Analytical and numerical bifurcation analysis of dislocation pattern formation of the Walgraef–Aifantis model

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© 2018 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.

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International Journal of Non-Linear Mechanics