A concise review of Ti64 nanoindentation and CPFEM modelling approaches
Source abstract
Abstract
Abstract Ti–6Al–4V’s dual-phase ( $$\upalpha +\upbeta $$ ) microstructure yields local mechanical behaviors that conventional tests cannot fully resolve. Instrumented nanoindentation delivers spatially resolved hardness, modulus, and creep at micron scales, while crystal plasticity finite-element modeling (CPFEM) translates these data into slip-system insights and component-scale predictions. This review unifies three decades of work on Ti64 nanoindentation and CPFEM, covering fundamental alloy chemistry and microstructures (equiaxed, bimodal, lamellar), advanced indentation protocols (continuous stiffness, spherical tips, high-throughput mapping), and CPFEM developments from phenomenological yield laws to gradient-enhanced, multi-phase simulations reproducing orientation-dependent responses. We spotlight four persisting challenges: defining representative volume elements, capturing size effects via length-scale physics, modeling $$\upalpha /\upbeta $$ interfacial constraints and Burgers orientation relationships, and managing experimental scatter for robust parameter inversion. We also survey emerging solutions—dense indent grids, in situ lattice-strain validation, and machine-learning-driven calibration—as pathways toward predictive, microstructure-sensitive design of Ti64 components. Graphical abstract
Source: crossref · Quality: candidate unverified · Source record ↗
Research context
OpenAlex topics: Metal and Thin Film Mechanics · Titanium Alloys Microstructure and Properties · Microstructure and mechanical properties
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (100+)
Show source-provided references
1. A. Gomez-Gallegos, P. Mandal, D. Gonzalez, N. Zuelli, P. Blackwell, Studies on titanium alloys for aerospace application. DDF 385, 419–423 (2018). https://doi.org/10.4028/www.scientific.net/ddf.385.41910.4028/www.scientific.net/ddf.385.419 ↗
2. K.K. Alaneme, A.A. Fatokun, S.R. Oke, P.A. Olubambi, Nanoindentation studies and analysis of the mechanical properties of TiNb$$2$$O$$_5$$ based composites. Manuf. Rev. 7, 19 (2020). https://doi.org/10.1051/mfreview/202001710.1051/mfreview/2020017 ↗
3. E. Rossi, C. Tromas, Z. Liu et al., Revealing new depths of information with indentation mapping of microstructures. MRS Bull. 50, 715–725 (2025). https://doi.org/10.1557/s43577-025-00919-610.1557/s43577-025-00919-6 ↗
4. V. Tuninetti, A.F. Jaramillo, G. Riu, C. Rojas-Ulloa, A. Znaidi, C. Medina, A.M. Mateo, J.J. Roa, Experimental correlation of mechanical properties of the Ti–6Al–4V alloy at different length scales. Metals 11(1), 104 (2021). https://doi.org/10.3390/met1101010410.3390/met11010104 ↗
5. T. Ohmura, M. Wakeda, Pop-in phenomenon as a fundamental plasticity probed by nanoindentation technique. Materials 14(8), 2021 (1879). https://doi.org/10.3390/ma1408187910.3390/ma14081879 ↗
6. M. Cabibbo, Instrumented nanoindentation tests applied to bulk metallic materials: from calibration issue to pile-up phenomena. Materials (Basel) 14(21), 6360 (2021). https://doi.org/10.3390/ma1421636010.3390/ma14216360 ↗
7. F. Han, B. Tang, H. Kou, J. Li, Y. Feng, Experiments and crystal plasticity finite element simulations of nanoindentation on Ti6Al4V alloy. Mater. Sci. Eng. A 625, 28–35 (2015)10.1016/j.msea.2014.11.090 ↗
