Analytical and numerical bifurcation analysis of dislocation pattern formation of the Walgraef–Aifantis model. (June 2018)

• Record Type: Journal Article
• Authors: Spiliotis, Konstantinos G.; Russo, Lucia; Siettos, Constantinos; Aifantis, Elias C.
• Is Part Of: International journal of non-linear mechanics. Volume 102(2018)
• Page Start: 41
• Page End: 52
• Publication Date: 2018-06
• Subjects: Dislocations -- Pattern formation -- Turing instabilities -- Numerical analysis; Nonlinear mechanics -- Periodicals; Mécanique non linéaire -- Périodiques; Nonlinear mechanics; Periodicals; 531
• DOI: 10.1016/j.ijnonlinmec.2018.03.002
• Languages: English
• ISSNs: 0020-7462

Abstract

Abstract: We analyze the pattern formation due to dislocations under cyclic loading resulting from the Walgraef–Aifantis model. The model consists of a set of partial differential equations of the reaction–diffusion type in the one dimensional finite space with two different diffusion-like coefficients, for the mobile (free to move when the applied resolved shear stress in the slip plane exceeds a certain threshold) and for the immobile (of slow movement or trapped) dislocations. We derive analytically the Turing spatial and Andronov–Hopf temporal instabilities emanating from the homogeneous solutions and construct the complete bifurcation diagram of the far-from-equilibrium spatio-temporal patterns, with respect to the applied stress and the size of the domain. Finally, we analyze the symmetric properties of all branches of both steady and oscillating far-from-equilibrium regimes. Highlights: We analyse analytically and numerically dislocation pattern formation. We derive analytically the Turing spatial and Andronov–Hopf instabilities We perform numerical bifurcation analysis to compute far-from-equilibirum patterns. We analyse the symmetric properties of the dislocation patterns.