Lilly Bell Kink -

This paper provides:

Geometric instabilities such as buckling, wrinkling, and kinking are central to the mechanics of slender structures. While classical Euler buckling has been extensively studied for straight beams, curved filaments exhibit richer behavior due to the coupling between curvature and external loads. The Lilly‑Bell kink—named after the characteristic bell‑shaped profile of the filament in which it was first observed (Lilly & Bell, 2002)—represents a distinct mode of failure where a localized, high‑curvature hinge forms, often accompanied by self‑contact (Figure 1).

Linearizing about the pre‑curved configuration (\theta_0(s) = \kappa_0 s) yields the critical load for the first buckling mode:

[ P_cr^(lin) = \fracEIR_0^2\bigl(\pi^2 - 1\bigr). ]

However, the LBK is a non‑linear localization that occurs before the global Euler buckling of the curved beam. By employing a weakly‑nonlinear expansion (see Appendix A) we obtain the kink criterion: lilly bell kink

[ \boxed \Gamma \equiv \fracP,\kappa_0EI > \Gamma_c = \frac2\pi, \tag1 ]

Equation (1) defines a dimensionless curvature‑load parameter (\Gamma). When (\Gamma) exceeds (\Gamma_c), the curvature concentrates into a narrow region, forming the kink.

After kink formation, the filament can be modeled as two elastica segments joined at a hinge of angle (\phi). The hinge is constrained by self‑contact:

[ \beginaligned &\mathbfr_1(s_k) = \mathbfr_2(s_k) \quad\text(position continuity)\ &\mathbft_1(s_k) \cdot \mathbft_2(s_k) = \cos\phi \quad\text(tangent jump)\ &\mathbfn_1(s_k) \cdot \mathbfn_2(s_k) = \mu ,\mathrmsgn(\dot\phi) \quad\text(Coulomb friction) , \endaligned ] This paper provides:

where (s_k) denotes the arc‑length of the kink, (\mu) the coefficient of friction, and (\mathbfn) the normal vector. Solving the coupled boundary‑value problem yields the post‑kink load‑deflection relationship:

[ P(\delta) = \frac2EIR_0^2,\frac1\bigl(1+\delta/L\bigr)^2, \Bigl[ 1 - \cos!\bigl(\phi(\delta)\bigr) \Bigr], \tag2 ]

where (\delta) is the imposed axial shortening.

The LBK can be detrimental (e.g., premature failure of fiber‑reinforced components) or beneficial (e.g., rapid shape change in soft actuators). However, the literature lacks a unified theoretical framework that predicts the onset, evolution, and post‑kink response of the LBK across material systems. Addressing this gap can enable intentional design of kinking mechanisms for: The LBK can be detrimental (e

Consider a slender filament of length (L), uniform cross‑section (A), and flexural rigidity (EI). The undeformed centerline follows a circular arc of radius (R_0) (intrinsic curvature (\kappa_0 = 1/R_0)). The filament is clamped at one end ((s = 0)) and loaded axially with a compressive force (P) at the other end ((s = L)).

The planar elastica governing equation (ignoring shear deformation) is

[ EI \fracd^2\thetads^2 + P\sin\theta = 0, ]

where (\theta(s)) is the angle between the tangent and the axial direction.