8. K. Kapoor, P. Ravi, R. Noraas, J.-S. Park, V. Venkatesh, M.D. Sangid, Modeling Ti6Al4V using crystal plasticity, calibrated with multi-scale experiments, to understand the effect of the orientation and morphology of the a and b phases on time dependent cyclic loading. J. Mech. Phys. Solids 146, 104192 (2021). https://doi.org/10.1016/j.jmps.2020.10419210.1016/j.jmps.2020.104192 ↗
9. Y. Chong, G. Deng, S. Gao, J. Yi, A. Shibata, N. Tsuji, Yielding nature and Hall–Petch relationships in Ti–6Al–4V alloy with fully equiaxed and bimodal microstructures. Scr. Mater. 172, 77–82 (2019). https://doi.org/10.1016/j.scriptamat.2019.07.01510.1016/j.scriptamat.2019.07.015 ↗
10. J.H. Luan, Z.B. Jiao, W.H. Liu, Z.P. Lu, W.X. Zhao, C.T. Liu, Compositional and microstructural optimization and mechanical-property enhancement of cast Ti alloys based on Ti–6Al–4V alloy. Mater. Sci. Eng. A 704, 91–101 (2017). ISSN 0921-5093. https://doi.org/10.1016/j.msea.2017.08.01810.1016/j.msea.2017.08.018 ↗
11. L.R. Zeng, H.L. Chen, X. Li, L.M. Lei, G.P. Zhang, Influence of alloy element partitioning on strength of primary a phase in Ti–6Al–4V alloy. J. Mater. Sci. Technol. 34(5), 782–787 (2018). https://doi.org/10.1016/j.jmst.2017.07.01610.1016/j.jmst.2017.07.016 ↗
12. L. Chai, J. Xia, K.L. Murty, X. Gu, J. Fan, Z. Yao, Revealing microstructural, textural, and hardness evolution of Ti6Al4V sheet cooled from sub-transus temperature at different rates. Metall. Mater. Trans. A 53(8), 3179–3193 (2022). https://doi.org/10.1007/s11661-022-06737-810.1007/s11661-022-06737-8 ↗
13. Y. Ji, T.W. Heo, F. Zhang, J.W. Saunders, M.R. Shankar, Y. Wang, Theoretical assessment on the phase transformation kinetic pathways of multi-component Ti alloys: application to Ti–6Al–4V. J. Phase Equilib. Diffus. 37(1), 53–64 (2016). https://doi.org/10.1007/s11669-015-0436-910.1007/s11669-015-0436-9 ↗
14. S.G. MacLeod, D. Errandonea, G.A. Cox, H. Cynn, D. Daisenberger, S.E. Finnegan, M.I. McMahon, K.A. Munro, C. Popescu, C.V. Storm, The phase diagram of Ti–6Al–4V at high-pressures and high-temperatures. J. Phys. Condens. Matter 33(15), 154001 (2021). https://doi.org/10.1088/1361-648X/abdffa10.1088/1361-648X/abdffa ↗
15. I.A. Naschetnikova, S.I. Stepanov, A.A. Redikultsev, V.Y. Yarkov, M.A. Zorina, M.L. Lobanov, Crystallographic features of phase transformations during the continuous cooling of a Ti6Al4V alloy from the single-phase $$\beta $$-region. Materials (Basel) 15(17), 5840 (2022). https://doi.org/10.3390/ma1517584010.3390/ma15175840 ↗
16. M.A. Galindo-Fernández, K. Mumtaz, P.E.J. Rivera-Díaz-del Castillo, E.I. Galindo-Nava, H. Ghadbeigi, A microstructure sensitive model for deformation of Ti–6Al–4V describing cast-and-wrought and additive manufacturing morphologies. Mater. Des. 160, 350–362 (2018). https://doi.org/10.1016/j.matdes.2018.09.02810.1016/j.matdes.2018.09.028 ↗
17. A. Attanasio, M. Gelfi, A. Pola, E. Ceretti, C. Giardini, Influence of material microstructures in micromilling of Ti6Al4V alloy. Materials (Basel) 6(9), 4268–4283 (2013). https://doi.org/10.3390/ma609426810.3390/ma6094268 ↗
18. R. Pederson, The Microstructures of Ti–6Al–4V and Ti–6Al–2Sn–4Zr–6Mo and Their Relationship to Processing and Properties. PhD Thesis, Luleå Tekniska Universitet, 2004. https://www.diva-portal.org/smash/record.jsf?pid=diva2:990777
