Boundary Value Problems for Euler-Bernoulli Planar Elastica. A Solution Construction Procedure

Abstract

We consider the problem of finding a curve minimizing the Bernoulli bending energy among planar curves of the same length, joining two fixed points and possibly carrying orientations at the endpoints (Euler elastica). We focus on the problem of constructing closed form elasticae for given boundary data and show that, rather than employing complicated numerical algorithms, it suffices to use easily available computer algebra systems to implement our procedure.

To this end, we first review some fundamental facts about the Euler-Bernoulli variational approach to the elastic rod. Our curves are only assumed to be stationary and not necessarily minimizers. Secondly, the Euler-Lagrange equations are expressed in terms of the curvature of the elasticae, what is used to compute their explicit parametrizations by means of the Jacobi elliptic functions. Lastly, we describe our approach to solving this problem under different boundary conditions, and the procedure is illustrated with numerous examples. We include the numerical code that we use in Appendix B.

Appendices

Appendix A: Jacobi Elliptic Functions

In this appendix we briefly review some basic facts about Jacobi elliptic functions which we have used earlier. Legendre elliptic integrals and Jacobian elliptic functions find many applications in dynamics, mechanics, electrostatics, conduction, field theory and mathematical physics. Elliptic integrals can be viewed as generalizations of the inverse trigonometric functions. Jacobi [21] wrote the classic treatise on elliptic functions because of the need to integrate second order kinetic energy equations.

Within the scope of this paper it will be enough to examine elliptic integrals of the first and second kind (for more details, see [7]). If we let $p$ satisfy $0 \leq p^{2} \leq 1$, the incomplete elliptic integral of the first kind and modulus$p$ is written as

$$ F ( \varphi ,p ) = \int ^{\varphi }_{0}\frac{\textrm{d} \theta }{\sqrt{1-p^{2}\sin ^{2}\theta }}= \int ^{x=\sin \varphi }_{0}\frac{ dt}{\sqrt{ ( 1-t^{2} ) ( 1-p^{2}t^{2} )}} , $$

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$0\leq \sin \varphi \leq 1$, $0\leq \varphi \leq \frac{\pi }{2}$. The complete elliptic integral of first kind, $K(p)$, can be obtained by setting the upper bound of the integral to its maximum range, i.e. $\sin \varphi = 1$ or $\varphi = \pi /2$ to give $K\equiv K ( p ) =F ( \frac{\pi }{2},p )$. Similarly, the incomplete elliptic integral of second kind of modulus p can be defined by

$$ E ( \varphi ,p ) = \int ^{\varphi }_{0}\sqrt{1-p^{2}\sin ^{2}\theta }\, d\theta = \int ^{x=\sin \varphi }_{0}\frac{\sqrt{1-p ^{2}t^{2}}}{\sqrt{1-t^{2}}}\,dt, $$

(50)

and the complete elliptic integral of second kind, $E(p)$, is obtained by setting the upper bound of integration to the maximum value to get $E\equiv E ( p ) =E ( \frac{\pi }{2},p )$.

Formally, the Jacobi elliptic functions are obtained by inversion of the elliptic integrals and can be considered as doubly periodic meromorphic functions over the complex plane. While there are twelve different types of Jacobian elliptic functions based on the number of poles and the upper limit on the elliptic integral, the three most popular are the copolar trio of Jacobi elliptic sine (of modulus $p$), $\mathit{sn}(u, p)$; Jacobi elliptic cosine (of modulus $p$), $\mathit{cn}(u, p)$; and the delta Jacobi elliptic function (of modulus $p$), $\mathit{dn}(u, p)$. Setting

$$ u=F ( \varphi ,p ) = \int ^{\varphi }_{0}\frac{d \theta }{\sqrt{1-p^{2}\sin ^{2}\theta }}, $$

(51)

they are defined, respectively, by

$$ \text{sn} ( u,p ) =\sin \varphi ,\qquad \text{cn} ( u,p ) =\cos \varphi , \qquad \text{dn} ( u,p ) =\sqrt{1-p ^{2}\sin ^{2}\varphi }=\Delta ( \varphi ) . $$

