Relaxation of the Non-Convex, Incremental Energy-Minimization Problem in Single-Slip Strain-Gradient Plasticity

Abstract:

We consider a variational formulation of gradient elasto-plasticity, as they arise in the incremental formulation of the plastic evolution problem, subject to a class of single-slip side conditions. Such side conditions typically render the associated boundary-value problems non-convex. We first show that, for a large class of plastic deformations, a given single-slip condition (specification of Burgers' vectors and slip planes) can be relaxed by introducing a lamination microstructure. This yields a relaxed side condition which allows for arbitrary slip in a prescribed family of slip planes. This relaxed model can be thought of as an aid to simulating macroscopic plastic behavior without the need to resolve arbitrarily fine spatial scales. We also discuss issues of existence of solutions for the relaxed model.

• Periodical: Key Engineering Materials (Volumes 651-653)
• Pages: 963-968
• DOI: https://doi.org/10.4028/www.scientific.net/KEM.651-653.963
• Online since: July 2015
• Authors: Keith Anguige*, Patrick W. Dondl
• Keywords: Calculus of Variations, Microstructure, Non-Convex Energies, Relaxation, Single Slip, Strain-Gradient Plasticity

References

[1] Hansen, H., 1990. Cold deformation microstrucutres. Materials science and technology 6(11), 1039-1047.

[2] Hughes, D. A., Liu, Q., Chrzan, D. C. and Hansen, N., 1997. Scaling of microstructural parameters: Misorientations of deformation induced boundaries. Acta Materialia 45(1), 105-112.

DOI: 10.1016/s1359-6454(96)00153-x

[3] Hughes, D. A., Liu, Q., Chrzan, D. C. and Hansen, N., 1998. Scaling of Misorientation Angle Distributions. Physical Review Letters 81(21), 4664-4667.

DOI: 10.1103/physrevlett.81.4664

[4] Jin, N.Y. and Winter, A. T., 1984a. Dislocation structures in cyclically deformed.

[1] copper crystals. Acta Metallurgica 32, 1173-1176.

[5] Nix, W. D., Gibeling J. C. and Hughes, D. A., 1985. Time-dependent deformation of metals. Metallurgical Transactions A 16(12), 2215-2226.

DOI: 10.1007/bf02670420

[6] Ball, J. M., and James, R. D. (1987). Fine Phase Mixtures as Minimizers of Energy. Archive for Rational Mechanics and Analysis, 100(1), 13-52.

DOI: 10.1007/bf00281246

[7] Dondl, P., Dmitrieva, O., Müller, S. and Raabe, D., 2009. Lamination microstructure in sheardeformed copper single crystals, Acta Materialia 57, 3439-3449.

DOI: 10.1016/j.actamat.2009.03.035

[8] Ortiz, M. and Repetto, E., 1999. Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47, 397-462.

DOI: 10.1016/s0022-5096(97)00096-3

[9] Kondo, K., 1952. On the geometrical and physical foundations of the theory of yielding. Proceedings Japan National Congress of Applied Mechanics, vol. 2, 41-47.

[10] Kondo, K., 1955. Non-Riemannian geometry of imperfect crystals from a macroscopic viewpoint. In: Kondo, K. (Ed. ), RAAG Memoirs of the Unifying Study of Basic Problems in Engineering and Physical Science by Means of Geometry, vol. 1. Gakuyusty Bunken Fukin-Kay, Tokyo.

[11] Nye, J. F., 1953. Some geometrical relations in dislocated solids. Acta Metall. 1, 153-162.

[12] Fokoua, L., Conti, S. and Ortiz, M., 2014. Optimal scaling laws for ductile fracture derived from strain-gradient microplasticity. J. Mech. Phys. Solids 62, 295-311.

DOI: 10.1016/j.jmps.2013.11.002

[13] Conti, S., and Ortiz, M., 2005. Dislocation Microstructures and the Effective Behavior of Single Crystals. Archive for Rational Mechanics and Analysis, 176(1), 103-147.

DOI: 10.1007/s00205-004-0353-2

[14] Mielke, A., and Müller, S., 2006. Lower semicontinuity and existence of minimizers in incremental finite-strain elastoplasticity. ZAMM, 86(3), 233-250.

DOI: 10.1002/zamm.200510245

[15] Anguige, K. and Dondl, P., 2013. Optimal energy scaling for a shear experiment in single-crystal plasticity with cross-hardening. Zeitschrift für angewandte Mathematik und Physik, online first.

DOI: 10.1007/s00033-013-0379-0

[16] Yalcinkaya, T., Brekelmans, W.A.M., Geers, M.G.D., 2011. Deformation patterning driven by rate dependent non-convex strain gradient plasticity. J. Mech. Phys. Solids. 59, 1-17.

DOI: 10.1016/j.jmps.2010.10.002

[17] Yalcinkaya, T., Brekelmans, W.A.M., Geers, M.G.D., 2012. Non-convex rate dependent strain gradient crystal plasticity and deformation patterning. International Journal of Solids and Structures 49, 2625-2636.

DOI: 10.1016/j.ijsolstr.2012.05.029

[18] Klusemann, B., Yalcinkaya, T., 2013. Plastic deformation induced microstructure evolution through gradient enhanced crystal plasticity based on a non-convex helmholtz energy. International Journal of Plasticity 48, 168-188.

DOI: 10.1016/j.ijplas.2013.02.012

[19] Klusemann, B., Yalcinkaya, T., Geers, M.G.D., Svendsen, B., 2013. Application of non-convex rate dependent gradient plasticity to the modeling and simulation of inelastic microstructure development and inhomogeneous material behavior. Computational Materials Science 80, 51-60.

DOI: 10.1016/j.commatsci.2013.04.016

[20] Anguige, K. and Dondl, P., 2014. Relaxation of the single-slip condition in strain-gradient plasticity. accepted for publication in Proc R Soc A.

DOI: 10.1098/rspa.2014.0098

[21] Devincre, B., Hoc, T., and Kubin, L. P., 2005. Collinear interactions of dislocations and slip systems. Materials Science and Engineering: A, 400-401, 182-185.

DOI: 10.1016/j.msea.2005.02.071

[22] Devincre, B., Kubin, L., and Hoc, T., 2007. Collinear superjogs and the low-stress response of fcc crystals. Scripta Materialia, 57(10), 905-908.

DOI: 10.1016/j.scriptamat.2007.07.026

[23] Hildebrand, F. and Miehe, C., 2012. Variational phase field modeling of laminate deformation microstructure in finite gradient crystal plasticity. Proc. Appl. Math. Mech. 12, 37-40.

DOI: 10.1002/pamm.201210011

[24] Anguige, K. and Dondl, P., forthcoming.