19. L. Huang, Z. Sun, J. Cao, Z. Yin, The formation and evolution of macrozone in Ti-6242S alloy during thermo-mechanical processing. J. Alloys Compd. 861, 158533 (2021). https://doi.org/10.1016/j.jallcom.2020.15853310.1016/j.jallcom.2020.158533 ↗
20. P. Bocchetta, L.-Y. Chen, J.D.C. Tardelli, A.C. Reis, F. Almeraya-Calderón, P. Leo, Passive layers and corrosion resistance of biomedical Ti–6Al–4V and $$\beta $$-Ti alloys. Coatings 11(5), 487 (2021). https://doi.org/10.3390/coatings1105048710.3390/coatings11050487 ↗
21. MatWeb, LLC, Titanium Ti–6Al–4V (Grade 5) Alloy, Annealed, Bar, AMS 4928, MIL T 9047, ASTM b348 (MatWeb, LLC, 2024). https://asm.matweb.com/search/specificmaterial.asp?bassnum=mtp641. Accessed 5 June 2025
22. V.A. Klimenov, V.V. Fedorov, M.S. Slobodyan et al., Microstructure and compressive behavior of Ti–6Al–4V alloy built by electron beam free-form fabrication. J. Mater. Eng. Perform. 29, 7710–7721 (2020). https://doi.org/10.1007/s11665-020-05223-910.1007/s11665-020-05223-9 ↗
23. F.J. Gil, M.P. Ginebra, J.M. Manero, J.A. Planell, Formation of $$\alpha $$-Widmanstätten structure: effects of grain size and cooling rate on the Widmanstätten morphologies and on the mechanical properties in Ti6Al4V alloy. J. Alloys Compd. 329(1–2), 142–152 (2001). https://doi.org/10.1016/S0925-8388(01)01571-710.1016/S0925-8388(01)01571-7 ↗
24. S. Dixit, B.B. Dash, D. Kumar, A. Bhattacharjee, S. Sankaran, Influence of phase morphology, static recrystallization, and crystallographic texture on room temperature tensile properties of Ti6Al4V alloy: comparison between post-tested equiaxed, bimodal, and lamellar microstructures. Mater. Sci. Eng. A 873, 144990 (2023). https://doi.org/10.1016/j.msea.2023.14499010.1016/j.msea.2023.144990 ↗
25. T. Seshacharyulu, S.C. Medeiros, W.G. Frazier, Y.V.R.K. Prasad, Microstructural mechanisms during hot working of commercial grade Ti6Al4V with lamellar starting structure. Mater. Sci. Eng. A 325(1–2), 112–125 (2002). https://doi.org/10.1016/S0921-5093(01)01448-410.1016/S0921-5093(01)01448-4 ↗
26. A.V. Dobromyslov, V.A. Elkin, Martensitic transformation and metastable $$\beta $$-phase in binary titanium alloys with $$d$$-metals of 4–6 periods. Scr. Mater. 44(6), 905–910 (2001). https://doi.org/10.1016/S1359-6462(00)00694-110.1016/S1359-6462(00)00694-1 ↗
27. A. Zafari, K. Xia, High ductility in a fully martensitic microstructure: a paradox in a Ti alloy produced by selective laser melting. Mater. Res. Lett. 6(11), 627–633 (2018). https://doi.org/10.1080/21663831.2018.152577310.1080/21663831.2018.1525773 ↗
28. ATI Materials, ATI 6-4 Titanium Alloy (Ti–6Al–4V) Technical Data Sheet (ATI Materials, 2022). https://www.atimaterials.com/Products/Documents/datasheets/titanium/alloyed/ati_6-4_tds_en_v1.pdf. Accessed 5 June 2025
29. AZo Network, Grade 29Ti6Al4V0.1Ru Alloy (UNS R56404) (AZo Network, 2013) https://www.azom.com/article.aspx?ArticleID=9373. Accessed 5 June 2025
30. A. Heldmann, M. Hoelzel, M. Hofmann, W. Gan, W. Schmahl, E. Griesshaber, T. Hansen, N. Schell, W. Pertry, Diffraction-based determination of single-crystal elastic constants of polycrystalline titanium alloys. J. Appl. Crystallogr. 52, 1144–1156 (2019). https://doi.org/10.1107/S160057671901072010.1107/S1600576719010720 ↗
