Spring clip

Discontinuous function with jump point at and jump height ${\displaystyle f(x)}$ ${\displaystyle x=s}$ ${\displaystyle [[f(s)]]}$

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 ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle x=s}$

${\displaystyle \lim _{x\to s-}f(x)=f^{-}(s)}$

and from above the right-hand limit value

${\displaystyle \lim _{x\to s+}f(x)=f^{+}(s)}$

predictable. Then the jump bracket is an abbreviation for the difference between these limit values ​​at the jump point:

${\displaystyle [[f(s)]]:=\lim _{x\to s+}f(x)-\lim _{x\to s-}f(x)=f^{+}(s)-f^{-}(s)}$

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

• Föppl clamp
• Heaviside function

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 .