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Computationally Efficient Protocols to Evaluate the Fatigue Resistance of Polycrystalline Materials
- 1. Computationally Efficient Protocols to Evaluate the Fatigue Resistance of Polycrystalline Materials Noah H. Paulson, Matthew W. Priddy, Surya R. Kalidindi, and David L. McDowell
- 2. Motivation *Welsch et al. (1994). Processing Options 2 Ti-64 Composition Titanium Aluminum Vanadium
- 3. Generate Microstructures CPFEM: Calculate local 𝜺 𝑝 field get FIP fields + FIP EVDs Evaluate HCF resistance HCF Evaluation Existing Approach Smith (2013). +UMAT 𝐻𝐶𝐹𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒: 𝐃 > 𝐂 > 𝐁 (0001) min: 0, max: 8 Basal texture 3
- 4. Material Property Representation SVE Concept Kanit, et al. (2003). Numerous samples are needed to capture the statistics of the properties of the material. Let us call these samples statistical volume elements (SVEs) 4 RVE SVE set vs.
- 5. Background Fatigue Indicator Parameters FIPs are a surrogate measure of driving force for fatigue crack formation and growth Critical Plane Approach • Fatemi-Socie Parameter 𝐹𝐼𝑃 𝐹𝑆 = ∆𝛾 𝑚𝑎𝑥 𝑝 2 1 + 𝑘 𝜎 𝑚𝑎𝑥 𝑛 𝜎 𝑦 max n 2 Crack formation due to intense shear along the slip band of Ti- 6Al-4V Le Biavant, et al. (2001). 5 Fatemi, et al. (1988).
- 6. Problem Statement Computational Burdens Hypothetical: Rank HCF resistance of the 12 heat treatments of Ti-64 𝐶𝑃𝑈𝑡𝑖𝑚𝑒 = 12 microstructures ∗ 100 SVEs microstructure ∗ 1.5 hours ∗ 4 processors 𝑆𝑉𝐸 = 𝟕𝟐𝟎𝟎 𝐡𝐨𝐮𝐫𝐬 A more efficient approach is needed to make computational fatigue analysis feasible for industrial applications 6
- 7. HCF Study MKS Approach MKS: Predict local 𝜺 𝑡𝑜𝑡. field Generate SVE set (Statistical Volume Element) get FIP fields + FIP EVDs Evaluate HCF resistance Estimate 𝜺 𝑝𝑙 from 𝜺 𝑡𝑜𝑡. 𝜺 𝑒𝑙 𝝈 𝜏 𝛼 Integrate flow rule 𝛾 𝛼 𝜺 𝑝𝑙 𝜺 𝑡𝑜𝑡. ≅ 𝜺 𝑒𝑙 7
- 8. The Materials Knowledge System (MKS) is a localization technique to estimate local response (e.g., 𝜀𝑖𝑗) given macroscopic applied condition MKS Framework 𝜺 𝒙 𝜺 𝑥 = 𝑰 − 𝑅 𝐻 𝜶 𝑟, 𝑛 𝑚 𝑥 + 𝑟, 𝑛 𝑑𝑛𝑑𝑟 + 𝑅 𝑅 𝐻 𝐻 𝜶 𝑟, 𝑟′ , 𝑛, 𝑛′ 𝑚 𝑥 + 𝑟, 𝑛 𝑚 𝑥 + 𝑟 + 𝑟′ , 𝑛′ 𝑑𝑛𝑑𝑛′ 𝑑𝑟𝑑𝑟′ − ⋯ 𝜺 𝑥 𝑚 𝑥, 𝑛 = 𝐿 𝑠 𝑀𝑠 𝐿 𝑄 𝐿(𝑛)𝑋𝑠(𝑥) Microstructure function: 𝜶 𝑟, 𝑛 = 𝐿 𝑡 𝑨 𝑡 𝐿 𝑄 𝐿 𝑛 𝜒𝑡 𝑟 Influence function: 𝑄 𝐿 𝑛 : orthonormal Fourier basis 𝑋𝑠(𝑥): indicator basis (Kalidindi 2012), (Adams 2012), (Kröner 1986), Yabansu (2014) 𝜺 𝒙 = 𝒂 𝒙 𝜺 𝒙 Influence Function 8
- 9. HCF Study MKS Approach MKS: Predict local 𝜺 𝑡𝑜𝑡. field Generate SVE set (Statistical Volume Element) get FIP fields + FIP EVDs Evaluate HCF resistance Calibrate MKS Influence Coefficients Generate SVE set for training FEM: Calculate local 𝜺 𝑡𝑜𝑡. field Estimate 𝜺 𝑝𝑙 from 𝜺 𝑡𝑜𝑡. 𝜺 𝑒𝑙 𝝈 𝜏 𝛼 Integrate flow rule 𝛾 𝛼 𝜺 𝑝𝑙 𝜺 𝑡𝑜𝑡. ≅ 𝜺 𝑒𝑙 9
- 10. 𝜺 𝑝𝑙 = 𝛼=1 𝑁 𝛾 𝛼 𝒔0 𝛼 ⨂𝒎0 𝛼 𝑠𝑦𝑚 Calculate 𝜺 𝑝𝑙 Calculate 𝜺 𝑝𝑙 Calculate 𝜏 𝛼 from 𝝈 𝜺𝑙𝑜𝑐𝑎𝑙 𝝈𝑙𝑜𝑐. 𝜺 𝑒𝑙 𝜺 𝑡𝑜𝑡𝑎𝑙 𝜺 𝑡𝑜𝑡𝑎𝑙 ≈ 𝜺 𝑒𝑙 𝝈 ≈ 𝑪𝜺 𝑡𝑜𝑡𝑎𝑙 𝛾 𝛼 = 𝛾0 𝜏 𝛼 − 𝜒 𝛼 − 𝜅 𝛼 𝐷 𝛼 𝑀 sgn 𝜏 𝛼 − 𝜒 𝛼 Integrate flow rule for 𝛾 𝛼 (1) (2) (3) (4) 10
- 11. HCF Study SVEs and Loading DREAM.3D input information – Grain size distribution • Avg. Grain Size: 43 elements – Misorientation distribution – Texture x y z ε t Fully-reversed cyclic loading • x-, y-, and z-direction uniaxial strain • Periodic boundary conditions 11
