Abstract

The post-buckling behavior of a beam that is subjected to lateral constraints is of relevance to a range of medical and engineering applications, such as endoscopic examination of internal organs, the insertion of a guidewire into an artery in-stent procedures, root growth, deep-drilling, and more. In this paper, we address a disconnect between the existing literature and the reality of these systems, in which the lateral constraints are flexible and experience nonlinear deformations. As a step towards bridging this gap, we consider a beam undergoing planar deformations that is laterally constrained by a nonlinear springy wall, i.e., a wall that is laterally pushed by the beam against a nonlinear spring. Based on a simplified mathematical model, we obtain closed-form analytical solutions, which provide valuable insights and intuition. For example, we show that important features of the behavior, such as the transition from point contact to line contact and switching to the next mode, are dictated solely by a non-dimensional force, regardless of all other parameters of the system, and that the full description of the behavior is possible by means of two non-dimensional quantities that describe the relative stiffness of the nonlinear spring compared to that of the beam. The results also highlight the fundamental differences between the behavior with a stiffening spring or with a softening spring, such as the number of attainable modes and the monotonicity of the overall force–displacement relation. These results are then validated by experiments.

1 Introduction

The post-buckling behavior of a linearly elastic slender beam that is subjected to lateral constraints has been a subject of interest since the 1950s. The first studies were motivated by the deep-drilling industry, which was looking for a better understanding of the buckling and post-buckling of rotary drilling strings and tubing in pumping wells [1,2]. In the last two decades, there has been a renewed interest in the subject due to its relevance to a wide range of applications that involve a fiber accessing an enclosed space or the growth of a slender structural element that is laterally constrained by its surrounding environment. Classic examples are medical procedures, such as endoscopic examination of internal organs or the insertion of a guidewire into an artery in-stent procedures, filopodia growth in living cells, the influence of delamination in composite materials, drilling of deep wells in the search for natural gas or oil reservoirs, and the insertion of fibers in industrial crimpers [312].

The post-buckling behavior of a laterally constrained beam may involve several distinct stages of deformation, each characterized by a different deformation shape (mode of deformation) and/or different contact characteristics. For example, eight distinct deformation stages were identified in the case of a fiber compressed inside a rigid tube [13], e.g., planar deformation of the beam with a single point contact (between the fiber and the rigid tube), spatial (3-D) deformation with one point contact that is associated with significant reduction of overall stiffness, spatial deformation with two points of contact, with a single line contact, with two line contacts, with point–line–point contact, etc. Importantly, the exact sequence of deformation, the evolution of contact characteristics, and the details of the overall force–displacement relation depend on all system parameters in an intriguing, and often non-intuitive, manner. The importance of understanding this complex behavior lies in the fact that it governs the function or functionality of the system. For example, the type of contact that develops between the artery wall and the inserted guidewire dictates the ability of the surgeon to control the movement of the leading end of the guidewire as well as the occurrence of damage to the artery; In some cases, the combination of significant contact and the application of a high level of force by the surgeon, e.g., in order to traverse an occluded vascular segment, may result in over-expansion of the artery and extreme cases even to the fracture of the guidewire [1417].

Almost all relevant literature has considered rigid constraints. For example, a beam undergoing planar (2-D) deformations that is constrained by two fixed parallel rigid walls [1831], a compressed fiber inside a rigid cylinder [4,3236], a fiber inserted into a curved rigid channel [3741], or a compressed beam subjected to discrete rigid constraints [4244]. In reality, however, the lateral constraints are often compliant (non-rigid). For example, the artery may change its inner diameter due to the internal force applied by the guidewire. Also, in filopodia growth, the lipid membrane that encapsulates the growing actin filaments is flexible and undergoes significant deformations [6,45,46]. The influence of flexible/compliant lateral constraints on the behavior of the system has been considered by only a few recent studies. In Ref. [47], a beam undergoing planar (2-D) deformations that is constrained by two parallel rigid walls, one is fixed and the other is moving against a linear spring, was studied. This study was later expanded in Ref. [48] to account for a deformable wall, modeled as a Winkler foundation, rather than a moving rigid wall, and involved analytical, numerical, and finite-element analyses. A static and dynamic analysis, validated by experiments, of post-buckled beams confined by movable and flexible constraints was presented in Ref. [49]. A similar mathematical model was applied in Ref. [50] to study, analytically and numerically, the case where the beam is simply supported. A beam undergoing planar deformations that are constrained by two parallel deformable walls was studied numerically in Ref. [51] and included also stability analysis. The spatial (3-D) deformations of a beam that is constrained by a “springy” cylinder were studied numerically in Ref. [52]. More recently, the post-buckling behavior of compressed elastic inside a flexible tube was studied theoretically and experimentally in Ref. [53]. Importantly, in all these studies, the flexible constraints were assumed to be linear elastic. Clearly, this is not the case in reality. For example, the resistance of the artery to deformations caused by an inserted guidewire is inherently nonlinear, or the lateral force applied by the soil to a growing root is far from linear. The main goal of the current contribution is to make a step toward bridging this gap.

In this paper, we study the influence of nonlinear flexible constraints on the post-buckling behavior of a slender beam. Due to the complexity of the problem, we consider, as a first step, the scenario of a beam undergoing planar (2-D) deformations that is laterally constrained by two parallel walls. The first wall is fixed and rigid, while the other is moving against a nonlinear spring (see schematic illustration in Fig. 1), i.e., the force–displacement relation of the spring (or the “spring function”) that resists the displacement of the moving wall is described by a nonlinear function. Our analysis is not limited to a specific type of spring nonlinearity, and the spring may be stiffening or softening, i.e., characterized by a spring function that is mathematically convex or concave, respectively. Our main goal is to obtain insights as well as intuition into the intriguing behavior of such systems and highlight the fundamental differences in the overall behavior of the spring is stiffening or softening. To this end, we formulate a simplified mathematical model that assumes small deformations (small rotations) of the compressed beam. Previous studies of laterally constrained beams have shown that this type of model produces reasonably accurate predictions while providing valuable insights through closed-form analytical expressions [18,47]. The results of the mathematical model are then validated by a series of experiments, and the limitations of the model are also discussed. Accordingly, the structure of the paper is as follows: Chapter 2 details the mathematical formulation of the model along with analytical insights and numerical examples. Chapter 3 presents the results of the experiments, which are compared to the predictions of the theoretical model. A summary of the results and main conclusions are provided in Chapter 4.

