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Elastic–Plastic Sinusoidal Waviness Contact Under Combined Normal and Tangential Loading

Abstract

The behavior of an elastic plastic contact between a deformable three-dimensional sinusoidal asperity and a rigid flat under combined normal and tangential loading is investigated using the finite element method. The sliding inception is determined by the maximum shear stress criterion. The resulting junction growth and static friction coefficient are investigated. It is found that for a general case, at the low dimensionless contact pressures, the static friction coefficient decreases sharply with increasing contact pressure. However, at the medium contact pressures, the static friction coefficient nearly approaches a constant value (around 0.23). Nevertheless, as the contact pressure further increases, the static friction coefficient keeps on reducing at a linear rate. The effects of material properties, geometric properties and critical shear strength on the static friction coefficient are also studied. An empirical expression for the static friction coefficient is provided.

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Abbreviations

A :

Area of contact

C :

Critical yield stress coefficient

E :

Elastic modulus

E′:

\(E/\left( {1 - \nu^{2} } \right)\)

f :

Spatial frequency (reciprocal of wavelength)

\(F_{\text{c}}\) :

Critical force (full stick condition)

\(F_{\text{cs}}\) :

Critical force (perfect slip condition)

\(F_{\text{n}}\) :

Normal preload

\(F_{\text{t}}\) :

Tangential load

h :

Height of sinusoidal surface from base

\(p^{*}\) :

Average pressure to cause complete contact (elastic)

\(p_{\text{ep}}^{*}\) :

Average pressure to cause complete contact (elastic–plastic)

\(\bar{p}\) :

Average pressure over the entire surface

\(S_{\text{y}}\) :

Yield strength

\(U_{x}\) :

Displacement in the x direction

\(\lambda\) :

Asperity wavelength

\(\varDelta\) :

Asperity amplitude

\(\varDelta_{\text{c}}\) :

Critical asperity amplitude

\(\psi\) :

Sinusoidal asperity parameter

\(\nu\) :

Poisson’s ratio

\(\tau_{\text{c}}\) :

Critical shear strength

\(\omega_{0}\) :

Interference under normal preload

\(\omega_{\text{c}}\) :

Critical interference (full stick condition)

\(\omega_{\text{cs}}\) :

Critical interference (perfect slip condition)

\(\mu_{\text{s}}\) :

Effective static friction coefficient

c:

Critical value at the onset of plastic deformation (full stick condition)

cs:

Critical value at the onset of plastic deformation (perfect slip condition)

ave:

Average value

max:

Maximum value

ep:

Elastic–plastic

JGH:

From model by Johnson, Greenwood and Higgson [1]

x :

In the x direction

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Correspondence to Robert L. Jackson.

Appendix

Appendix

The critical values for sinusoidal contact under normal load in the perfect slip condition are given in [33]:

$$\omega_{\text{cs}} = \left( {\frac{{\pi CS_{\text{y}} }}{2\pi f}} \right)^{2} \frac{1}{\varDelta }$$
(22)
$$F_{\text{cs}} = \frac{1}{6\pi }\left( {\frac{1}{{\varDelta f^{2} E^{\prime } }}} \right)^{2} \left( {\frac{C}{2}S_{\text{y}} } \right)^{3}$$
(23)

where \(= 1.295 { \exp }\left( {0.736 \nu } \right)\), \(\omega_{\text{cs}}\) is the critical interference under perfect slip condition and \(F_{\text{cs}}\) is the critical force under the perfect slip condition.

The critical values for a sinusoidal surface in normal load under the full stick condition are given by [36] as

$$\omega_{\text{c}} = \omega_{\text{cs}} \left( {6.82\upsilon - 7.83\left( {\nu^{2} + 0.0586} \right)} \right)$$
(24)
$$F_{c} = F_{cs} \left( {8.88\upsilon - 10.13\left( {\nu^{2} + 0.089} \right)} \right)$$
(25)

where \(\omega_{\text{c}}\) is the critical interference under full stick condition, and \(F_{\text{c}}\) is the critical force under the full stick condition.

The term \(\frac{{\bar{p}}}{{p_{\text{ep}}^{*} }}\) in Eq. (20) can then be expressed as:

$$\frac{{\bar{p}}}{{p_{\text{ep}}^{*} }} = \frac{{\bar{p} A_{n} }}{{p_{\text{ep}}^{*} A_{n} }} = \frac{{F_{\text{n}} }}{{\left( {p_{\text{ep}}^{*} A_{n} /F_{\text{c}} } \right) F_{\text{c}} }} = \left( {\frac{{F_{\text{c}} }}{{p_{\text{ep}}^{*} A_{n} }}} \right)\frac{{F_{\text{n}} }}{{ F_{\text{c}} }}$$
(26)

where \(p_{\text{ep}}^{*}\) is the contact pressure to cause complete contact for elastic–plastic case.

By substituting Eq. (26) into Eq. (20), then Eq. (20) becomes:

$$\mu_{\text{s}} = \left[ {1.848 \coth \left( {6.5 \varphi^{2/3} \left( {\left( {\frac{{F_{\text{c}} }}{{p_{\text{ep}}^{*} A_{n} }}} \right)\frac{{F_{\text{n}} }}{{ F_{\text{c}} }}} \right)^{1/3} } \right) - 0.184 \varphi^{{\frac{1}{4}}} \left( {\left( {\frac{{F_{\text{c}} }}{{p_{\text{ep}}^{*} A_{n} }}} \right)\frac{{F_{\text{n}} }}{{ F_{\text{c}} }}} \right)^{{\frac{1}{8}}} - 1.482} \right] \left[ {\left( {\frac{{\tau_{\text{c}} }}{{S_{\text{y}} }}} \right)^{2} + 2\frac{{\tau_{\text{c}} }}{{S_{\text{y}} }}} \right]$$
(27)

This equation is alternative version that is a function of \(\frac{{F_{\text{n}} }}{{ F_{\text{c}} }}\).

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Wang, X., Xu, Y. & Jackson, R.L. Elastic–Plastic Sinusoidal Waviness Contact Under Combined Normal and Tangential Loading. Tribol Lett 65, 45 (2017). https://doi.org/10.1007/s11249-017-0827-7

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Keywords

  • Contact mechanics
  • Sinusoidal asperity
  • Elastic–plastic
  • Sliding
  • Static friction