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- 1. An Eulerian Cutting Model for Unidirectional Fiber-Reinforced Polymers A Ph.D. Defense Presentation 10/29/201 6 Title Student: Shengqi Zhang Committee: John Strenkowski Mark Pankow Kara Peters Mohammed Zikry GSR: Jagannadham Kasichainula
- 2. Contents 1. Introduction 2. Numerical approach 3. Verification 4. Sensitivity analysis 5. Conclusion 10/29/201 6
- 3. Fiber-Reinforced Polymers (FRP)1.1 • Strength / light weight • Fatigue resistance • Corrosion resistance • Design flexibility
- 4. Machining of FRP 10/29/201 6 1.2 Issues Subsurface damage Fiber-pullout Fiber buckling Matrix crushing Matrix cracking Delamination Tool wear Iliescu et al. 2010Bhatnagar et al. 2004
- 5. Machining of FRP Experiments Time-delay Expensive Material cost Instrument noise/error Lack of detailed insight Too many parameters 10/29/201 6 1.2 Simulations Cheap & fast Repeatability Entire knowledge Different levels of modeling Design insights Iliescu et al. 2010 Zenia et al. 2015
- 6. Simulation Details Expensive Short cutting length Only possible for 2-D Fibers are round Does not account imperfections 10/29/201 6 1.3 Micro-mechanics Rao et al. 2007
- 7. Simulation Averaged material properties Permits longer cutting length The only affordable way for 3-D analyses 10/29/201 6 1.3 Soldani et al. 2009 Usui et al. 2014 Equivalent Homogeneous Material Lasri et al. 2009
- 8. Eulerian formulation Accommodates large deformations Fixed geometry No need for an expensive contact algorithm. Efficient mesh refinement only at points of interests. 10/29/201 6 1.4
- 9. Contents 1. Introduction 2. Numerical approach 3. Verification 4. Sensitivity analysis 5. Conclusion 10/29/201 6 2.1 Problem description 2.2 Damage mechanics 2.3 Fiber bending and curvature 2.4 Governing diff. eqns 2.5 Lagrangian phase 2.6 Eulerian phase 2.7 Data structure 2.8 Crush regularization 2
- 10. Problem description 10/29/201 6 2.1 2-D plane stress analysis Tool as boundary condition Clearance angle not considered Chips not considered after formation Homogeneous equivalent material Zhang et al. 2001