Fig. 1
Schematic description of the system at different loading stages. The upper wall moves, parallel to the fixed wall, against a nonlinear spring: (a) initial state, when P = 0, (b) point contact between the beam and the wall, and (c) line contact. The moving wall and the fixed surface at the bottom are always parallel; thus, the displacement of the wall is y = D−D0.
Fig. 1
Schematic description of the system at different loading stages. The upper wall moves, parallel to the fixed wall, against a nonlinear spring: (a) initial state, when P = 0, (b) point contact between the beam and the wall, and (c) line contact. The moving wall and the fixed surface at the bottom are always parallel; thus, the displacement of the wall is y = D−D0.
Close modal

2 Formulation of the Problem and Theoretical Analysis

In this chapter, we study the system shown in Fig. 1 by means of a mathematical model, assuming small deformations of the beam. This assumption enables analytical derivations and insights, yet limited by its nature to small strains and rotations. The system involves a straight beam that is clamped at both ends and lies on a rigid, flat, and fixed surface. The beam, of length L0 and cross-section moment of inertia I, is made from a material with Young's modulus E. For specificity, and without loss of generality, we consider a rectangular cross section of width b and height t, i.e., I=bt3/12. We note that the geometry of the cross section may affect the details of the contact mechanics between the beam and the walls; the influence of these effects on the overall behavior is assumed to be small and therefore neglected. The beam is subjected to a compressive (axial) force, P, that results in horizontal displacement Δ of the left end. A moving wall, which is parallel to the flat surface, is positioned at distance D0 from it. The beam is constrained between the two surfaces, which remain parallel to each other; yet, the upper wall may be pushed (by the beam) against a nonlinear spring. The force–displacement relation of a nonlinear spring is described as
F(y)=F0f(y¯);y¯=y/D0
(1)
where F is the force in the nonlinear spring, y is the change of length of the spring (and is also the displacement of the moving wall), F0 is a positive constant which sets the scale of the spring stiffness such that df/dy¯|y¯=0=1, where f is a (nonlinear) monotonously increasing function. Note that the case of a non-monotonic function, f(y¯), which results in multiple equilibria for the same external load (that is applied on the spring) as well as in snap-through events, is not considered here.

Loading of the structure is performed by a gradual increase of Δ. At some point, the beam buckles, and point contact forms between the beam and the upper wall. This point contact generates a lateral force, which is equal to the spring force. From symmetry considerations, the point of contact is located at the mid-span. Therefore, we examine the behavior of one-fold and then extend it to the entire beam. Increasing the horizontal displacement may eventually lead to the formation of line contact, where a beam segment, of finite length, makes contact with the wall. Further increase of the displacement may result in instability and therefore switching to the next mode or even transition to point-contact again. Immediately after switching to the next mode, the contact characteristics may correspond to either point contact or line contact. Note that, by symmetry considerations, point contact between the beam and the moving wall also leads to point contact between the beam and the fixed surface at the bottom. The same applies to line contact.

Assuming that the beam undergoes small deformations (small displacements and moderate rotations), the governing (equilibrium) equation takes the form of
w,xxxx+k2w,xx=0;k=PEI
(2)
where w(x) describes the deflection function of the fold, and (),x, (),xx,… denote first derive, second derivative, etc. of () with respect to x. The corresponding boundary conditions are dictated by the contact conditions. Let us introduce the following non-dimensional quantities associated with the external (compressive) force, P, and the horizontal displacement, Δ:
ζ=kL02π,η=3L0π2t2Δ
(3)
where
Δ=εxL0+n0Hw,x2dx
(4)
Here, H is the fold length, εx=P/Ebt, and n is the mode number (the number of folds is 2n). Once w(x) is obtained by solving Eq. (2), the horizontal displacement, Δ, may be calculated using Eq. (4). Finally, the energy stored in the system, U, includes the energy stored in the spring as well as the axial compressions and bending of the beam
U=12EAL0εx2+2n0H12EI(w,xx)2dx+0yF(y)dy
(5)
where A is the beam cross-section area. Next, we analyze the various equilibrium configurations based on the earlier relations.

2.1 Buckling of a Clamped–Clamped Segment.

For a straight clamped–clamped segment of length c, buckling occurs at
kc=2π
(6)

Thus, the beam first buckles at ζ=1, corresponding to c=L0. After buckling takes place, contact forms between the mid-span of the beam and the moving wall, and point-contact forms.