- 12. HCF Study MKS Results (α-Ti Basal Texture) ε11 mean error: 0.22%, ε11 max error: 1.3% 𝑒𝑟𝑟 ≡ 𝜀𝑖𝑗 𝐹𝐸𝑀 − 𝜀𝑖𝑗 𝑀𝐾𝑆 𝜀𝑖𝑗 𝐹𝐸𝑀 12
- 13. HCF Study MKS Results (α-Ti Basal Texture) 𝜺 𝑡𝑜𝑡𝑎𝑙 → 13
- 14. HCF Study Results 14 Gumbel distribution
- 15. HCF Study Results 15 Gumbel distribution • New protocol 240X faster than traditional protocols • Traditional Protocol: 1.5 hours on 4 processors per SVE • New Protocol: 90 seconds on 1 processor per SVE
- 16. • Protocols have been developed to evaluate the HCF and LCF resistance of polycrystalline materials 𝛾 𝛼 = 𝛾 𝛼 𝜏 𝛼 , 𝜺 𝑝𝑙 = 𝛼=1 𝑁 𝛾 𝛼 𝑷 𝛼 • HCF Study: New protocol 240X faster than traditional protocols HCF/LCF Study Conclusions 16
- 17. Acknowledgements Also thanks to Donald S. Shih (Boeing), Yuksel C. Yabansu (GT), Dipen Patel (GT), and David Brough (GT) GOALIFunding provided by:
- 18. Appendix
- 19. References • Alharbi HF, Kalidindi SR. Int J Plasticity 2015;66:71. • Adams BL, Kalidindi SR, Fullwood DT. Microstructure Sensitive Design for Performance Optimization: Elsevier Science, 2012. • Bunge HJ, Moris PR. Texture Analysis in Materials Science: Butterworth & Co, 1982 • Fast T, Kalidindi SR. Acta Mater 2011;59:4595. • Kalidindi SR. ISRN Mater Sci 2012;2012:13. • Kröner E. J Mech Phy Solids 1977;25:137. • Landi G, Niezgoda SR, Kalidindi SR. Acta Mater 2010;58:2716. • Przybyla C., Prasannavenkatesan R., Salajegheh N., McDowell D.L. Microstructure-sensitive modeling of high cycle fatigue. International Journal of Fatigue, Vol. 32, Iss. 3, (2010) pg. 512-525 • Przybyla C.P., McDowell D.L. Simulation-based extreme value marked correlations in fatigue of advanced engineering alloys. Procedia Engineering, Vol. 2, Iss. 1, (2010) pg. 1045-1056 • Smith BD. Masters Thesis 2013. • Yabansu YC, Patel DK, Kalidindi SR. Acta Mater 2014;81:151. 19
Editor's Notes
- Numerous possibilities for material microstructure and properties It is expensive to experimentally evaluate new materials for strength, fatigue life, etc. Computational tools provide a more efficient way to explore the space of materials!
- In recent publications fatigue life of various materials have been compared using a computational approach Talk about this procedure in moderate detail Unfortunately, CPFEM is a major computational effort. In (Smith 2013) 5 materials were represented by 100 SVEs each. Each of these SVEs would likely take several hours on a supercomputer to calculate. In this work we propose a procedure to compute these plastic strains hundreds of times faster. This approach allows for the comparison of many more materials with the elimination of the CPFEM bottleneck.
- For each microstructure we want to perform simulations and evaluate some homogenized property, in this case the HCF resistance Traditionally we would generate a representative volume element (RVE), i.e. an instantiation of microstructure large enough that it captures the statistical variation of the property of interest (if we made the RVE larger, the homogenized property would not change). Unfortunately this is a problem because big RVEs require even bigger computational resources The other option is to build a set of numerous smaller microstructure instantiations called statistical volume elements (SVEs). Each SVE takes much less time to evaluate so this approach is more computationally efficient.