- 11. Damage model (1) 10/29/201 6 2.2 𝜎1 𝜎2 𝜏6 = 1 𝐷 𝐸1 𝜈21 𝐸1 0 𝜈12 𝐸2 𝐸2 0 0 0 𝐷𝐺12 𝜖1 𝜖2 𝛾6 𝐷 = 1 − 𝑣12 𝜈21 o Fiber tension ( 𝜎1 ≥ 0): 𝑓ft = 𝜎1/𝑋𝑡 2 o Fiber compression ( 𝜎1 < 0): 𝑓fc = 𝜎1/𝑋𝑐 2 o Matrix tension ( 𝜎2 ≥ 0): 𝑓mt = 𝜎2/𝑌𝑡 2 + 𝜏6/𝑆 𝐿 2 o Matrix compression ( 𝜎2 < 0): 𝑓mc = 𝜎2 2𝑆 𝑇 2 + 𝑌 𝐶 2𝑆 𝑇 2 − 1 𝜎2 𝑌 𝐶 + 𝜏6 𝑆 𝐿 2 Effective stresses Hashin’s criteria 𝑟𝑖 = 𝐵𝑖 1 − 1 max 𝑓𝑖 − 𝑟𝑖, 1 𝑑𝑖 = min 1, 𝐴𝑖 𝑟𝑖 𝑟𝑖 + 1 (𝐴𝑖 − 1) 𝑖 ∈ ft, fc, mt, mc Damage evolution
- 12. Damage model (2)2.2 𝜎1 𝜎2 𝜏6 = 1 𝐷 1 − 𝑑 𝑓 𝐸1 1 − 𝑑 𝑓 1 − 𝑑 𝑚 𝜈21 𝐸1 0 1 − 𝑑 𝑓 1 − 𝑑 𝑚 𝜈12 𝐸2 1 − 𝑑 𝑚 𝐸2 0 0 0 𝐷 1 − 𝑑 𝑠 𝐺12 𝜖1 𝜖2 𝛾6 𝐷 = 1 − 1 − 𝑑 𝑓 1 − 𝑑 𝑚 𝜈12 𝜈21 𝑑 𝑠= 1 − 1 − 𝑑 𝑓𝑡 1 − 𝑑 𝑓𝑡 1 − 𝑑 𝑓𝑡 1 − 𝑑 𝑓𝑡 Damaged material Validation examples 𝑑𝑖 = min 1, 𝐴𝑖 𝑟𝑖 𝑟𝑖 + 1 (𝐴𝑖 − 1) 𝑟𝑖 = 𝐵𝑖 1 − 1 max 𝑓𝑖 − 𝑟𝑖, 1 𝜖1 = ±5 × 104 /𝑠 𝜖2 = 𝛾6 = ±5 × 104 /𝑠
- 13. Fiber bending and rotation2.3 Qi et al. 2015 𝑓 𝑚𝑐 = 𝑓𝑚𝑐 1 + 𝛽 𝜅 𝜅 = cos −𝜃 + 𝜓 𝜕𝜓 𝜕𝑥 + sin −𝜃 + 𝜓 𝜕𝜓 𝜕𝑦 Contradicts with Hashin’s criteria Experiments needed for transverse shear Affects the entire workpiece Issue: 𝜃 dependence Idea : track bending Method: curvature Rao et al. 2007 Bhatnagar et al. 2004
- 14. Governing differential equations 10/29/201 6 2.4 Momentum: 𝜕𝑣𝑖 𝜕𝑡 + 𝑣𝑗 𝜕𝑣𝑖 𝜕𝑥𝑗 = 1 𝜌 𝜕𝜎𝑖𝑗 𝜕𝑥𝑗 Strain: 𝜕 𝜕𝑡 𝜖1 𝜖2 𝛾6 + 𝑣𝑖 𝜕 𝜕𝑥𝑖 𝜖1 𝜖2 𝛾6 = 𝑻 𝜕𝑣 𝑥/𝜕𝑥 𝜕𝑣 𝑦/𝜕𝑦 𝜕𝑣 𝑦/𝜕𝑥 + 𝜕𝑣 𝑥/𝜕𝑦 Rotation: 𝜕𝜓 𝜕𝑡 + 𝑣𝑖 𝜕𝜓 𝜕𝑥𝑖 = 1 2 𝜕𝑣 𝑦 𝜕𝑥 − 𝜕𝑣 𝑥 𝜕𝑦 Damage: 𝜕𝑟𝑖 𝜕𝑡 + 𝑣𝑗 𝜕𝑟𝑖 𝜕𝑥𝑗 = 𝐵𝑖 1 − 1 max 𝑓𝑖 − 𝑟𝑖, 1 𝜕𝜙 𝜕𝑡 + 𝑣𝑖 𝜕𝜙 𝜕𝑥𝑖 = 𝑆 𝜕𝜙 𝐿 𝜕𝑡 = 𝑆 𝜕𝜙 𝐸 𝜕𝑡 + 𝑣𝑖 𝜕𝜙 𝜕𝑥𝑖 = 0 Lagrangian phase Eulerian phase 𝜕𝜙 𝜕𝑡 = 𝜕𝜙 𝐿 𝜕𝑡 + 𝜕𝜙 𝐸 𝜕𝑡 Operator Split
- 15. Lagrangian phase 10/29/201 6 Momentum 𝜕𝑣𝑖 𝜕𝑡 = 1 𝜌 𝜕𝜎𝑖𝑗 𝜕𝑥𝑗 Strain 𝜕 𝜕𝑡 𝜖1 𝜖2 𝛾6 = 𝑻 𝜕𝑣 𝑥/𝜕𝑥 𝜕𝑣 𝑦/𝜕𝑦 𝜕𝑣 𝑦/𝜕𝑥 + 𝜕𝑣 𝑥/𝜕𝑦 Rotation 𝜕𝜓 𝜕𝑡 + 𝑣𝑖 𝜕𝜓 𝜕𝑥𝑖 = 1 2 𝜕𝑣 𝑦 𝜕𝑥 − 𝜕𝑣 𝑥 𝜕𝑦 Damage 𝜕𝑟𝑖 𝜕𝑡 = 𝐵𝑖 1 − 1 max 𝑓𝑖 − 𝑟𝑖, 1 Simplified friction model Explicit integration w. Mass scaling 4-node quads, 2x2 integration pts Numerical damping w. viscosity Standard FEM Shape functions Algebra 2.5
- 16. Eulerian phase 10/29/201 6 𝜕Φ 𝜕𝑡 + 𝑣𝑖 𝜕Φ 𝜕𝑥𝑖 = 0 Ω 𝑒 𝛿Φ 𝜕Φ 𝜕𝑡 dΩ + Ω 𝑒 𝛿Φ𝑣𝑖 𝜕Φ 𝜕𝑥𝑖 dΩ = 0 Γ 𝑒 𝛿ΦΦ𝑣𝑖 𝑛𝑖dΓ − Ω 𝑒 Φ 𝜕 𝜕𝑥𝑖 𝛿Φ𝑣𝑖 dΩ Discretization Same mesh w. Lagrangian phase 3rd order Runge-Kutta explicit time integration Invoked by intervals 𝛿Φ FEM Upwind Ω 𝑒 𝛿Φ𝑣𝑖 𝜕Φ 𝜕𝑥𝑖 dΩ Ω 𝑒 𝜕 𝜕𝑥𝑖 𝛿ΦΦ𝑣𝑖 dΩ 2.6
