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An Exactly Solvable Phase-Field Theory of Dislocation Dynamics, Strain Hardening and Hysteresis in Ductile Single Crystals
An exactly solvable phase-field theory of dislocation dynamics, strain
hardening and hysteresis in ductile single crystals is developed. The theory
accounts for: an arbitrary number and arrangement of dislocation lines over a
slip plane; the long-range elastic interactions between dislocation lines; the
core structure of the dislocations resulting from a piecewise quadratic Peierls
potential; the interaction between the dislocations and an applied resolved
shear stress field; and the irreversible interactions with short-range
obstacles and lattice friction, resulting in hardening, path dependency and
hysteresis. A chief advantage of the present theory is that it is analytically
tractable, in the sense that the complexity of the calculations may be reduced,
with the aid of closed form analytical solutions, to the determination of the
value of the phase field at point-obstacle sites. In particular, no numerical
grid is required in calculations. The phase-field representation enables
complex geometrical and topological transitions in the dislocation ensemble,
including dislocation loop nucleation, bow-out, pinching, and the formation of
Orowan loops. The theory also permits the consideration of obstacles of varying
strengths and dislocation line-energy anisotropy. The theory predicts a range
of behaviors which are in qualitative agreement with observation, including:
hardening and dislocation multiplication in single slip under monotonic
loading; the Bauschinger effect under reverse loading; the fading memory
effect, whereby reverse yielding gradually eliminates the influence of previous
loading; the evolution of the dislocation density under cycling loading,
leading to characteristic `butterfly' curves; and others
Multi-Phase Equilibrium of Crystalline Solids
A continuum model of crystalline solid equilibrium is presented in which the
underlying periodic lattice structure is taken explicitly into account. This
model also allows for both point and line defects in the bulk of the lattice
and at interfaces, and the kinematics of such defects is discussed in some
detail. A Gibbsian variational argument is used to derive the necessary bulk
and interfacial conditions for multi-phase equilibrium (crystal-crystal and
crystal-melt) where the allowed lattice variations involve the creation and
transport of defects in the bulk and at the phase interface. An interfacial
energy, assumed to depend on the interfacial dislocation density and the
orientation of the interface with respect to the lattices of both phases, is
also included in the analysis. Previous equilibrium results based on nonlinear
elastic models for incoherent and coherent interfaces are recovered as special
cases for when the lattice distortion is constrained to coincide with the
macroscopic deformation gradient, thereby excluding bulk dislocations. The
formulation is purely spatial and needs no recourse to a fixed reference
configuration or an elastic-plastic decomposition of the strain. Such a
decomposition can be introduced however through an incremental elastic
deformation superposed onto an already dislocated state, but leads to
additional equilibrium conditions. The presentation emphasizes the role of
{configurational forces} as they provide a natural framework for the
description and interpretation of singularities and phase transitions.Comment: 32 pages, to appear in Journal of the Mechanics and Physics of Solid
Dislocation microstructures and strain-gradient plasticity with one active slip plane
We study dislocation networks in the plane using the vectorial phase-field
model introduced by Ortiz and coworkers, in the limit of small lattice spacing.
We show that, in a scaling regime where the total length of the dislocations is
large, the phase field model reduces to a simpler model of the strain-gradient
type. The limiting model contains a term describing the three-dimensional
elastic energy and a strain-gradient term describing the energy of the
geometrically necessary dislocations, characterized by the tangential gradient
of the slip. The energy density appearing in the strain-gradient term is
determined by the solution of a cell problem, which depends on the line tension
energy of dislocations. In the case of cubic crystals with isotropic elasticity
our model shows that complex microstructures may form, in which dislocations
with different Burgers vector and orientation react with each other to reduce
the total self energy
Intermittency in crystal plasticity informed by lattice symmetry
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.Comment: 13 pages, 4 figure
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