Modeling of Particle Emission During Dry Orthogonal Cutting

Abstract

Because of the risks associated with exposure to metallic particles, efforts are being put into controlling and reducing them during the metal working process. Recent studies by the authors involved in this project have presented the effects of cutting speeds, workpiece material, and tool geometry on particle emission during dry machining; the authors have also proposed a new parameter, named the dust unit (D _u), for use in evaluating the quantity of particle emissions relative to the quantity of chips produced during a machining operation. In this study, a model for predicting the particle emission (dust unit) during orthogonal turning is proposed. This model, which is based on the energy approach combined with the microfriction and the plastic deformation of the material, takes into account the tool geometry, the properties of the worked material, the cutting conditions, and the chip segmentation. The model is validated using experimental results obtained during the orthogonal turning of 6061-T6 aluminum alloy, AISI 1018, AISI 4140 steels, and grey cast iron. A good agreement was found with experimental results. This model can help in designing strategies for reducing particle emission during machining processes, at the source.

Appendices

Appendix A: The proposed particle emission model solution algorithm

The proposed particle emission model D _u (Eq 9) is a function of cutting conditions, work material properties, and tool geometry. It also includes the shear stress, the deformation, the shear force, and the variation of shear stress as a function of the temperature, all being variables which are not easy to determine. Zaghbani and Songmenehave proposed a predictive force temperature model and a solution algorithm for the high speed milling of ductile materials (Ref 43). This oblique cutting model is transformed into a predictive model and solution algorithm for orthogonal cutting. The variables are described, and then used to obtain the final equation for particle emission (Eq 9).

The analytical expressions for the shear strain and shear strain rate (Eq 15 and 17) in the primary shear zone are obtained from the modified Oxley shear plan theory developed by Tounsi (Ref 49).

Thermal and Mechanical Properties

• ρ Workpiece density in kg/m³

• K _p Workpiece thermal conductivity in W/m

• C _p Workpiece specific heat in J/kg K

• K_t Tool thermal conductivity in W/m

• C_t Tool specific heat in J/kg K

• T₀Room temperature in K

• T_m Material melting temperature in K

Tool Geometry

• α Rake angle

Cutting Parameters

• f Feed in mm/rev

• b Width of cut in mm

• V_c Cutting velocity in m/s

Needleman-Lemonds Constitutive Equation

• $m_{1} ,\,m_{2}, \,{\text{and}}\,n$

• $\dot{\upgamma }_{t} \,,\dot{\upgamma }_{0} \,,\upgamma_{0} ,\,{\text{and}}\,\upalpha_{\text{NL}}$

Variables

• A Zvoykin constant for the shear angle

• R_a Average roughness of the tool rake face

• V₀ Reference cutting velocity in m/s

• β_max Maximum segmentation coefficient

• β_c Segmentation coefficient

• E_A Particle activation energy

Algorithm

Calculate the shear angle ϕ using Zvorikyn formulae:

$$ \upphi = A + {\frac{\upalpha - \uplambda }{2}} $$

(11)

Calculate the contact length C _l

$$ C_{\text{l}} = {\frac{{h\,{ \sin }\,\uptheta }}{{{ \sin }\,\upphi \,\cos (\uptheta + \upalpha - \upphi )}}} $$

(12)

Calculate the ratio C _h

$$ C_{\text{h}} = {\frac{{{ \sin }\,\upphi }}{{{ \cos }(\upphi - \upalpha )}}} $$

(13)

Calculate the segmentation density η_S

$$ \upeta_{\text{S}} = {\frac{1}{{C_{\text{l}} }}} $$

(14)

Calculate the average shear strain in the Primary Shear Zone

$$ \bar{\upgamma }_{\text{AB}} = {\frac{{{ \cos }\,\upalpha }}{{{ \cos }(\upphi - \upalpha ){ \sin }\,\upphi }}} $$

(15)

Calculate the coefficient B ₀

$$ B_{0} = \sqrt {{\frac{{K_{p} \upgamma_{\text{AB}} }}{Vf}}} $$

(16)

