A line-tension model of dislocation networks on several slip planes

Conti, Sergio; Gladbach, Peter

doi:10.1016/j.mechmat.2015.01.013

Cited by 8 publications

(7 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, relaxation phenomena may be present, and straight dislocations with certain Burgers vectors and orientations may spontaneously decompose into several parallel dislocations, and in some cases a zig-zag structure is optimal, see [11,12,14]. The relaxation formula for (2.2) involves the H 1 -elliptic envelope…”

Section: Variational Models Of Dislocations and Plasticity In Crystalsmentioning

confidence: 99%

“…Numerical results were first presented in [26,28], in situations in which individual dislocations are resolved. A mathematical analysis of the model was performed in [21,22] for the scalar case in which all dislocations have the same Burgers vector, in [11,13] for the vectorial situation with multiple Burgers vectors, and in [12] for dislocations localized to two parallel planes. The asymptotic analysis results in a line tension model associated to a Caccioppoli partition of the type (1.2), where the energy density ψ is obtained by the H 1 -elliptic envelope of the pre-logarithmic factor ψ 0 , as defined in (2.3).…”

Section: Variational Models Of Dislocations and Plasticity In Crystalsmentioning

confidence: 99%

See 1 more Smart Citation

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

Conti

Garroni

Müller

2016

Boll Unione Mat Ital

Self Cite

View full text Add to dashboard Cite

We consider target-space homogenization for energies defined on partitions parametrized\ud by a discrete lattice B ⊂ RN. For a smallσ > 0, the variable is a piecewise constant\ud function taking values in σB, and the energy depends on the jumps and their orientation. In\ud the limit as σ → 0 we obtain a homogenized functional defined on functions of bounded\ud variation. This result is relevant in the study of dislocation structures in plastically deformed\ud crystals. We review recent literature on the topic and propose our limiting effective energy\ud as a continuum model for strain-gradient plasticity

show abstract

Section: Variational Models Of Dislocations and Plasticity In Crystalsmentioning

confidence: 99%

Section: Variational Models Of Dislocations and Plasticity In Crystalsmentioning

confidence: 99%

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

Conti

Garroni

Müller

2016

Boll Unione Mat Ital

Self Cite

View full text Add to dashboard Cite

show abstract

“…For the intermediate case, the limit energy of a mollified straight jump w b,ν,ε = b1 {x·ν>0} * ν ε is (1 − β)ϕ short (b, ν) + βϕ long (b, ν). The double relaxation is achieved by first replacing the straight jump by a large-scale exterior microstructure and then replacing all straight jumps in the microstructure by interior microstructures at scale h. For an example of a non-trivial optimal two-scale microstructure, see [5]. The proof of the theorem will appear in [6].…”

Section: The Limit Energy For Multiple Planesmentioning

confidence: 99%

A Phase‐field Model of Dislocations on Parallel Slip Planes

Gladbach¹

2015

Proc Appl Math and Mech

Self Cite

View full text Add to dashboard Cite

We establish a line-tension energy for dislocation networks concentrated in finitely many parallel slip planes as the limit of a Peierls-Nabarro energy. Dislocations in different planes interact depending on their distance. At intermediate distances a two-scale dislocation microstructure is optimal.We extend the Peierl-Nabarro phase-field energy model for dislocations in [1] to a system of slip fields on a number of parallel planes. This energy features two relevant length scales, the lattice spacing ε and the distance between neighboring planes h. To leading order as ε → 0, the energy converges to a line-tension functional concentrated on jump lines of the slip fields, the dislocations. The limit functional is characterized by an energy density, which measures the macroscopic elastic energy induced by a dislocation line. Straight lines are prone to spontaneous generation of microstructure, leading to a relaxation in the variational limit. The energy density for parallel dislocation lines in closely spaced planes contains interactions between the lines, leading to complex microstructures.

show abstract

“…Macroscopic line-tension models have been rigorously derived from the KCO model for single slip [28,29], for multipleslip [16,19] and for multi-planar slip [20]. A remarkable result of the analysis of Conti et al [19] is that the effective self-energy may be strictly smaller than the classical self-energies of straight dislocations (cf., for example [11]) as a result of fine reconstruction of the dislocation line.…”

Section: Introductionmentioning

confidence: 99%

The Line-Tension Approximation as the Dilute Limit of Linear-Elastic Dislocations

Conti

Garroni

Ortiz

2015

Arch Rational Mech Anal

Self Cite

119

View full text Add to dashboard Cite

We prove that the classical line-tension approximation for dislocations in crystals, that is, the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the Γ -limit of regularized linear-elasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length ψ(b, t), which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length ψ 0 (b, t) obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation of the classical energy down to its H 1 -elliptic envelope.

show abstract

A line-tension model of dislocation networks on several slip planes

Cited by 8 publications

References 18 publications

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

A Phase‐field Model of Dislocations on Parallel Slip Planes

The Line-Tension Approximation as the Dilute Limit of Linear-Elastic Dislocations

Contact Info

Product

Resources

About