A view of the relation between the continuum theory of lattice defects and non-euclidean geometry in the linear approximation

Wit, R. de

doi:10.1016/0020-7225(81)90073-2

Cited by 83 publications

(58 citation statements)

References 5 publications

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“…The Q tensor is thus the covariant derivative of a tensor g='~', where g~,~, is symmetrie but is n o t a metric tensor. In particular, for the (~) state shown in Now we have already seen that S,,a,~, is due to dislocations, while DEWIT [8] has shown that K[g,~,]s, is due to disclinations. Since neither is present in It is clear that this does not give the same value as that obtained from Eq.…”

Section: (38)mentioning

confidence: 90%

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The relationship between concepts in Materials Science and Sosmology

Marcinkowski

1981

Acta Physica

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A number of eoneepts eommon to both the fields of Materials Seienee ah4 Astrophysies have been elueidate4. In particular, the eurvature and unboundedness of the universe is associated with the idea of surfaee energy. On the other hand, the geometries associated with black holes and preeipitate partieles are shown to be basically similar, and can be represented in terms of dislocations. Finally, the idea of a dis]ocation is extended to the fourdimensiona] space-time continuum. IntroduetionThe understanding of the meehanieal behavior of matter is based largely on geometrieal eoneepts [1]. The same holds true for an understanding of the universe as embodied in Einstein's general theory of relativity [2]. In both eases, the most sophistieated mathematies available for the preeise formulation of these geometrieal eoneepts is that of differential geometry. It would therefore appear that deeper links should exist between the diseiplines of Materials Seienee, on the one hand, and astronomy, on the other. In particular, we shall attempt to show how the eoneept of surfaee tension can aeeount for the unboundedness of the universe. In addition, we shall also show that the distortions in spaee eaused by large eoneentrations of matter, i.e. blaek holes and those whieh oeeur in the vieinity of preeipitates in solids, are geometrieally identieal and can be represented in terms of disloeations. The relationship between surface energy and eurvatureConsider the linear ehain of plus-minus eharges of magnitude e separated by nearest-neighbor distanees r. The eleetrostatie foree between any nearestneighbor pair is simply d Fe = ----,(1) r ~ Aaa Physiea Acadendae Seientiarum Hungarieae 50, 1981

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Section: (38)mentioning

confidence: 90%

“…Thus the torsion and Q tensors lead to the same effectŸ dŸ content for small distortions. In the latter case the dislocations have been termed quasi [8] For completeness, Fig. 4a shows the distortion caused by a biaxial contraction such as might occur by localized cooHng of by a phase transformation.…”

Section: (38)mentioning

confidence: 99%

The relationship between concepts in Materials Science and Sosmology

Marcinkowski

1981

Acta Physica

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“…Kinematics of generalized continuum formulations that consider more general defects, such as disclinations and point-defects (vacancies and/or interstitial atoms in the crystal lattice), besides dislocations have been considered already by [16][17][18][19]. A comprehensive account of the geometrically linearized version of the continuum theory of general defects in crystal lattices is outlined in [20]. Other aspects such as the gauge theory of dislocations as treated, e.g., by [21] or nonsingular stress and strain fields of dislocations and disclinations, see [22][23][24][25], are exciting topics of related research activities.…”

Section: Introductionmentioning

confidence: 99%

On generalized crystal‐plasticity based on defect‐densities in a second‐order continuum

Steinmann

2013

GAMM-Mitteilungen

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Single crystals display various defects that may potentially act as obstacles to further evolution of inelastic deformations, i.e. to further plastic flow. These are translational defects in terms of dislocations, rotational defects in terms of disclinations, and (dilatational) pointdefects in terms of lattice vacancies or interstitial atoms. A formulation of generalized crystalplasticity is proposed that incorporates the densities of these defects in order to capture the hardening of the material. In particular the inclusion of the defect densities other than the dislocation density requires to root the formulation in a second-order continuum description.

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“…The different fields which generate the non-zero curvature tensor can be identified with the disclination density (see also the comments by de Wit 1981;ClejaTigoiu and Maugin 2000), the torsion is a measure of the dislocation density, while the third order tensor Q as measure of non-metric property of the connection, represents the extra-matter of the defect theory (Kröner 1990(Kröner , 1992Kröner and Lagoudas 1992). The defects density tensors are introduced as the appropriate measure of the deviation of the plastic distortion and plastic connection from the compatibility relationships.…”

Section: Introductionmentioning

confidence: 99%

Elasto-plastic materials with lattice defects modeled by second order deformations with non-zero curvature

Cleja-Ţigoiu

2010

Int J Fract

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The paper deals with elasto-plastic materials, whose structural damage is produced by lattice defects, within the constitutive framework of finite elasto-plasticity with second order deformations. For second order deformations, which are represented by distortions and connections, we adopt the decomposition rule into second order elastic and plastic components. The split of the plastic connection into independent components, which involve Bilby's connection type and a second order tensor, called disclination, plays a fundamental role in defining kinematical variables. The lattice defects are modeled by dislocations and disclinations, which are introduced within the finite deformation formalism and based on the differential geometry arguments. The constitutive framework is compatible with the free energy imbalance, postulated in the configuration with torsion, in such a form to include the presence of the disclinations. The model includes macro forces, which are power conjugated with the appropriate rate of the second order elastic deformations, and micro forces which are related with the plastic behaviour, as well as with the disclinations. The micro balance equations which are associated with the presence of disclinations are derived from a virtual power principle relative to the disclination mechanism. As a consequence of the principle of the free energy

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A view of the relation between the continuum theory of lattice defects and non-euclidean geometry in the linear approximation

Cited by 83 publications

References 5 publications

The relationship between concepts in Materials Science and Sosmology

The relationship between concepts in Materials Science and Sosmology

On generalized crystal‐plasticity based on defect‐densities in a second‐order continuum

Elasto-plastic materials with lattice defects modeled by second order deformations with non-zero curvature

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