2.2 Point Contact.

The general solution of Eq. (2) is:
w(x)=Asin(kx)+Bcos(kx)+Cx+D
(7)
The boundary conditions associated with point contact are
w(0)=w,x(0)=w,x(H)=0;2nEIw,xxx(H)=F
(8)
Plugging Eq. (7) into the first three boundary conditions in Eq. (8), we find that
w(x)=A(kxsin(kx)tan(Hk2)(1cos(kx)))
(9)
and the end displacement, w(H), is:
w(H)=Az(k);z(k)=2(kH2tan(Hk2))
(10)
Plugging Eq. (9) into the fourth boundary condition in Eq. (8), we have that
2nEIAk3=F(y)
(11)
where
y=Az(k)D0
(12)
Before we proceed, it is instructive to examine the above relations at the onset of buckling. Recall that Eq. (11) is valid only after the beam buckles, namely for kHπ. In the limit case of kHπ+, we have from Eq. (10) that z(k). In order to satisfy Eq. (11), the end displacement, Az(k), must be finite, implying that A1/z(k), i.e., A0. This means that, in the limit case of kHπ+, the left-hand side of Eq. (11) vanishes, implying that F(y)=0, and therefore, the end displacement, Az(k), is equal to D0, as expected. Rearranging Eq. (12), we have that
A=D0z(yD0+1)
(13)
and therefore substituting Eqs. (13) into (11) gives:
α(y¯+1)=f(y¯)
(14)
where y¯=y/D0 and α is a non-dimensional parameter defined as
α=2nEID0F0k3z
(15)
Since α is a monotonously increasing function of the load (through k), one may consider it as a non-dimensional loading parameter. The left side of Eq. (14) describes a linear relation in terms of y¯ with slope α, whereas the right-hand side constitutes a nonlinear function of y¯. These are schematically illustrated in Fig. 2. In the case of a stiffening spring, this equation leads to a single solution in terms of the spring displacement. On the other hand, for a softening spring, two solutions may be possible. Typically (except special cases that are discussed later), we are interested in the solution corresponding to the smaller spring displacement, since α is gradually increased from zero. We denote by y¯* the abovementioned solution for y¯ obtained from Eq. (14). Note that y¯* depends only on the non-dimensional parameter α and the functional form of the nonlinear force–displacement relation of the spring. Plugging the solution y¯* into Eq. (13), we find that
A=D0z(y¯*+1)
(16)
from which, using Eq. (9), we obtain the solution for the deflection w¯=w/D0:
w¯(x)=1z(y¯*+1)(sin(kx)+tan(Hk2)(1cos(kx))kx)
(17)
Fig. 2
Schematic illustration of the two sides of Eq. (14) and the corresponding solution y*¯, for stiffening and softening springs
Fig. 2
Schematic illustration of the two sides of Eq. (14) and the corresponding solution y*¯, for stiffening and softening springs
Close modal
Introducing Eqs. (3) into (17), and using H=L0/2n, we conclude with
w¯=(y¯*+1)ψ(x¯,ζ)
(18)
where x¯=x/H, and
ψ(x¯,ζ)=1z(πζnx¯sin(πζnx¯)tan(πζ2n)(1cos(πζnx¯)))
(19)
is the deflection shape function, satisfying the first three boundary conditions in Eq. (8), i.e., ψ(0,ζ)=ψ,x¯(0,ζ)=ψ,x¯(1,ζ)=0, along with ψ(1,ζ)=1. This shape function is illustrated in Fig. 3 for various values of ζ. Finally, note that vertical lengths are normalized by D0 while horizontal lengths by H.
Fig. 3
The deflection shape function ψ(x¯,ζ), for values of ζ=1,43,53,2, associated with a point contact. The deflection shape function for line contact is indifferent to ζ and corresponds to ψ with ζ=2.
Fig. 3
The deflection shape function ψ(x¯,ζ), for values of ζ=1,43,53,2, associated with a point contact. The deflection shape function for line contact is indifferent to ζ and corresponds to ψ with ζ=2.
Close modal
Calculating η by plugging Eqs. (18), (15), and (4) into (3), we have that
η=ζ2+3D~02(ζ2(2+1cos2(πζ2n))6nζπtan(πζ2n))A¯2
(20)
where D~0=D0/t. The displacement of the moving wall can be calculated by substituting x¯=1 in (18) to obtain
y¯=A¯2(πζ2ntan(πζ2n))1
(21)
Finally, the energy stored in the system is obtained from Eq. (5) by using Eqs. (18) and (21):
UΠ=ζ4+12nD¯02ζ3π(πζ2ncos2(πζ2n)tan(πζ2n))A¯2+24D¯02πα¯0y¯f(y¯)dy¯
(22)
where Π=23(π2t)2L03EI is the energy scale set by the compressive energy, and α¯=2nEID0F0 is a non-dimensional quantity, which includes all parameters of the system, and describes the relevant stiffness of the beam relative to the stiffness of the nonlinear spring. Thus, for a given spring function f(y¯), α¯ dictates the behavior. This is further discussed below.

2.3 Transition From Point-Contact to Line-Contact Configuration.

The transition from point-contact to line-contact configuration takes place when the internal bending moment at the contact point vanishes, i.e.,
w,xx(H)=0
(23)
Note that by symmetry, this condition implies also that w,xx(0)=0. Applying this condition to Eq. (17), we conclude that the transition to line contact takes place at
ζ=2n
(24)

Interestingly, the above results do not depend on the characteristics of the spring. On the other hand, the gap between the two walls and the horizontal displacement Δ (or η) do depend on the spring characteristics and can be found by plugging Eqs. (24), (4) into (10).

2.4 Line Contact Configuration.

After line contact has formed, further increase of the end displacement results in spreading the folds against the walls. The boundary conditions for each fold remain identical to Eq. (8) and include also Eq. (23). Note that, now, the length of the fold is unknown; thus, the relation H=L0/2n cannot be used.

Accordingly, using Eq. (17), the solution for the deflection is
w¯(x)=1z(y¯*+1)(2πζL0xsin(2πζL0x)tan(πζHL0)(1cos(2πζL0x)))
(25)
where H is unknown, but can be found from requirement (Eq. (23)) which gives
H=L0ζ
(26)
and describes the relation between the applied compressive force and the length of the fold. By plugging Eqs. (26) into (25), we obtain the solution for the deflection
w¯=(y¯*+1)ψ(x¯,2)
(27)
where ψ(x¯,2)=sin(2πx¯)2πx¯ is the deflection shape function (Eq. (19)) associated with ζ=2, i.e., the deflection shape function at the transition from point contact to line contact. Contrary to point contact, the deflection shape function in line-contact configurations is indifferent to the load, ζ. Similar to point-contact configuration, Eq. (27) may be used to calculate the horizontal displacement and energy stored in the system, namely:
η=ζ2+18nζD~02A¯2
(28)
y¯=2πA¯1
(29)
UΠ=ζ4+12nD~02ζ3πA¯2+24D~02πα¯0y¯f(y¯)dy¯
(30)

2.5 Switching to the Next Mode.

Transition to the next mode occurs when condition (Eq. (6)) is satisfied. We consider three special (particular, yet common) cases:

  • Only one contact segment exists.

  • All contact segments have the same length.

  • In the symmetric case, where the contact segments in the middle have a double length than the contact segments at the two ends.

Next, we combine conditions Eqs. (6) and (26). In the first case, the longest contact segment is c=L02nH. Therefore, we have from Eq. (6) that
ζ=1+2n
(31)
In the second case, c=L02nH2n+1, thus using Eq. (6), we have that
ζ=1+4n
(32)
Lastly, in the third case, the longest contact segment is c=L02nH, leading to
ζ=4n
(33)
The results of the two limit cases (Eqs. (31) and (32)) indicate that, in reality, switching to the next mode takes place at:
1+2nζ1+4n
(34)

Interestingly, the earlier results are indifferent to the characteristics of the nonlinear spring.