- 17. Data structure All variables at nodes Multiple nodal values 10/29/201 6 Velocity at nodes Others at intgr. pts Extrapolation Interpolation Averaging Lagrangian Eulerian 2.7
- 18. Crush regularization 𝜖1 = 𝛼 𝜆 − 1 𝜖1 + 𝐸1 𝑋𝑐 − 𝜖1 𝜖2 = 𝛼 𝜆 − 1 𝜖2 + 𝐸2 𝑌𝑐 − 𝜖2 𝛾6 = 𝛼 𝜆 − 1 𝐺12 𝑆 𝐿 − 𝛾6 − 𝛾6 − 𝐺12 𝑆 𝐿 − 𝜖1 − 𝜖2 𝜕𝜆 𝜕𝑡 + 𝑣𝑖 𝜕𝜆 𝑥𝑖 = 𝜆 2 − 𝜆 𝜕𝑣𝑖 𝜕𝑥𝑖 Issue Damage layer Method Validation: 1-D test Aux. var: 𝜖 modifier: 2.8
- 19. 2. Numerical Approach 10/29/201 6 Questions? 2.1 Problem description 2.2 Damage mechanics 2.3 Fiber bending and curvature 2.4 Governing diff. eqns 2.5 Lagrangian phase 2.6 Eulerian phase 2.7 Data structure 2.8 Crush regularization 2
- 20. Contents 1. Introduction 2. Numerical approach 3. Verification 4. Sensitivity analysis 5. Conclusion 10/29/201 6 3.1 Parameters 3.2 Damage modes 3.3 Fiber bending 3.4 Force and damage 3.5 Damage initiation 3.6 Damage completion 3.7 Cutting force vs fiber angle 3
- 21. Cutting and modeling parameters Material Cutting Assumptions 3.1 𝑑𝑖 = min 1, 𝐴𝑖 𝑟𝑖 𝑟𝑖 + 1 (𝐴𝑖 − 1) 𝑟𝑖 = 𝐵𝑖 1 − 1 max 𝑓𝑖 − 𝑟𝑖, 1
- 22. 10/29/201 6 Damage modes: 𝜃 = 30° Soldani et al. 2011 3.2 Fiber tension Fiber compression Matrix tension Matrix compression Matrix tension 2 ms -> 0.52 x DOC 4 modes Shear zone
- 23. 10/29/201 6 Soldani et al. 2011 3.2 Fiber tension Fiber compression Matrix tension Matrix compression Damage modes: 𝜃 = 60° 2 ms -> 0.52 x DOC 4 modes Fiber direction affects the shear zone
- 24. 10/29/201 6 3.2 Fiber tension Fiber compression Matrix tension Matrix compression Damage changes with 𝜃 Matrix modes are dominating No need to plot all modes Use 𝑑 𝑠 instead 𝑑 𝑠 = 1 − 1 − 𝑑ft 1 − 𝑑f𝑐 1 − 𝑑mt 1 − 𝑑mc Damage modes: 𝜃 = 90°
- 25. 10/29/201 6 Fiber bending Santiuste et al. 2010 Model prediction 10 ms -> 2.6 x DOC3.3
- 26. Forces and damage3.4 A B C D A B C D
- 28. 3.5 𝑑 𝑠 = 1 − 1 − 𝑑ft 1 − 𝑑f𝑐 1 − 𝑑mt 1 − 𝑑mc 𝜃 = 30° 𝜃 = 60° 𝜃 = 90° 𝜃 = 120° 𝜃 = 150° Chip initiation
- 29. 10/29/201 6 Chip completion3.6 𝜃 = 30° 𝜃 = 90° 𝜃 = 120° 𝜃 = 150° 𝜃 = 60°
