown in Figs. 3 and 4 for fiber of r = 0.61 mm and cylinder of R = 55 mm, we show similar plots for the other four fibers in Figs. 12 and 13, respectively.

Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal

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