2.6 Numerical Examples.

In this section, we present numerical results for a stiffening spring and a softening spring. Unless stated otherwise, all line-contact results assume the symmetric case. Further, for the softening spring, the figures include all possible modes. On the other hand, recalling that the number of modes is theoretically unlimited with a stiffening spring, we show in the figures only the first few modes for systems with a stiffening spring. In order to examine the differences between the behavior with a stiffening or with a softening spring, we consider particular spring functions, f(y¯), which are chosen merely for demonstration purposes. One may obtain similar results for other particular cases. For the stiffening spring, we consider the following force–displacement relation
f(y¯)=exp(y¯)1
(35)
while the softening spring is characterized by
f(y¯)=1exp(y¯)
(36)
By plugging these relations into Eq. (14), we find from Eq. (16) that
y¯*+1=1αLambertW(1αexp((1+1α)))
(37)
for the stiffening spring, and that
y¯*+1=1α+LambertW(1αexp(11α))
(38)
for the softening spring. Above, LambertW is the Lambert W (or product logarithm) function. The results for the stiffening spring and for the softening spring are presented in Figs. 4 and 5, respectively. These figures describe the behavior in terms of the relations between η, ζ, y¯, and U. The results in both figures correspond to the same values of relative stiffness α¯=9104 and initial gap D~0=3. The first and most apparent difference between the behavior with a stiffening spring compared to that of the softening spring is the number of modes that can be attained. With the softening spring, the highest mode that can be attained is n=2. Therefore, Fig. 5 includes two distinguished curves, where each corresponds to a different mode. On the other hand, with a stiffening spring, the number of modes is unlimited. This is well exemplified in Fig. 2 which shows that a solution y¯* can be found for any value of α for the stiffening system. On the contrary, for the softening system, a solution y¯* exists only for a limited range of α that is smaller than some threshold. For clarity, we show in Fig. 4 results only for the first three modes. The second notable difference between the behavior with a softening spring compared to that of a stiffening spring is that, with a stiffening spring, the force–displacement relation (ζ versus η) is always monotonously increasing, in all modes. On the other hand, the force–displacement relation with a softening spring is non-monotonous in the highest attainable mode, nmax. This non-monotonous behavior prevents the force to reach the transition force, ζ=4nmax. Thus, in the highest attainable mode, the increase of η cannot lead to mode transition. It should be noted that, in the last mode, transition to line contact may (or may not) be possible, depending on the specific properties of the system, and in particular the value of α¯ and the details of f(y¯). In the example presented in Fig. 5, the system does reach line contact in the highest attainable mode (n=2). Significant differences between the behaviors with a stiffening spring or with a softening spring are also observed when comparing curves of the same mode. Here, for the same horizontal displacement, η, the external force, ζ, is larger while the displacement of the wall, y¯, is smaller with a stiffening spring. These observations are rather intuitive. However, interestingly, the external force at the onset of transition (from point contact to line contact or to the next mode) is indifferent to the stiffness of the spring (Eqs. (24) and (33)). Consequently, at transition, the horizontal displacement, η, and the wall displacement, y¯, are both smaller with a stiffening spring. In addition, the transition from point contact to line contact is “smooth” in the sense that all curves are continuous at the point of transition. On the other hand, transitions to the next mode involve abrupt changes associated with changing from one equilibrium configuration to a different equilibrium configuration. For example, in a displacement control experiment (“hard device”) where η is gradually increased, transition to the next mode is accompanied by an abrupt and simultaneous decrease of the external force and the displacement of the wall. In addition, the transition to the next mode is accompanied by a decrease in the stored energy, U, which means that the system releases energy due to the transition. Thus, hysteresis is expected to be observed in a complete loading–unloading cycle, where the loading path differs from the unloading path. As mentioned earlier, transition to the next mode is associated with an abrupt change of the equilibrium configuration. Interestingly, this new equilibrium configuration may involve either point contact (see, for example, transition to mode n=2 in Figs. 4 and 5) or line contact (see transition to mode n=3 in Fig. 4).
Fig. 4
The behavior predicted by the theoretical model with a stiffening spring (Eq. (35)) characterized by α¯=9⋅10−4 and D~0=3: (a) ζ as a function of η, (b) ζ as a function of y¯, (c) y¯ as a function of η, and (d) U/∏ as a function of η. Line contact is distinguished from point contact using dashed lines. Results for line contact assume the symmetric case. Different curves in the same plot correspond to different modes.
Fig. 4
The behavior predicted by the theoretical model with a stiffening spring (Eq. (35)) characterized by α¯=9⋅10−4 and D~0=3: (a) ζ as a function of η, (b) ζ as a function of y¯, (c) y¯ as a function of η, and (d) U/∏ as a function of η. Line contact is distinguished from point contact using dashed lines. Results for line contact assume the symmetric case. Different curves in the same plot correspond to different modes.
Close modal
Fig. 5
The behavior predicted by the theoretical model with a softening spring (Eq. (36)) characterized by α¯=9⋅10−4 and D~0=3: (a) ζ as a function of η, (b) ζ as a function of y¯, (c) y¯ as a function of η, and (d) U/∏ as a function of η. Line contact is distinguished from point contact using dashed lines. Results for line contact assume the symmetric case. Different curves in the same plot correspond to different modes.
Fig. 5
The behavior predicted by the theoretical model with a softening spring (Eq. (36)) characterized by α¯=9⋅10−4 and D~0=3: (a) ζ as a function of η, (b) ζ as a function of y¯, (c) y¯ as a function of η, and (d) U/∏ as a function of η. Line contact is distinguished from point contact using dashed lines. Results for line contact assume the symmetric case. Different curves in the same plot correspond to different modes.
Close modal

