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1

Generalized continuum with defects and asymmetric stresses

Boratyński W., Teisseyre R.

Acta Geophysica Polonica

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2004

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Vol. 52, nr 2

185-195

EN

Consistent theory is presented of continuum with defect distribution: dislocation and disclination densities and the densities of rotation nuclei of two kinds. Special attention is paid to rotation and twist motions, and the two approaches to the definition of the twist-bend tensor are combined. The elastic and self fields of stresses and strains become asymmetric, while the total fields remain symmetric as required by the compatibility conditions. However, the tensor of incompatibility becomes asymmetric. The dislocation-stress relations and the equations of motion for symmetric and asymmetric parts of stresses are given and the wave equations for spin and twist are derived. Some applications are shortly discussed.

2

Continuum with self-rotation nuclei: Evolution of defect fields and equations of motion

Teisseyre R., Boratyński W.

Acta Geophysica Polonica

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2002

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Vol. 50, nr 2

223-229

EN

Asymmetric theory of elastic continuum with dislocations, disclinations and nuclei of rotations is extended; we study the evolution and flow of the defect fields and arrive at the equations of motion for the symmetric and antisymmetric parts. The total fields can be represented by its elastic and self parts and the respective equations can be split into its self part prevailing on the fracture plane and into its total part describing seismic radiation field in a surrounding space. Special attention is paid to the rotation and twist motions.

3

On generators of the group of projective collineations

Boratyński W.

Demonstratio Mathematica

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2001

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Vol. 34, nr 1

123-134

EN

The problem of decomposition of a projective collineation into special harmonic homologies i.e. homologies with fundamental hyperplanes containing the fixed points, is considered. We prove that for every harmonic homology of the n- dimensional projective space there exists J< 3 such that the harmonic homology is a product of j harmonic homologies from the given class.

4

Projective collineation as a product of special harmonic homologies

Boratyński W.

Demonstratio Mathematica

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1999

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Vol. 32, nr 4

823-828

EN

The problem of decomposition of a projective collineation into special harmonic homologies i.e. homologies with the fixed center and homologies with fundamental hyperspaces containing the fixed points, is considered. We prove that for every harmonic homology of the n-dimensional projective space there exists j < 3 such that the ha.rmonic homology is a product of j harmonic homologies from the given class.

5

Rozkład przekształcenia rzutowego przestrzeni P2 (F) na pewne szczególne homologie harmoniczne

Boratyński W.

Zeszyty Naukowe. Geometria i Grafika Inżynierska / Politechnika Śląska

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1998

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z. 2

63-69

PL

W artykule zajmiemy się rozkładem dowolnego przekształcenia rzutowego przestrzeni P2 (F) na pewne szczególne homologie harmoniczne, a mianowicie homologie o zadanym środku lub homologie o osi przechodzącej przez dany punkt. Udowodnimy, że dowolna homologia harmoniczna przestrzeni (P do 2) (F) jest złożeniem co najwyżej pięciu takich homologii. Ponieważ każde przekształcenie rzutowe przestrzeni P2 (F) jest złożeniem co najwyżej trzech homologii harmonicznych, stąd otrzymujemy rozkład dowolnego przekształcenia rzutowego przestrzeni P2 (F) na wspomniane wyżej szczególne homologie harmoniczne.

EN

The problem of a projective collineation onto special harmonic homologies I.e. homologies with the fixed centre and homologies with fundamental lines containing the fixed point, is considered. It is proved that every harmonic homology of two dimensional projective space (P 2) (F) is a product of at most five harmonic homologies from the given class.