On the Planar Elastica, Stress, and Material Stress

Abstract

We revisit the classical problem of the planar Euler elastica with applied forces and moments, and present a classification of the shapes in terms of tangentially conserved quantities associated with spatial and material symmetries. We compare commonly used director, variational, and dynamical systems representations, and present several illustrative physical examples. We remark that an approach that employs only the shape equation for the tangential angle obscures physical information about the tension in the body. * harmeet@vt.edu † hannaj@vt.edu 1 Material forces and closely related quantities, which in our one-dimensional system have just one component, appear under many names in the literature, including Eshelbian force, quasimomentum, pseudomomentum, (Kelvin) impulse, and configurational force [3][4][5][6][7][8][9][10][11][12]. In the present case, one can derive everything from consideration of conventional force balance and its projection onto the tangents of the body, but the concept of material force is useful as a descriptive term, and also corresponds to an important symmetry of the Lagrangian. We generally prefer the term "spatial" to "conventional", but try to avoid it in this note because of its alternative meaning as "non-planar" in the context of rods embedded in three dimensions rather than two.arXiv:1706.03047v4 [physics.class-ph]

Configurational forces and nonlinear structural dynamics

“…While from a theoretical point of view, an analytical definition of the transition value p tr (α) is not feasible (within a dynamic framework characterized by strong nonlinearities), from a practical point of view, the transition can be evaluated by analyzing λ(τ ) at large values of the dimensionless time τ . This can be done for different values of load p and inclination α, as the result of an integration of the equations of motion (41), where dissipative terms are included. Note that the introduction of the viscous damping ratio ζ, essential to correctly capture the experiments that will be reported in the next section, also provides a regularization useful for the stability of the numerical integration.…”

“…The value of the friction coefficient µ has been calibrated through an optimal matching between experimentally measured and theoretical predicted trajectories of the lumped mass (shown in the next section). The numerical integration is accomplished by means of the function NDSolve in Mathematica (v. 11) considering the options MaxStepSize → 10 −3 , StartingStepSize → 10 −8 , Method → IndexReduction, 5 and the small 5 The option Method → IndexReduction is only needed in the numerical integration during the large rotation regime, namely, in solving the DAE system (41). Such an option implies that a differentiation of the algebraic equation is performed during the numerical integration.…”

“…so that the nonlinear DAE system (41) can be reduced to the following system of two nonlinear differential equations…”

Exact and approximate mechanisms for pure bending of sheets

“…where x(s) is the position of the body, n and m(s) are the local contact force and moment, and the constant vectors P and J are the conserved force and torque [51]. As the body is only loaded at its ends, the force balance (2) indicates that the contact force n is a constant.…”

“…To understand localization, we need to determine how each of these quantities scale with curvature. For a simple Euler elastica, perturbative analysis of the shape equation [51] near a circular solution seems to imply that d P dκ ∼ κ 0 , but we reserve further analysis for future work. There appear to be few studies focused on the onset of plastic localization in forming operations where bending is the predominant deformation mode.…”