Evaluation of Conformal and Non-Conformal Contact Parameters Using Digital Photoelasticity

Abstract

Experimental studies to exploit photoelastic data of conformal geometries to extract contact parameters are non-existent because closed-form stress field equations were not available until recently. In this paper, the explicit equations recently reported in the literature for a flat punch with rounded edges are generalized so that a single set of equations can be used for a flat punch with rounded edges and Hertzian contacts with arbitrary radii of curvatures. The generality of the governing equations is verified by plotting isochromatics for conformal and non-conformal contact situations. A generic method to evaluate unknown contact parameters from the whole-field isochromatic data for conformal and non-conformal geometries is implemented. The methodology is initially verified using theoretically generated isochromatic data and is then used to experimentally evaluate two contact situations. In view of high-fringe gradient zones, the suitability of various digital photoelastic methods is compared. A novel four-step phase shifting technique is proposed in which isochromatic and isoclinic data can be evaluated using the minimum number of images.

Appendix 1

Parameters u₁, u₂, u₃, u₆, v₁, and v₆ used in equations (10)–(15) are functions of x, y, a and b and are defined as

$$ {u}_1\left(x,y\right)=\frac{\sqrt{2\sqrt{c_1}+2{c}_2}}{2} $$

(A.1)

$$ {u}_2\left(x,y\right)=\frac{\ln \left({\gamma}_1\right)}{2} $$

(A.2)

$$ {u}_3\left(x,y\right)=\frac{\ln \left({\gamma}_2\right)}{2} $$

(A.3)

$$ {u}_6\left(x,y\right)=\frac{x\left({x}^2+{y}^2-{a}^2+\sqrt{c_1}\right)}{\sqrt{2{c_1}^{1.5}+2{c}_1{c}_2}} $$

(A.4)

$$ {v}_1\left(x,y\right)=-\operatorname{sgn}(x)\frac{\sqrt{2\sqrt{c_1}-2{c}_2}}{2} $$

(A.5)

$$ {v}_2\left(x,y\right)={\theta}_1-{\theta}_2 $$

(A.6)

$$ {v}_3\left(x,y\right)={\theta}_3-{\theta}_4 $$

(A.7)

$$ {v}_6\left(x,y\right)=-\frac{y\left({x}^2+{y}^2+{a}^2-\sqrt{c_1}\right)}{\sqrt{2{c_1}^{1.5}+2{c}_1{c}_2}} $$

(A.8)

The additional functions used in equations (A.1)–(A.8) are defined as

$$ {\gamma}_1=\frac{\sqrt{c_3}+{c}_5+\sqrt{2{c}_5}\sqrt{\sqrt{c_3}+{c}_4}}{\sqrt{c_3}+{c}_5-\sqrt{2{c}_5}\sqrt{\sqrt{c_3}+{c}_4}} $$

(A.9)

$$ {\gamma}_2=\frac{\sqrt{c_3}+1/{c}_5+\sqrt{2/{c}_5}\sqrt{\sqrt{c_3}+{c}_4}}{\sqrt{c_3}+1/{c}_5-\sqrt{2/{c}_5}\sqrt{\sqrt{c_3}+{c}_4}} $$

(A.10)

$$ {\theta}_1=\arcsin \left[\frac{\sqrt{2}}{2}\sqrt{\sqrt{c_3}-{c}_4},\frac{\sqrt{2}}{2}\sqrt{\sqrt{c_3}+{c}_4}+\sqrt{c_5}\right] $$

(A.11)

$$ {\theta}_2=\arcsin \left[\frac{\sqrt{2}}{2}\sqrt{\sqrt{c_3}-{c}_4},\frac{\sqrt{2}}{2}\sqrt{\sqrt{c_3}+{c}_4}-\sqrt{c_5}\right] $$

(A.12)

$$ {\theta}_3=\arcsin \left[\frac{\sqrt{2}}{2}\sqrt{\sqrt{c_3}-{c}_4},\frac{\sqrt{2}}{2}\sqrt{\sqrt{c_3}+{c}_4}+\sqrt{1/{c}_5}\right] $$

(A.13)

$$ {\theta}_4=\arcsin \left[\frac{\sqrt{2}}{2}\sqrt{\sqrt{c_3}-{c}_4},\frac{\sqrt{2}}{2}\sqrt{\sqrt{c_3}+{c}_4}-\sqrt{1/{c}_5}\right] $$

(A.14)

With the terms c_1,- c₅ defined as

$$ {c}_1=\left({\left(a+x\right)}^2+{y}^2\right)\left({\left(a-x\right)}^2+{y}^2\right) $$

(A.15)

$$ {c}_2={x}^2-{y}^2-{a}^2 $$

(A.16)

$$ {c}_3=\frac{{\left(a-x\right)}^2+{y}^2}{{\left(a+x\right)}^2+{y}^2} $$

(A.17)

$$ {c}_4=\frac{a^2-{x}^2-{y}^2}{{\left(a+x\right)}^2+{y}^2} $$

(A.18)

$$ {c}_5=\frac{a-b}{a+b} $$

(A.19)