31. M. Kalyan Phani, A. Kumar, T. Jayakumar, W. Arnold, K. Samwer, Mapping of elasticity and damping in an $$\alpha {+}\beta $$ titanium alloy through atomic force acoustic microscopy. Beilstein J. Nanotechnol. 6, 767–776 (2015). https://doi.org/10.3762/bjnano.6.7910.3762/bjnano.6.79 ↗
32. UnionFab, Ti6Al4V Density (UnionFab, 2024). https://www.unionfab.com/blog/2024/03/ti6al4v-density. Accessed 5 June 2025
33. J. Luo, M. Li, W. Yu, H. Li, The variation of strain rate sensitivity exponent and strain hardening exponent in isothermal compression of Ti6Al4V alloy. Mater. Des. 31(2), 741–748 (2010). https://doi.org/10.1016/j.matdes.2009.09.05510.1016/j.matdes.2009.09.055 ↗
34. E. Alabort, P. Kontis, D. Barba, K. Dragnevski, R.C. Reed, On the mechanisms of superplasticity in Ti6Al4V. Acta Mater. 105, 449–463 (2016). https://doi.org/10.1016/j.actamat.2015.12.00310.1016/j.actamat.2015.12.003 ↗
35. B.R. Anne, Y. Okuyama, T. Morikawa, M. Tanaka, Activated slip systems in bimodal Ti6Al4V plastically deformed at low and moderately high temperatures. Mater. Sci. Eng. A 798, 140211 (2020). https://doi.org/10.1016/j.msea.2020.14021110.1016/j.msea.2020.140211 ↗
36. S.L. Semiatin, T.R. Bieler, Effect of texture and slip mode on the anisotropy of plastic flow and flow softening during hot working of Ti–6Al–4V. Metall. Mater. Trans. A 32, 1787–1799 (2001). https://doi.org/10.1007/s11661-001-0155-110.1007/s11661-001-0155-1 ↗
37. R.B.M. Ameen, D.W. Mohammed, Y.-L. Chiu, I.P. Jones, The strength of Ti–6Al–4V investigated using micro-pillars. Micromachines 16(3), 293 (2025). https://doi.org/10.3390/mi1603029310.3390/mi16030293 ↗
38. V. Tuninetti, G. Gilles, O. Milis, T. Pardoen, A.-M. Habraken, Anisotropy and tension compression asymmetry modeling of the room temperature plastic response of Ti6Al4V. Int. J. Plast. 67, 53–68 (2015). https://doi.org/10.1016/j.ijplas.2014.10.00310.1016/j.ijplas.2014.10.003 ↗
39. G.G. Yapici, I. Karaman, Z.-P. Luo, Mechanical twinning and texture evolution in severely deformed Ti6Al4V at high temperatures. Acta Mater. 54(14), 3755–3771 (2006). https://doi.org/10.1016/j.actamat.2006.04.00710.1016/j.actamat.2006.04.007 ↗
40. W.-S. Lee, M.-T. Lin, The effects of strain rate and temperature on the compressive deformation behaviour of Ti–6Al–4V alloy. J. Mater. Process. Technol. 71(2), 235–246 (1997). https://doi.org/10.1016/S0924-0136(97)00080-010.1016/S0924-0136(97)00080-0 ↗
41. A.A. Salem, S.R. Kalidindi, R.D. Doherty, Strain hardening of titanium: role of deformation twinning. Acta Mater. 51(14), 4225–4237 (2003). https://doi.org/10.1016/S1359-6454(03)00239-810.1016/S1359-6454(03)00239-8 ↗
42. T.A. Venkatesh, B.P. Conner, S. Suresh, A.E. Giannakopoulos, An experimental investigation of fretting fatigue in Ti–6Al–4V: the role of contact conditions and microstructure. Metall. Mater. Trans. A 32, 1131–1146 (2001). https://doi.org/10.1007/s11661-001-0124-810.1007/s11661-001-0124-8 ↗
43. A.M. Beese, Microstructure and mechanical properties of AM builds, in Thermo-mechanical Modeling of Additive Manufacturing. ed. by M. Gouge, P. Michaleris (Butterworth-Heinemann, Oxford, 2018), pp.81–92. https://doi.org/10.1016/B978-0-12-811820-7.00007-010.1016/B978-0-12-811820-7.00007-0 ↗
