Nonlinear repulsive force between two solids with axial symmetry

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 $\mathcal{F}\propto z^n$, $1<n<\infty$, where $\mathcal{F}$ 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 $z_{1,2}=x^{\alpha }/R_{1,2}^{\alpha -1}+y^{\alpha }/({\mathit{mR}}_{1,2}){}^{\alpha -1}$, where $z_{1,2}⩾0$ refers to the compression of bodies $1$, $2$, $\alpha >1$, $m>0$, $x,y⩾0$. 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 $F=a{z}^n$, where $n\equiv 1+1/\alpha$, and $z\equiv z_1+z_2$ is the overlap and we present an expression for $a=f(E,\sigma ,m,\alpha ,R_1,R_2)$ with $E$ and $\sigma$ as Young’s modulus and the Poisson ratio, respectively. For $\alpha =2,\infty$, 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 $\alpha$ and $m$.

Figure 1

(a) Cross-sectional areas along the $x$-$y$ plane for various values of $\alpha$, $m=1$. (b) Cross-sectional areas obtained for the same cases as in (a) but for $m>1$, where $m$ is eccentricity.

Figure 2

Undistorted radius of curvature of one of the bodies is shown as curve a, the modification of the contact surface of the object deformed by $z_1$ is shown as curve b, and curve c shows the curvature at the origin after the transverse expansion, indicated by ${\rho }_{\mathrm{new}}$ [see Eq. (2)], has taken place.

Figure 3

Plot showing how the exponent $n$ in the force law depends on $\alpha$ in Eq. (1). For $n=2$ one obtains Hertz’s law, whereas for $n\to \infty$ one obtains the force between two disks as discussed in the text.

Figure 4

Schematic of how grains satisfying two different force laws on opposite ends may be constructed. Note that in this case $h=R_1+R_2+a$.