Calculate the shear rate in the Primary Shear Zone

$$ \dot{\bar{\upgamma }}_{\text{AB}} = {\frac{{2V\,{ \cos }\,\upalpha }}{{e_{\text{psz}} \,{ \cos }(\upphi - \upalpha )}}} $$

(17)

If $\dot{\bar{\upgamma }}_{\text{AB}} \le \dot{\bar{\upgamma }}_{\text{t}}$, then

Solve this equation:

$$ \bar{\uptau }_{\text{AB}} = \uptau_{0} \left( {1 + {\frac{{\bar{\upgamma }_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{n}}}} \left( {1 + {\frac{{\bar{\upgamma }_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{{m_{1} }}}}} \left[ {1 - {\frac{{\uptau_{\text{AB}} \upgamma_{\text{AB}} }}{{\uprho_{p} C_{p} }}}{\frac{0.9}{{1 + 1.329B_{0} }}}} \right] $$

(18)

Get shear stress in the Primary Shear Zone τ_AB

Else solve this equation:

$$ \bar{\uptau }_{\text{AB}} = \uptau_{0} \left( {1 + {\frac{{\bar{\upgamma }_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{n}}}} \left( {1 + {\frac{{\bar{\upgamma }_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{{m_{1} }}}}} \left[ {1 - {\frac{{\uptau_{\text{AB}} \upgamma_{\text{AB}} }}{{\uprho_{p} C_{p} }}}{\frac{0.9}{{1 + 1.329B_{0} }}}} \right]\left( {1 - {\frac{{\bar{\upgamma }_{\text{t}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{{m_{1} }}} - {\frac{1}{{m_{2} }}}}} $$

(19)

Get the shear stress in the Primary Shear Zone τ_AB

Calculate the temperature in the Primary Shear Zone T _AB

$$ T_{\text{AB}} = T_{0} + {\frac{{\uptau_{\text{AB}} \upgamma_{\text{AB}} }}{{\uprho_{p} C_{p} }}}{\frac{0.9}{{1 + 1.329B_{0} }}} $$

(20)

Calculate the shearing force F _sh

$$ F_{\text{sh}} = {\frac{{\uptau_{\text{AB}} fb}}{{{ \sin }\,\upphi }}} $$

(21)

Calculate the chip segmentation coefficient β using Xie formulae (Ref 42):

$$ \upbeta = - {\frac{\sqrt 3 }{m}}\left[ {\upmu + {\frac{0.9(\partial \uptau /\partial T)}{{\uprho_{p} C_{p} (1 + 1.328B_{0} )}}}} \right]\left[ {\upmu \upgamma + 1 - {\frac{{0.664B_{0} }}{{1 + 1.328B_{0} }}}} \right] $$

(22)

Calculate the dust unit D _u using Eq 9.

Appendix B: The Flow Localization Parameter β

The proposed flow localization parameter β of the model (Eq 4) developed by Xie et al. (Ref 42) is a function of the cutting conditions, work material properties, and tool geometry.

Thermal and Mechanical Properties

• μ Strain hardening parameter

• m Strain rate sensitivity

Algorithm

Calculate the average shear strain in the Primary Shear Zone (Eq 15)

Calculate the coefficient B ₀ (Eq 16)

Calculate the shear rate in the Primary Shear Zone (Eq 17)

Calculate the Needleman-Lemonds constitutive equation:

$$ \left\{ {\begin{array}{*{20}c} \begin{aligned} \left( {1 + {\frac{{\dot{\bar{\upgamma }}_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right) = \left( {{\frac{{\bar{\uptau }_{\text{AB}} }}{{g(\bar{\uptau }_{\text{AB}} )}}}} \right)^{{m_{1} }} \to \dot{\bar{\upgamma }}_{\text{AB}} \le \dot{\bar{\upgamma }}_{\text{t}} \hfill \\ \left( {1 + {\frac{{\dot{\bar{\upgamma }}_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)\left( {1 + {\frac{{\dot{\bar{\upgamma }}_{\text{t}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{{m_{2} }}{{m_{1} }}} - 1}} = \left( {{\frac{{\bar{\uptau }_{\text{AB}} }}{{g(\bar{\uptau }_{\text{AB}} )}}}} \right)^{{m_{2} }} \to \dot{\bar{\upgamma }}_{\text{AB}} \ge \dot{\bar{\upgamma }}_{\text{t}} \hfill \\ \end{aligned} \hfill \\ {g(\bar{\uptau }_{\text{AB}} ) = \uptau_{0} \left( {1 + {\frac{{\bar{\upgamma }_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{n}}}} \left[ {1 - \upalpha_{\text{NL}} \left( {T - T_{0} } \right)} \right]} \hfill \\ \end{array} } \right. $$