44. N. Singh, P. Hameed, R. Ummethala, G. Manivasagam, K.G. Prashanth, J. Eckert, Selective laser manufacturing of Ti-based alloys and composites: impact of process parameters, application trends, and future prospects. Mater. Today Adv. 8, 100097 (2020). https://doi.org/10.1016/j.mtadv.2020.10009710.1016/j.mtadv.2020.100097 ↗
45. H.P. Tang, P. Zhao, C.S. Xiang, N. Liu, L. Jia, Ti–6Al–4V orthopedic implants made by selective electron beam melting, in Titanium in Medical and Dental Applications. ed. by F.H. Froes, M. Qian. Woodhead Publishing Series in Biomaterials. (Woodhead Publishing, Cambridge, 2018), pp.239–249. https://doi.org/10.1016/B978-0-12-812456-7.00011-110.1016/B978-0-12-812456-7.00011-1 ↗
46. I. Golovin, Nanoindentation and mechanical properties of materials at submicro- and nanoscale levels: recent results and achievements. Phys. Solid State 63, 1–41 (2021). https://doi.org/10.1134/S106378342101010810.1134/S1063783421010108 ↗
47. W.C. Oliver, G.M. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564–1583 (1992). https://doi.org/10.1557/JMR.1992.156410.1557/JMR.1992.1564 ↗
48. B. SridharBabu, A. Kumaraswamy, B. AnjaneyaPrasad, Numerical analysis of effect of friction on mechanical properties of Ti–6Al–4V alloy through nanoindentation. Procedia Eng, 64, 1048–1055 (2013). https://doi.org/10.1016/j.proeng.2013.09.18210.1016/j.proeng.2013.09.182 ↗
49. I. Sen, S. Roy, M.F.-X. Wagner, Indentation response and structure–property correlation in a bimodal Ti6Al4V alloy. Adv. Eng. Mater. 19, 1700298 (2017). https://doi.org/10.1002/adem.20170029810.1002/adem.201700298 ↗
50. Y. Wen, L. Xie, Z. Wang, L. Wang, W. Lu, L.-C. Zhang, Nanoindentation characterization on local plastic response of Ti–6Al–4V under high-load spherical indentation. J. Mater. Res. Technol. 8(4), 3434–3442 (2019). https://doi.org/10.1016/j.jmrt.2019.06.00910.1016/j.jmrt.2019.06.009 ↗
51. H.C. Kaushik, S.I. Shakil, B.S. Amirkhiz, M. Mohammadi, E. Asadi, M. Haghshenas, A. Hadadzadeh, Indentation strain rate sensitivity of laser-powder bed fused and electron beam melted Ti6Al4V. Vacuum 195, 110690 (2022). https://doi.org/10.1016/j.vacuum.2021.11069010.1016/j.vacuum.2021.110690 ↗
52. R. Li, L. Riester, T.R. Watkins, P.J. Blau, A.J. Shih. Metallurgical analysis and nanoindentation characterization of Ti6Al4V workpiece and chips in high-throughput drilling. Mater. Sci. Eng. A 472(1–2), 115–124 (2008). ISSN 0921-5093. https://doi.org/10.1016/j.msea.2007.03.05410.1016/j.msea.2007.03.054 ↗
53. F. Han, B. Tang, X. Yan, L. Liu, Y. Lu, Z. Hu, Indentation pileup behavior of Ti–6Al–4V alloy: experiments and nonlocal crystal plasticity finite element simulations. Metall. Mater. Trans. A 48, 2051–2061 (2017). https://doi.org/10.1007/s11661-016-3946-010.1007/s11661-016-3946-0 ↗
54. A. Hadadzadeh, E. Asadi, S.I. Shakil, B.S. Amirkhiz, M. Mohammadi, M. Haghshenas, Indentation-derived mechanical properties of Ti–6Al–4V: laser-powder bed fusion versus electron beam melting. Mater. Lett. 301, 130273 (2021). https://doi.org/10.1016/j.matlet.2021.13027310.1016/j.matlet.2021.130273 ↗
55. N. Randall, M. Conte, G. Mohanty, J. Michler, J. Schwiedrzik, High temperature nanoindentation up to 800 $$\circ $$C for characterizing high temperature properties of materials, in Nanomechanical Testing in Materials Research and Development VI. ed. by K. Durst. ECI Symposium Series. Conference Chapter (2017)
