Instantaneous Plane Stress Observation and Numerical Simulation During Wear in an Initial Line Contact

Abstract

The photoelastic technique was applied to capture in situ plane stress in a block-on-cylinder contact. This method allows researchers to qualitatively visualize the distribution of principal stress difference during wear processes. Additionally, numerical simulation of the evolution of wear was conducted. The conjugate gradient method together with fast Fourier transform was employed to solve the pressure distribution. Based on the computed pressure, the block wear was solved with Archard’s wear law. The subsurface stress was computed using the influence coefficient method. The simulated results of wear and stress were compared with the experimental results; the similarities and differences between the results highlight the usefulness and limitations of the numerical method. However, the photoelastic technique together with the numerical method casts light on the nature of wear.

Appendices

Appendix A

The surface deflection distribution produced by a given contact pressure p(x) is shown in Eq. (1). As the ‘rigid-body’ approach δ in Eq. (5) can be determined based on the load balance condition, the constant C in Eq. (1) can be merged with it. Equation (1) can be expressed as the Green function:

$$u= - \int_{{{I_g}}} {K\left( {x - x^{\prime}} \right)p\left( {x^{\prime}} \right){\text{d}}{\kern 1pt} x^{\prime}} ,$$

(A.1)

where I_g is the grid area and K(x) represents the surface displacement distribution produced by a unit concentrated normal force acting at the origin. Comparing Eq. (A.1) with (1), the kernel K(x) is given by

$$K\left( x \right)=\frac{2}{{\pi {E^*}}}\ln \left| x \right|+\frac{\mu }{{2{G^*}}}H\left( x \right),$$

(A.2)

where H(x) is the Heaviside step function. Under the assumption of stepwise constant distribution of contact pressure, the influence coefficient K_i−j is given by

$${K_{i - j}}=\int_{{{x_j} - {{\Delta x} \mathord{\left/ {\vphantom {{\Delta x} 2}} \right. \kern-0pt} 2}}}^{{{x_j}+{{\Delta x} \mathord{\left/ {\vphantom {{\Delta x} 2}} \right. \kern-0pt} 2}}} {\frac{2}{{\pi {E^*}}}\ln \left| {x^{\prime} - {x_i}} \right|+\frac{\mu }{{2{G^*}}}H\left( {x^{\prime} - {x_i}} \right){\text{d}}{\kern 1pt} x^{\prime}} .$$

(A.3)

Integrating Eq. (A.3) yields an explicit formula for K_i−j, which can be expressed as

$${K_{i - j}}=\frac{2}{{\pi {E^*}}}\left[ {f\left( {\left| {{x_i} - {x_j}} \right|+{{\Delta x} \mathord{\left/ {\vphantom {{\Delta x} 2}} \right. \kern-0pt} 2}} \right) - f\left( {\left| {{x_i} - {x_j}} \right| - {{\Delta x} \mathord{\left/ {\vphantom {{\Delta x} 2}} \right. \kern-0pt} 2}} \right)} \right]+\frac{{\mu \Delta x}}{{2{G^*}}}H\left( {{x_j} - {x_i}} \right),$$

(A.4)

where $f\left( x \right)=x\left( {\ln \left| x \right| - 1} \right)$.

Appendix B

$$\begin{gathered} N_{{i - j}}^{{rs}}\left( {{z_k}} \right)={n^{rs}}\left( {{x_i} - {x_j}+\Delta x/2,{z_k}} \right) - {n^{rs}}\left( {{x_i} - {x_j} - \Delta x/2,{z_k}} \right) \hfill \\ T_{{i - j}}^{{rs}}\left( {{z_k}} \right)={t^{rs}}\left( {{x_i} - {x_j}+\Delta x/2,{z_k}} \right) - {t^{rs}}\left( {{x_i} - {x_j} - \Delta x/2,{z_k}} \right), \hfill \\ \end{gathered}$$

(B.1)

where

$$\begin{gathered} {n^{xx}}\left( {x,z} \right)=\frac{1}{\pi }\left[ {\frac{{xz}}{{{x^2}+{z^2}}} - \arctan \left( {\frac{x}{z}} \right)} \right] \hfill \\ {n^{zz}}\left( {x,z} \right)=\frac{1}{\pi }\left[ { - \frac{{xz}}{{{x^2}+{z^2}}} - \arctan \left( {\frac{x}{z}} \right)} \right] \hfill \\ {n^{xz}}\left( {x,z} \right)=\frac{1}{\pi }\frac{{{z^2}}}{{{x^2}+{z^2}}} \hfill \\ {t^{xx}}\left( {x,z} \right)=\frac{1}{\pi }\left[ { - \frac{{{z^2}}}{{{x^2}+{z^2}}} - \ln \left( {{x^2}+{z^2}} \right)} \right] \hfill \\ {t^{zz}}\left( {x,z} \right)=\frac{1}{\pi }\frac{{{z^2}}}{{{x^2}+{z^2}}} \hfill \\ {t^{xz}}\left( {x,z} \right)=\frac{1}{\pi }\left[ {\frac{{xz}}{{{x^2}+{z^2}}} - \arctan \left( {\frac{x}{z}} \right)} \right]. \hfill \\ \end{gathered}$$