Nonlinear repulsive force between two solids with axial symmetry

Abstract

We modify the formulation of Hertz contact theory between two elastic half-solids with axial symmetry and show that these modifications to Hertz's original framework allow the development of force laws of the form F ∝ z n , 1 < n < ∞, where F is the force and z is the distance between the centers of the two solids. The study suggests that it may be possible to design physical systems that can realize such force laws. We let the half-solids be characterized by radii of curvatures R 1 and R 2 and invoke a factor m > 0 to describe any aspect ratio in the two bodies, all being valid near the contact surface. We let the x-y plane be the contact surface with an averaged pressure across the same as opposed to a pressure profile that depends on the contact area of a nonconformal contact as originally used by Hertz. We let the z axis connect the centers of the masses and define z 1,2 = x α /R α−1 1,2 + y α /(mR 1,2 ) α−1 , where z 1,2 0 refers to the compression of bodies 1, 2, α > 1, m > 0, x,y 0. The full cross section can be generated by appropriate reflections using the first quadrant part of the area. We show that the nonlinear repulsive force is F = az n , where n ≡ 1 + 1/α, and z ≡ z 1 + z 2 is the overlap and we present an expression for a = f (E,σ,m,α,R 1 ,R 2 ) with E and σ as Young's modulus and the Poisson ratio, respectively. For α = 2,∞, to similar geometry-dependent constants, we recover Hertz's law and the linear law, describing the repulsion between compressed spheres and disks, respectively. The work provides a connection between the contact geometry and the nonlinear repulsive law via α and m.

Breather Solutions of the Discrete p-Schrödinger Equation

“…In the same way (using the reflectional symmetry of (4)), if limn→−∞ an = 0 then an+1 and an have opposite signs for all n ≤ −n0 if n0 ≥ 0 is sufficiently large. Inequality (53) follows similarly from theorem 2, after elementary computations based on the decay estimate (27). One can fix C = q 1−α in (27) and observe that q −1 |u0| ≤ q provided |n0| is large enough, which yields (53).…”

“…Inequality (53) follows similarly from theorem 2, after elementary computations based on the decay estimate (27). One can fix C = q 1−α in (27) and observe that q −1 |u0| ≤ q provided |n0| is large enough, which yields (53).…”

“…This limiting procedure is a first step towards the rigorous justification of previous formal continuum approximations [17,26,13]. Moreover, it is interesting to note that the case p ≈ 2 is physically sound in the context of granular chain models [16,27,19].…”

Characterizing traveling-wave collisions in granular chains starting from integrable limits: The case of the Korteweg–de Vries equation and the Toda lattice

“…The mechanical, and more specifically dynamic, properties of these systems are governed by the stress propagation at the contact between neighboring particles. This confers to the overall system a highly nonlinear response dictated, in the case of particles with an elliptical or spherical contact, by the discrete Hertzian law of contact interaction [21][22][23]. Geometry and/or material anisotropy between particles composing the systems allows for the observation of interesting dynamic phenomena deriving from the interplay of discreteness and nonlinearity of the problem (i.e.…”

Wave propagation in one-dimensional microscopic granular chains

“…The dynamic response of granular crystals is governed by highly nonlinear interactions between neighboring particles, which is determined by their contact geometry (e.g., Hertzian). More generally, by modifying the contact geometry and particle arrangements (i.e., changing the stress-strain relation and the direction of contact forces) [8], granular crystals can be designed to behave as highly flexible wave guides [9], focusing lenses [10] and filtering materials [11]. These features of granular crystals have inspired their use in engineering applications, such as shock mitigation [12][13][14][15], acoustic rectification [16], and logic elements [17].…”