On the Lagrangian and instability of medium with defects |
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Authors: | V Kobelev |
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Institution: | 1.University of Siegen,Siegen,Germany |
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Abstract: | The article presents the Lagrangian of defects in the solids, equipped with bending and warp. The deformation of such elastic
medium with defects is based on Riemann-Cartan geometry in three dimensional space. In the static theory for the media with
dislocations and disclinations the possible choice of the geometric Lagrangian yield the equations of equilibrium. In this
article, the assumed expression for the free energy leading is equal to a volume integral of the scalar function (the Lagrangian)
that depends on metric and Ricci tensors only. In the linear elastic isotropic case the elastic potential is a quadratic function
of the first and second invariants of strain and warp tensors with two Lame, two mixed and two bending constants. For the
linear theory of homogeneous anisotropic elastic medium the elastic potential must be quadratic in warp and strain. The conditions
of stability of media with defects are derived, such that the medium in its free state is stable. With the increasing strain
the stability conditions could be violated. If the strain in material attains the critical value, the instability in form
of emergence of new topological defects occurs. The medium undergoes the spontaneous symmetry breaking in form of emerging
topological defects. |
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Keywords: | |
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