The angle $\textrm{am} ( u,p ) =\varphi$ is called the Jacobi amplitude. When the precise value of $p$ is not important we can suppress the dependence upon $p$. In particular, $\textrm{sn} ( u )$ and $\textrm{cn} ( u )$ are periodic functions of period $4K$, while the period of $\textrm{dn} ( u )$ is $2K$. The remaining nine Jacobi functions are defined in terms of these three functions [7]. It is clear from the definitions that Jacobi elliptic functions generalize trigonometric and hyperbolic functions since

$$\begin{aligned} \text{sn} ( 0 )=0, \qquad \text{cn} ( 0 ) =1, \qquad \text{dn} ( 0 ) =1,\qquad \text{am} ( 0 ) =0, \\ \text{sn} ( u,0 ) =\sin u,\qquad \text{cn} ( u,0 ) =\cos u,\qquad \text{dn} ( u,0 ) =1 , \\ \text{sn} ( u,1 )=\tanh u,\qquad \text{cn} ( u,1 ) =\text{dn} ( u,1 ) = \text{sech}\,u, \\ \text{sn} ( -u ) =-\text{sn}\, u,\qquad \text{cn} ( -u ) =\text{cn} \,u,\qquad \text{dn} ( -u ) = \text{dn}\,u, \qquad \text{am} ( -u ) =-\text{am}\,u. \end{aligned}$$

It is not surprising that these functions enjoy properties and show algebraic relations between them which are similar to those known between the singly periodic trigonometric functions. Indeed, squares of copolar functions all have trig-type relationships. For example

$$\begin{aligned} \text{sn}^{2}u+\text{cn}^{2}u=1,\qquad \text{dn}^{2}u-p^{2} \text{cn}^{2}u=1-p^{2}= \bigl( p^{\prime } \bigr) ^{2},\qquad p^{2} \text{sn}^{2}u+\text{dn}^{2}u=1. \end{aligned}$$

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Another important property of the Jacobi elliptic functions is that their derivatives are always constant multiples of the products of the copolar functions. For instance, the relation

$$ \frac{du}{d\varphi }=\frac{1}{\sqrt{1-p^{2}\sin ^{2}\varphi }}=\frac{1}{\text{dn}\,u} $$

is derived from (51), and since $\textrm{sn}\, u =\sin \varphi$, we have

$$ \frac{d}{du}\textrm{sn}\,u=\cos \varphi \frac{d\varphi }{du}=\textrm{cn}\,u\,\textrm{dn}\,u. $$

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Similarly

$$\begin{aligned} \frac{d}{du}\textrm{cn}\,u=-\textrm{sn}\,u\,\textrm{dn}\,u,\qquad \frac{d}{du}\textrm{dn}\,u=-p^{2}\textrm{sn}\,u\,\textrm{cn}\,u,\qquad \frac{d}{du}\textrm{am}\,u=\textrm{dn}\,u. \end{aligned}$$

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Appendix B: Mathematica™ Code to Complete the Procedure

Here, Mathematica™’s FindRoot command will be used as a main tool to support the construction of specific solutions to the different boundary value problems for planar elasticae analyzed in Sect. 3. Note first that Mathematica™ uses $p^{2}$ instead of $p$ to denote the modulus of Jacobi elliptic functions. For example, the Jacobi sine function should be called within Mathematica™ by using $\textrm{JacobiSn}$$(u,p^{2})$ rather than by $\textrm{JacobiSn}( u,p)$.

Now, without loss of generality, we may suppose that $L=1$, that $\gamma (0)=P$ and $\gamma (1)=Q$ are points on the $x$-axis, and that the vectors $\gamma '(0)=\mathbf{v}$ and $\gamma '(1)=\mathbf{w}$ are identified with the angles they make with $x$-axis at the points $P$ and $Q$, respectively. For further simplification, we can assume that the maximum of the curvature $\kappa _{o}$ is attained at $s=a$ and the last noteworthy point of the curvature is obtained at $s=1-b$, being $[0,1]$ the interval of definition of our curve in this case. Then, the following lines of code are defined in Mathematica™ by using formulae (33) to (42) in Sect. 2.2. We remark, however, that these lines are written for illustrative purposes and they are not optimized from the point of view of code programming.

