Second Order Phase Transition and Universality of Self-Buckled Elastic Slender Columns

Self-buckling is an interesting phenomenon that is easily found around us, either in nature or in objects made by human. Palm fronds which initially directed upward when they were short and turned into bending after appreciably longer is an example of the self-buckling phenomenon. We report here that the self-buckling of columns can be treated as a process of second order phase transition by considering the straight column as disorder state, the bending column as order state, and the temperature as the inverse of column length. The critical temperature corresponds to the inverse of critical length for buckling, 1/Lcr, and the deviation angle made by column free end relative to vertical direction satisfies a scaling relationship with a scaling power of 0.485. Changing of the column geometry from the vertically upward to the bending state can be considered as a transition from disorder state to order state.


Introduction
Initially, most paddy leaves grow up vertically, but after a certain height, they start to bend. We observe the similar phenomenon in the pandanus leaves or other leaves having similar shape (long and slim). The similar phenomenon is also observed in hairs growing from bald or nearly bald heads or skins. The hairs, initially grow straight and then bend after reaching a certain length.
The banana's young leaf initially grows up and stays vertically in a shape like a cylinder. But, when the leaf sheet opens, the leaf suddenly bends. The palm's young leaf also shows the same mechanism, initially grows vertically upward and then bends when the leaf opens. Those are all very common phenomena around us, but most of us have likely ignored them and considered neither interesting scientific role involved. At this work we will show that the physical roles underlying such phenomena are very attractive. Surprisingly, such phenomena mimic a kind of phase transition.
To demonstrate such phenomena more viable, we can use a sheet clamped between two vertical plates. The sheet is then moved up progressively through the clamp. We clearly observe the sheet initially directs vertically. But, by increasing the height of the sheet above the clamp, the sheet suddenly bends having reached a certain height, after which the sheet continuously bends (Figure 1(a)). This process is similar to bending of paddy or pandanus leaves when reaching a certain length. In Fig. 1(b) we show the variation of bending profiles of the paper sheets having different lengths fixed together at a vertical frame. Simetrical bending formation was clearly observed. columns. Bending of a slender column is a process of self-buckling. It is then interesting to reconstruct the bending evolution of the sheet after experiencing a self-buckling and this attempt is rarely discussed by authors.
In fact, many common phenomena around us have challenged researchers. Most discussions regarding the self-buckling were merely focused on the boundary problems. At the present work, we extract many attractive phenomena in the self-buckling process that likely rarely considered. For this purpose, we use the discrete form of the bending equation of slender columns, rather than the continuum form, and the calculation can be easily processed numerically. The discrete equation can be applied for describing bending profiles of slender columns having arbitrary bending angles (either small or large bending).
This approach might be compared to the direct approach, which is based on the deformable curve model as reported by Bîrsan et al [9,10]. This is an efficient approach for analyzing elastic beams and rods with a complex internal structure (functionally graded, composite, nonhomogeneous, etc.). Linul et al [11][12][13] and Berto et al [14,15] have applied this method to describe bending of various composites (inhomogeneous) beams. This method is also important for describing the bending of materials where the mechanical properties can be controlled. Study of the change in the mechanical behavior of materials when exposed to radiation (neutron, ion, or electron beams) was reported by Zeyad [16].
In the present work, however, we were able to relate the shape of the self-buckled sheet with a phase transition phenomenon. In addition, we also identified a universal equation for describing the bending for all homogeneous columns and proved the famous critical column height for buckling, Lcr = (7.8373YI/gA) 1/3 (Y = elastic modulus, I = area moment,  = density, A = cross-section area, and g = the acceleration of gravitation) [17], in a strongly different approach. This is not a hard topic and might be not a very new one.

Method
At present we restrict our consideration to inextensible columns without twisting. This assumption is very acceptable for geometry like a sheet. The equilibrium equation is , with M  flexion moment, R the curvature which can be expressed as , t the unit vector along the centerline, and n the unit normal vector perpendicular to t . We choose t and n to govern the x-y plane (x axis to right and y 5 axis to vertical upward). The M  directs parallel to the z axis and after performing a scalar product with z ê , the bending equation reads

Discretization
We need to transform Eq. (1) into a discrete form so that it can be easily processed numerically.
This equation is a general equation for arbitrary columns, either homogeneous or inhomogeneous.
When the force acting on the rod is only its own weight, Equation (6) has been used by us to determine the elastic modulus of slender beams by processing the bending image of the beam. This is possible since, although the equation has been derived for the column, it also applies for beams. The method is also potential for estimating the glass transition temperature of polymeric materials [20]. Very precise glass transition temperatures of several polymers have been obtained.

Numerical Procedures
Let us first restrict on the case when the column is bent by its own weight. We will calculate angles of all segments by firstly fixing the angle for the first segment. The problem is, the boundary condition is applied to the fixed segment (the last segment), instead of the 1 st segment. Therefore, eventually, after performing calculations the angle made by all segments, the calculated angle of the fixed segment is no longer satisfies the boundary condition. If this result happens, the initial angle applied to the first segment was wrong and we must try another angle for the first segment until the calculated angle of the last segment is equal to the boundary condition. This approach looks tedious, but we can simplify the solving process by using iteration. The criterion for stopping the iteration is , with bc is the bending angle of the N-th (fixed segment) and  is a small number which depends on the accuracy we are intending. In simulation we used  = 0.001 rad.

