Relationship between subsurface damage and surface roughness of glass BK7 in rotary ultrasonic machining and conventional grinding processes

Abstract

Subsurface damage (SSD) induced during the abrasive machining process considerably influences the technological application of the optical components. However, to date, there is no rapid and effective method to inspect the depth of SSD. For the purpose of precise and nondestructive evaluation of the SSD depth generated in rotary ultrasonic machining (RUM) and conventional grinding (CG) processes, a theoretical model, ground on indentation fracture mechanics of brittle material, was proposed by analyzing the correlation between the median and lateral crack systems aroused by a sharp indenter. It was found that the SSD depth was nonlinear monotone increasing with square of surface roughness (SR), namely, SSD = χSR² + l. Utilizing this model, the SSD depth could be quickly and precisely predicted through the SR (p–v value.) of the machined surface, geometrical features of the abrasive, and the material mechanical properties. To validate the feasibility of this method, both RUM and CG tests were conducted on the BK7 glass specimens with a Sauer Ultrasonic 20. Subsequently, the SSD of these specimens was exposed with the polishing–etching technique. The measurement results of SSD depth were consistent with the prediction values of this model, which reflected the feasibility of using this model to rapidly and accurately predict the SSD depth.

Appendix

Solutions for the Boussinesq stress field due to a normal concentrated load P in spherical coordinates (r,θ,φ) of an elastic half-space are shown in Fig. 8. The stress components were:

$$ \left\{ {\begin{array}{*{20}{c}} {{{\sigma }_{{rr}}}=\frac{P}{{2\pi {{r}^{2}}}}\left( {\frac{{1-2\nu }}{{\left( {1+\cos \varphi } \right)}}-3{{{\sin }}^{2}}\varphi \cos \varphi } \right)} \\ {{{\sigma }_{{\theta \theta }}}=\frac{P}{{2\pi {{r}^{2}}}}\left( {1-2\nu } \right)\left( {\cos \varphi -\frac{1}{{1+\cos \varphi }}} \right)} \\ {{{\sigma }_{{zz}}}=-\frac{P}{{2\pi {{r}^{2}}}}\cdot 3{{{\cos }}^{3}}\varphi } \\ {{{\tau }_{{rz}}}=-\frac{P}{{2\pi {{r}^{2}}}}\cdot 3{{{\cos }}^{2}}\varphi \sin \varphi } \\ {{{\tau }_{{r\theta }}}={{\tau }_{{z\theta }}}=0} \\ \end{array}.} \right. $$

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The magnitudes and directions of the principal stress components could be provided by suitable tensor transformations. The three principal normal stresses could be written as:

$$ \left\{ \matrix{ {\sigma_{{11}}} = \frac{{{\sigma_{{rr}}} + {\sigma_{{zz}}}}}{2} + \sqrt {{{{\left( {\frac{{{\sigma_{{rr}}} - {\sigma_{{zz}}}}}{2}} \right)}^2} + \tau_{{rz}}^2}} \hfill \\ {\sigma_{{22}}} = {\sigma_{{\theta \theta }}} \hfill \\ {\sigma_{{33}}} = \frac{{{\sigma_{{rr}}} + {\sigma_{{zz}}}}}{2} - \sqrt {{{{\left( {\frac{{{\sigma_{{rr}}} - {\sigma_{{zz}}}}}{2}} \right)}^2} + \tau_{{rz}}^2}} \hfill \\ }<!endgathered> \right. $$

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where σ ₂₂ is perpendicular to the symmetry plane everywhere. σ ₁₁ and σ ₃₃ are contained in the symmetry plane θ = constant. The angle between the specimen surface and σ ₁₁ and σ ₃₃ was given by

$$ \tan 2\alpha = \frac{{{\tau_{{rz}}}}}{{{\sigma_{{zz}}} - {\sigma_{{rr}}}}}. $$

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