Dislocation microstructures and strain-gradient plasticity with one active slip plane

Conti, Sergio; Garroni, Adriana; Müller, Stefan

doi:10.1016/j.jmps.2015.12.008

Cited by 12 publications

(9 citation statements)

References 42 publications

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“…Dislocations are topological defects of the metal lattice, [33,37,38], which play an important role in the effect of plastic slip i.e., the relative slip of atom layers which result in a permanent deformation of the metal lattice. Additionally to phenomenologically derived models (see, for example, [17,25,27] and references therein), there has been extensive research in the mathematical community to derive macroscopic plasticity models from microscopic or mesoscopic dislocation models (see [7,[10][11][12][13]15,[20][21][22]30,34,36]).…”

Section: Introductionmentioning

confidence: 99%

Strain-Gradient Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

Ginster¹

2019

SIAM J. Math. Anal.

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In this paper a we derive by means of Γ-convergence a macroscopic strain-gradient plasticity from a semi-discrete model for dislocations in an infinite cylindrical crystal. In contrast to existing work, we consider an energy with subquadratic growth close to the dislocations. This allows to treat the stored elastic energy without the need to introduce an ad-hoc cut-off radius. As the main tool to prove a complementing compactness statement, we present a generalized version of the geometric rigidity result for fields with non-vanishing curl. A main ingredient is a fine decomposition result for L 1 -functions whose divergence is in certain critical Sobolev spaces.If β is a gradient then this estimates reduces to the geometric rigidity result in [29]. This estimate allows us to control the rotational invariance of the energy in order to obtain a compactness result for a sequence of suitably rescaled elastic strains which induce a uniformly bounded rescaled elastic energy, Theorem 1.2. The proof of the generalized (nonlinear) Korn's inequality in [20] and [30] rely on a fine estimate due to Bourgain and Brézis (see [3,4] and also [5]). It states that an L 1 -function in two dimensions, whose divergence is in H −2 , is already in H −1 andFor our mixed-growth situation we prove a corresponding result, namely we show that an L 1 -function whose divergence is in the space H −2 + W −2,p , for 1 < p < 2, belongs to the space H −1 + W −1,p . Corresponding estimates hold, Theorem 1.4.The paper is organized as follows. In Subsection 1.1 we introduce notation and the mathematical setting of the problem. The main results are presented in Subsection 1.2. In Section 2 we prove the generalized Bourgain-Brézis decomposition result which we use in Section 3 to show the generalized geometric rigidity result in the mixed growth case. Finally, the proof of the Γ-convergence result can be found in Section 4. Setting of the ProblemLet Ω ⊆ R 2 be a simply-connected, bounded domain with Lipschitz boundary representing the cross section of an infinite cylindrical crystal. The set of (normalized) minimal Burgers vectors for the given crystal is denoted by S = {b 1 , b 2 } for two linearly independent vectors b 1 , b 2 ∈ R 2 . Moreover, set S = span Z S = {λ 1 b 1 + λ 2 b 2 : λ 1 , λ 2 ∈ Z} to be the the set of (renormalized) admissible Burgers vectors. Let ε > 0 the interatomic distance for the given crystal. We define the set of admissible dislocation densities as a subset of the R 2 -valued Radon measures M(Ω; R 2 ), namelywhere we assume that ρ ε satisfies 1. lim ε→0 ρ ε /ε s = ∞ for all fixed s ∈ (0, 1) and

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Section: Introductionmentioning

confidence: 99%

Strain-Gradient Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

Ginster¹

2019

SIAM J. Math. Anal.

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“…The situation of line singularities is much richer as one has to take into account the possibility of complex dislocation structures. As shown with an explicit example in [15], this may need a further relaxation beyond the one leading from ψ 0 to ψ rel 0 and the emergence of microstructures related to the so called cell structures for dislocations (see Fig. 3).…”

Section: Variational Models Of Dislocations and Plasticity In Crystalsmentioning

confidence: 99%

“…We remark that if u ∈ S BV ( ; R N ) takes a finite number of values then necessarily ∇u = 0 almost everywhere. In [15] we have constructed an example in which the relaxation formula for g(A) requires a non trivial microstructure, as illustrated in Fig. 3.…”

Section: Cell Problemmentioning

confidence: 99%

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

Conti

Garroni

Müller

2016

Boll Unione Mat Ital

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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

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“…Thus, whereas pair annihilation, b − b → 0, and monopole splitting, 2b → b + b, lower the energy, monopole pairing, b + b → 2b, increases the energy. This lack of weak lowersemicontinuity has far-reaching consequences for microstructural evolution, as the crystal can lower its energy, or relax, through microstructural rearrangements involving annihilation, splitting, network formation and other mechanisms [34].…”

Section: Dislocation Reactionsmentioning

confidence: 99%

A line-free method of monopoles for 3D dislocation dynamics

Deffo

Ariza

Ortiz

2019

Journal of the Mechanics and Physics of Solids

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We develop an approximation scheme for three-dimensional dislocation dynamics in which the dislocation line density is concentrated at points, or monopoles. Every monopole carries a Burgers vector and an element of line. The monopoles move according to mobility kinetics driven by elastic and applied forces. The divergence constraint, expressing the requirement that the monopoles approximate a boundary, is enforced weakly. The fundamental difference with traditional approximation schemes based on segments is that in the present approach an explicit linear connectivity, or 'sequence', between the monopoles need not be defined. Instead, the monopoles move as an unstructured point set subject to the weak divergence constraint. In this sense, the new paradigm is 'line-free', i. e., it sidesteps the need to track dislocation lines. This attribute offers significant computational advantages in terms of simplicity, robustness and efficiency, as demonstrated by means of selected numerical examples.

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Dislocation microstructures and strain-gradient plasticity with one active slip plane

Cited by 12 publications

References 42 publications

Strain-Gradient Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

Strain-Gradient Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

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

A line-free method of monopoles for 3D dislocation dynamics

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