- 30. Cutting forces vs Fiber angle3.7 Good trend No clearance face Experimental error Need longer simulation time Complex macroscopic behavior Wang et al. 1995 300 250 200 150 100 50 0 Force (N/mm) ~100 x DOC 2.6 x DOC
- 31. 3. Verification 10/29/201 6 Questions? 3.1 Parameters 3.2 Damage modes 3.3 Fiber bending 3.4 Forces and damage 3.5 Damage initiation 3.6 Damage completion 3.7 Cutting force vs fiber angle 3
- 32. Contents 1. Introduction 2. Numerical approach 3. Verification 4. Sensitivity analysis 5. Conclusion 10/29/201 6 4.1 Convergence 4.2 Elementary effects 4.3 Machining conditions 4
- 33. Convergence study 4.1 Mesh refinement 4.2 Mass scale factor 4.3 Eulerian interval 10/29/201 6 4.1 Accurate Expensive Inaccurate Efficient
- 34. Mesh refinements 10/29/201 6 4.1.1 1900 6 𝜇𝑚 8600 3.3 𝜇𝑚 3968 4.5 𝜇𝑚
- 36. Mass scaling4.1.2 Δ𝑡 𝐿 = 𝑙 𝑚𝑖𝑛 𝜌𝑠 𝑚𝑎𝑠𝑠 𝐸1 Fast Slow Large Δ𝑡 Small Δ𝑡
- 38. Elementary effects 10/29/201 6 4.2 22 model parameters Analyze effect of each parameter 3-levels for each parameter (Low – mid – high) 322 = 31,381,059,609 model evaluations! Need a better method
- 39. Elementary effects 10/29/201 6 4.2 22 model parameters (𝑘 = 22) Factorial sampling of 𝒙 (Morris 1991) 4 samples of each 𝑒𝑖 𝒙 Mean, standard deviation of 𝑒𝑖 𝒙 𝑒𝑖 𝒙 = 𝑦 𝑥1, 𝑥2, … , 𝑥𝑖−1, 𝑥𝑖 + Δ, 𝑥𝑖+1, … , 𝑥 𝑘 − 𝑦(𝒙) Δ 𝑥𝑖 ∈ [0, 1] Definition 𝑦 = 𝑦 𝒙 Scalar model response 𝑘-dimensional input vector Approx. of 𝜕𝑦(𝒙) 𝜕𝑥𝑖
- 40. Elementary effects 10/29/201 6 4.2 Linear scale --- measured properties Log scale --- assumed parameters Maximum subsurface damage depth as 𝑦
- 44. Elementary effects4.2 Strong dependency on fiber angle Least sensitive for 𝜃 = 30° Most nonlinear for 𝜃 = 60° More operators are functional at higher fiber angles
- 45. Influence of machining conditions 4.1 Depth of cut 4.2 Rake angle 4.3 Nose radius 10/29/201 6 4.3
- 46. Influence of machining conditions 10/29/201 6 4.3.1 Depth of cut
- 47. Influence of machining conditions 10/29/201 6 4.3.2 Rake angle
- 48. Influence of machining conditions 10/29/201 6 4.3.3 Tool nose radius
- 49. Influence of machining conditions 10/29/201 6 4.3.4 Consitent trend for 𝜃 = 30° Inconsistent for 𝜃 = 60° and 𝜃 = 90° Different failure and chip formation mechanisms Relative nose radius = nose radius/DOC Sharp Blunt