(23)

where n is the hardening coefficient, α_NL is the coefficient for thermal softening, τ₀ is the elastic average shear stress, $\bar{\uptau }$ is the average shear stress, $\dot{\bar{\upgamma }}_{\text{t}}$ is the transition shear strain rate, m ₁ and m ₂ are the coefficients of sensitivity to the strain rate in the low and high regimes, respectively, and $\dot{\bar{\upgamma }}_{0}$ is the reference shear strain rate.

The shear stress in the Primary Shear Zone will be

$$ \bar{\uptau}_{\text{AB}} = \left\{ {\begin{array}{*{20}c} {\uptau_{0} \left( {1 + {\frac{{\bar{\upgamma }_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{n}}}} \left( {1 + {\frac{{\dot{\bar{\upgamma }}_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{{m_{1} }}}}} \left[ {1 - \upalpha_{\text{NL}} \left( {T - T_{0} } \right)} \right] \to \dot{\bar{\upgamma }}_{\text{AB}} \le \dot{\bar{\upgamma }}_{\text{t}} } \hfill \\ {\uptau_{0} \left( {1 + {\frac{{\bar{\upgamma }_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{n}}}} \left( {1 + {\frac{{\dot{\bar{\upgamma }}_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{{m_{2} }}}}} \left( {1 + {\frac{{\dot{\bar{\upgamma }}_{\text{t}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{{m_{1} }}} - {\frac{1}{{m_{2} }}}}} \left[ {1 - \upalpha_{\text{NL}} \left( {T - T_{0} } \right)} \right] \to \dot{\bar{\upgamma }}_{\text{AB}} \ge \dot{\bar{\upgamma }}_{\text{t}} } \hfill \\ \end{array} } \right. $$

(24)

If $\dot{\bar{\upgamma }}_{\text{AB}} \le \dot{\bar{\upgamma }}_{\text{t}}$, then

Calculate the variation of shear stress in the Primary Shear Zone as

$$ {\frac{{\partial{\bar{\uptau}}_{\text{AB}} }}{\partial T}} = \uptau_{0} \left( {1 + {\frac{{\bar{\upgamma }_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{n}}}} \left( {1 + {\frac{{\dot{\bar{\upgamma }}_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{{m_{1} }}}}} \left[ { - \upalpha_{\text{NL}} } \right] $$

(25)

Get the shear stress in the Primary Shear Zone $\uptau_{\text{AB}}$

Else Calculate the variation of shear stress in the Primary Shear Zone as

$$ {\frac{{\partial\bar{\uptau}_{\text{AB}} }}{\partial T}} = \uptau_{0} \left( {1 + {\frac{{\bar{\upgamma }_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{n}}}} \left( {1 + {\frac{{\dot{\bar{\upgamma }}_{\text{AB}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{{\frac{1}{{m_{2} }}}}} \left( {1 + {\frac{{\dot{\bar{\upgamma }}_{\text{t}} }}{{\bar{\upgamma }_{0} }}}} \right)^{{\left( {{\frac{1}{{m_{1} }}} - {\frac{1}{{m_{2} }}}} \right)}} \left[ { - \upalpha_{\text{NL}} } \right] $$

(26)

Get the shear stress in the Primary Shear Zone $\uptau_{\text{AB}}$

Calculate the chip segmentation coefficient β using Xie (Ref 42) formulae (Eq 22).