- “Fatigue crack formation in alpha/beta Ti alloys is primarily associated with the development of crystallographic facets at the grain scale” [Pryzbyla 2010]
- 7200 Hours is approximately 10 months
- 𝑚 𝑥,𝑛 (microstructure function): probability density of finding local state 𝒏+𝒅𝒏 at 𝒙+𝒅𝒙 𝑚 𝑥,𝑛 𝑑𝑥𝑑𝑛: probability of finding local state 𝒏+𝒅𝒏 at 𝒙+𝒅𝒙 𝜶 𝑟,𝑛 (influence function): contribution to local response at current spatial location from local state 𝒏±𝒅𝒏 existing at spatial location 𝒓+𝒅𝒓 away 𝐻: set of all possible distinct local states in materials system 𝜞 𝑟 a has singularity at r = 0, and the convergence of the series is sensitive to the choice of 𝑪 𝑹 . Instead of using 𝑪 𝑹 the MKS localization relationships are calibrated through a linear regression of microstructures and their local responses. Once calibrated, the MKS predicts the response of any microstructure in the materials system with low computational cost Both microstructure function and influence function can be expressed as linear combination of orthonormal basis functions in local state and spatial location Indicator function is chosen for spatial location to make regular grid. This is needed so that DFT may be used to decouple convolution in MKS series summation The contribution to local strain in the influence function decreases with an increasing distance r. The influence coefficients in the image decay very quickly This is consistent as the material system is a two phase composite with low contrast (1.5) for the elastic moduli between the two phases Talk about computational issues with the local state space for polycrystals, explain why some complicated mathematics may be required…
- Question from Dr. Georges Cailletaud: You have used a voxelated mesh, how do you know that this is a good assumption in terms of the eventual prediction of FIPs? Will the non-smooth surface be an issue? How about the representation of small grains? Matthew Priddy’s Answer: I believe Craig Przybyla looked at element size, but I am not sure if anyone has looked at element type or grain boundary shapes. I have seen a couple of papers in the literature that look at staircase (voxelated) versus smooth grain boundaries, but I don't believe I read anything definitive that made me believe the voxelated mesh wasn't sufficient. Question from Dr. Georges Cailletaud: How come you don’t have a free surface for your BCs? From my understanding fatigue cracks often initiate from the surface as the stress state is more extreme. Matthew Priddy’s Answer: “In titanium alloys, there is a transition from surface to subsurface fatigue crack initiation between 10^6 and 10^9 cycles (Przybyla thesis, page 36). Subsurface fatigue crack formation has also been seen on pyramidal planes in Ti-64 (Przybyla, 172). He also looked at surface nucleation (removing the PBCs) and found that they could be in competition with the subsurface formation. But, there is a question here of whether we are simulating a large enough volume for the surface effects to be accurate. I think that might be the more relevant item in this discussion. In order to accurately capture the surface effects, you would need a larger volume, I believe.”
