Modeling the Supporting Surface Curve by a Linear Function in Stiffness Calculations
Abstract Generated abstract
This paper examines the use of a linear approximation for the bearing surface curve in stiffness calculations for rough surface contact, instead of the more common power law relation between relative contact area and relative approach. The authors derive expressions for load, relative approach, and actual contact area for elastic contact and for plastic contact with hardening, using asperity contact relations and integration over contacting asperities. A comparison with calculations based on the power law model indicates that the linear model gives somewhat smaller approach values for ground, polished, milled, and planed surfaces, although the discrepancy may be reduced by adjusting the linear coefficient.
Full Text
Modeling the Curve of the Supporting Surface by a Linear Function in Stiffness Calculations
G. M. Kharach, V. I. Maksak
Presented by the scientific seminar of the Department of Strength of Materials
When determining the deformation of a rough surface, a power-law dependence of the relative contact area \(\eta\) on the relative approach \(\varepsilon\) is usually used:
\[ \eta = A_r / A_c = b \varepsilon^\nu, \tag{1} \]
where
\(A_r\) is the actual contact area,
\(A_c\) is the contour area,
\(b, \nu\) are constants.
However, with a linear approximation of the supporting curve, the calculations are represented in a simpler form, and for some cases this is physically justified.
\[ \eta = b \varepsilon . \tag{2} \]
To calculate the actual contact area, as in [1], one can use the formula
\[ N = \int_0^{n_r} N_i \, dn_r . \tag{3} \]
For this purpose, we represent dependence (1) in the form
\[ \frac{\Delta \bar A_r n_r}{A_c} = b \varepsilon, \tag{4} \]
where
\(\Delta \bar A_r\) is the average size of a unit contact spot,
\(n_r\) is the number of contacting asperities.
Elastic Contact
For elastic contact of a sphere with an absolutely rigid surface, according to the Hertz formula,
\[ \Delta A_{ri} = \pi r h_{\max} \varepsilon_i, \tag{5} \]
here
\(r\) is the radius of asperities of the rough surface,
\(h_{\max}\) is the maximum height of irregularities.
In this connection it is necessary to note that, under elastic deformation,
\[ \Delta A_{ri}\simeq 0.5\,\Delta \overline{A}_r,\quad \text{i.e.} \]
\[ \Delta A_{ri}=A_c b\varepsilon/2n_r. \tag{6} \]
Taking (5) and (6) into account, we obtain
\[ n_r=A_c b\varepsilon/2\pi r h_{\max}\varepsilon_i \]
or
\[ n_r=A_c bx/2\pi r h_{\max}(\varepsilon-x). \tag{7} \]
Hence
\[ dn_r=\frac{A_c b\varepsilon}{2\pi r h_{\max}}\cdot \frac{dx}{(\varepsilon-x)^2}. \tag{8} \]
Substituting (8) into (3), with allowance for
\[ N_i=1.33\,\frac{r^{1/2}h_{\max}^{3/2}E}{1-\mu^2}\,\varepsilon_i^{3/2}, \tag{9} \]
we obtain
\[ N=\frac{1.33\,A_c b h_{\max}^{1/2}E\varepsilon^{3/2}} {\pi(1-\mu^2)r^{1/2}}. \tag{10} \]
Here \(\mu\) is Poisson’s ratio,
\(E\) is the modulus of elasticity.
The relative approach \(\varepsilon\) can be determined from equality (10):
\[ \varepsilon= \left[ \frac{0.75\,\pi(1-\mu^2)r^{1/2}N} {b h_{\max}^{1/2} E A_c} \right]^{2/3}. \tag{11} \]
Substituting (11) into (2), we obtain the actual contact area
\[ A_r= \left[ \frac{0.75\,\pi(1-\mu^2)r^{1/2}b^{1/2}A_c^{1/2}N} {h_{\max}^{1/2}E} \right]^{2/3} \tag{12} \]
or
\[ A_r= \left(\frac{rbA_c}{h_{\max}}\right)^{1/3} \left[ \frac{0.75\,\pi(1-\mu^2)N}{E} \right]^{2/3}. \tag{13} \]
Plastic contact in the presence of hardening
In the case under consideration, the force per unit asperity as a function of approach is determined according to [1] by
\[ N_i=2^{\vartheta/2}\pi H_y h_{\max}^{\vartheta/2}r^{\,2-\vartheta/2}\varepsilon_i^{\vartheta/2}, \tag{14} \]
where \(\vartheta\) is a coefficient characterizing the properties of the material,
\(H_y\) is the Meyer hardness.
After substituting into (3) the values (14) and
\[ dn_r=\frac{A_c b\varepsilon}{2\pi r h_{\max}}\cdot \frac{dx}{(\varepsilon-x)^2}, \]
we obtain
\[ N=\frac{2^\omega H_y h_{\max}^{\omega}A_c b\varepsilon^{\omega+1}} {\omega r^\omega}. \tag{15} \]
Hence we obtain the relative approach
\[ \varepsilon= \left( \frac{\omega r^\omega N} {2^\omega b h_{\max}^{\omega}A_cH_y} \right)^{\frac{1}{1+\omega}}. \tag{16} \]
and the actual contact area
\[ A_r=\left(\frac{\omega}{H_y}\right)^{\frac{1}{1+\omega}} \left(\frac{r b A_c}{2h_{\max}}\right)^{\frac{\omega}{1+\omega}} . \tag{17} \]
Here
\[ \omega=\frac{\vartheta}{2}-1 . \]
Comparison of calculations obtained when modeling the curve of the supporting surface by linear and power-law (1) functions
For the supporting surface (1), the relative approach is determined by the formula
\[ \varepsilon_y= \left[ \frac{1.5\pi(1-\mu^2)r^{1/2}N} {b_y K_2 h_{\max}^{1/2} A_c E} \right]^{\frac{2}{2\nu+1}} . \tag{18} \]
The ratio of (11) and (18) has the form
\[ \frac{\varepsilon}{\varepsilon_y} = \left(\frac{0.75}{b}\right)^{2/3} \left(\frac{b_y K_2}{1.5}\right)^{\frac{2}{2\nu+1}} . \tag{19} \]
For ground surfaces one may take [1]
\[ b=1.6\div 2.0;\quad b_y=4\div 6;\quad \nu=3;\quad K_2=0.68, \quad \text{then}\quad \varepsilon/\varepsilon_y=0.7\div 0.75 . \]
For polished surfaces, with \(b=2.8;\ b_y=5\div 10;\ \nu=3;\ \kappa_2=0.65\),
\[ \varepsilon/\varepsilon_y=0.6\div 0.65 . \]
For milled and planed surfaces, with \(b=1\div 1.4\), \(b_y=1\div 4;\ \nu=2;\ \kappa_2=0.8\),
\[ \varepsilon/\varepsilon_y=0.65\div 0.91 . \]
The quantities \(b_y\), \(\nu\), and \(\kappa_2\) were taken from the tables in [1], while the quantity \(b\) was determined from the longest segment of the supporting curve with a constant intensity of its variation.
In conclusion, it should be noted that, when the curve of the supporting surface is modeled by a linear function, the calculated value of the approach proves to be somewhat smaller than with the usual modeling by a power-law function. The discrepancy in the calculations can be reduced by correcting the value of \(b\).
References
- N. B. Demkin. The Actual Contact Area of Hard Surfaces. Publishing House of the Academy of Sciences of the USSR, 1962.