Abstract
In this paper, we show that a strain-gradient plasticity
model arises as the Γ -limit of a nonlinear semi-discrete dislocation
energy. We restrict our analysis to the case of plane elasticity,
so that edge dislocations can be modelled as point singularities of
the strain field.
A key ingredient in the derivation is the extension of the rigidity
estimate [9, Theorem 3.1] to the case of fields β : U ⊂ R2 → R2×2
with nonzero curl. We prove that the L2-distance of β from a
single rotation matrix is bounded (up to a multiplicative constant)
by the L2-distance of β from the group of rotations in the plane,
modulo an error depending on the total mass of Curlβ. This
reduces to the classical rigidity estimate in the case Curlβ = 0.
model arises as the Γ -limit of a nonlinear semi-discrete dislocation
energy. We restrict our analysis to the case of plane elasticity,
so that edge dislocations can be modelled as point singularities of
the strain field.
A key ingredient in the derivation is the extension of the rigidity
estimate [9, Theorem 3.1] to the case of fields β : U ⊂ R2 → R2×2
with nonzero curl. We prove that the L2-distance of β from a
single rotation matrix is bounded (up to a multiplicative constant)
by the L2-distance of β from the group of rotations in the plane,
modulo an error depending on the total mass of Curlβ. This
reduces to the classical rigidity estimate in the case Curlβ = 0.
Original language | English |
---|---|
Pages (from-to) | 1365-1396 |
Number of pages | 32 |
Journal | Indiana University Mathematics Journal |
Volume | 63 |
Issue number | 5 |
DOIs | |
Publication status | Published - 2014 |