Configuration (mechanics)

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In continuum mechanics, a configuration is the mapping of a body from the Euclidean space of our perception into an abstract Euclidean vector space . In this way the physical body becomes accessible to a mathematical description. It is the basis for kinematics , the formulation of laws of nature and laws of matter . The configuration must not be confused with the body, nor with its movement or deformation. It should only represent this faithfully at all times. However, its definition already includes the continuum hypothesis, according to which the properties of a real body are continuously distributed over space.

body

A body of our perception is a delimited area of ​​space filled with matter. Several attempts have been made to depict the body as a section of the universe, which is formally complex. Instead, it should be cut out of the universe according to the cutting principle for local analyzes, whereby the cutting edges can be chosen quite arbitrarily. In this way, the body gets a surface that should be smooth in sections. The body may have inner surfaces so holes, but should be coherent between these surfaces and fill the space evenly with matter. If the properties of the matter in the body are the same under the same conditions, then the body is homogeneous , otherwise inhomogeneous . If different phases occur, one speaks of heterogeneous bodies.

Continuum hypothesis

To explain the continuum hypothesis, the body should macroscopically consist of only one material. The body is cut into partial bodies using the cutting principle. The total volume is then divided into the partial volumes in such a way that the total volume is the sum of the individual volumes ${\displaystyle V}$ ${\displaystyle {v}_{i}>0}$

${\displaystyle V=\sum _{i}v_{i}}$.

Each of the partial bodies has a mass and analogous to the volume, the total mass of the body should be the sum of the masses of the partial bodies: ${\displaystyle {m}_{i}>0}$ ${\displaystyle M}$

${\displaystyle M=\sum _{i}m_{i}}$.

Each part of the body can have a medium density

${\displaystyle \rho _{i}={\frac {m_{i}}{v_{i}}}>0}$

and other physical properties, e.g. B. a temperature can be assigned. An assumption that has proven itself in many practical problems is that the properties change only slightly from one volume to the next, i.e. a steady course of the properties, e.g. B. the density, results in the body. This is the continuum hypothesis. It is also said that the properties are smeared over the body. In the limit value , the sum of the masses of the sub-bodies is converted into a volume integral: ${\displaystyle v_{i}\rightarrow 0}$

${\displaystyle M=\lim _{n\to \infty \atop v_{i}\to 0}\sum _{i=1}^{n}\rho _{i}v_{i}=\int _{V}\rho \,\mathrm {d} V}$.

The calculability of this integral presupposes that the material points themselves are evenly distributed over space, i.e. H. the volume is a compact set of points.

Definition of the configuration

Let a body be a compact, piecewise smoothly bounded set of particles and a Euclidean vector space . Then every figure is ${\displaystyle K}$ ${\displaystyle P}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \kappa }$

${\displaystyle {\begin{array}{llll}\kappa :&K&\rightarrow &V\subset \mathbb {R} ^{3}\\&P&\mapsto &{\vec {x}}\end{array}}}$

a configuration. In order to make physical sense, this figure must

• be continuous so that previously neighboring points remain neighboring,
• be continuously differentiable so that smooth parts remain smooth,
• uniquely reversible so that the body cannot penetrate itself and
• get the orientation , since reflections of material bodies in nature are not possible.

The reference system is chosen so that the vector has the same coordinates as the particle in the space of our perception. In parlance, the image is often simply referred to as the configuration and volume of the body. ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle V}$

Special configurations

Reference configuration

The reference configuration is a time-fixed configuration that is used to name the material points. The name of a material point is then the vector . The reference configuration does not need to be assumed by the material body at any time. For a square body of any shape, e.g. B. the unit square as a reference configuration. ${\displaystyle P}$ ${\displaystyle {\vec {x}}}$

Output configuration

The initial configuration describes the initial state of the material body at a specified point in time . Because this state has been assumed once, this configuration describes an object of our intuition. In mechanical engineering , the construction position is mostly used for this. Analyzes based on forming processes can also use the formed state of the component. The initial configuration can also serve as a reference configuration and is then also referred to as such. ${\displaystyle t_{0}}$

Current configuration

The movement of a material body is described by a continuous sequence of time-dependent configurations . The time-dependent configuration is called the instantaneous configuration . ${\displaystyle \kappa \left(t\right),t\geq t_{0}}$ ${\displaystyle \kappa \left(t\right)}$

Intermediate configuration

In the material theory dealing with large deformations, intermediate configurations are introduced in order to formulate material models on them. These intermediate configurations are achieved through a local relief, which can be imagined as a volume element of the body being cut free and relieved, whereby it changes its shape. After the relief, neighboring volume elements in Euclidean space will no longer fit together. It is also said that the distortions are incompatible , which means that there is no field of motion from which the distortions can be derived.

Representative volume element

Possible RVE for fiber-reinforced plastic

If the material is inhomogeneous, such as fiber-reinforced plastic, the described homogenization can be difficult. Ultimately, one wants to model an inhomogeneous material with homogeneously distributed properties in order to make it accessible to an efficient analysis. The pragmatic approach is to choose the smallest volume that (presumably) provides a value comparable to the whole body when measured for its properties. This smallest volume is called the representative volume element (RVE) and one imagines that the whole body is made up of copies of this RVE. There are often different options for choosing the RVE - as the picture shows - and which is the best choice is measured by how well the model reflects the macroscopic material behavior and whether this is possible with reasonable effort.

See also

• Continuum mechanics
• Deformation gradient
• deformation
• Differentiable manifold

literature

• H. Giesekus: Phenomenological Rheology: An Introduction. Springer, Berlin 1994, ISBN 3-540-57513-8 , books.google.de/books?isbn=3540575138
• K. Willner: Continuum and contact mechanics: synthetic and analytical representation. Springer, Berlin 2003, ISBN 3-540-43529-8 , books.google.de/books?isbn=3540435298
• H. Altenbach: Continuum Mechanics: Introduction to the material-independent and material-dependent equations . 2nd edition, Springer Vieweg, Berlin 2012, ISBN 978-3-642-24118-5
• H. Bertram: Axiomatic Introduction to Continuum Mechanics. Bibliographisches Institut Wissenschaftsverlag, Mannheim 1989, ISBN 3-411-14031-3 .
• P. Haupt: Continuum Mechanics and Theory of Materials . 2nd edition, Springer, Berlin 2002, ISBN 978-3-540-43111-4

Individual evidence

1. ↑ ^{a b} A. Bertram, p. 67. The smoothness is needed so that the divergence theorem applies
2. ↑ H. Giesekus, p. 10
3. ↑ A. Bertram, p. 70
4. ↑ H. Giesekus, p. 10, Eq. 2.1
5. ↑ K. Willner, p. 56