Abstract
We develop a nonlinear, three-dimensional phase field model for crystal plasticity which accounts for the infinite and discrete symmetry group \(G\) of the underlying periodic lattice. This generates a complex energy landscape with countably-many \(G\)-related wells in strain space, whereon the material evolves by energy minimization under the loading through spontaneous slip processes inducing the creation and motion of dislocations without the need of auxiliary hypotheses. Multiple slips may be activated simultaneously, in domains separated by a priori unknown free boundaries. The wells visited by the strain at each position and time, are tracked by the evolution of a \(G\)-valued discrete plastic map, whose non-compatible discontinuities identify lattice dislocations. The main effects in the plasticity of crystalline materials at microscopic scales emerge in this framework, including the long-range elastic fields of possibly interacting dislocations, lattice friction, hardening, band-like vs. complex spatial distributions of dislocations. The main results concern the scale-free intermittency of the flow, with power-law exponents for the slip avalanche statistics which are significantly affected by the symmetry and the compatibility properties of the activated fundamental shears.
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Notes
As for instance in [18], we do not include here any interface-penalizing gradient term in the free-energy functional because in all our simulations we keep the typical distance \(H\) between interpolation nodes always greater than or equal to the minimal physically relevant length \(h\) in the continuum, which in the present context is taken of the order the lattice cell spacing. To model plastic behavior of crystals at scales \(H\) larger than \(h\), besides including a gradient term in the phase field potential, the Schmidt tensors in (1) must also be suitably re-scaled (see for instance [35]). The corresponding scaling imposed on the potential \(\psi_{\mu}\) determines whether in the limit \(H\gg h\) the yield and Peierls stresses vanish or stay finite. In what follows we only consider plasticity at microscales with \(H\simeq h\).
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We acknowledge financial support from the Italian PRIN Contract 200959L72B004 and from a SAES Getters - Politecnico di Milano research contract.
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Biscari, P., Urbano, M.F., Zanzottera, A. et al. Intermittency in Crystal Plasticity Informed by Lattice Symmetry. J Elast 123, 85–96 (2016). https://doi.org/10.1007/s10659-015-9548-z
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DOI: https://doi.org/10.1007/s10659-015-9548-z