Finite element modeling of reduced grain boundaries elasticity in nanocrystalline metals
DOI:
https://doi.org/10.7242/1999-6691/2025.18.2.14Keywords:
grain, grain boundary, nanocrystalline metals, representative volume elements, weakening coefficient, Young's modulusAbstract
Nanocrystalline metals consist of two distinct phases: the crystalline phase, namely grains, and the intercrystalline phase, which includes grain boundaries, triple junctions, and quadruple nodes. Weakening of elasticity in the intercrystalline phase of nanocrystalline metals, especially in grain boundaries, causes a decrease in the overall elastic modulus. Consequently, studying the elastic behavior and calculating the elasticity of grain boundaries are critical to the understanding of nanocrystalline metals. The purpose of this research is to model the elasticity of grain boundaries in nanocrystalline metals and calculate it. For this purpose, five different metal samples with different crystalline structures are considered. For each sample, three representative volume elements (RVEs) with different grain sizes and constant grain boundary thickness are modeled. The behavior of the crystalline phase is assumed to be elastic with cubic symmetry, while the behavior of grain boundaries is assumed to be elastically isotropic. Uniaxial tension is then simulated using a finite element analysis to calculate the Young's modulus of the RVEs. The weakening coefficient for grain boundaries is obtained through this analysis. To verify the validity of this coefficient, the Young's modulus of the simulated RVEs is compared with the Young's modulus extracted from molecular dynamics simulation and experiments reported in the literature.
Downloads
References
Forrest R.M., Lazar E.A., Goel S., Bean J.J. Quantifying the differences in properties between polycrystals containing planar and curved grain boundaries. Nanofabrication. 2022a. Vol. 7. P. 11–23. DOI: 10.37819/nanofab.007.250
Valat-Villain P., Durinck J., Renault P.O. Grain Size Dependence of Elastic Moduli in Nanocrystalline Tungsten. Journal of Nanomaterials. 2017a. Vol. 2017. DOI: 10.1155/2017/3620910
Latapie A., Farkas D. Effect of grain size on the elastic properties of nanocrystalline α-iron. Scripta Materialia. 2003a. Vol. 48, no. 5. P. 611–615. DOI: 10.1016/S1359-6462(02)00467-0
Wang N., Wang Z., Aust K.T., Erb U. Effect of grain size on mechanical properties of nanocrystalline materials. Acta Metallurgica et Materialia. 1995a. Vol. 43, no. 2. P. 519–528. DOI: 10.1016/0956-7151(94)00253-E
Pan Z., Li Y., Wei Q. Tensile properties of nanocrystalline tantalum from molecular dynamics simulations. Acta Materialia. 2008a. Vol. 56, no. 14. P. 3470–3480. DOI: 10.1016/j.actamat.2008.03.025
Xu W., Dávila L.P. Size dependence of elastic mechanical properties of nanocrystalline aluminum. Materials Science and Engineering: A. 2017a. Vol. 692. P. 90–94. DOI: 10.1016/j.msea.2017.03.065
Zheng L., Xu T.-D. Method for determining the elastic modulus at grain boundaries for polycrystalline materials. Materials Science and Technology. 2004a. Vol. 20, no. 5. P. 605–609. DOI: 10.1179/026708304225012017
Zhu L., Zheng X. Influence of interface energy and grain boundary on the elastic modulus of nanocrystalline materials. Acta Mechanica. 2010a. Vol. 213, no. 3/4. P. 223–234. DOI: 10.1007/s00707-009-0263-3
Lai W.M., Rubin D., Krempl E. Introduction to continuum mechanics. Butterworth-Heinemann, 2009a
Diana V. Anisotropic Continuum-Molecular Models: A Unified Framework Based on Pair Potentials for Elasticity, Fracture and Diffusion-Type Problems. Archives of Computational Methods in Engineering. 2023a. Vol. 30, no. 2. P. 1305–1344. DOI: 10.1007/s11831-022-09846-0
Yeheskel O., Chaim R., Shen Z., Nygren M. Elastic moduli of grain boundaries in nanocrystalline MgO ceramics. Journal of Materials Research. 2005a. Vol. 20, no. 3. P. 719–725. DOI: 10.1557/JMR.2005.0094