- 50. Influence of machining conditions 10/29/201 6 4.3.5 𝜃 = 30° 1 2 0.169 mm 0.323 mm
- 51. Conclusion Fiber bending is important Damage propagation paths Strong fiber angle dependence Subsurface damage is influenced by the cutting conditions 10/29/201 6 5 Major findings Future work Numerical & analytical bending models ALE approaches Uncertainty quantification Parallel computing
- 53. 10/29/2016 Coordinate transformation 3.6 Stress in the X-Y coordinate 𝜎𝑥 𝜎 𝑦 𝜏 𝑠 = 𝑇 −1 𝜎1 𝜎2 𝜏6 Constitutive model in the fiber coordinate 𝜎1 𝜎2 𝜎6 = 1 𝐷 1 − 𝑑 𝑓 𝐸1 1 − 𝑑 𝑓 1 − 𝑑 𝑚 𝜈21 𝐸1 0 1 − 𝑑 𝑓 1 − 𝑑 𝑚 𝜈12 𝐸2 1 − 𝑑 𝑚 𝐸2 0 0 0 𝐷 1 − 𝑑 𝑠 𝐺12 𝜖1 𝜖2 𝛾6 Strain in the fiber coordinate 𝜖1 𝜖2 1 2 𝛾6 = 𝑇 𝜖 𝑥 𝜖 𝑦 1 2 𝛾𝑠 𝑇 = 𝑚2 𝑛2 2𝑚𝑛 𝑛2 𝑚2 −2𝑚𝑛 −𝑚𝑛 𝑚𝑛 𝑚2 − 𝑛2 𝑚 = cos 𝜋 − 𝜃 𝑛 = sin(𝜋 − 𝜃)
- 54. Elementary effects 10/29/201 6 22 model parameters 500 random samples 4 samples selected using Monte Carlo 𝑒𝑖 𝒙 = 𝑦 𝑥1, 𝑥2, … , 𝑥𝑖−1, 𝑥𝑖 + Δ, 𝑥𝑖+1, … , 𝑥 𝑘 − 𝑦(𝒙) Δ 0 0 0 1 0 0 1 1 0 1 1 1 𝒙 0 1 0 1 1 0 1 0 0 1 0 1 𝑥𝑖 ∈ [0, 1] 0.2 1 0 0.8 1 0 0.8 0.4 0 0.8 0.4 0.6 Δ = 0.6 𝒙 𝑏𝑎𝑠𝑒 = 0.2 0.4 0 𝑒1 𝑒3 𝑒2 Definition Factorial sampling: example All random processes
Editor's Notes
- Committee members More space between title letters
- larger
- Fuel savings 70% Lighter than steel, 40% lighter than aluminum The 787 has composite fuselage, wings, and most other airframe components VW XL1
- Damage---roughness/life Fluorescent dye---UV
- Increasing costs
- Damage variables Shows only the fiber damage
- How the load is applied
- Single material point Different levels of validation
- For EHM Go back for fmc Validated later Only affects large strain zone
- Solved individually at each time step
- What are fi’s and ri’s
- Check colors
- Auxiliary variable for material loss during crushing
- Larger font size for the section title
- Where does data come from
- Line size is fine
- Line size is fine
- Explain oscillations
- Damage initiation plots
- Temp strainrate
- Looks ugly
- Explain \mu Optimize the visualization---not clear
- Larger legend, larger line weights
- Include references for the images?