Example 1

Clamped ends.

We know that there exist both wavelike and orbitlike solutions in this case. To fix ideas we consider the following example. For both types of elasticae we choose $n=2$, length $L=1$, and assume that the endpoints $P$ and $Q$ are separated by a $\mbox{distance} (P,Q )=0.4$. Suppose that the tangent vectors at those points make angles $\pi /2$ and $5\pi /6$ with the line $\overrightarrow{\mathit{PQ}}$, respectively. Following the arguments of Sect. 3.1, we first choose an arbitrary parameter $n$ and, then, determine $\{p,a,b\}$. For $n=2$ the next code computes the appropriate parameters $\{p,a,b\}$ and then plots the corresponding clamped wavelike elastica.

The last instruction includes a matrix M (not defined here) whose role is rotating the curve so that the $\overrightarrow{\mathit{PQ}}$ line is sent onto the $x$-axis and the tangent vector $\gamma '(0)$ onto $\mathbf{v}$. In the orbitlike case, the above output parameters $\{p,a,b\}$ would become

Figures 14 and 15 plot these two elasticae and their corresponding curvature functions, illustrating also the meaning of $a$, $b$ and $n$ as explained in Sect. 3.1. Recall that $a$ is measured positively from $s=0$ to the right, while $b$ is measured positively from $s=L=1$ to the left.

Fig. 14

Plots of the wavelike elastic curve $\gamma$ (left) and its curvature function $\kappa$ (right), defined by the parameters $(n,a,b,p)$ obtained from the Mathematica™ code as described in Example 1

Fig. 15

Plots of the orbitlike elastic curve $\gamma$ (left) and its curvature function $\kappa$ (right), defined by the parameters $(n,a,b,p)$ obtained from the Mathematica™ code as described in Example 1. Note that the same curve is obtained for the parameter values $n=2$, $a<0$, $b>0$ and also for $n=1$, $a'>0$, $b'>0$. The green lines does not belong to the trace of the solution curves (Color figure online)

Example 2

Partially clamped endpoint.

With the same data as in Example 1, suppose now that the tangent vector at the right endpoint $Q=\gamma (1)$ is left unprescribed. The natural boundary condition (17) imposes $\kappa (1)=0$ and, therefore, the elastica has to be wavelike. The definitions of $b$ and $n$ given in Sect. 3 imply that $b=0$ and $n$ must be odd. So the solutions to this kind of problem can be accomplished by slightly changing the code of the previous Example 1: one just need to set $b=0$ and choose $n$ odd from the beginning. In this way, the boundary condition on the second angle $\psi _{1}$ in (47) becomes redundant. Thus, the remaining parameters $(a,p)$ (which were used to get the curve labelled as $i)$ in Fig. 11) are obtained, via the description given in Sect. 3.2, with the following code (this time we have chosen $n=1$).

Example 3

Unprescribed tangents at the endpoints (pinned elasticae).

Once more, we use the same data of Example 1, but we assume now that both tangent vectors at the endpoints are left unprescribed. From (27) we know that $\kappa (0)=\kappa (1)=0$ and the solution curves are also wavelike elasticae. Moreover, we can also say that the solution curves will be symmetrical because there is no difference between the roles of the endpoints $P$ and $Q$ in the variational problem. Again, we have $b=0$ and $n$ odd, and, due to the mentioned symmetry, $n$ and $a$ also satisfy the relation $a=1/ (n+1 )$. Therefore, the parameters $n$, $a$ and $b$ are known from the beginning, so we just need to use the first condition in (47) to get the remaining parameter $p$, as indicated in Sect. 3.3. For instance, choosing $n=3$, the following code gives us the value of $p$ in our example and corresponds to the curve $\mathit{iv})$ in Fig. 12

Example 4

Closed elasticae.

