The Relation Between Subsurface Stresses and Useful Wear Life in Sliding Contacts

Abstract

Equations for quantifying the subsurface shear stress in dry point contact are utilized to obtain the value and location of the maximum subsurface shear stress. A series of experiments using a pin-on-disk tribometer is conducted on run-in specimens made of steel, brass, and aluminum, and the weight loss and wear rate of the specimen are measured. The results reveal a correlation between the depth of the maximum subsurface shear stress obtained from the model and the measured wear rate. It is shown that at the onset of failure, the friction coefficient suddenly increases. This increase affects the location of maximum subsurface shear stress by pushing it toward the surface and producing wear particles. SEM images of all three friction-pair tested reveal that the size of the wear particles is directly related to the applied load.

Graphical abstract

Appendix

1.1 Governing Equations for Substress Field in Cylindrical and Spherical Contacts

1.1.1 Cylindrical Contact

A schematic view of the contact of two elastic cylindrically shaped bodies is shown in Fig. 14 in Appendix. The contact area of two cylinders is a rectangle of length L and width of 2b. The half-width of b is given by [24]:

Fig. 14

Cylindrical contact

$$b=\sqrt{\frac{4F\left[\frac{1-{\upsilon }_{1}^{2}}{{E}_{1}}+\frac{1-{\upsilon }_{2}^{2}}{{E}_{2}}\right]}{\pi L\left(\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}\right)}}$$

(A1)

where E₁ and E₂ are moduli of elasticity, ν₁ and ν₂ are the Poisson’s ratios for cylinders 1 and 2, respectively, and L is the length of the contact area.

The equations for calculation of subsurface stress in a half-space loaded by a normal pressure p(s) and shear loading q(s) are [10]:

$${\sigma }_{xx}=-\frac{2z}{\pi }\int \frac{p\left(s\right){\left(x-s\right)}^{2}{\text{d}}s}{{\left\{{\left(x-s\right)}^{2}+{z}^{2}\right\}}^{2}}-\frac{2}{\pi }\int \frac{q\left(s\right){\left(x-s\right)}^{3}{\text{d}}s}{{\left\{{\left(x-s\right)}^{2}+{z}^{2}\right\}}^{2}}$$

(A2)

$${\sigma }_{zz}=-\frac{2{z}^{3}}{\pi }\int \frac{p\left(s\right){\text{d}}s}{{\left\{{\left(x-s\right)}^{2}+{z}^{2}\right\}}^{2}}-\frac{2{z}^{2}}{\pi }\int \frac{q\left(s\right)\left(x-s\right)ds}{{\left\{{\left(x-s\right)}^{2}+{z}^{2}\right\}}^{2}}$$

(A3)

$${\tau }_{xz}=-\frac{2{z}^{2}}{\pi }\int \frac{p\left(s\right)\left(x-s\right)ds}{{\left\{{\left(x-s\right)}^{2}+{z}^{2}\right\}}^{2}}-\frac{2z}{\pi }\int \frac{q\left(s\right){\left(x-s\right)}^{2}{\text{d}}s}{{\left\{{\left(x-s\right)}^{2}+{z}^{2}\right\}}^{2}}$$

(A4)

where for semi-elliptic pressure distribution in cylindrical contact, p(s) and q(s) are given by:

$$p(s)=\frac{2F}{\pi bL}\sqrt{1-\frac{{(x-s)}^{2}}{{b}^{2}}}$$

(A5)

$$q(s)=\frac{2\mu F}{\pi bL}\sqrt{1-\frac{{(x-s)}^{2}}{{b}^{2}}}$$

(A6)

where F is the applied load and μ is the friction coefficient. Maximum pressure ${p}_{0}=\frac{2F}{\pi bL}$ occurs at x = 0.