Kim T.- Y., Dolbow J.E., Fried E. Numerical study of the grain-size dependent Young’s modulus and Poisson’s ratio of bulk nanocrystalline materials. International Journal of Solids and Structures. 2012a. Vol. 49, no. 26. P. 3942–3952. DOI: 10.1016/j.ijsolstr.2012.08.023
Borokinni A.S., Akinola A.P., Layeni O.P., Fadodun O.O. A new strain-gradient theory for an isotropic plastically deformed polycrystalline solid body. Mathematics and Mechanics of Solids. 2018a. Vol. 23, no. 9. P. 1333–1344. DOI: 10.1177/1081286517720842
Gurtin M.E., Murdoch A.I. A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis. 1975a. Vol. 57. P. 291–323. DOI: 10.1007/BF00261375
Zhou J., Li Y., Zhu R., Zhang Z. The grain size and porosity dependent elastic moduli and yield strength of nanocrystalline ceramics. Materials Science and Engineering: A. 2007a. Vol. 445/446. P. 717–724. DOI: 10.1016/j.msea.2006.10.005
Sanders P.G., Eastman J.A., Weertman J.R. Elastic and tensile behavior of nanocrystalline copper and palladium. Acta Materialia. 1997a. Vol. 45, no. 10. P. 4019–4025. DOI: 10.1016/S1359-6454(97)00092-X
Schiøtz J., Di Tolla F.D., Jacobsen K.W. Softening of nanocrystalline metals at very small grain sizes. Nature. 1998a. Vol. 391, no. 6667. P. 561–563. DOI: 10.1038/35328
Suryanarayana C., Froes F.H. The structure and mechanical properties of metallic nanocrystals. Metallurgical Transactions A. 1992a. Vol. 23. P. 1071–1081. DOI: 10.1007/BF02665039
Korn D., Morsch A., Birringer R., Arnold W., Gleiter H. Measurements of the elastic constants, the specific heat and the entropy of grain boundaries by means of ultra-fine grained materials. Le Journal de Physique Colloques. Vol. 49. 1988a. P. 769–779. DOI: 10.1051/jphyscol:1988596
Trelewicz J.R., Schuh C.A. Grain boundary segregation and thermodynamically stable binary nanocrystalline alloys. Physical Review B. 2009a. Vol. 79, no. 9. P. 094112. DOI: 10.1103/PhysRevB.79.094112
Wei Y., Su C., Anand L. A computational study of the mechanical behavior of nanocrystalline fcc metals. Acta Materialia. 2006a. Vol. 54, no. 12. P. 3177–3190. DOI: 10.1016/j.actamat.2006.03.007
Shimokawa T., Nakatani A., Kitagawa H. Grain-size dependence of the relationship between intergranular and intragranular deformation of nanocrystalline Al by molecular dynamics simulations. Physical Review B. 2005a. Vol. 71, no. 22. P. 224110. DOI: 10.1103/PhysRevB.71.224110
Kuleyev I.I., Kuleyev I.G. Dynamic Properties and Focusing of Phonons in Metallic and Dielectric Crystals of Cubic Symmetry. Review 1. Physics of Metals and Metallography. 2023a. Vol. 124, S1. P. S2–S31. DOI: 10.1134/S0031918X23601993
Lord E.A., Mackay A.L. Periodic minimal surfaces of cubic symmetry. Current Science. 2003a. P. 346–362. DOI: 10.2307/24108665
Hürlimann T. Index notation in mathematics and modelling language LPL: theory and exercices. Fribourg, Switzerland: Department of Informatics University of Fribourg, 2007a
Povey R.G. Voigt transforms. 2023a. URL: https://rhyspovey.com/science/voigt.pdf
Ben-Ari M. A tutorial on euler angles and quaternions. Israel, 2014a. URL: https://raw.githubusercontent.com/motib/mathematics/master/quaternions/quaternion-tutorial.pdf
Diebel J. Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix. 2006a. Vol. 58, no. 15/16. P. 1–35.
Hahn H., Mondal P., Padmanabhan K.A. Plastic deformation of nanocrystalline materials. Nanostructured Materials. 1997a. Vol. 9, no. 1–8. P. 603–606. DOI: 10.1016/S0965-9773(97)00135-9
Shunmugesh K., Raphel A., Unnikrishnan T.G., Akhil K.T. Finite element modelling of carbon fiber reinforced with vespel and honey-comb structure. Materials Today: Proceedings. 2023a. Vol. 72. P. 2163–2168. DOI: 10.1016/j.matpr.2022.08.301
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Computational Continuum Mechanics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