Finally, let us take a quick look at non-constant curvature closed elasticae. As it has been said before, they are, in fact, clamped curves with $P=Q$ whose tangent vectors and curvatures coincide at the endpoints. This kind of symmetry is translated to the construction scheme of Sect. 3.4 by setting $a=b$, which implies in turn that $n$ has to be a multiple of 4. Moreover, as the initial point of a closed curve can be chosen arbitrarily, we choose $P$ as any point where the curvature attains its maximum. That is to say, we can set $a=b=0$ from the beginning and, once $n$ has been chosen, it only remains to determine the parameter $p$ by using the first condition in (47). The second eight-shaped curve in Fig. 13 is obtained by taking $n=8$ and using the value for $p$ returned by the following code

The parameter $p$ is known as the shape parameter because it has the same value for all the eight-shaped closed elastic curves, regardless of their length ($L$) or of the number of times they are covered ($n/4$).

Appendix C: A Mathematical Digression

Assume we want to find the minima of the bending energy ℰ given by (6) within a family of curves, $\widehat{\varOmega }$, satisfying given boundary conditions and constraints. Considered as a purely mathematical variational problem, a major issue is the analysis of the existence and classification of the least energy elasticae. For curves in the plane $\mathbb{R}^{2}$, it is clear that straight lines are global minima of ℰ. When boundary conditions are imposed that preclude straight segments, the existence of a minimum is no longer guaranteed. Let us see a few examples. Consider first the bending energy (6) acting on the space of closed planar curves, $\widehat{\varOmega }$ (curves with the same end points $P= Q$ and same tangents at the endpoints). Observe that this time we are not fixing the length of the curves. Then, there are no minimizers of ℰ in $\widehat{\varOmega }$, since the energy of circles of increasing radii tends to zero, but they are not straight segments. What is more, there are no stationary curves in the space $\widehat{\varOmega }$ of closed curves with variable length [27]. As another example, consider now minimization of ℰ acting on $\widehat{\varOmega }=\varOmega$ as given in (5) with $P\neq Q$ and no constraint on the length of the curves. If the tangents at one or both endpoints are prescribed (clamped curves), stationary curves can be described in terms of two parameters, the interesting case being when at least one of the endpoint tangent vectors does not point in the direction of the straight line between the two endpoints. In this case, the straight line is excluded, the minimum is not reached and there is no global minimizer [28]. However, if the tangents at the endpoints are not prescribed and the length of the curves is either bounded or fixed, there always exists a minimizer of the bending energy among curves joining two given points [20]. Therefore, in general, additional constraints are needed in order to guarantee the existence of non-trivial minimizers. Frequently, in the case of Bernoulli’s elasticae, to achieve this goal the length of the curves is prescribed, but other constraints can also be used as, for example: prescribed enclosed area for closed curves in the plane [2]; confined closed curves in the plane [9]; variable length graphs [29] with constraints precluding the straight lines in $\mathbb{R}^{2}$, etc. An alternative way to guarantee the existence of minimizers of the bending energy (6) on $\varOmega$ is to add a penalty to ℰ: $\mathcal{E}+\lambda \mathcal{L}$, ℒ being the length of the curve. In this case, if $\lambda >0$ (variable length with positive tension), there is a global minimum in the space of closed curves [25]. In general, showing the existence of minimizers ℰ within a given space of curves $\widehat{\varOmega }$ is a non-trivial mathematical task. The stability problem of curvilinear configurations of flexible rods, either based on exact analytical solutions or resorting to numerics, is also of great importance in structural analysis, architecture and engineering. In this respect, it is a remarkable fact that already in 1906 M. Born [5, 26] showed that an elastic arc is stable if it does not contain inflection points.

Finally, we observe that the bending energy has appeared in many other different contexts, being not only fundamental in Mechanics, but it is also important in connection with other physical systems. Thus, for example, elasticae can be used to provide closed form solutions for the Helfrich-Canham model for biomembranes in Biophysics [13, 19, 32], explicit examples of worldsheets for the Kleinert-Polyakov action in string theory [3, 13], invariant solutions for the vortex filament equation in Fluid Dynamics [10, 18, 22], and soliton solutions of the non-linear cubic Schrödinger equation [17]. Moreover, the bending energy also inspired the modern theory of mathematical splines, a mathematical model of the mechanical spline used for shipbuilding and related applications. It also plays a role in image inpainting, image processing and computer vision [1, 8, 31]. Hence, elastic curve theory encompasses a broad range of mathematically and physically fruitful ideas and lies at the intersection of many different areas.