An elastic–plastic contact model of ellipsoid bodies

Abstract

A theoretical model for the elastic–plastic contact of ellipsoid bodies is presented in this paper. Relation of the contact parameters, such as the mean contact pressure, the contact area and the contact load as a function of the contact interference are modeled in the three different contact regimes: elastic, elastic–plastic and fully plastic. The model is verified by experimental results and is compared with published theoretical models. Very good agreement between the present model and the experimental results are found compared to the prediction of the other contact models.

Appendices

Appendix A: Contact area and load of elastic elliptical contact

From the theory of elasticity, the maximum contact pressure p _o, the semi-major contact ellipse radius b and the interference of an asperity ω can be written as [12]

$$ p_{0} = \frac{3} {2}p = \frac{{3P}} {{2A}} = \frac{{3P}} {{2\pi ab}} $$

(A.1)

$$ b = {\left[ {\frac{{3{\bf E}(e)PR_{{\text{m}}} }} {{\pi E(1 - e^{2} )}}} \right]}^{{1/3}} $$

(A.2)

$$ \omega = \frac{{2{\bf K}(e)}} {\pi }{\left[ {\frac{{\pi (1 - e^{2} )}} {{4{\bf E}(e)R_{{\text{m}}} }}} \right]}^{{1/3}} {\left( {\frac{{3P}} {{4E}}} \right)}^{{2/3}} $$

(A.3)

where p is the mean contact pressure, P is the contact load, a is the semi-minor radius of the elliptic contact area and E is the effective elastic modulus as defined in equation (A.4):

$$ \frac{1} {E} = \frac{{1 - v^{2}_{1} }} {{E_{1} }} + \frac{{1 - v^{2}_{2} }} {{E_{2} }} $$

(A.4)

The ratio of the semi-minor and semi-major contact ellipse radii, a and b, are related to the principal relative radii, R _x and R _y , as:

$$ \frac{a} {b} = {\left\{ {\frac{{{\bf E}(e)}} {{{\bf K}(e) + \frac{{R_{x} }} {{R_{y} }}{\left[ {{\bf K}(e) - {\bf E}(e)} \right]}}}} \right\}}^{{0.5}} $$

(A.5)

K(e) and E(e) are the complete elliptic integrals of the first and second kind, respectively:

$$ {\bf K}(e) = {\int_0^{\pi /2} {(1 - e^{2} {\text{sin}}^{2} \varphi )^{{ - 0.5}} \,{\text{d}}\varphi } } $$

(A.6)

$$ {\bf E}(e) = {\int_0^{\pi /2} {(1 - e^{2} {\text{sin}}^{2} \varphi )^{{0.5}} \,{\text{d}}\varphi } } $$

(A.7)

The elastic contact area A _e and the contact load P _e can be expressed in terms of the contact interference ω by combining equations (A.1), (A.2) and (A.3):

$$ A_{{\text{e}}} = {\left[ {\frac{{{\bf E}(e)}} {{{\bf K}(e)(1 - e^{2} )^{{0.5}} }}} \right]}2\pi R_{{\text{m}}} \omega $$

(A.8)

$$ P_{{\text{e}}} = {\left[ {\frac{{\pi {\bf E}(e)^{{0.5}} }} {{2{\bf K}(e)^{{1.5}} (1 - e^{2} )^{{0.5}} }}} \right]}\frac{{4{\sqrt 2 }}} {3}ER^{{0.5}}_{{\text{m}}} \omega ^{{1.5}} $$

(A.9)

Appendix B: Fully plastic contact experiment

In Ref. [23] the measurements of the contact parameter in the fully plastic contact regime were performed on copper and aluminum. Results showed that for both materials the behavior of the mean contact pressure and the contact area are similar. Experiments were done on a pin-on-disk machine where the flat specimen (SiC) was firmly mounted on a disk and the pin of the tester was replaced by a sphere specimen. A dead load system was used to apply the normal load.

The plastic contact area was measured by a new novel technique which so-called the matching and stitching method. In this method a measurement of a large area can be done by matching and stitching from sequence measurements as shown schematically in figure B.1. Figure B.2a demonstrates a typical result of the matching and stitching of plastic deformation of a sphere after compression. The change of the profile is presented in figure B.2b. Here, the ‘before’ profile is calculated based on the known initial sphere radius and the two points reference at the undeformed sphere.