The History of the Planar Elastica: Insights into Mechanics and Scientific Method

Abstract

Euler's formula for the buckling of an elastic column is widely used in engineering design. However, only a handful of engineers will be familiar with Euler's classic paper De Curvis Elasticis in which the formula is derived. In addition to the Euler Buckling Formula, De Curvis Elasticis classifies all the bent configurations of elastic roda landmark in the development of a rational theory of continuum mechanics. As a historical case study, Euler's work on elastic rods offers an insight into some important concepts which underlie mechanics. It sheds light on the search for unifying principles of mechanics and the role of analysis. The connection between results obtained from theory and those obtained from experiments on rods, highlights two different approaches to scientific discovery, which can be traced back to Bacon, Descartes and Galileo. The bent rod also has an analogy in dynamics, with a pendulum, which highlights the crucial distinctions between initial value and boundary value problems and between linear and nonlinear differential equations. In addition to benefiting from the overview which a historical study provides, the particular problem of the elastica offers students of science and engineering a clear elucidation of the connection between mathematics and real-world engineering, issues which still have relevance today.

Competition between curls and plectonemes near the buckling transition of stretched supercoiled DNA

“…10), which contains 3D twisted configurations, can be continued past b 1 up to point b̄ where the rod is twistless, planar, and self-intersecting. This well-known solution is refered to as the homoclinic planar elastica (see [39] for a review), and has the following properties: $$\text{Lk}=1,\text{Wr}=1,\text{Tw}=0$$ $$X=L-4\sqrt{K_0/f}$$ $$E=\frac{1}{2}{\int }_0^L{K}_0{\kappa }^{2}false(Sfalse)dS-fX=8\sqrt{K0f}-fL$$ $$xfalse(Sfalse)=S-2\sqrt{K_0/f}tanhfalse(S/\sqrt{K_0/f}false)$$ $$yfalse(Sfalse)=2\sqrt{K_0/f}/coshfalse(S/\sqrt{K_0/f}false)$$ see also Refs. [40, 41].…”

Stability Analysis of Nonlocal Elastic Columns with Initial Imperfection

“…so as to reduce the complexity of the boundary-value problem defined by (9), (13), and (18) and to capture the intrinsic geometrical nonlinearity at the same time. This simplification would introduce some small errors, as will be verified later by comparison with classical elastic solution.…”

“…In the light of the mathematical consistency in the way the mechanical model is formulated, hereinafter, we will restrict our attention to the elastica theory [11,12], which entails using the exact expression of curvature. The first study of the deformation of an elastica under loading was probably due to James Bernoulli although it was Euler who established the basic theory [11,13]. By taking advantage of Euler's celebrated work, considerable efforts have been devoted to the theory with applications extended from classical fields to statistical mechanics (see [14] and the references therein), biomechanics [15], and ocean engineering [16,17].…”

Loading paths for an elastic rod in contact with a flat inclined surface

“…Formulated during the eighteenth century by Euler and Bernoulli, the elastica is an established model for the large deflections of long slender rods; see Levien (2008) and Goss (2009) for historical perspectives. Consequently, the formulation of the boundary value problem set out in this paper follows a well trodden path, but we mention here Frisch-Fay (1962) and Batista (2013) where we find related formulations, and the constrained problems considered in Plaut et al (1999) and Domokos et al (1997) .…”