Inverse analysis of the residual stress in laser-assisted milling


In laser-assisted milling, higher temperature in shear zone softens the material potentially resulting in a shift of mean residual stress, which significantly affects the damage tolerance and fatigue performance of product. In order to guide the selection of laser and cutting parameters based on the preferred mean residual stress, inverse analysis is conducted by predicting residual stress based on guessed process parameters, which is defined as the forward problem, and applying iterative gradient search to find process parameters for next iteration, which is defined as the inverse problem. An analytical inverse analysis is therefore proposed for the mean residual stress in laser-assisted milling. The forward problem is solved by analytical prediction of mean residual stress after laser-assisted milling. The residual stress profile is predicted through the calculation of thermal stress, by treating laser beam as heat source, and plastic stress by first assuming pure elastic stress in loading process, then obtaining true stress with kinematic hardening followed by the stress relaxation. The variance-based recursive method is applied to solve inverse problem by updating process parameters to match the measured mean residual stress. Three cutting parameters including depth of cut, feed per tooth, and cutting speed, and two laser parameters including laser-tool distance and laser power, are updated with respected to the minimization of resulting residual stress and measurement in each iteration. Experimental measurements are referred on the laser-assisted milling of Ti–6Al–4 V and Si3N4. The percentage difference between experiments and predictions is less than 5% for both materials, and the selection is completed within 50 loops.

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ρ :


ξ :

Absorption ratio

α :

Thermal diffusivity

θ :

Rotation angle

ε p :

Peak strain

μ :

Friction coefficient

ϕ :

Shear angle

c p :

Specific heat

Cs :

Side cutting edge angle

d :

Grain size

d a :

Axial depth of milling

F c :

Cutting force

F t :

Tangential force

f z :

Feed per tooth

G s :

Shear modulus

G :

Gain coefficient

h :

Heat transfer coefficient

h p :

Plastic modulus

k AB :

Flow stress

K :

Thermal conductivity

K intensity :

Intensity distribution coefficient

K n :

Kalman gain matrix

L AB :

Shear length

L :

Laser-tool distance

P :

Laser power

P thrust :

Normal plowing force

P cut :

Cutting plowing force

P n :

Simulation covariance matrix

Q act :

Activation energy

R g :

Gas constant

R :

Error covariance matrix

R t :

Tool radius

r :

Radius of laser spot

r corner :

Corner edge radius

T laser :

Preheating temperature

T 0 :

Environment temperature

T m :

Melting temperature

t c :

Cutting depth

V f :

Feed rate

V r :

Cutting speed

w :

Cutting width

X drex :

Recrystallized volume fraction


  1. 1.

    Feng, Y., et al., Analytical and numerical predictions of machining-induced residual stress in milling of Inconel 718 considering dynamic recrystallization. 2018(51388): p. V004T03A023

  2. 2.

    Schlauer C, Peng RL, Odén M (2002) Residual stresses in a nickel-based superalloy introduced by turning. Mater Sci Forum 404-407:173–178

    Article  Google Scholar 

  3. 3.

    Dudzinski D et al (2004) A review of developments towards dry and high speed machining of Inconel 718 alloy. Int J Mach Tool Manu 44:439–456

    Article  Google Scholar 

  4. 4.

    Sharman ARC, Hughes JI, Ridgway K (2006) An analysis of the residual stresses generated in Inconel 718™ when turning. J Mater Process Technol 173(3):359–367

    Article  Google Scholar 

  5. 5.

    Madariaga A et al (2014) Analysis of residual stress and work-hardened profiles on Inconel 718 when face turning with large-nose radius tools. Int J Adv Manuf Technol 71(9–12):1587–1598

    Article  Google Scholar 

  6. 6.

    Outeiro JC et al (2008) Analysis of residual stresses induced by dry turning of difficult-to-machine materials. CIRP Ann Manuf Technol 57(1):77–80

    Article  Google Scholar 

  7. 7.

    Le Coz, G., et al., Residual stresses after dry machining of Inconel 718, experimental results and numerical simulation. 2010

    Google Scholar 

  8. 8.

    Ulutan D, Erdem Alaca B, Lazoglu I (2007) Analytical modelling of residual stresses in machining. J Mater Process Technol 183(1):77–87

    Article  Google Scholar 

  9. 9.

    Fergani O et al (2014) Analytical modeling of residual stress and the induced deflection of a milled thin plate. Int J Adv Manuf Technol 75(1–4):455–463

    Article  Google Scholar 

  10. 10.

    Peng FY et al (2016) Analytical modeling and experimental validation of residual stress in micro-end-milling. Int J Adv Manuf Technol 87(9–12):3411–3424

    Article  Google Scholar 

  11. 11.

    Zhou R, Yang W (2016) Analytical modeling of residual stress in helical end milling of nickel-aluminum bronze. Int J Adv Manuf Technol 89(1–4):987–996

    Google Scholar 

  12. 12.

    Huang X, Zhang X, Ding H (2017) An enhanced analytical model of residual stress for peripheral milling. Procedia CIRP 58:387–392

    Article  Google Scholar 

  13. 13.

    Pan Z et al (2017) Heat affected zone in the laser-assisted milling of Inconel 718. J Manuf Process 30:141–147

    Article  Google Scholar 

  14. 14.

    Feng Y et al (2018) Analytical prediction of temperature in laser-assisted milling with laser preheating and machining effects. Int J Adv Manuf Technol

  15. 15.

    Oxley PLB (1989) Mechanics of machining, an analytical approach to assessing machinability. ELLIS HORWOOD LIMITED, p 242

  16. 16.

    Pan Z et al (2017) Turning induced residual stress prediction of AISI 4130 considering dynamic recrystallization. Mach Sci Technol 22(3):507–521

    Article  Google Scholar 

  17. 17.

