# Difference between revisions of "Kelvin's medium"

From Department of Theoretical and Applied Mechanics

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U = C({\bf n}_{1}\cdot{\bf n}_{2}) | U = C({\bf n}_{1}\cdot{\bf n}_{2}) | ||

</math>, | </math>, | ||

− | where <math>C</math> is an interaction constant, <math>\bf{n}_1</math>, <math>\bf{n}_2</math> are the unit vectors bound to the solids. The interaction torque | + | where <math>C</math> is an interaction constant, <math>\bf{n}_1</math>, <math>\bf{n}_2</math> are the unit vectors bound to the solids. The interaction torque has the form: |

::<math> | ::<math> | ||

− | {\bf M}_{1} = {\bf n}_{1}\times\frac{\partial U}{\partial {\bf n}_{1}} = | + | {\bf M}_{1} = {\bf n}_{1}\times\frac{\partial U}{\partial {\bf n}_{1}} = C({\bf n}_{1}\times{\bf n}_{2}) |

</math> | </math> | ||

− | Then the motion equation of the k-th particle | + | Then the motion equation of the k-th particle is as follows: |

::<math> | ::<math> | ||

J\ddot{\bf \phi}_{k} = C(({\bf n}_{k}\times{\bf n}_{k+1}) + ({\bf n}_{k}\times{\bf n}_{k-1})) | J\ddot{\bf \phi}_{k} = C(({\bf n}_{k}\times{\bf n}_{k+1}) + ({\bf n}_{k}\times{\bf n}_{k-1})) |

## Revision as of 03:47, 30 May 2016

Kelvin's one-dimensional medium is a chain consisting of solids interacting via a pair torque potential. In this example, the solids are visualized by rods.
The solids interact via the torque potential:

- ,

where

is an interaction constant, , are the unit vectors bound to the solids. The interaction torque has the form:Then the motion equation of the k-th particle is as follows: