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A 3D analytical model for residual stress in flank milling process

  • S.Q. Wang
  • J.G. LiEmail author
  • C.L. HeEmail author
  • Z.Y. Xie
ORIGINAL ARTICLE
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Abstract

Residual stress in metal cutting is critical due to its significant influence on the work performance of the workpiece. A 3D analytical model of residual stress appropriate for flank milling is established in this paper which takes both mechanical effect and thermal effect into consideration. Firstly, the 3D mechanical stress component is calculated considering its instantaneous and intermitted properties in the milling process. Elastic semi-infinite space contact mechanics and coordinate transformation are employed in the calculation process. Subsequently, the temperature distribution in the workpiece and corresponding thermal stress component are acquired based on the time-varying transient moving heat source model in the milling process. The plastic stress component is calculated based on the radial return method, in which all the stress components are updated during the plastic loading process. Finally, the measurement experiments for milling temperature and residual stress are performed to validate the theoretical models proposed in this work. The fine consistency between the prediction and the experiment demonstrates the accuracy of the prediction model. According to the calculation results of the proposed theoretical model, the plowing effect induced by the squeeze of the cutting edge on the work material is the major source of the residual stress.

Keywords

Residual stress Flank milling process Radial return method Thermal-mechanical load 

Nomenclature

ae

cutting width (mm)

ap

cutting depth (mm)

As

area of shear plane

Aγ

rake face

c

specific heat capacity of work material

ct

specific heat capacity of tool material

c1, c2

coefficients of hardening model

C1

heat transfer ratio into the workpiece from the primary deformation zone

C2

heat transfer ratio into the workpiece from the third deformation zone

\( \overline{CA} \)

plow effect length (mm)

dap

length of the cutting microelement (mm)

proportional loading coefficient

\( \mathrm{d}{\varepsilon}_{ij}^{\mathrm{p}} \)

plastic strain increment

dσij

stress increment

E

Young’s modulus (GPa)

f

von-Misses yield function

fz

feed per tooth (mm)

F

cutting resultant force (N)

Fc

friction force on the rake face (N)

Fnc

normal force on the rake face (N)

Fns

normal force on the shear plane (N)

Fs

shear force on the shear plane (N)

h

undeformed chip thickness (mm)

i

cutting edge inclination (°)

J2

second invariant

k

iteration numbers of Newton-Raphson method

k0

material shear flow stress (MPa)

K

thermal diffusivity of the work material (mm2/s)

Kt

thermal diffusivity of the tool material (mm2/s)

L

is the length of the arc heat source (mm)

m

friction coefficient between the cutting tool and workpiece surface

m0

temperature softening coefficient

M

number of microelements

M0

number of stress release steps

\( \hat{n} \)

unit normal vector on the yield surface

nr

rotation speed (m/min)

Nf

number of the tool teeth

O-x0y0z0

workpiece coordinate system defined on the machined surface

O-x1y1z1

workpiece coordinate system defined on the cutting edge

O-xnynzn

normal plane coordinate system

O-xnsynszns

shear plane coordinate system

O-xpypzp

plowing coordinate system

p(t)

stress load produced by surface temperature (MPa)

Pn

normal plane

Ps

cutting plane

Pshear

shear plane

\( {q}_{\mathrm{total}}^{\mathrm{fit}} \)

fitted heat flux density function (J/s)

q(θ,t)

heat flux density at angle θ and time t (J/s)

Q1

stress coordinate conversion matrix from O-xnsynszns to O-xnynzn

Q2

stress coordinate conversion matrix from O-xnynzn to O-x1y1z1

Q3

stress coordinate conversion matrix from O-x1y1z1 to O-x0y0z0

Q4

stress coordinate conversion matrix from O-xpypzp to O-xnynzn

re

cutting edge radius (μm)

R

tool radius (mm)

Rp

radius of the circular fan field centered (mm)

sij

deviation stress tensor (MPa)

sn

deviation stress in the nth loading step (MPa)

sn+1

deviation stress in the (n + 1)th loading step (MPa)

\( {\boldsymbol{s}}_{n+1}^{\mathrm{T}} \)

trail deviation stress in the (n + 1)th loading step (MPa)

t

time (s)

T

cutting temperature (°C)

Tol

Newton-Raphson tolerance

Tm

melt temperature (K)

