Investigating the Stored Deformation Energy Distribution in a Polycrystalline Metal using a Dislocation Density-based Crystal Viscoplasticity Theory
Authors
Abstract
The stored deformation energy in the dislocation structures in a polycrystalline metal can provide a sufficient driving force to move grain boundaries during annealing. In this paper, a thermodynamically-consistent three-dimensional, finite-strain and dislocation density-based crystal viscoplasticity constitutive theory has been developed to describe the distribution of stored energy and dislocation density in a polycrystalline metal. The developed constitutive equations have been numerically implemented into the Abaqus finite element package via writing a user material subroutine. The simulations have been performed using both the simple Taylor model and the full micromechanical finite element model. The theory and its numerical implementation are then verified using the available data in literature regarding the physical experiments of the single crystal aluminum. As an application of the developed constitutive model, the relationship between the stored energy and the strain induced grain boundary migration in aluminum polycrystals has been investigated by the Taylor model and also, the full finite element model. The obtained numerical results indicated that the Taylor model could not precisely simulate the distribution of the stored deformation energy within the polycrystalline microstructure; consequently, the strain induced grain boundary migration. This is due to the fact that the strain induced grain boundary migration in a plastically deformed polycrystalline microstructure is principally dependent on the spatial distribution of the stored deformation energy rather than the overall stored energy value.
Keywords
1. Humphreys, M., and Hatherly, F., Recrystallization and Related Annealing Phenomena, Second Edition, Elsevier, 2004.
2. Bever, M., Holt, D., and Titchener, A., “The Stored Energy of Cold Work”, Progress in Materials Science, Vol. 17, pp. 5-177, 1973.
3. Rosakis, P., Rosakis, A., Ravichandran, G., and Hodowany, J., “A Thermodynamic Internal Variable Model for the Partition of Plastic Work into Heat and Stored Energy in Metals”, Journal of the Mechanics and Physics of Solids, Vol. 48, pp. 581-607, 2000.
4. Benzerga, A., Brechet, Y., Needleman, A., and derGiessen, E. V., “The Stored Energy of Cold Work: Predictions from Discrete Dislocation Plasticity”, Acta Materialia, Vol. 53, pp. 4765-4779, 2005.
5. Anand, L., Gurtin, M. E., and Reddy, B. D., “The Stored Energy of Cold Work, Thermal Annealing, and other Thermodynamic Issues in Single Crystal Plasticity at Small Length Scales”, International Journal of Plasticity, Vol. 64, pp. 1-25, 2015.
6. McBride, A., Bargmann, S., and Reddy, B., “A Computational Investigation of a Model of Single-crystal Gradient Thermoplasticity that Accounts for the Stored Energy of Cold Work and Thermal Annealing”, Computational Mechanics, Vol. 55, pp. 755-769, 2015.
7. Anand, L., “Single-crystal Elasto-viscoplasticity: Application to Texture Evolution in Polycrystalline Metals at Large Strains”, Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp. 5359-5383, 2004.
8. Jafari, M., Jamshidian, M., and Ziaei-Rad, S., “A Finite-deformation Dislocation Density-based Crystal Viscoplasticity Constitutive Model for Calculating the Stored Deformation Energy”, International Journal of Mechanical Sciences, Vol. 128-129, pp. 486-498, 2017.
9. Gurtin, M. E., “A Finite-deformation, Gradient Theory of Single-crystal Plasticity with Free Energy Dependent on the Accumulation of Geometrically Necessary Dislocations”, International Journal of Plasticity, Vol. 26, pp. 1073-1096, 2010.
10. Popova, E., Staraselski, Y., Brahme, A., Mishra, R., and Inal, K., “Coupled Crystal Plasticity Probabilistic Cellular Automata Approach to Model Dynamic Recrystallization in Magnesium Alloys”, International Journal of Plasticity, Vol. 66, pp. 85-102, 2015.
11. Kalidindi, S., Bronkhorst, C. A., and Anand, L., “Crystallographic Texture Evolution in Bulk De- formation Processing of FCC Metals”, Journal of the Mechanics and Physics of Solids, Vol. 40, pp. 537-569, 1992.
12. Stojakovic, D., Doherty, R., Kalidindi, S., and Landgraf Fernando J. G., “Thermomechanical Processing for Recovery of Desired ⟨001 ⟩ Fiber Texture in Electric Motor Steels”, Metallurgical and Materials Transactions A, Vol. 39, pp. 1738-1746, 2008.
13. Lele, S. P., and Anand, L., “A Large-deformation Strain-Gradient Theory for Isotropic Viscoplastic Materials”, International Journal of Plasticity, Vol. 25, pp. 420-453, 2009.
14. Fried, E., and Gurtin, M. E., “Dynamic Solid-solid Transitions with Phase Characterized by an Order Parameter”, Physica D: Nonlinear Phenomena, Vol. 72, pp. 287-308, 1994.
15. Gurtin, M. E., Anand, L., and Lele, S. P., “Gradient Single-crystal Plasticity with Free Energy Dependent on Dislocation Densities”, Journal of the Mechanics and Physics of Solids, Vol. 55, pp. 1853-1878, 2007.