- My Comments: The HCF resistance for both textures is highest for z-axis loading (direction 1) because the material is stiffer in that direction (both have strong basal texture such that the c-axes of many grains align with the loading direction) therefore the maximum plastic strain range is much lower. Question by Dr. Georges Cailletaud: While the plastic strain range is lower for loading in the z-direction, the normal stress will be higher. How then are you sure that with the FS-FIP that z-axis loading will display a better response? My Answer to question: While the normal stress will increase for z-axis loading, (maybe from 850-1000 MPa), the plastic strain range will decrease by orders of magnitude. Therefore the FS-FIP will still be lower.
- My Comments: The HCF resistance for both textures is highest for z-axis loading (direction 1) because the material is stiffer in that direction (both have strong basal texture such that the c-axes of many grains align with the loading direction) therefore the maximum plastic strain range is much lower. Question by Dr. Georges Cailletaud: While the plastic strain range is lower for loading in the z-direction, the normal stress will be higher. How then are you sure that with the FS-FIP that z-axis loading will display a better response? My Answer to question: While the normal stress will increase for z-axis loading, (maybe from 850-1000 MPa), the plastic strain range will decrease by orders of magnitude. Therefore the FS-FIP will still be lower.
- To determine the stress, each slip system of the material is then evaluated for plastic deformation, which is dislocation slip in this case. The flow rule used to describe the plastic shear strain rate of each slip system is a power-law flow rule. The drag, threshold, and back stress describe the non-directional and directional hardening aspects of the material, respectively. And the McCauley brackets require the numerator to be greater than zero for gamma dot to be a non-zero value. Now, Ti-64 can either be made up of primary-alpha grains, which are solely an HCP crystal structure, or it can be made up of alpha and beta regions, which is a combination of HCP and BCC crystal structures. This model can handle either. If the grain is deemed a “colony” grain, then we consider it as a homogenized lamellar structure similar to the one shown. The Burger’s orientation relation (BOR) is used to describe the BCC crystal orientation relative to the HCP crystal orientation. Additionally, the CRSS for the basal and selective prismatic slip systems are increased. And the diameter term in the threshold stress is modified to reflect the lath width of the lamellar structure. *Power-law Flow Rule. Backstress evolves according to an Armstrong-Frederick direct hardening/dynamic recovery relation. *Threshold stress is combination of a Hall-Petch strengthening term and a softening term. The Hall-Petch diameter is the mean slip distance in the alpha-phase, whether it is pure alpha or a colony structure. The softening term follows a dynamic recovery. *Drag stress is a function of the CRSS and the initial threshold stress. *The CRSS values are strengthened for colony grains. *The CRSS in compression is also modified to account for the tension-compression asymmetry that has been observed, experimentally. Possibly due to prismatic dislocations dissociating into pyramidal planes.