56. J.M. Wheeler, D.E.J. Armstrong, W. Heinz, R. Schwaiger, High temperature nanoindentation: the state of the art and future challenges. Curr. Opin. Solid State Mater. Sci. 19(6), 354–366 (2015)10.1016/j.cossms.2015.02.002 ↗
57. Y.L. Shen, On the viscoelastic drift behavior during nanoindentation. Front. Mater. 9, 900088 (2022)10.3389/fmats.2022.900088 ↗
58. N.M. Everitt, M.I. Davies, J.F. Smith, High temperature nanoindentation—the importance of isothermal contact. Philos. Mag. 91(7–9), 1221–1244 (2010). https://doi.org/10.1080/14786435.2010.49674510.1080/14786435.2010.496745 ↗
59. J.M. Wheeler, J. Michler, Elevated temperature, nano-mechanical testing in situ in the scanning electron microscope. Rev. Sci. Instrum. 84(4), 045103 (2013). https://doi.org/10.1063/1.479582910.1063/1.4795829 ↗
60. B. Merle, G. Tiphéne, G. Kermouche, Extending nanoindentation testing toward extreme strain rates and temperatures for probing materials evolution at the nanoscale. MRS Bull. 50, 705–714 (2025)10.1557/s43577-025-00918-7 ↗
61. S. Verma, P. Sarkar, P. Pant, Thermal drift in room temperature nanoindentation experiments: measurement and correction. J. Mater. Res. 36(17), 3436–3444 (2021). https://doi.org/10.1557/s43578-021-00407-810.1557/s43578-021-00407-8 ↗
62. J.C. Trenkle, C.E. Packard, C.A. Schuh, Hot nanoindentation in inert environments. Rev. Sci. Instrum. 81(7), 073901 (2010). https://doi.org/10.1063/1.343663310.1063/1.3436633 ↗
63. S. Chattaraj, P. Pant, H. Nanavati, Inter-relationships between mechanical properties of glassy polymers from nanoindentation and uniaxial compression. Polymer 144, 128–141 (2018). https://doi.org/10.1016/j.polymer.2018.04.04010.1016/j.polymer.2018.04.040 ↗
64. S. Korte, W.J. Clegg, Micropillar compression of ceramics at elevated temperatures. Scr. Mater. 60(9), 807–810 (2009). https://doi.org/10.1016/j.scriptamat.2009.01.02910.1016/j.scriptamat.2009.01.029 ↗
65. K.V. Rajulapati, M.M. Biener, J. Biener, A.M. Hodge, Temperature dependence of the plastic flow behavior of tantalum. Philos. Mag. Lett. 90(1), 35–42 (2009). https://doi.org/10.1080/0950083090335689310.1080/09500830903356893 ↗
66. J.M. Wheeler, P. Brodard, J. Michler, Elevated temperature, in situ indentation with calibrated contact temperatures. Philos. Mag. 92(25–27), 3128–3141 (2012). https://doi.org/10.1080/14786435.2012.67464710.1080/14786435.2012.674647 ↗
67. T.P. Weihs, J.B. Pethica, Monitoring time-dependent deformation in small volumes. MRS Online Proc. Libr. 239, 325 (1991). https://doi.org/10.1557/PROC-239-32510.1557/PROC-239-325 ↗
68. K.W. Siu, A.H.W. Ngan, The continuous stiffness measurement technique in nanoindentation intrinsically modifies the strength of the sample. Philos. Mag. 93(5), 449–467 (2013). https://doi.org/10.1080/14786435.2012.72223410.1080/14786435.2012.722234 ↗
69. P.S. Phani, W.C. Oliver, A direct comparison of high temperature nanoindentation creep and uniaxial creep measurements for commercial purity aluminum. Acta Mater. 111, 31–38 (2016)10.1016/j.actamat.2016.03.032 ↗
70. Nanoscience Instruments, Nanoindentation (Nanoscience Instruments, 2025). https://www.nanoscience.com/techniques/nanoindentation/. Accessed 1 Oct 2025