    Song X et al (2013) Numerical comparison of iterative ensemble Kalman filters for unsaturated flow inverse modeling. Vadose Zone J

  18. 18.

    Gua Y et al (2003) Micro-indentation and inverse analysis to characterize elastic-plastic graded materials. Mater Sci Eng A345:223–233

    Article  Google Scholar 

  19. 19.

    Xie T et al (2015) An inverse analysis to estimate the endothermic reaction parameters and physical properties of aerogel insulating material. Appl Therm Eng 87:214–224

    Article  Google Scholar 

  20. 20.

    Cuellar KJQ, Silveira JLL (2017) Analysis of torque in friction stir welding of aluminum alloy 5052 by inverse problem method. J Manuf Sci Eng 139

  21. 21.

    Agmell M, Ahadi A, Ståhl J-E (2014) Identification of plasticity constants from orthogonal cutting and inverse analysis. Mech Mater 77:43–51

    Article  Google Scholar 

  22. 22.

    Denkena B et al (2015) Inverse determination of constitutive equations and cutting force Modelling for complex tools using Oxley’s predictive machining theory. Procedia CIRP 31:405–410

    Article  Google Scholar 

  23. 23.

    Ning J et al (2018) Inverse determination of Johnson–Cook model constants of ultra-fine-grained titanium based on chip formation model and iterative gradient search. Int J Adv Manuf Technol 99:1131–1140

    Article  Google Scholar 

  24. 24.

    Pan Z et al (2017) Microstructure-sensitive flow stress modeling for force prediction in laser assisted milling of Inconel 718. Manuf Rev 4:6

    Google Scholar 

  25. 25.

    Faghidian SA (2015) Inverse determination of the regularized residual stress and eigenstrain fields due to surface peening. J Strain Anal 50(2):84–91

    Article  Google Scholar 

  26. 26.

    Buljak V et al (2017) Estimation of residual stresses by inverse analysis based onexperimental data from sample removal for “small punch” tests. Eng Struct 136:77–86

    Article  Google Scholar 

  27. 27.

    Feng Y et al (2018) Inverse analysis of the cutting force in laser-assisted milling on Inconel 718. Int J Adv Manuf Technol 96(1):905–914

    Article  Google Scholar 

  28. 28.

    Feng Y et al (2019) Inverse analysis of the tool life in laser-assisted milling. Int J Adv Manuf Technol

  29. 29.

    Feng Y et al (2018) Inverse analysis of Inconel 718 laser-assisted milling to achieve machined surface roughness. Int J Precis Eng Manuf 19(11):1611–1618

    Article  Google Scholar 

  30. 30.

    Hedberg GK (2013) Laser assisted milling of difficult to machine materials. Purdue University

  31. 31.

    Hedberg GK, Shin YC (2015) Laser assisted milling of Ti-6Al-4V ELI with the analysis of surface integrity and its economics. Lasers Manuf Mater Process 2(3):164–185

    Article  Google Scholar 

  32. 32.

    Shen X, Lei S (2010) Experimental study on operating temperature in laser-assisted milling of silicon nitride ceramics. Int J Adv Manuf Technol 52(1–4):143–154

    Google Scholar 

  33. 33.

    Rahim EA et al (2015) A prediction of laser spot-to-cutting tool distance in laser assisted micro milling Inconel 718. Adv Mater Process Technol 1(3–4):529–541

    Google Scholar 

  34. 34.

    Pan Z et al (2017) Force modeling of Inconel 718 laser-assisted end milling under recrystallization effects. Int J Adv Manuf Technol 92(5):2965–2974

    Article  Google Scholar 

  35. 35.

    Pan, Z, et al., Turning force prediction of AISI 4130 considering dynamic recrystallization. 2017(50725): p. V001T02A040

  36. 36.

    Pan Z, Feng Y, Liang SY (2017) Material microstructure affected machining: a review. Manuf Rev 4:5

    Google Scholar 

  37. 37.

    Pan Z et al (2016) Prediction of machining-induced phase transformation and grain growth of Ti-6Al-4 V alloy. Int J Adv Manuf Technol 87(1–4):859–866

    Article  Google Scholar 

  38. 38.

    Feng Y, Pan Z, Liang SY (2018) Temperature prediction in Inconel 718 milling with microstructure evolution. Int J Adv Manuf Technol 95(9–12):4607–4621

    Article  Google Scholar 

  39. 39.

    Feng Y et al (2019) Residual stress prediction in ultrasonic vibration–assisted milling. Int J Adv Manuf Technol 104(5):2579–2592

    Article  Google Scholar 

  40. 40.

    Mirkoohi E, Bocchini P, Liang SY (2018) An analytical modeling for process parameter planning in the machining of Ti-6Al-4V for force specifications using an inverse analysis. Int J Adv Manuf Technol 98(9):2347–2355

    Article  Google Scholar 

  41. 41.

    Mirkoohi E, Bocchini P, Liang SY (2019) Inverse analysis of residual stress in orthogonal cutting. J Manuf Process 38:462–471

    Article  Google Scholar 

  42. 42.

    Mirkoohi E, Bocchini P, Liang SY (2018) An analytical modeling for designing the process parameters for temperature specifications in machining. Preprints

  43. 43.

    Feng Y et al (2019) Residual stress prediction in laser-assisted milling considering recrystallization effects. Int J Adv Manuf Technol

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Correspondence to Yixuan Feng.

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Feng, Y., Hung, TP., Lu, YT. et al. Inverse analysis of the residual stress in laser-assisted milling. Int J Adv Manuf Technol 106, 2463–2475 (2020).

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  • Residual stress
  • Inverse analysis
  • Laser-assisted milling
  • Ti–6Al–4V
  • Si3N4
  • Iterative gradient search