Experiments
We have inspected the evolution of the sheet shape as the function of its height. The first attempt used an A4-sized photocopy paper ( = 0.297 m length, w = 0.21 m width, and  = 10 -4 m thickness) of mass density per unit area  = 70 gsm (0.07 kg/m 2 ) or the mass density per unit length  = w = 0.0147 kg/m. The area moment of the paper is I = w 3 /12. The reported elastic modulus of paper is around Y = 3 GPa [21,22]. The predicted critical height of this paper derived from the Euler-Bernoulli equation is 0.143 m.
Input sheet height  when the length is very high (1). The bending decreases when the length is reduced. Bending with 1 positive occurs when L > 0.157 m (states 1 to 4), and bending with 1 when occurs when 0.14.1 m < L < 0.157 m (states 4 to 9). By carefully varying the paper height, we observed that the critical height where the paper suddenly directs straight upward. The starting height for shaping straight vertical is exactly the same as the critical buckling height 0.1418 m. The 1 is unchanged when L just is shorter than Lcr.

Low temperatures
Order state    . Only one 1 that produces N = -90 o for a certain paper height. This value corresponds to equilibrium shape of the paper. Curve was plotted with respect to 1/L (correspondence to temperature in phase transition). In phase transition, the order state of zero occurs at high temperature, while in our case, the free end angle of zero occurs at high 1/L (short L). The change in Fig. 4(b) is very similar to second order phase transition, where the first phase (order) corresponds to bending state (1/L < 1/Lcr) and the second phase with the critical exponential t = 0.485 (could be rounded to 0.5). We will simply prove the evidence of this scaling relationship as follows.
By carefully inspecting Fig. 4(b), it is clear that the curve is very similar to curve of order of parameter in second order phase transition where 1/L plays a role as temperature and 1   +90 (in degree) is the order parameter. Therefore, using the Landau theory for the second order phase transition we may approximate the bending column free energy as [23] Similarly, taking the first order in "temperature" (inverse of column length) expansion and keeping the parameter  to nearly constant, we have The order parameter that minimizing the free energy satisfies resulting, (11) which is similar to the expression in Eq. (7).
The zero temperature state corresponds to 0 . In this case, the column free end directs vertically downward. This state can be considered as the most ordered state. When temperature is increased or L is reduced, the angle of the free end decreases and becomes directly upward when cr L L / 1 / 1 → , and then stays zero when

Universality for Self-Buckling Column
We will show that Eq. Since we can arbitrarily select a as long as still produce a large number of segments, let us select a' so that for all materials. Therefore, the segment length must satisfy Based on this selection, the recursive Eq. (10) can be re-expressed as Equation (12) is independent of material, since  is constant, therefore it is a universal equation.
The angle made by the fixed end is  From Eq. (15), we clearly can conclude that when the length of the slender column having uniform Y, I, and  is scaled with L0 then the bending profile of all columns coincide.
We need only one bending profile for all columns when the fixed end boundary is identical.
14 Figure 5 shows the comparison of bending profile of mica plastic sheet and buffalo brandingname paper at the same length in the scale of characteristic length. By comparing both papers bending at three lengths: 2 L , 2.3 L , and 2.5 L , it is clearly observed that they show precisely identical bending. Let us look back at Eq. (13) which can be written as

cm cm
For columns having constant  and Y, the critical length for buckling is solely controlled by area moment, and the area moment itself depends on the column geometry. If the geometry of the column can be changed even the length is kept to be constant, the phase transition can also be generated. It is likely a mechanism that controls the bending profile of tree leaves having a geometry similar to slender columns such as banana leaves, palm leaves, and pandanus leaves. Discussion of the leaf as cantilever column has been reported by Pasini and Mirjalili [24] and Ennos et al [25]. Their discussions were focused on the effect of petiole on the leaf bending, and neither related their discussion with phase transition.
The banana leaves (Figure 6(a)) start to grow in nearly cylindrical shape (rolled sheet), the cross section of which is illustrated in Fig. 6(b) with an effective radius r and the . After a few days, the cylinder opens and the shape changes into the sheet and the area moment is around 12 / ' 3  w I = (we approximate the shape with a rectangular sheet), with w and  are the leaf width and thickness, respectively.
The cross section area of the young banana leaf is r 2 . Since the sheet thickness is , the width of the sheet after opening becomes   / 2 r w = . Therefore, the area moment after the leaf opens is approximately , and by using Eq. (14) we obtain (16) It is clear from Eq. (16), the leaves bend larger when the the radius to width ratio is smaller.
We can observe in many images available in the internet or directly in the garden, that wider leaves bend larger than slim leaves. A Similar mechanism also happens in bending of palm frond, where the young one shapes like a vertical rod (the area moment is large). After few days, the leaves open to decrease the area moment, so that the critical length for buckling is surpassed, and the frond becomes bending. For special case when the load weight is much larger than the column weight, Eq. (1) approaches the Euler strut as discussed by Bobnar et al [26]. However, if the load at the free end is absent, Eq. (1) estimates precisely the same critical length for buckling as have been well understood. Proves of these two statements are shown in Appendix 2.

Conclusion
We have demonstrated clearly that self-buckling of inextensible slender columns has a strong relationship with the second phase transition. The "critical temperature" for the phase transition is represented by the inverse of the critical height for buckling, 1/Lcr, and the deviation angle made by column free end relative to vertical direction satisfies a scaling relationship 485 .
. We have also introduced a universal equation applied to all slender columns having homogeneous elastic modulus, area moment, and mass density.
Using this universal equation, we re-derived the famous critical buckling height of that has been derived two centuries ago using Bessel function.
Finally, we introduced a new characteristic length , and if the column length is scaled with this characteristic length, the bending profile of all columns that are positioned at the same fix angle are precisely coincided. This scale unifies all kinds of inextensible and homogeneous slender columns.

Appendix 1
We can write Eq.

Appendix 2
The discrete form of bending equation as given by Eq. (1)