In order to gain more insights, we show in Figs. 6 and 7 a comparison between the behaviors of systems with a stiffening spring, a softening spring, and a linear spring. In each of these figures, the three springs have the same relative stiffness, α¯, whereas in Fig. 6 we consider α¯=0.2, which is larger than the one examined in Figs. 4 and 5, and in Fig. 7, it is α¯=5103, which is smaller than in Figs. 4 and 5. With α¯=0.2, the maximum attainable mode with a softening spring or with a linear spring is n=1. On the other hand, as mentioned earlier, the number of modes with a stiffening spring is theoretically unlimited. Still, for clarity, we show in Fig. 6 only the first mode for the three springs. Similar to Figs. 4 and 5, the force–displacement relation is monotonously increasing with a stiffening (and also linear) spring, but is non-monotonous with a softening spring. Interestingly, with the softening spring, the decrease in force is so significant that the system experiences a transition from line contact back point contact. This unique feature cannot occur with a stiffening spring or even with a linear spring. As expected, the onset of this “reverse” transition takes place at the same level of force, ζ=2, as for the “forward” transition (from point contact to line contact); yet, the reverse transition corresponds to larger horizontal displacement, η, and larger wall displacement, y¯. The latter is associated with the second intersection (the solution for larger y¯*) in Fig. 2.

Fig. 6
A comparison between the behavior with a stiffening spring (Eq. (35)), softening spring (Eq. (36)), and a linear spring, all having the same value of α¯=0.2 and D~0=3: (a) ζ as a function of η, (b) ζ as a function of y¯, and (c) y¯ as a function of η. Line contact is distinguished from point contact using dashed lines. Results for line contact assume the symmetric case.
Fig. 6
A comparison between the behavior with a stiffening spring (Eq. (35)), softening spring (Eq. (36)), and a linear spring, all having the same value of α¯=0.2 and D~0=3: (a) ζ as a function of η, (b) ζ as a function of y¯, and (c) y¯ as a function of η. Line contact is distinguished from point contact using dashed lines. Results for line contact assume the symmetric case.
Close modal
Fig. 7
A comparison between the behavior with a stiffening spring (Eq. (35)), softening spring (Eq. (36)), and a linear spring, all having the same value of α¯=5⋅10−3 and D~0=3: (a) ζ as a function of η, (b) ζ as a function of y¯, and (c) y¯ as a function of η. Line contact is distinguished from point contact using dashed lines. Results for line contact assume the symmetric case. Note that the three curves (with a stiffening, linear, or softening spring) practically overlap in the first two modes. Also, the plot includes only two curves in mode n = 4 since the system cannot reach mode n = 4 with a softening spring.
Fig. 7
A comparison between the behavior with a stiffening spring (Eq. (35)), softening spring (Eq. (36)), and a linear spring, all having the same value of α¯=5⋅10−3 and D~0=3: (a) ζ as a function of η, (b) ζ as a function of y¯, and (c) y¯ as a function of η. Line contact is distinguished from point contact using dashed lines. Results for line contact assume the symmetric case. Note that the three curves (with a stiffening, linear, or softening spring) practically overlap in the first two modes. Also, the plot includes only two curves in mode n = 4 since the system cannot reach mode n = 4 with a softening spring.
Close modal

With α¯=5103, the maximum attainable mode is n=3 with a softening spring and n=4 with a linear spring. These are illustrated in Fig. 7 along with the four first modes of the stiffening spring. In the lower modes, and in particular the first two modes, the behavior associated with each of the three springs is practically similar. The reason is that the high (initial) stiffness of the springs (small values of α¯) results in small displacements, y¯, and therefore small changes in the instantaneous stiffness of the nonlinear springs. Only at higher modes, where the wall displacement is large enough to influence the instantaneous spring stiffness, deviations between the three systems are observed.

3 Experiments

In this section, we present the results of experiments that examine the influence of the nonlinear spring characteristics on the behavior of the laterally constrained compressed beam. The experimental system, shown in Fig. 8, consists of a rigid aluminum frame that forms the fixed wall (bottom surface in Fig. 1) and provides a base for the moving wall. The moving wall, which is made of aluminum as well, is connected through sliding bearings that allow the wall to move only parallel to the fixed surface. A nonlinear spring connects the moving wall and the rigid frame, such that when the wall is pushed by the beam the spring is compressed. If necessary, the initial length of the spring, which dictates D0, is set by a weight. A beam made from polycarbonate is placed on the fixed wall and clamped on both sides by aluminum plates which prevent rotations. At one end (bottom of the beam in Fig. 8), the aluminum plates are rigidly connected to the aluminum frame while at the other end, the beam is connected to a carrier which can slide toward the fixed end. This is done by a cylindrical rod (connector) that is connected to the moving arm of the Instron machine. The weight of the carrier is 35 grams and is negligible compared to the forces acting on the beam. Note that the beam is not “inserted” into the “compliant channel” through a sliding sleeve; thus, a so-called configurational (or Eshelby-like) force is not generated [54,55]. All beams in the experiment have a free length of 115 mm, a width of 12 mm, and a thickness of 0.8 mm. During the experiment, the axial displacement, Δ, and the axial force acting on the beam, P, were measured using the Instron machine. The distance between the fixed wall and the moving wall is measured using a laser extensometer system (LE-05). Six different experiments were carried out with three pairs of nonlinear springs. Each pair consists of a stiffening spring and a softening spring that have the same initial stiffness: (i) 3.5N/mm, (ii) 2.4N/mm, and (iii) 1.7N/mm. These specific values were chosen to highlight differences in the response between systems with a softening and a stiffening spring, as discussed below. The force–displacement relation of each of the six springs was measured separately in a preliminary experiment (see Appendix  A).

Fig. 8
The experimental system: (a) front view of the experiment setup and (b) a zoom-in view of the apparatus.
Fig. 8
The experimental system: (a) front view of the experiment setup and (b) a zoom-in view of the apparatus.
Close modal

3.1 Experimental Results.

The results of the six experiments are shown in Figs. 914. In each of these figures, three plots describe the response in terms of the (non-dimensional) force, ζ, the axial displacement, η, and the displacement of the wall, y¯. In each of these plots, the measured (experimental) curve is compared with the theoretical prediction of the mathematical model presented in Chapter 2. In addition, we show three snapshots from the experiment, which correspond to the points indicated in the ζη plot. Overall, the results of the experiments demonstrate a good quantitative agreement with the theoretical model. In what follows, we highlight some of the fundamental differences that were observed in the behavior with a softening or a stiffening spring, even if both have the same initial stiffness. In addition, we discuss the limitations of the theoretical model, which assumes small deformations.