By substituting Eqs. (A5) and (A6) into Eqs. (A2) to (A4) and integrating the results, the following expressions for the subsurface stresses are obtained:

$$\begin{aligned} \sigma _{{xx}}^{{cy}} (x,z) = & \left. {\frac{{2F}}{{\pi bL}}\left[ {\frac{{ - 2z}}{{\pi b}}{\text{sin}}^{{ - 1}} \left( {\frac{{s - x}}{b}} \right) + \frac{{2z^{2} + b^{2} }}{{\pi b\sqrt {z^{2} + b^{2} } }}{\text{tan}}^{{ - 1}} \left( {\frac{{\sqrt {z^{2} + b^{2} } \left( {s - x} \right)}}{{bz\sqrt {1 - \frac{{\left( {x - s} \right)^{2} }}{{b^{2} }}} }}} \right) - \frac{{z(s - x)}}{{\pi \sqrt {1 - \frac{{\left( {x - s} \right)^{2} }}{{b^{2} }}} (\frac{{\left( {z^{2} + b^{2} } \right)\left( {s - x} \right)^{2} }}{{b^{2} \left( {1 - \frac{{\left( {x - s} \right)^{2} }}{{b^{2} }}} \right)}} + z^{2} )}}} \right]} \right|_{{ - b}}^{b} \\ & + \left. {\frac{{2\mu F}}{{\pi bL}}\left[ \begin{gathered} \frac{{\left( {z^{4} + \left( { - z^{2} - b^{2} } \right)\left( {2z^{2} + b^{2} } \right) + 2b^{2} z^{2} + b^{4} } \right){\text{ln}}\left( {\frac{{\sqrt {b^{2} - \left( {s - x} \right)^{2} } + \sqrt {z^{2} + b^{2} } }}{{\left| {\sqrt {b^{2} - \left( {s - x} \right)^{2} } - \sqrt {z^{2} + b^{2} } } \right|}}} \right)}}{{2\pi b\sqrt {z^{6} + 3b^{2} z^{4} + 3b^{4} z^{2} + b^{6} } }} - \frac{{(2z^{2} + b^{2} )ln\left( {\frac{{\left| {\sqrt {b^{2} - \left( {s - x} \right)^{2} } - \sqrt {z^{2} + b^{2} } } \right|}}{{\sqrt {b^{2} - \left( {s - x} \right)^{2} } + \sqrt {z^{2} + b^{2} } }}} \right)}}{{\pi b\sqrt {z^{2} + b^{2} } }} \hfill \\ + \frac{{\left( { - z^{4} - \left( { - z^{2} - b^{2} } \right)\left( {2z^{2} + b^{2} } \right) - 2b^{2} z^{2} - b^{4} } \right)}}{{2\pi b\left( {z^{2} + b^{2} } \right)\sqrt {b^{2} - \left( {s - x} \right)^{2} } + 2\pi b\sqrt {(z^{2} + b^{2} )^{3} } }} + \frac{{\left( { - z^{4} - \left( { - z^{2} - b^{2} } \right)\left( {2z^{2} + b^{2} } \right) - 2b^{2} z^{2} - b^{4} } \right)}}{{2\pi b\left( {z^{2} + b^{2} } \right)\sqrt {b^{2} - \left( {s - x} \right)^{2} } - 2\pi b\sqrt {(z^{2} + b^{2} )^{3} } }} - \frac{{2\sqrt {b^{2} - \left( {s - x} \right)^{2} } }}{{\pi b}} \hfill \\ \end{gathered} \right]} \right|_{{ - b}}^{b} \\ \end{aligned}$$

(A7)

$$\begin{aligned} \sigma _{{zz}}^{{cy}} = & \left. {\frac{{2F}}{{\pi bL}}\left[ { - \frac{b}{{\pi \sqrt {z^{2} + b^{2} } }}{\text{tan}}^{{ - 1}} \left( {\frac{{\sqrt {z^{2} + b^{2} } \left( {s - x} \right)}}{{bz\sqrt {1 - \frac{{\left( {x - s} \right)^{2} }}{{b^{2} }}} }}} \right) - \frac{{z(s - x)}}{{\pi \sqrt {1 - \frac{{\left( {x - s} \right)^{2} }}{{b^{2} }}} (\frac{{\left( {z^{2} + b^{2} } \right)\left( {s - x} \right)^{2} }}{{b^{2} \left( {1 - \frac{{\left( {x - s} \right)^{2} }}{{b^{2} }}} \right)}} + z^{2} )}}} \right]} \right|_{{ - b}}^{b} \\ & + \left. {\frac{{2\mu F}}{{\pi bL}}\left[ {\frac{{z^{2} \left( {z^{2} + b^{2} } \right){\text{ln}}\left( {\frac{{\sqrt {b^{2} - \left( {s - x} \right)^{2} } + \sqrt {z^{2} + b^{2} } }}{{\left| {\sqrt {b^{2} - \left( {s - x} \right)^{2} } - \sqrt {z^{2} + b^{2} } } \right|}}} \right)}}{{2\pi b\sqrt {z^{6} + 3b^{2} z^{4} + 3b^{4} z^{2} + b^{6} } }} + \frac{{z^{2} ln\left( {\frac{{\left| {\sqrt {b^{2} - \left( {s - x} \right)^{2} } - \sqrt {z^{2} + b^{2} } } \right|}}{{\sqrt {b^{2} - \left( {s - x} \right)^{2} } + \sqrt {z^{2} + b^{2} } }}} \right)}}{{\pi b\sqrt {z^{2} + b^{2} } }} - \frac{{z^{2} }}{{2\pi b\left( {\sqrt {b^{2} - \left( {s - x} \right)^{2} } + \sqrt {z^{2} + b^{2} } } \right)}} - \frac{{z^{2} }}{{2\pi b(\sqrt {b^{2} - \left( {s - x} \right)^{2} } - \sqrt {z^{2} + b^{2} } )}}} \right]} \right|_{{ - b}}^{b} \\ \end{aligned}$$

