Elliptic integral solutions of spatial elastica of a thin straight rod bent under concentrated terminal forces

In this article we solve in closed form a system of nonlinear differential equations modelling the elastica in space of a thin, flexible, straight rod, loaded by a constant thrust at its free end. Common linearizations of strength of materials are of course not applicable any way, because we analyze great deformations, even if not so large to go off the linear elasticity range. By passing to cylindrical coordinates ρ, θ, z, we earn a more tractable differential system evaluating ρ as elliptic function of polar anomaly θ and also providing z through elliptic integrals of I and III kind. Deformed rod’s centerline is then completely described under both tensile or compressive load. Finally, the planar case comes out as a degeneracy, where the Bernoulli lemniscatic integral appears. KEYWORD: Elliptic integrals, Spatial elastica, Nonlinear differential equations, Linear elasticity NOTE: this is the post print version of the paper appeaered on Meccanica 41 (2006) 519–527 doi:10.1007/s11012006-9000-3 1 A literature’s background As early as 1691 Jakob Bernoulli proposed the problem to determine the deformed centerline (planar “elastica”) of a thin, homogeneous, straight and flexible rod under forces and couples applied at its end. Whereas Galilei (Discorsi e dimostrazioni matematiche intorno a due nuove scienze, 1638) and Mariotte (Traité du mouvement des eaux et des autres corps fluides, 1686) had investigated the strength of beams, he studied the geometry of their deflection. In such a way, some time later, he established (Curvatura laminae elasticae, 1694) elastica’s differential equation and met the same elliptic integral ∫ ( 1− γ4 )−1/2 dγ of lemniscate rectification. He integrated it by series and proved the required line, sometimes referred as lintearia, to be shaped as a cross section of a horizontal flexible cylinder filled with water with free surface belonging to the line of thrust. The subject was also treated by his brother Johann at lessons XLIV (De curvatura lintei a fluido incumbente) and XLV, (Constructio curvae linteariae) of his Lectiones Mathematicae de methodo integralium . . . annis 1691 & 1692, issued in 1742. Daniel Bernoulli, Johann’s son, had obtained in 1738 an integral expression of potential energy stored in a bent rod: in 1742 he guessed it would attain such a shape as to minimize the functional of squared curvature. Accordingly, Euler dealt elastica as an isoperimetric problem of Calculus of Variations, De curvis elasticis, 1744, arrived at elastica’s ODE, and -mainly interested in the geometrical forms of elastic curves-identified nine forms of them. The subject attracted also C. Maclaurin, who realized the elastica had to be connected to elliptic integrals, see A treatise on fluxions, 1742, at § 927: The construction of the elastic curve, and of other figures, by the rectification of the conic sections. In 1757 Euler wrote a paper, Sur la force des colonnes, concerning the buckling of columns again, where he approached the critical load through a simplified expression to the elastica’s curvature. In 1770 Lagrange, Sur la figure des colonnes, did not limit himself to a calculation of critical loads already discussed by Euler, but went on to investigate the deflections when the load exceeds its critical value, using the exact curvature, and integrating by series his elastica’s ODE. Euler went over the problem, always in the field of bending, again in 1770 and 1775: for all this historical concern, see [4], [5] and [18]. Some analytic solutions to elastic planar curves through elliptic integrals of I and II kind, can be read at [17], [39] and [14]. The link between elliptic functions and elastica was deemed so close, to induce to repre∗Via Negroli, 6 20136 Milano [email protected] †Dipartimento di Matematica per le scienze economiche e sociali, Università di Bologna [email protected]