Fig. 9
Experimental results (blue curve) compared with small-deformation analysis (black curve) for a system with a stiffening spring characterized by an initial stiffness of α¯=0.11 and D0=2mm. (a) ζ versus η, (b) ζ versus y¯, and (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing line contact right before transition to mode n = 2. (f) Snapshot from the experiment showing mode n = 2 with point contact.
Fig. 9
Experimental results (blue curve) compared with small-deformation analysis (black curve) for a system with a stiffening spring characterized by an initial stiffness of α¯=0.11 and D0=2mm. (a) ζ versus η, (b) ζ versus y¯, and (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing line contact right before transition to mode n = 2. (f) Snapshot from the experiment showing mode n = 2 with point contact.
Close modal
Fig. 10
Experimental results (blue curve) compared with small-deformation analysis (black curve) for a system with a softening spring characterized by an initial stiffness of α¯=0.11 and D0=2mm: (a) ζ versus η, (b) ζ versus y¯, and (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing line contact. (f) Snapshot from the experiment showing line contact.
Fig. 10
Experimental results (blue curve) compared with small-deformation analysis (black curve) for a system with a softening spring characterized by an initial stiffness of α¯=0.11 and D0=2mm: (a) ζ versus η, (b) ζ versus y¯, and (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing line contact. (f) Snapshot from the experiment showing line contact.
Close modal
Fig. 11
Experimental results (blue curve) compared with small-deformation analysis (black curve) for a system with a stiffening spring characterized by an initial stiffness of α¯=0.16 and D0=2mm. (a) ζ versus η. (b) ζ versus y¯. (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing line contact right before transition to mode n = 2. (f) Snapshot from the experiment showing mode n = 2 with point contact.
Fig. 11
Experimental results (blue curve) compared with small-deformation analysis (black curve) for a system with a stiffening spring characterized by an initial stiffness of α¯=0.16 and D0=2mm. (a) ζ versus η. (b) ζ versus y¯. (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing line contact right before transition to mode n = 2. (f) Snapshot from the experiment showing mode n = 2 with point contact.
Close modal
Fig. 12
Experimental results (blue curve) compared with small-deformation analysis (black curve) for a system with a Softening spring characterized by an initial stiffness of α¯=0.16 and D0=2mm: (a) ζ versus η, (b) ζ versus y¯, and (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing line contact. (f) Snapshot from the experiment showing that the system returns to point contact while y¯ and η are larger compared to snapshot (e).
Fig. 12
Experimental results (blue curve) compared with small-deformation analysis (black curve) for a system with a Softening spring characterized by an initial stiffness of α¯=0.16 and D0=2mm: (a) ζ versus η, (b) ζ versus y¯, and (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing line contact. (f) Snapshot from the experiment showing that the system returns to point contact while y¯ and η are larger compared to snapshot (e).
Close modal
Fig. 13
Experimental results (blue curve) compared with small-deformation analysis (black curves; Mode 2, illustrated with a broken line, is shown for reference) for a system with a stiffening spring characterized by an initial stiffness of α¯=0.23 and D0=2mm. (a) ζ versus η, (b) ζ versus y¯, and (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing line contact. (f) Snapshot from the experiment showing line contact.
Fig. 13
Experimental results (blue curve) compared with small-deformation analysis (black curves; Mode 2, illustrated with a broken line, is shown for reference) for a system with a stiffening spring characterized by an initial stiffness of α¯=0.23 and D0=2mm. (a) ζ versus η, (b) ζ versus y¯, and (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing line contact. (f) Snapshot from the experiment showing line contact.
Close modal
Fig. 14
Experimental results (blue curve) compared with small-deformation analysis (black curve) for a system with a Softening spring characterized by an initial stiffness of α¯=0.23 and D0=3mm. (a) ζ versus η, (b) ζ versus y¯, and (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing point contact. (f) Snapshot from the experiment showing point contact.
Fig. 14
Experimental results (blue curve) compared with small-deformation analysis (black curve) for a system with a Softening spring characterized by an initial stiffness of α¯=0.23 and D0=3mm. (a) ζ versus η, (b) ζ versus y¯, and (c) y¯ versus η. (d) Snapshot from the experiment showing point contact. (e) Snapshot from the experiment showing point contact. (f) Snapshot from the experiment showing point contact.
Close modal

Figures 9 and 10 show the results for systems with stiffening and softening springs, respectively, having the same initial stiffness F0/D0=3.5(N/mm), corresponding to α¯=0.11, with initial gap of D0=2(mm). According to the theoretical model, the number of modes with the stiffening spring characterized by α¯=0.11 is unlimited, irrespectively to the value of D0. On the other hand, the theoretical model predicts that the system with the softening spring cannot switch to mode n=2 (the highest mode that can be attained is n=1). Indeed, our experiments suggest that the system with a softening spring cannot switch to the second mode, even at relatively large values of η, since the force–displacement curve (ζη) is non-monotonous and experiences a decline in the force after reaching a value of ζ2.2, as predicted by the theory. This value of force enables line contact, but is not sufficient for switching to the next mode. This is well observed in the snapshots (df) of Fig. 10, which show the evolution of line contact at forces ζ>2. Contrary to the system with a softening spring, the experiments with a stiffening spring show that the system switches to mode n=2. This occurs at a force ζ3. According to the theoretical model, such value of force at the transition from n=1 to n=2 corresponds to case (i) (see Sec. 2.5) where only one contact segment exists. This is well demonstrated in snapshot (e) of Fig. 9, which shows that only one line-contact segment has formed (in the middle of the beam). Once the system switches to mode n=2, it returns to a point-contact configuration, as predicted by the theoretical model.