16. Lee, M., Lim, H., Adams, B., Hirth, J., and Wagoner, R., “A Dislocation Density-based Single Crystal Constitutive Equation”, International Journal of Plasticity, Vol. 26, pp. 925-938, 2010.
17. Hosford, W., Fleischer, R., and Backofen, W., “Tensile Deformation of Aluminum Single Crystals at Low Temperatures”, Acta Materialia, Vol. 8, pp. 187-199, 1960.
18. Nouri, N., Ziaei-Rad, V., and Ziaei-Rad, S., “An Approach for Simulating Microstructures of Polycrystalline Materials”, Computational Mechanics, Vol. 52, pp. 181-192, 2012.
2. Bever, M., Holt, D., and Titchener, A., “The Stored Energy of Cold Work”, Progress in Materials Science, Vol. 17, pp. 5-177, 1973.
3. Rosakis, P., Rosakis, A., Ravichandran, G., and Hodowany, J., “A Thermodynamic Internal Variable Model for the Partition of Plastic Work into Heat and Stored Energy in Metals”, Journal of the Mechanics and Physics of Solids, Vol. 48, pp. 581-607, 2000.
4. Benzerga, A., Brechet, Y., Needleman, A., and derGiessen, E. V., “The Stored Energy of Cold Work: Predictions from Discrete Dislocation Plasticity”, Acta Materialia, Vol. 53, pp. 4765-4779, 2005.
5. Anand, L., Gurtin, M. E., and Reddy, B. D., “The Stored Energy of Cold Work, Thermal Annealing, and other Thermodynamic Issues in Single Crystal Plasticity at Small Length Scales”, International Journal of Plasticity, Vol. 64, pp. 1-25, 2015.
6. McBride, A., Bargmann, S., and Reddy, B., “A Computational Investigation of a Model of Single-crystal Gradient Thermoplasticity that Accounts for the Stored Energy of Cold Work and Thermal Annealing”, Computational Mechanics, Vol. 55, pp. 755-769, 2015.
7. Anand, L., “Single-crystal Elasto-viscoplasticity: Application to Texture Evolution in Polycrystalline Metals at Large Strains”, Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp. 5359-5383, 2004.
8. Jafari, M., Jamshidian, M., and Ziaei-Rad, S., “A Finite-deformation Dislocation Density-based Crystal Viscoplasticity Constitutive Model for Calculating the Stored Deformation Energy”, International Journal of Mechanical Sciences, Vol. 128-129, pp. 486-498, 2017.
9. Gurtin, M. E., “A Finite-deformation, Gradient Theory of Single-crystal Plasticity with Free Energy Dependent on the Accumulation of Geometrically Necessary Dislocations”, International Journal of Plasticity, Vol. 26, pp. 1073-1096, 2010.
10. Popova, E., Staraselski, Y., Brahme, A., Mishra, R., and Inal, K., “Coupled Crystal Plasticity Probabilistic Cellular Automata Approach to Model Dynamic Recrystallization in Magnesium Alloys”, International Journal of Plasticity, Vol. 66, pp. 85-102, 2015.
11. Kalidindi, S., Bronkhorst, C. A., and Anand, L., “Crystallographic Texture Evolution in Bulk De- formation Processing of FCC Metals”, Journal of the Mechanics and Physics of Solids, Vol. 40, pp. 537-569, 1992.
12. Stojakovic, D., Doherty, R., Kalidindi, S., and Landgraf Fernando J. G., “Thermomechanical Processing for Recovery of Desired ⟨001 ⟩ Fiber Texture in Electric Motor Steels”, Metallurgical and Materials Transactions A, Vol. 39, pp. 1738-1746, 2008.
13. Lele, S. P., and Anand, L., “A Large-deformation Strain-Gradient Theory for Isotropic Viscoplastic Materials”, International Journal of Plasticity, Vol. 25, pp. 420-453, 2009.
14. Fried, E., and Gurtin, M. E., “Dynamic Solid-solid Transitions with Phase Characterized by an Order Parameter”, Physica D: Nonlinear Phenomena, Vol. 72, pp. 287-308, 1994.
15. Gurtin, M. E., Anand, L., and Lele, S. P., “Gradient Single-crystal Plasticity with Free Energy Dependent on Dislocation Densities”, Journal of the Mechanics and Physics of Solids, Vol. 55, pp. 1853-1878, 2007.
16. Lee, M., Lim, H., Adams, B., Hirth, J., and Wagoner, R., “A Dislocation Density-based Single Crystal Constitutive Equation”, International Journal of Plasticity, Vol. 26, pp. 925-938, 2010.
17. Hosford, W., Fleischer, R., and Backofen, W., “Tensile Deformation of Aluminum Single Crystals at Low Temperatures”, Acta Materialia, Vol. 8, pp. 187-199, 1960.
18. Nouri, N., Ziaei-Rad, V., and Ziaei-Rad, S., “An Approach for Simulating Microstructures of Polycrystalline Materials”, Computational Mechanics, Vol. 52, pp. 181-192, 2012.
)