On the dynamics of continuous distributions of dislocations

Binz, Ernst; Schwarz, Günter; Wenzelburger, Jan

doi:10.1090/qam/1827812

Cited by 2 publications

(2 citation statements)

References 17 publications

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“…and N is the outward-pointing unit normal vector on the boundary ∂M. This is known as the Helmholtz decomposition [24,25,40]. Let ψ ∈ C ∞ (M; R 3 ) be a component of the exact part and let Θ ∈ D(M; R 3 ) be the other component in the non-exact part.…”

Section: Helmholtz Decomposition and Variational Formulationmentioning

confidence: 99%

“…dislocation and disclination, and the torsion and curvature in the affine connection [15,16,17,18]. This theory was developed further by Noll [19], Wang [20], de Wit [21], Le and Stumpf [22,23], Wenzelburger [24] and Binz et al [25]. Yavari and Goriely reformulated the theories using modern differential geometry and derived analytical solutions to the nonlinear boundary value problems [26,27,28].…”

Section: Introductionmentioning

confidence: 99%

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Geometrical Modelling and Numerical Analysis of Dislocaion Mechanics

Kobayashi¹,

Tarumi²

2022

Preprint

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This study undertakes the mathematical modelling and numerical analysis of dislocations within the framework of differential geometry. The fundamental configurations, i.e. reference, intermediate and current configurations, are expressed as the Riemann-Cartan manifold, which equips the Riemannian metric and Weitzenböck connection. The torsion 2-form on the intermediate configuration is obtained through the Hodge duality of the dislocation density and the corresponding bundle isomorphism is subjected to the Helmholtz decomposition. This analysis introduces the boundary condition for plastic deformation. Cartan first structure equation and stress equilibrium equation are solved numerically using weak form variational expressions and isogeometric analysis. The numerical analysis carried out for this study reveals the distribution of plastic deformation fields around screw and edge dislocations for the first time. It also demonstrates stress fields around dislocations of which the distant fields show full agreement with the classical Volterra theory, while at the same time eliminating the singularity otherwise introduced at the dislocation by classical methods. The stress fields include several characteristic features due to the geometrical nonlinearity included therein. We also demonstrate that free surfaces affect both plastic and elastic deformation, but in different ways. The mathematical framework of this study is applicable to an arbitrary configuration of dislocations.

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Section: Helmholtz Decomposition and Variational Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Geometrical Modelling and Numerical Analysis of Dislocaion Mechanics

Kobayashi¹,

Tarumi²

2022

Preprint

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Geometrical frustration in nonlinear mechanics of screw dislocation

Kobayashi,

Tarumi

2024

R. Soc. Open Sci.

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The existence of stress singularities and reliance on linear approximations pose significant challenges in comprehending the stress field generation mechanism around dislocations. This study employs differential geometry and calculus of variations to mathematically model and numerically analyse screw dislocations. The kinematics of the dislocation are expressed by the diffeomorphism of the Riemann–Cartan manifold, which includes both the Riemannian metric and affine connection. The modelling begins with a continuous distribution of dislocation density, which is transformed into torsion τ through the Hodge duality. The plasticity functional is constructed by applying the Helmholtz decomposition to bundle isomorphism, which is equivalent to the Cartan first structure equation for the intermediate configuration B . The current configuration is derived by the elastic embedding of B into the standard Euclidean space ℝ 3 . The numerical analysis reveals that the elastic stress fields effectively eliminate the singularity along the dislocation line and exhibit excellent conformity with Volterra’s theory beyond the dislocation core. Geometrical frustration is the direct source of dislocation stress fields, as demonstrated through the multiplicative decomposition of deformation gradients. By leveraging the mathematical properties of the Riemann–Cartan manifold, we demonstrate that the Ricci curvature determines the symmetry of stress fields. These results substantiate a long-standing mathematical hypothesis: the duality between stress and curvature.

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On the dynamics of continuous distributions of dislocations

Cited by 2 publications

References 17 publications

Geometrical Modelling and Numerical Analysis of Dislocaion Mechanics

Geometrical Modelling and Numerical Analysis of Dislocaion Mechanics

Geometrical frustration in nonlinear mechanics of screw dislocation

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