The results for the next pair of experiments, with stiffening and softening springs having an initial stiffness of F0/D0=2.4(N/mm), corresponding to α¯=0.16 with initial gap of D0=2(mm), are shown in Figs. 11 and 12, respectively. With the stiffening spring, the behavior is qualitatively similar to that with α¯=0.11 (see Fig. 9). For example, in both cases, the force monotonously increases in mode n=1, starting with point-contact and evolving to line contact (see snapshots (d) and (e) in Fig. 11). In addition, switching to mode n=2 takes place at a force ζ3, corresponding to case (i) (see Sec. 2.5) where only one contact segment exists. This is well demonstrated in snapshot (e) of Fig. 11, which shows that only one line–contact segment has formed (in the middle section). Still, one may observe significant quantitative differences between the behavior of the system with α¯=0.11 and α¯=0.16. In particular, the force–displacement curve in Fig. 11 is significantly flatter compared to the one in Fig. 9. As a result, for the same axial displacement, η, the force, ζ, is smaller for α¯=0.16 compared to α¯=0.11, whereas the displacement of the wall, y¯, is larger. A direct consequence is that the transition to line contact as well as switching to the next mode occurs at larger displacements η and y¯. The behavior of the system with a softening spring (Fig. 12) is fundamentally different from the one with a stiffening spring: The force–displacement curves with the softening spring are non-monotonous, as predicted by the theoretical model, and switching to mode n=2 is not possible. These observations appeared also with the stiffer spring (α¯=0.11, Fig. 10). Nevertheless, with α¯=0.16, the contact between the beam and the walls evolves from point contact to line contact and back to point contact, while η and y¯ monotonously increase. This unique behavior is predicted by the theoretical model and is well demonstrated in the snapshots of Fig. 12.

The last set of experimental results is shown in Figs. 13 and 14, for stiffening and softening springs, respectively, having an initial stiffness of F0/D0=1.7[N/mm], corresponding to α¯=0.23. The most notable observation is that the system with the stiffening spring does not switch to mode n=2, contrary to the previous cases (with α¯=0.16,0.11). This result of the experiment does not agree with the theoretical prediction of the small-deformation analysis. We recall that according to that theory, the number of modes, with a stiffening spring, is unlimited. It is evident, however, that such behavior is not expected to occur in reality. The reason for the discrepancy is that high-modes eventually lead to large deformations, for which the small-deformation theory discussed earlier is not valid. In particular, in the results shown in Fig. 13, it is evident that switching to the mode n=2 (which requires ζ>3) must be accompanied with very large displacements, which are not accounted for by the small-deformation analysis. It is noted that similar discrepancies between the small-deformation model and the actual behavior have been observed before [47] and were indeed confirmed using a large-deformation (elastica) model, which accounts for geometrical nonlinearities. Therefore, these results suggest that although the small-deformation model presented in this paper may be useful and insightful, one should be careful in applying it to scenarios where the small-deformation assumption does not hold. The behavior of the system with the softening spring (with the same initial stiffness α¯=0.23) is shown in Fig. 14. Two fundamental differences are observed compared to the system with the stiffening spring. The first is that relations ζη and ζy¯ are not monotonous, as predicted by the theoretical model. This is similar to what have been observed in the two previous cases of the softening springs (α¯=0.16,0.11). The main difference is that, here, the beam always maintains point contact and does not form line contact with the walls, as predicted by the theoretical model.

4 Discussion and Conclusions

In this paper, we have studied analytically, numerically, and experimentally the post-buckling behavior of a compressed beam constrained by a nonlinear “springy” wall (a rigid wall that moves against a nonlinear spring). The main goal of the theoretical analysis was to obtain analytical insights and intuition; Thus, a simplified mathematical model, that assumes small deformations of the compressed beam was adopted, and special attention was given to the influence of the nonlinear characteristics of the constraining spring on the overall behavior. In particular, differences (in the post-buckling behavior of the beam) when constrained by a stiffening spring or a softening spring were studied. Further, the results of the theoretical model were validated experimentally, showing good agreement with the theoretical model; The results of these experiments highlight the importance and usefulness of the analytical insights which provide valuable intuition. Yet, these experiments also demonstrate the limitations of the model, namely small deformations of the compressed beam, as further discussed below.

The central insight provided by the theoretical model is that the main features of the behavior, such as transition from point contact to line contact and switching to the next mode, are dictated solely by the non-dimensional force, ζ, regardless of all other parameters of the system, e.g., the initial gap, the stiffness of the spring or whether it is softening or stiffening. For example, transition from point contact to line contact occurs at ζ=2n (n is the mode number) irrespective of any other parameter. Also, switching to the next mode occurs between 1+2nζ1+4n, where the two extremes, ζ=2n+1 and ζ=4n+1, correspond to the case where only one line–contact segment forms or if all line–contact segments have the same length, respectively. These insights are extremely useful, as one can immediately understand/predict the mode number and contact characteristics by knowing merely the level of applied force. Still, quantitative characterization of the system response, including the horizontal displacement, η, the distance between the constraining walls, D¯ (or y¯), and the deformed geometry of the compressed beam, requires all geometrical and material properties of the system. Interestingly, we have found that this description is possible by means of closed-form analytical expressions that use two non-dimensional quantities: (i) the relative (initial) compliance of the nonlinear spring, α¯, which roughly describes the ratio between the stiffness of the beam and the stiffness of the constraining nonlinear spring, and (ii) the “spring function” (normalized to unit initial stiffness) of the force–displacement curve characterizing the nonlinear spring, which is mathematically convex or concave depending on whether the spring is stiffening or softening. In other words, for a prescribed external force ζ, one can use Eq. (14) to calculate y¯, and in turn Eq. (13) to determine the “deformation amplitude,” A¯. This calculation requires only α¯ and f(y¯). Once y¯ and A¯ are computed, the entire description is possible by means of closed-form analytical expressions in terms of these non-dimensional quantities.

