Dislocation nucleation and work hardening in anti-plane constrained shear

Analytical solution of plane constrained shear problem for single crystals within continuum dislocation theory

“…The analogous problem of anti-plane shear at nonzero resistance [15] suggests that the solution of (29) is symmetric, i.e.,…”

“…We know that, for the variational problem of this type, there exists a threshold value γ en such that when γ < γ en no dislocations are nucleated and β = 0 [15]. Near the threshold value the dislocation density must be small so that the last term in (12) can be neglected.…”

“…Due to the boundary conditions β should change its sign on the interval (0, h). The one-dimensional theory of dislocation pile-ups [16] as well as the solution of the analogous anti-plane shear problem [15] suggest to seek the minimizer in the form…”

Plastic deformation of bicrystals within continuum dislocation theory

“…Assuming zero-dissipation, we employ a minimizing sequence [3] to derive an energetic threshold for dislocation nucleation which reads for the particular cases of pure shear γ en = 4k| sin ϕ|/hbρ s | cos 2ϕ| and uniaxial extension ε en = 4k sin ϕ/hbρ s | sin 2ϕ|. The combined solution depending on angles ϕ and θ (again inversely proportional to h) is presented in [4].…”

Plane constrained uniaxial extension of a single crystal strip

“…Based on the analysis of a minimizing sequence [2] we obtain the energetic threshold en = 2k hbρ s | sin ϕ| | sin 2ϕ| which being exactly the same of that the single slip and clearly showing the size effect typical to problems of crystal plasticity. We found out that for symmetric double slip system case β l (y) = −β r (y) = β(y) for y ∈ (0, h).…”