Spring clip
The jump bracket [[…]] is a simplified notation for the jump height of a function at a point of discontinuity , see graphic.
In the continuum mechanics such discontinuities occur, for example
- at force introduction points, where forces are introduced over a large area and suddenly drop to zero at the edge of the introduction point,
- at material boundaries, where the density is discontinuous, or
- in shock waves , where the speed can jump.
These situations are quite common and to some extent ubiquitous, so that they should generally not be ignored. The resulting terms in the equations , for example Reynolds' transport theorem, can be written briefly and legibly using the jump brackets.
definition
For the definition a real-valued function is considered, which has a point of discontinuity at this point . When approaching from below, the limit value on the left is assumed
and from above the right-hand limit value
predictable. Then the jump bracket is an abbreviation for the difference between these limit values at the jump point:
The functions can be the components of vector or tensor fields , which is why the jump bracket can also be generalized to vector or tensor arguments.
See also
literature
- WH Müller: Forays through the continuum theory . Springer, 2011, ISBN 978-3-642-19869-4 .
- H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .
- P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2010, ISBN 978-3-642-07718-0 .