The gauge theory of dislocations: conservation and balance laws

Lazar, Markus; Anastassiadis, Charalampos

doi:10.1080/14786430802255653

Cited by 26 publications

(36 citation statements)

References 35 publications

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“…The plastic distortion is a well-known quantity in dislocation theory and in Mura's theory of eigenstrain. Nowadays, this field can be understood as a tensorial gauge field in the framework of dislocation gauge theory (Lazar and Anastassiadis, 2008;Lazar, 2010). For dislocations, the incompatibility tensors are the dislocation density and dislocation current tensors (e.g., Holländer (1962); Kosevich (1979); Lazar (2011a)).…”

Section: Field Identities and Equations Of Motionmentioning

confidence: 99%

Distributional and regularized radiation fields of non-uniformly moving straight dislocations, and elastodynamic Tamm problem

Lazar

Pellegrini

2016

Journal of the Mechanics and Physics of Solids

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This work introduces original explicit solutions for the elastic fields radiated by non-uniformly moving, straight, screw or edge dislocations in an isotropic medium, in the form of time-integral representations in which acceleration-dependent contributions are explicitly separated out. These solutions are obtained by applying an isotropic regularization procedure to distributional expressions of the elastodynamic fields built on the Green tensor of the Navier equation. The obtained regularized field expressions are singularity-free, and depend on the dislocation density rather than on the plastic eigenstrain. They cover non-uniform motion at arbitrary speeds, including faster-than-wave ones. A numerical method of computation is discussed, that rests on discretizing motion along an arbitrary path in the plane transverse to the dislocation, into a succession of time intervals of constant velocity vector over which time-integrated contributions can be obtained in closed form. As a simple illustration, it is applied to the elastodynamic equivalent of the Tamm problem, where fields induced by a dislocation accelerated from rest beyond the longitudinal wave speed, and thereafter put to rest again, are computed. As expected, the proposed expressions produce Mach cones, the dynamic build-up and decay of which is illustrated by means of full-field calculations.

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Section: Field Identities and Equations Of Motionmentioning

confidence: 99%

Distributional and regularized radiation fields of non-uniformly moving straight dislocations, and elastodynamic Tamm problem

Lazar

Pellegrini

2016

Journal of the Mechanics and Physics of Solids

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“…The force between dislocations is the so-called Peach-Koehler force [53,54,55]. In the gauge theory of dislocations the gauge invariant formulation of the Peach-Koehler force has been derived by Lazar and Anastassiadis [56]. Kadić and Edelen [11] and Edelen and Lagoudas [12] just derived an expression which is not gauge invariant because they used the canonical Eshelby stress tensor instead of the gauge invariant one.…”

Section: Screw Dislocationmentioning

confidence: 99%

The gauge theory of dislocations: Static solutions of screw and edge dislocations

Lazar¹,

Anastassiadis²

2009

Philosophical Magazine

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We investigate the T (3)-gauge theory of static dislocations in continuous solids. We use the most general linear constitutive relations bilinear in the elastic distortion tensor and dislocation density tensor for the force and pseudomoment stresses of an isotropic solid. The constitutive relations contain six material parameters. In this theory both the force and pseudomoment stresses are asymmetric. The theory possesses four characteristic lengths ℓ 1 , ℓ 2 , ℓ 3 and ℓ 4 which are given explicitely. We first derive the three-dimensional Green tensor of the master equation for the force stresses in the translational gauge theory of dislocations. We then investigate the situation of generalized plane strain (anti-plane strain and plane strain). Using the stress function method, we find modified stress functions for screw and edge dislocations. The solution of the screw dislocation is given in terms of one independent length ℓ 1 = ℓ 4 . For the problem of an edge dislocation, only two characteristic lengths ℓ 2 and ℓ 3 arise with one of them being the same ℓ 2 = ℓ 1 as for the screw dislocation. Thus, this theory possesses only two independent lengths for generalized plane strain. If the two lengths ℓ 2 and ℓ 3 of an edge dislocation are equal, we obtain an edge dislocation which is the gauge theoretical version of a modified Volterra edge dislocation. In the case of symmetric stresses we recover well known results obtained earlier.

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“…which is also known as translational gauge-invariance (see Lazar [76,77,78] Table 6). In reality however, the orientation of the crystallites matters.…”

Section: (Fgi)mentioning

confidence: 99%

A fourth-order gauge-invariant gradient plasticity model for polycrystals based on Kröner’s incompatibility tensor

Ebobisse

Neff

2019

Mathematics and Mechanics of Solids

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In this paper we derive a novel fourth order gauge-invariant phenomenological model of infinitesimal rate-independent gradient plasticity with isotropic hardening and Kröner's incompatibility tensor inc(ε p ) := Curl [(Curl ε p ) T ], where ε p is the symmetric plastic strain tensor. Here, gauge-invariance denotes invariance under diffeomorphic reparametrizations of the reference configuration, suitably adapted to the geometrically linear setting. The model features a defect energy contribution which is quadratic in the tensor inc(ε p ) and it contains isotropic hardening based on the rate of the plastic strain tensorε p . We motivate the new model by introducing a novel rotational invariance requirement in gradient plasticity, which we call micro-randomness, suitable for the description of polycrystalline aggregates on a mesoscopic scale and not coinciding with classical isotropy requirements. This new condition effectively reduces the increments of the non-symmetric plastic distortionṗ to their symmetric counterpartε p = symṗ. In the polycrystalline case, this condition is a statement about insensitivity to arbitrary superposed grain rotations. We formulate a mathematical existence result for a suitably regularized non-gauge-invariant model. The regularized model is rather invariant under reparametrizations of the reference configuration including infinitesimal conformal mappings. Appendix 322 Aifantis writes in [4, p. 218]: "... In conformity with established results -that the plastic strain rateεp is a state variable, rather than the strain εp itself."3 The plastic spin in the finite deformation flow theory of plasticity is defined as the skew-symmetric part of the so-called plastic distortion rate i.e., Wp := skew(ḞpF −1 p ) , while its counter-part in the small strain theory is simply skew(ṗ) , where p is the non-symmetric infinitesimal plastic distortion.

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The gauge theory of dislocations: conservation and balance laws

Cited by 26 publications

References 35 publications

Distributional and regularized radiation fields of non-uniformly moving straight dislocations, and elastodynamic Tamm problem

Distributional and regularized radiation fields of non-uniformly moving straight dislocations, and elastodynamic Tamm problem

The gauge theory of dislocations: Static solutions of screw and edge dislocations

A fourth-order gauge-invariant gradient plasticity model for polycrystals based on Kröner’s incompatibility tensor

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