# Continuum (physics)

A continuum (Latin: continuus for "connected", plural: the continua and continuums) denotes something that follows one another without interruption. In physics, a quantity is continuous if with every possible value all values ​​are possible in a sufficiently small environment . Such a set of values ​​is called a continuum . In contrast to this, a value is discrete if, apart from it, no further value from a sufficiently small area is possible (see also the grid model ). Nevertheless, discrete values ​​are often calculated as a continuum and vice versa, because the error from the approximation is often significantly smaller than the numerical error of the calculation.

Mathematically, the terms open set , connected set or continuous function correspond to physical language usage, depending on the context .

For example the binding energies of the hydrogen atom

${\ displaystyle E_ {nlm} = - {\ frac {\ text {Ry}} {n ^ {2}}}, \ qquad {\ text {Ry}} = 13 {,} 6 \ {\ text {eV} }, \ qquad n \ in \ {1,2, \ dotsc \}, \ qquad l \ in \ {0,1, \ dotsc, n-1 \}, \ qquad m \ in \ {- l, -l +1, \ dotsc, l \}}$

discrete, the energies of the ionized electron-proton pair, on the other hand, which expire with pulses and for large distances , non-negative and continuous, ${\ displaystyle \ mathbf {p} _ {\ text {e}}}$${\ displaystyle \ mathbf {p} _ {\ text {p}}}$

${\ displaystyle E (\ mathbf {p} _ {\ mathrm {e}}, \ mathbf {p} _ {\ mathrm {p}}) = {\ frac {\ mathbf {p} _ {\ mathrm {e} } ^ {2}} {2 \, m _ {\ mathrm {e}}}} + {\ frac {\ mathbf {p} _ {\ mathrm {p}} ^ {2}} {2 \, m _ {\ mathrm {p}}}}, \ qquad \ mathbf {p} _ {\ mathrm {e}}, \ mathbf {p} _ {\ mathrm {p}} \ in \ mathbb {R} ^ {3}}$

Likewise, the places and times that a particle can traverse are continuous, while the places at which the atoms (more precisely their ion cores) are in a crystal lattice are discrete.

Since physical approximations and measured variables are prone to errors, it can depend on the accuracy of the observation or the measurement whether a variable is viewed as discrete or continuous. For example, the solar wind is considered continuous, whereas cosmic rays are considered discrete.

## Continuum in matter

Matter is viewed as a continuum for many calculations. This does not necessarily mean that matter always behaves continuously at the atomic level, rather it means that there is a representative volume element that represents the properties on the desired size scale, this must be much smaller than the dimensions of the object under consideration. Since matter is made up of atoms , this is always the case when all dimensions of an object are much larger than an atom. For example, the properties of a steel beam can be derived completely from its shape and the properties of the steel material with the accuracy required for technical applications . It is not necessary to consider the individual position of individual atoms.

If a mass density in a body depends on the location, then this dependency is considered to be continuous if the mass density does not make any jumps in the size scale under consideration and takes on the level of a representative volume element between two locations every value in the interval between the values ​​at these two locations .

## Continuum in mechanics

Continuum mechanics often uses the model of a continuum. In the case of a crack that can occur in a continuum or between two continuums, only compressive forces, but no tensile forces, can generally be transmitted; Due to the toothing / friction, shear stresses can often also be transferred (to a lesser extent).

In research, multiscale models of continuum mechanics are often used, which assume the existence of a representative volume element which is so small that it has constant loads ; this is often already achieved with a size 3–5 times smaller than the object under consideration.

The continuum mechanics is described by Cauchy's stress tensor with 6 independent components.

## Cosserat continuum

The Cosserat continuum, named after the brothers Eugène and François Cosserat , is based on micropolar elasticity and goes beyond the classical continuum theory: here one assumes a non-symmetrical stress tensor with 9 independent components and a moment stress tensor with 9 components.

The Cosserat continuum is used when the inhomogeneities are of the same order of magnitude as the dimensions of the structure.

## literature

• Arnold Sommerfeld: Mechanics of Deformable Media , Leipzig: Becker & Erler, 1945. - Lectures on theoretical physics; Volume 2 (6th edition, Harri Deutsch, Thun 1992, ISBN 3-87144-375-1 )

## Individual evidence

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Wiktionary: Continuum  - explanations of meanings, word origins, synonyms, translations
2. a b c Duden. Retrieved April 16, 2017 .
3. What does continuum mean. Retrieved April 16, 2017 .
4. a b H. Schaefer: The cosserat continuum . In: Wiley Online Library (Ed.): ZAMM-Journal of Applied Mathematics and Mechanics / Journal for Applied Mathematics and Mechanics . tape 47 , no. 8 , 1967, p. 485-498 , doi : 10.1002 / zamm.19670470802 ( wiley.com ).
5. a b c Euripides Papamichos: 03 Continua with microstructure: Cosserat Theory . ( alertgeomaterials.eu [PDF]). 03 Continua with microstructure: Cosserat Theory ( Memento of the original from January 12, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.
6. Euripides Papamichos: Continua with microstructure: Cosserat theory . In: Taylor & Francis (Eds.): European Journal of Environmental and Civil Engineering . tape 14 , no. 8-9 , 2010, pp. 1011-1029 , doi : 10.1080 / 19648189.2010.9693277 ( tandfonline.com [PDF]).