(A8)

$$\begin{aligned} \sigma _{{xz}}^{{cy}} (x,z) = & \left. {\frac{{2F}}{{\pi bL}}\left[ { - \frac{{\sqrt {z^{2} + b^{2} } }}{{\pi b}}{\text{tan}}^{{ - 1}} \left( {\frac{{\sqrt {z^{2} + b^{2} } \left( {s - x} \right)}}{{bz\sqrt {1 - \frac{{\left( {x - s} \right)^{2} }}{{b^{2} }}} }}} \right) + \frac{z}{{\pi b}}{\text{sin}}^{{ - 1}} \left( {\frac{{s - x}}{b}} \right)} \right]} \right|_{{ - b}}^{b} \\ & + \left. {\frac{{2\mu F}}{{\pi bL}}\left[ {\frac{{ - 2z}}{{\pi b}}{\text{sin}}^{{ - 1}} \left( {\frac{{s - x}}{b}} \right) + \frac{{2z^{2} + b^{2} }}{{\pi b\sqrt {z^{2} + b^{2} } }}{\text{tan}}^{{ - 1}} \left( {\frac{{\sqrt {z^{2} + b^{2} } \left( {s - x} \right)}}{{bz\sqrt {1 - \frac{{\left( {x - s} \right)^{2} }}{{b^{2} }}} }}} \right) - \frac{{z(s - x)}}{{\pi \sqrt {1 - \frac{{\left( {x - s} \right)^{2} }}{{b^{2} }}} (\frac{{\left( {z^{2} + b^{2} } \right)\left( {s - x} \right)^{2} }}{{b^{2} \left( {1 - \frac{{\left( {x - s} \right)^{2} }}{{b^{2} }}} \right)}} + z^{2} )}}} \right]} \right|_{{ - b}}^{b} \\ \end{aligned}$$

(A9)

1.2 Spherical Contact

A general schematic of contact of two elastic bodies is illustrated in Fig. 15 in Appendix. The radius of the spherical contact area resulting in a semi-elliptic pressure distribution which is formed due to the applied loading is calculated from the theory of elasticity as [24]:

Fig. 15

Spherical contact

$$a=\sqrt[3]{\frac{3F\left[\frac{1-{\upsilon }_{1}^{2}}{{E}_{1}}+\frac{1-{\upsilon }_{2}^{2}}{{E}_{2}}\right]}{4(\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}})}}$$

(A10)

where E₁ and E₂ are moduli of elasticity for spheres 1 and 2 and ν₁ and ν₂ are the Poisson’s ratios.

For the contact with a sphere on the flat plate, the flat plate is considered as a sphere with an infinitely large radius (R₁ = ∞).

On the surface where z = 0, the expression for pressure components reduces to the following.

$${P}_{zz}=\frac{3F}{2\pi {a}^{3}}\sqrt{{a}^{2}-{r}^{2}}$$

(A11)

$${P}_{xz}=\frac{3\mu F}{2\pi {a}^{3}}\sqrt{{a}^{2}-{r}^{2}}$$

(A12)

$$r=\sqrt{{x}^{2}+{y}^{2}} <a$$

(A13)

where μ is the friction coefficient, F is the applied load, and a is the contact radius. The maximum contact pressure ${p}_{0}=\frac{3F}{2\pi {a}^{2}}$ occurs at the center of circular contact r = 0. Figure 16 in Appendix shows a schematic view of the pressure distribution.

