Machining process modeling of green alumina ceramics in orthogonal cutting and fragmentation process

Abstract

Theories on mechanical model and discrete element model (DEM) of machining green ceramics are sought to reveal the material removal mechanisms and cutting force. Theoretical models are based on brittle fracture mechanics, equilibrium, and minimum chipping energy principle etc., and in the model, friction on flank face of tools and material’s rebound deformation are considered. DEM is established based on macro-mechanical properties of green alumina ceramics. Orthogonal cutting experiments were conducted to validate the mechanical model and DEM. Results show that the material removal mechanism of green alumina ceramics contains both crushing failure and chipping failure. The experiments validate good agreement with mechanical model and DEM in regard to chip formation and tangential cutting force. Friction force during machining green alumina ceramics contributes more than 50% of tangential cutting force. It is the reason that there exists more serious flank wear of cutting tool.

Appendix

So the resultant F_t generated by the tensile stress is calculated as F_p in Eq. 1, i.e.,

$$ \left\{\begin{array}{c}{\overrightarrow{F}}_t=w{\int}_o^{2a}\left(\overrightarrow{i} sin\theta +\overrightarrow{j} cos\theta \right){\sigma}_t Rd\theta =w{\sigma}_tR\left[\left(\overrightarrow{j}\mathit{\sin }2a-\overrightarrow{i}\left(\mathit{\cos}2a-1\right)\right)\right]\\ {}R=\frac{\overline{BD}}{2\mathit{\sin}\ a}=\frac{a_p}{2{\mathit{\sin}}^2\ a}\\ {}\mathrm{where}\kern0.5em \overline{BD}=\frac{a_p}{\mathit{\sin}\ a}\end{array}\right. $$

(21)

where R and a are the radius and half angle of arc boundary BD of chunk-like chips respectively, $\overline{BD}$ stands for the length of line BD, and a_p means the depth of cut. So, the value of F_t is rewritten, i.e.,

$$ {\displaystyle \begin{array}{c}{F}_t=\left|{\overrightarrow{F}}_t\right|=w{\sigma}_t\frac{a_p}{2{\mathit{\sin}}^2\ a}\sqrt{{\mathit{\sin}}^2(2a)+{\left(\mathit{\cos}2a-1\right)}^2}\\ {}=w{\sigma}_t\frac{a_p}{\sin a}\end{array}} $$

(22)

According to the assumption above, the equilibrium of moments in terms of F_p and F_t at point B before the chunk-like chips detachment is calculated:

$$ \left\{\begin{array}{c}{F}_t\times \overline{BC}-{F}_p\times \overline{AB}=0\\ {}\overline{BC}=\frac{a_p}{2\sin a}\\ {}\overline{AB}=\frac{d}{2\mathit{\sin}\frac{\alpha_o+{\theta}_t}{2}}+\frac{a_p}{\sin a}\mathit{\cos}\left(\frac{\alpha_o+{\theta}_t}{2}+a\right)\end{array}\right. $$

(23)

Combining Eqs. 22–23, besides the ratio of d to depth of cut a_p is always non-negative, the ratio is given:

$$ \left\{\begin{array}{c}\frac{d}{a_p}=\frac{\frac{-2\mathit{\sin}\left(\frac{\alpha_o+{\theta}_t}{2}\right)\mathit{\cos}\left(\frac{\alpha_o+{\theta}_t}{2}+a\right)}{\mathit{\sin}\ a}+\sqrt{\Delta}}{2}\\ {}\Delta =\frac{4{\mathit{\sin}}^2\left(\frac{\alpha_o+{\theta}_t}{2}\right)}{{\mathit{\sin}}^2a}\left[{\mathit{\cos}}^2\left(\frac{\alpha_o+{\theta}_t}{2}+a\right)+\frac{\sigma_t}{\sigma_p}\right]\end{array}\right. $$

(24)

To get the ratio of d to depth of cut a_p, half angle a of arc boundary BD of chunk-like chips should be obtained. According to the minimum energy principle [28], a can be determined:

$$ \left\{\begin{array}{c}\frac{\partial (E)}{\partial a}=0\\ {}E={F}_xL\\ {}{F}_x={K}_{pi}\left|\overrightarrow{i}\bullet \overrightarrow{F_p}\right|+{F}_f={k}_{pi}w{\sigma}_p{a}_p\frac{d}{a_p}+{F}_f\end{array}\right. $$

(25)

where E is the energy of chipping, F_x and F_f are the tangential cutting force and friction force respectively during processing, and F_f is one of main parts of tangential cutting force, which is introduced on the interface between flank face of cutting tools and surfaces of green ceramics, and it will be studied in the later section including friction coefficient. L is the cutting distance. k_pi is the force factor of F_p in horizontal directions, 0 < k_pi ≤ 1, which is related with material properties. k_pi is also in response to the analysis of rebound deformation during processing in Section 2.1.