The immediate observation regarding the system behavior with a softening spring or with a stiffening spring is that the force–displacement relations are both quantitatively and qualitatively different. Unsurprisingly, with a stiffening spring, the distance between the constraining walls (or gap y¯) is smaller compared to the softening spring for the same value of η, and the corresponding force ζ is larger. Another important result obtained from the theoretical model is that the overall force–displacement relation (ζη) is always monotonously increasing with a stiffening or linear spring, in all modes. On the other hand, with a softening spring, this force–displacement relation may be non-monotonous in the highest attainable mode, nmax. Consequently, with a softening spring, the level of force cannot reach the threshold required for mode transition, e.g., ζ=2nmax+1, in the case of one line–contact segment, and the system cannot switch to a higher mode. Moreover, the decrease in force may even lead to a change back from the line-contact configuration to the point–contact configuration. The non-monotonicity in the overall force–displacement relation stems from a competition between the (decreasing) resistance of the (softening) spring and the instantaneous stiffness of the beam (relating between the external axial force, P, and the lateral displacement of the beam). It is noted that the abovementioned stiffness of the beam depends on the beam deformation. This becomes even more prominent when the beam experiences large rotations that may even evolve to S-shape of the deformed beam geometry. These effects are associated with geometrical nonlinearity that is not accounted for in the current model and therefore may lead to discrepancy between the predictions of the model and the real behavior. Interestingly, the picture is different in the case of a compressed beam on a nonlinear Winkler foundation, where non-monotonicity may stem from local softening due to localized deformation [56]. On the contrary, with a stiffening spring, the theoretical model predicts that the number of attainable modes is unlimited. The reason is that the force–displacement relation (ζη) is always monotonously increasing, and therefore, the force, ζ, can always reach the required level for switching to the next mode by increasing the displacement η. Evidently, this may be accompanied by relatively large displacements (in terms of η and/or y¯), which violates the assumption of small rotations assumed by the theoretical model. Indeed, the results of our experiments suggest that once the beam is subjected to large deformations, the accuracy of the model significantly reduces. In particular, the force–displacement relation (ζη) may be non-monotonous even with a stiffening spring, preventing the system from switching to the next mode. It should be noted, that a similar behavior has been observed in a previous study [47], which considered a linear constraining spring. There, also, experiments have shown that the number of possible modes is limited (contrary to the predictions of a small-deformation model). This discrepancy between the experimental and theoretical results was resolved by formulating a mathematical model that accounts for large rotations of the beam. This highly nonlinear model has indeed confirmed that the overall force–displacement relation may be non-monotonous at the highest attainable mode, as observed in the experiments. Still, based on the results of our experiments and the results obtained with a linear spring [47], the analytical insights of the small-deformation model are found to be insightful, relevant to a wide range of parametric values, and provide reasonably accurate predictions in a wide range of practical scenarios. Developing a model that accounts for large rotations of the beam combined with nonlinear deformations of the lateral constraints is therefore expected to be necessary only for very small (relative) stiffness of the spring. Pursuing such model, and in particular gaining general insights from it, is a natural extension of the current work, but poses a significant challenge. Perhaps, a practical approach would be to generate numerical “maps,” based on a large-deformation model, which provide an estimation of the error due to the small-deformation assumption [47]. Finally, the analysis of 3D scenarios, such as a beam constrained inside a flexible cylinder that undergoes nonlinear deformations, is another desired and challenging extension.

Acknowledgment

This work was supported by the Israel Science Foundation (Grant No. ISF 1598/21).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

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

Appendix A: The Nonlinear Springs in the Experiments

In this section, we provide additional details regarding the nonlinear springs that were used in the experiments. Three pairs of nonlinear springs were used. Each pair consisted of a stiffening spring and a softening spring that have the same initial stiffness: (i) 3.5N/mm, (ii) 2.4N/mm, and (iii) 1.7N/mm. All springs are made of Acrylonitrile Butadiene Styrene (ABS) and were manufactured by 3-D printing. The design of the softening springs is inspired by the influence of geometrical imperfection on the buckling behavior of columns [57] (Fig. 15). The initial stiffness and the resulting softening behavior are dictated by the magnitude of the geometrical imperfection as well as the thickness, length, and width of the beam. The geometrical imperfection was introduced by designing the initial (as manufactured) geometry of the beam to be non-straight. For convenience, we chose a quadratic shape that is determined by the off-location at the mid-span. In order to produce a stiffening spring, we incorporated holes of varying sizes in the printed specimen (Fig. 16). Each of these holes closes at a different level of force resulting in a nonlinear stiffening force–displacement relation. The measured force–displacement relations for each of the six springs that were used in the experiments (see Figs. 15 and 16) are shown in Fig. 17.

Fig. 15
Drawings of the three softening springs: (a) 1.7N/mm, (b) 2.4N/mm, and (c) 3.5N/mm. The (out of plane) depth of all specimens is b=4(mm).
Fig. 15
Drawings of the three softening springs: (a) 1.7N/mm, (b) 2.4N/mm, and (c) 3.5N/mm. The (out of plane) depth of all specimens is b=4(mm).
Close modal
Fig. 16
Drawings of the three stiffening springs: (a) 1.7N/mm, (b) 2.4N/mm, and (c) 3.5N/mm. The (out of plane) depth of all specimens is b=6(mm).
Fig. 16
Drawings of the three stiffening springs: (a) 1.7N/mm, (b) 2.4N/mm, and (c) 3.5N/mm. The (out of plane) depth of all specimens is b=6(mm).
Close modal
Fig. 17
Force–displacement relations of the six springs (three plots of three pairs): (a) 1.7N/mm, (b) 2.4N/mm, and (c) 3.5N/mm
Fig. 17
Force–displacement relations of the six springs (three plots of three pairs): (a) 1.7N/mm, (b) 2.4N/mm, and (c) 3.5N/mm
Close modal

Appendix B: The Nonlinear Springs in the Experiments

In this appendix, we present a magnified version (Figs. 1820) of the snapshots from the experiments shown in Figs. 914.

Fig. 18
Snapshots from Figs. 9 and 10(α¯=0.11), magnified for convenience
Fig. 18
Snapshots from Figs. 9 and 10(α¯=0.11), magnified for convenience
Close modal
Fig. 19
Snapshots from Figs. 11 and 12(α¯=0.16), magnified for convenience
Fig. 19
Snapshots from Figs. 11 and 12(α¯=0.16), magnified for convenience
Close modal
Fig. 20
Snapshots from Figs. 13 and 14(α¯=0.23), magnified for convenience
Fig. 20
Snapshots from Figs. 13 and 14(α¯=0.23), magnified for convenience
Close modal

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