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2 Scholle M.

2 1998

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A balance theorem and its applications to a continuum theory of dislocations and plasticity

Scholle M.

Zeszyty Naukowe Politechniki Świętokrzyskiej. Mechanika

1998

Z. 66

209-224

Noether's theorem is a wellknown tool to draw connections between Lie symmetry groups of a Lagrangian and homogenous balance equations of the volume-type of the associated system. Helmholtz's equation for vortex motion in an ideal fluid, however, is a balance equation of a different type as compared with those occuring in Noether's theorem: It is of the areatype; it is outside the scope of Noether's theorem. An alternative symmetry theorem has been established which connects non-Lie symmetries with homogenous balance equations of the area-type. Within this scheme Helmholtz's equation can be derived. More generally, this alternative concept is adequate to proceed towards the dynamics of line-shaped objects like vortex lines and dislocations in crystals. Consequently, dislocation dynamics can also be formulated within this framework: Based on analogies with the wellknown Lagrangian for fluids I propose Lagrangians for solid bodies in different situations, a.g. in elasticity and plasticity based on dislocation dynamics including thermomechanical effects (dissipation).

System symmetries and inverse variational problems in continuum theory

Scholle M.

Archives of Mechanics

1998

Vol. 50, nr 3

599-611

The aim of the conventional Inverse Problem in Lagrange formalism is to find a Lagrangian, the associated Euler-Lagrange equations of which are equivalent to a given set of partial differential equations of a physical system. In contrast, I am dealing with a different type of an inverse problem. I look for a Lagrangian which is associated with a given set of balance equations. My approach is based on general relations between symmetry groups (geometrical and gauge symmetries) and its associated balance equations. I follow two different mathematical lines: The first one is Noether's theorem: Universal Lie symmetry groups like translations (spatial and temporal), rotations and Galilei transformation are connected with the fundamental conservation laws for energy, linear monumentum, angular monumentum and center of mass motion. All of these balances are of the "volume-type". The second line takes account of a relationship between non-Lie symmetry groups (e.g. regauging of potentials) and balances of the "area-type". These are physically associated with line-shaped objects like vortex lines and dislocations. Following both lines in an inverse manner I derive the relevant symmetry properties of a yet unknown Lagrangian for a given set of balance equations of volume- and area-types. Consequently, a rough scheme for the analytical structure of the Lagrangian can be given. As an example, a Lagrangian for the elstic deformation of a body with eigenstresses due to fixed dislocations is constructed.