71. L.A. Lumper, G.J. Schaffar, M. Sommerauer, V. Maier-Kiener, In situ microscopy methods for imaging high-temperature microstructural processes—exploring the differences and gaining new potentials. Mater. Sci. Eng. A 887, 145738 (2023). https://doi.org/10.1016/j.msea.2023.14573810.1016/j.msea.2023.145738 ↗
72. C.A. Schuh, C.E. Packard, A.C. Lund, Nanoindentation and contact-mode imaging at high temperatures. J. Mater. Res. 21(3), 725–736 (2006)10.1557/jmr.2006.0080 ↗
73. B. Merle, G. Tiphéne, G. Kermouche, Extending nanoindentation testing toward extreme strain rates and temperatures for probing materials evolution at the nanoscale. MRS Bull. 50(6), 705–714 (2025). https://doi.org/10.1557/s43577-025-00918-710.1557/s43577-025-00918-7 ↗
74. K. Li, Y. Bu, H. Wang, Advances on in situ TEM mechanical testing techniques: a retrospective and perspective view. Front. Mater. 10, 1207024 (2023). https://doi.org/10.3389/fmats.2023.120702410.3389/fmats.2023.1207024 ↗
75. S. Mohan, A.L. Pilchak, S.R. Kalidindi, Use of spherical nanoindentation protocols to study the anisotropic mechanical response of alpha–beta single colonies in Ti6Al4V alloy. Metall. Mater. Trans. A 55(10), 3737–3745 (2024). https://doi.org/10.1007/s11661-024-07844-910.1007/s11661-024-07844-9 ↗
76. T. Pasang et al., Directionally-dependent mechanical properties of Ti6Al4V manufactured by electron beam melting (EBM) and selective laser melting (SLM). Materials 14(13), 3603 (2021). https://doi.org/10.3390/ma1413360310.3390/ma14133603 ↗
77. L.Y. Chen, J.C. Huang, C.H. Lin, C.T. Pan, S.Y. Chen, T.L. Yang, D.Y. Lin, H.K. Lin, J.S.C. Jang, Anisotropic response of Ti–6Al–4V alloy fabricated by 3D printing selective laser melting. Mater. Sci. Eng. A 682, 389–395 (2017). https://doi.org/10.1016/j.msea.2016.11.06110.1016/j.msea.2016.11.061 ↗
78. M.E. Maja, O.E. Falodun, B.A. Obadele, S.R. Oke, P.A. Olubambi, Nanoindentation studies on tin nanoceramic reinforced Ti6Al4V matrix composite. Ceram. Int. 44(4), 4419–4425 (2018). https://doi.org/10.1016/j.ceramint.2017.12.04210.1016/j.ceramint.2017.12.042 ↗
79. L.A. Ávila Calderón, B. Graf, B. Rehmer, T. Petrat, B. Skrotzki, M. Rethmeier, Characterization of Ti6Al4V fabricated by multilayer laser powder-based directed energy deposition. Adv. Eng. Mater. 24, 2101333 (2022). https://doi.org/10.1002/adem.20210133310.1002/adem.202101333 ↗
80. M. Kunert, O. Kienzle, B. Baretzky, Hardness depth profile of a carbon-implanted Ti6Al4V alloy and its relation to composition and microstructure. J. Mater. Res. 16, 23–29 (2001). https://doi.org/10.1557/JMR.2001.031910.1557/JMR.2001.0319 ↗
81. M. Haghshenas, O. Totuk, M. Masoomi, S. Thompson, N. Shamsaei, Small-scale mechanical properties of additively manufactured Ti–6Al–4V, in Proceedings of the 28th Annual International Solid Freeform Fabrication Symposium an Additive Manufacturing Conference, 2017 (2017)
82. W.D. Nix, H. Gao, Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46(3), 411–425 (1998). https://doi.org/10.1016/S0022-5096(97)00068-510.1016/S0022-5096(97)00068-5 ↗
83. B. SridharBabu, A. Kumaraswamy, B. AnjaneyaPrasad, Effect of indentation size and strain rate on nanomechanical behavior of Ti–6Al–4V alloy. Trans. Indian Inst. Met. 68, 143–150 (2015). https://doi.org/10.1007/s12666-014-0438-z10.1007/s12666-014-0438-z ↗