Tr

environment temperature (K)

V

cutting speed (m/min)

Vf

feed speed (m/min)

Vs

chip flow velocity (m/min)

z

coordinate position along tool axis (mm)

zj

axial cutting depth at a certain point of the jth cutting edge (mm)

α

Coefficient of linear expansion (um/m-K)

βn

normal friction angle (°)

γ

fan field angle (°)

γn

normal rake angle (°)

η

angle between the spline-line and bottom surface of the build-up region (°)

ηc

chip flowing angle (°)

ηs

shear flow angle (°)

θ

angle position of a certain point on cutting edge (°)

θ1

fan field angle (°)

θd

maximum milling angle (°)

θi

angle between F and Pn (°)

θn

angle between x-axis and projection of F (°)

\( \kappa \left({\overline{\varepsilon}}_{\mathrm{p}}\right) \)

yield stress of material (MPa)

\( \kappa \left({\overline{\varepsilon}}_{\mathrm{p},n}\right) \)

yield stress of material in nth loading step (MPa)

\( \kappa \left({\overline{\varepsilon}}_{\mathrm{p},n+1}\right) \)

yield stress of material in (n + 1)th loading step (MPa)

λ

thermal conductivity of work material (W/m-K)

λt

thermal conductivity of tool material (W/m-K)

εij

strain tensor

\( {\varepsilon}_{ij}^{\mathrm{p}} \)

strain tensor

\( {\varepsilon}_{ij}^{\mathrm{r}} \)

residual strain tensor

\( {\varepsilon}_n^{\mathrm{p}} \)

plastic strain in nth loading step

\( {\varepsilon}_{n+1}^{\mathrm{p}} \)

plastic strain in (n + 1)th loading step

\( {\overline{\varepsilon}}_{\mathrm{p}} \)

equivalent plain strain

\( {\overline{\varepsilon}}_{\mathrm{p},n+1}^k \)

equivalent plastic strain in kth iteration step and (n + 1)th loading step

\( {\overline{\varepsilon}}_{\mathrm{p},n+1}^{k+1} \)

equivalent plastic strain in (k + 1)th iteration step and nth loading step

ν

Poisson’s ratio

ρ

density of work material (g/cm3)

ρ t

density of tool material (g/cm3)

σij

stress tensor (MPa)

\( {\sigma}_{ij}^{\mathrm{f}} \)

stress tensor to be released (MPa)

\( {\sigma}_{ij}^{\mathrm{mech}} \)

mechanical stress tensor by microelement (MPa)

\( {\sigma}_{ij}^{\mathrm{p}} \)

stress tensor by plow effect (MPa)

\( {\sigma}_{ij}^{\mathrm{r}} \)

residual stress tensor (MPa)

\( {\sigma}_{ij}^{\mathrm{s}} \)

stress tensor by shear effect (MPa)

\( {\sigma}_{ij}^{\mathrm{t}\_\mathrm{mech}} \)

total mechanical stress tensor (MPa)

\( {\sigma}_{ij}^{\mathrm{therm}} \)

thermal stress tensor (MPa)

\( {\sigma}_{ij}^{\mathrm{total}} \)

total stress tensor (MPa)

\( {\sigma}_{ij}^{\mathrm{T}} \)

trail stress tensor (MPa)

σij, n

stress tensor in nth loading step (MPa)

σij, n + 1

stress tensor in (n + 1)th loading step (MPa)

σs

normal stress on the shear plane (MPa)

\( {\overline{\sigma}}_{\mathrm{s}} \)

initial yield stress (MPa)

σw

normal load on the plow zone (MPa)

τs

shear stress on the shear plane (MPa)

τs x

decomposed component of τs along xns (MPa)

τs y

decomposed component of τs along yns (MPa)

τw

tangential load on the plow zone (MPa)

ϕ(t)

entrance angle of cutter rotating at time t (°)

ϕj

angular position at a certain coordinate and time of jth cutting edge (°)

ϕn

normal shear angle (°)

ϕp

angle between two adjacent cutter teeth (°)

Ψ

inclination angle of the diffraction vector

Notes

Acknowledgments

The authors would like to thank L.H. Kong at Harbin Institute of Technology for his careful operations in the experiments.

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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mechatronics EngineeringHarbin Institute of TechnologyHarbinChina

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