Influence of Dislocations on the Deformability of Metallic Sheets

Abstract:

Within the constitutive framework adopted here, the plastic distortion is described by multislips in the appropriate crystallographic system, the dislocation densities $\rho^{\alpha}$ and hardening variables $\zeta^{\alpha}$ in the $\alpha-$slip system are the internal variables involved in the model. The rate type boundary value problem at time $t$ leads to an appropriate variational equality to be satisfied by the velocity field when the current state of the body is known. Numerical solutions are analyzed in a tensile problem when only two physical slip systems are activated in the single (fcc) crystal sheet. The slip directions are in the plane of the sheet, while the normals to the slip planes are spatially represented. At the initial moment the distribution of the dislocation density is localized in a central zone of the sheet and in the tensile problem no geometrical imperfection has been introduced. The plane stress state is compatible with the rate type constitutive formulation of the model. The FEM is applied for solving the variational problem in the actual configuration, together with a temporal discretization of the differential system to update the current state in the sheet. The activation condition, which is formulated in terms of Schmid's law, allows us to describe the spread of the plastically deformed zone on the sheet.

• Periodical: Key Engineering Materials (Volumes 554-557)
• Pages: 99-109
• DOI: https://doi.org/10.4028/www.scientific.net/KEM.554-557.99
• Online since: June 2013
• Authors: Sanda Cleja-Ţigoiu, Raisa Paşcan
• Keywords: Dislocation Density, Finite Crystal Plasticity, Hardening Variables, Local Evolution Equations, Metallic Sheets, Non-Local Evolution Equations, Numerical Simulation, Update Algorithm, Variational Problem

References

[1] L. Bortoloni and P. Cermelli: Journal of Elasticity 76 (2004), p.113

[2] M. Kuroda: Int. J. Solid Struct. 48 (2011), p.3382

[3] M. Kuroda and V. Tvergaard: Int. J. Solid Struct. 56 (2008), p.2573

[4] S. Cleja-Ţigoiu and E. Soós: Applied Mechanics Reviews 43 (1990), p.131

[5] S. Cleja-Ţigoiu: European Journal of Mechanics - A/Solids (1996), p.761

[6] S. Cleja-Ţigoiu: Rendiconti del seminario matematico, Univ. Pol. Torino, 58 (2000), p.69

[7] S. Cleja-Ţigoiu and R. Paşcan: Mathematics and Mechanics of Solids (2012).

DOI: 10.1177/1081286512439060

[8] C. Teodosiu and J.L. Raphanel, in Large Plastic Deformations, Fundamental Aspects and Applications to Metal Forming, edited by Teodosiu, Raphanel and Sidoroff, Brookfiels, Rotterdam (1993).

DOI: 10.1201/9780203749173

[9] C. Teodosiu and F. Sidoroff: Int. J. Engn. Sci. 14 (1976), p.165

[10] T.C.T. Ting, Anisotropic Elasticity (Oxford University Press 1996).

[11] S. Ha, K. Kim: Mathematics and Mechanics of Solids, (2011), p.651

[12] S. Cleja-Ţigoiu, R. Paşcan and N. Stoicuţă, in Inverse problems and computational mechanics, edited by L. Marin, L. Munteanu, V. Chiroiu, Ed. Academiei Române (2011)

[13] T.J.R. Hughes, The Finite Element Method. Linear Static and Dynamic Finite Element Analysis (Prentice-Hall, Inc., Englewood Cliffs, New Jersey 07632, 1987).

DOI: 10.1177/058310247700900109

[14] L.J. Segerlind, Applied Finite Element Analysis (John Wiley and Sons, New York 1984).