Fig. 16

Schematic view of pressure distribution in spherical contact

As a result of loading, subsurface stresses are created inside the contacting bodies. Subsurface stresses for spherical contact of isotropic bodies for normal loading are given by [16]:

$${\sigma }_{xx}=\frac{3F}{2\pi {a}^{3}}\left[2(1+\upsilon )zarcsin\left(\frac{{J}_{1}}{r}\right)-\left(1+2\upsilon \right)\sqrt{{a}^{2}-{{J}_{1}}^{2}}-\frac{za\sqrt{{{J}_{2}}^{2}-{a}^{2}}}{{{J}_{2}}^{2}-{{J}_{1}}^{2}}\right]$$

(A14)

$${\sigma }_{zz}=-\frac{3F}{2\pi {a}^{3}}\left[\frac{\sqrt[3]{{(a}^{2}-{{{J}_{1}}^{2})}^{2}}}{{{J}_{2}}^{2}-{{J}_{1}}^{2}}\right]$$

(A15)

$${\sigma }_{xz}=\left|\frac{{\sigma }_{zz}-{\sigma }_{xx}}{2}\right|$$

(A16)

where parameters of J₁ and J₃ are denoted as:

$${J}_{1}=\frac{1}{2}\left[\sqrt{{\left(r+a\right)}^{2}+{z}^{2}}-\sqrt{{\left(r-a\right)}^{2}+{z}^{2}}\right]$$

(A17)

$${J}_{2}=\frac{1}{2}\left[\sqrt{{(r+a)}^{2}+{z}^{2}}+\sqrt{{(r-a)}^{2}+{z}^{2}}\right]$$

(A18)

Subsurface stresses for spherical contact of isotropic bodies for shear loading are given by [16]:

$${\sigma{^\prime}}_{xx}=\frac{3F}{2\pi {a}^{3}}(\mu cos\theta )r\left[\left(1+\upsilon \right)\left(-\mathrm{arcsin}\left(\frac{{J}_{1}}{r}\right)+\frac{{J}_{1}}{{r}^{2}}\sqrt{{r}^{2}-{{J}_{1}}^{2}}\right)+z{{J}_{1}}^{2}\frac{\sqrt{{a}^{2}-{{J}_{1}}^{2}}}{{r}^{2}({{J}_{2}}^{2}-{{J}_{1}}^{2})}\right]$$

(A19)

$${\sigma {^\prime}}_{zz}=-\frac{3F}{2\pi {a}^{3}}(\mu cos\theta )\left[z{a}^{2}{J}_{1}\frac{\sqrt{{{J}_{2}}^{2}-{r}^{2}}}{{{J}_{2}}^{2}({{J}_{2}}^{2}-{{J}_{1}}^{2})}\right]$$

(A20)

$${\sigma {^\prime}}_{xz}=\frac{3F}{4\pi {a}^{3}}\mu \left[3\mathrm{zarcsin}\left(\frac{{J}_{1}}{r}\right)-2\sqrt{{a}^{2}-{{J}_{1}}^{2}}-z{a}^{2}\frac{\sqrt{{r}^{2}-{{J}_{1}}^{2}}}{{{J}_{1}}^{2}\left({{J}_{2}}^{2}-{{J}_{1}}^{2}\right)}-{e}^{-2i\theta }z{{J}_{1}}^{3}\frac{\sqrt{{r}^{2}-{{J}_{1}}^{2}}}{{r}^{2}\left({{J}_{2}}^{2}-{{J}_{1}}^{2}\right)}\right]$$

(A21)

where θ is the polar angle. In the case when a normal, as well as shear loading, is applied, the subsurface stresses are calculated as:

$${\sigma }_{xx}^{sp}={\sigma }_{xx}+{{\sigma }^{^{\prime}}}_{xx}$$

(A22)

$${\sigma }_{zz}^{sp}={\sigma }_{zz}+{{\sigma }^{^{\prime}}}_{zz}$$

(A23)

$${\sigma }_{xz}^{sp}={\sigma }_{xz}+{\sigma{^{\prime}}}_{xz}$$

(A24)