84. S. Demirci, T. Dikici, R. Dalmis, Nanoindentation and corrosion behavior of additively manufactured Ti–6Al–4V alloy for biomaterial applications. J. Mater. Eng. Perform. 33(5), 2133–2145 (2024)10.1007/s11665-023-08139-2 ↗
85. R. Courant, Variational methods for the solution of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 49, 1–23 (1943)10.1090/S0002-9904-1943-07818-4 ↗
86. F. Roters, P. Eisenlohr, L. Hantcherli, D.D. Tjahjanto, T.R. Bieler, D. Raabe, Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: theory, experiments, applications. Acta Mater. 58(4), 1152–1211 (2010). https://doi.org/10.1016/j.actamat.2009.10.05810.1016/j.actamat.2009.10.058 ↗
87. D. Peirce, R.J. Asaro, A. Needleman, An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metall. 30, 1087–1119 (1982)10.1016/0001-6160(82)90005-0 ↗
88. R.J. Asaro, A. Needleman, Texture development and strain hardening in rate dependent polycrystals. Acta Metall. 33, 923–953 (1985)10.1016/0001-6160(85)90188-9 ↗
89. S.V. Harren, H. Devé, R.J. Asaro, Shear band formation in plane strain compression. Acta Metall. 36, 2435–2480 (1988)10.1016/0001-6160(88)90193-9 ↗
90. S.V. Harren, R.J. Asaro, Nonuniform deformations in polycrystals and aspects of the validity of the Taylor model. J. Mech. Phys. Solids 37, 191–232 (1989)10.1016/0022-5096(89)90010-0 ↗
91. R. Becker, Analysis of texture evolution in channel die compression I. Effects of grain interaction. Acta Metall. Mater. 39(6), 1211–1230 (1991). https://doi.org/10.1016/0956-7151(91)90209-J10.1016/0956-7151(91)90209-J ↗
92. R. Becker, J.F. Butler, H. Hu, Analysis of an aluminum single crystal with unstable initial orientation (001) [110] in channel die compression. Metall. Trans. A 22, 45–58 (1991). https://doi.org/10.1007/BF0335094810.1007/BF03350948 ↗
93. D. Peirce, R.J. Asaro, A. Needleman, Material rate dependence and localized deformation in ductile single crystals. Acta Metall. 31, 1051–1076 (1983)10.1016/0001-6160(83)90014-7 ↗
94. S.R. Kalidindi, C.A. Bronkhorst, L. Anand, Crystallographic texture evolution in bulk deformation processing of FCC metals. J. Mech. Phys. Solids 40(3), 537–569 (1992). https://doi.org/10.1016/0022-5096(92)80003-910.1016/0022-5096(92)80003-9 ↗
95. A.M. Cuitino, M. Ortiz, Computational modelling of single crystals. Model. Simul. Mater. Sci. Eng. 1(3), 225–263 (1993). https://doi.org/10.1088/0965-0393/1/3/00110.1088/0965-0393/1/3/001 ↗
96. L. Anand, M. Kothari, A computational procedure for rate-independent crystal plasticity. J. Mech. Phys. Solids 44(4), 525–558 (1996). https://doi.org/10.1016/0022-5096(96)00001-410.1016/0022-5096(96)00001-4 ↗
97. H. Gao, Y. Huang, Geometrically necessary dislocation and size-dependent plasticity. Scr. Mater. 48(2), 113–118 (2003). https://doi.org/10.1016/S1359-6462(02)00329-910.1016/S1359-6462(02)00329-9 ↗
98. L.P. Evers, W.A.M. Brekelmans, M.G.D. Geers, Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. Int. J. Solids Struct. 41(10), 5209–5230 (2004)10.1016/j.ijsolstr.2004.04.021 ↗
99. S.R. Kalidindi, Modeling anisotropic strain hardening and deformation textures in low stacking fault energy FCC metals. Int. J. Plast. 17, 837–860 (2001)10.1016/S0749-6419(00)00071-1 ↗
100. S.R. Kalidindi, Incorporation of deformation twinning in crystal plasticity models. J. Mech. Phys. Solids 46, 267–290 (1998)10.1016/S0022-5096(97)00051-3 ↗