Numerical study of the size effect and nucleation center concentration on the elastic properties of copper
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Abstract
This article presents the results of a numerical study on the influence of representative volume size and the concentration of initial nucleation centers on the elastic properties of polycrystalline copper, taking into account the anisotropy of its microstructure. The modeling was carried out in a voxel-based system using the finite element method within a numerical analysis environment, applying periodic boundary conditions. This approach enables the simulation of an infinite material structure and avoids edge effects. Three-dimensional geometric models of cubic specimens ranging in size from 5 to 20 voxels were constructed at various concentrations of crystallization points (from 5% to 25%). For each configuration, 100 independent statistical realizations with randomly distributed initial points were performed. This made it possible to obtain reliable estimates of the mean values and variances of the key mechanical properties. The effective values of Young’s modulus, shear modulus, and Poisson’s ratios were calculated depending on geometric parameters and microstructural variability. The distributions of the obtained values were analyzed using histograms and statistical parameters of the normal distribution. The results showed that increasing the size of the representative volume and the number of crystallization points reduces result variability and leads to convergence towards a stationary mean value. The maximum deviation of Young’s modulus across all realizations did not exceed 1.91%, indicating high stability and reproducibility of the calculated properties. It is shown that accounting for anisotropy through an appropriate grain orientation model enables significantly more accurate simulation of polycrystalline material behavior under various loading conditions. The proposed approach can be useful for improving the accuracy of numerical predictions of elastic properties in copper-based components and optimizing technological processes that shape the microstructure. Furthermore, the results can be applied in the design of structural elements with predictable stiffness characteristics, taking into account the internal structure and its influence on the macroscopic behavior of the material.
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References
[1] R. Kumar, M.K. Gupta, S.K. Rai, V. Panwar, Grain size responsive uniaxial tensile behavior of polycrystalline nanocopper under different temperatures and strain rates, Multidiscipline Modeling in Materials and Structures. 19 (2023) 507–521. https://doi.org/10.1108/MMMS-09-2022-0187
[2] G.W. Nieman, J.R. Weertman, R.W. Siegel, Mechanical Behavior of Nanocrystalline Cu and Pd, J. Mater. Res. 6 (1991) 1012–1027. https://doi.org/10.1557/JMR.1991.1012
[3] M.Y. Shen, J.R. Lloyd, Modeling the Copper Microstructure and Elastic Anisotropy and Studying Its Impact on Reliability in Nanoscale Interconnects, Mech. Adv. Mater. Mod. Process. 3 (2017) 6. https://doi.org/10.1186/s40759-017-0021-z
[4] Y. Geng, Y. Ban, B. Wang, X. Li, K. Song, Y. Zhang, Y. Jia, B. Tian, Y. Liu, A. Volinsky, Review of Microstructure and Texture Evolution with Nanoscale Precipitates for Copper Alloys, J. Mater. Res. Technol. 9 (2020) 11918–11934. https://doi.org/10.1016/j.jmrt.2020.08.055
[5] T. Böhlke, A. Bertram, Asymptotic Values of Elastic Anisotropy in Polycrystalline Copper for Uniaxial Tension and Compression, Int. J. Solids Struct. 91 (2016) 123–134.
[6] Z. Li, K. Chen, X. Zhu, Y. Chen, M. Wang, Y. Shen, J. Shi, J. Wang, Z. Wang, In-Situ Fabrication, Microstructure and Mechanical Performance of Nano Iron-Rich Precipitate Reinforced Cu and Cu Alloys, Metals. 12 (2022) 1453. https://doi.org/10.3390/met12091453
[7] M. Xie, W. Huang, H. Chen, L. Gong, W. Xie, H. Wang, B. Yang, Microstructural Evolution and Strengthening Mechanisms in Cold-Rolled Cu–Ag Alloys, J. Alloys Compd. 851 (2021) 156893. https://doi.org/10.1016/j.jallcom.2020.156893
[8] L. Ladani, J. Razmi, T.C. Lowe, Manufacturing of High Conductivity, High Strength Pure Copper with Ultrafine Grain Structure, J. Manuf. Mater. Process. 7 (2023) 137. https://doi.org/10.3390/jmmp7040137
[9] C. Saldana, A.H. King, S. Chandrasekar, Thermal Stability and Strength of Deformation Microstructures in Pure Copper, Acta Mater. 60 (2012) 4107–4116. https://doi.org/10.1016/j.actamat.2012.04.022
[10] C.F. Gu, C.H.J. Davies, Thermal Stability of Ultrafine-Grained Copper During High Speed Micro-Extrusion, Mater. Sci. Eng. A. 527 (2010) 1791–1799. https://doi.org/10.1016/j.msea.2009.11.005
[11] V.M. Senkivskyi, Z.I. Pikh, N.V. Yavorska, M. Lewandowska, Modeling of Fatigue Crack Growth in Shape Memory Alloys, Sci. J. Ternopil Natl. Tech. Univ. 98 (2020) 5–14 https://doi.org/10.33108/visnyk_tntu2020.03.005
[12] S.D. Tekade, K.N. Patil, Physical properties of Cu nanoparticles: A molecular dynamics study, in: Proceedings of the International Conference on Nanotechnology (ICN), 2013, pp. 1–5.. https://doi.org/10.1016/j.matchemphys.2014.04.030
[13] Z. Liu, S. Wu, Y. Li, Mechanical Behavior Analysis of Porous Materials Using Monte Carlo Simulation, Appl. Sci. 12 (2022) 575. https://doi.org/10.3390/app12020575
[14] L. Wang, X. Liu, J. Zhang, Microstructure-based numerical modeling of elastic anisotropy in polycrystalline copper, Mater. Charact. 202 (2023) 113028. https://doi.org/10.1016/j.matchar.2023.113142
[15] A. Vázquez, A. Alonso, C. Castro, J. León, Voxel-Based Modeling for the Mechanical Characterization of 3D-Printed Structures, Materials. 14 (2021) 5670. https://doi.org/10.3390/ma14195670
[16] H.M. Ledbetter, E.R. Naimon, Elastic Properties of Metals and Alloys, II. Copper, J. Phys. Chem. Ref. Data. 3 (1974) 897–935.