Finite Strain Theory - Transformation of A Surface and Volume Element

Transformation of A Surface and Volume Element

To transform quantities that are defined with respect to areas in a deformed configuration to those relative to areas in a reference configuration, and vice versa, we use Nanson's relation, expressed as

da~\mathbf{n} = J~dA~\mathbf{F}^{-T}\cdot \mathbf{N} \,\!

where is an area of a region in the deformed configuration, is the same area in the reference configuration, and is the outward normal to the area element in the current configuration while is the outward normal in the reference configuration, is the deformation gradient, and .

The corresponding formula for the transformation of the volume element is

dv = J~dV \,\!
Derivation of Nanson's relation
To see how this formula is derived, we start with the oriented area elements

in the reference and current configurations:

d\mathbf{A} = dA~\mathbf{N} ~;~~ d\mathbf{a} = da~\mathbf{n} \,\!

The reference and current volumes of an element are

dV = d\mathbf{A}^{T}\cdot d\mathbf{L} ~;~~ dv = d\mathbf{a}^{T} \cdot d\mathbf{l} \,\!

where .

Therefore,

d\mathbf{a}^{T} \cdot d\mathbf{l}= dv = J~dV = J~d\mathbf{A}^{T}\cdot d\mathbf{L} \,\!

or,

d\mathbf{a}^{T} \cdot \mathbf{F}\cdot d\mathbf{L} = dv = J~dV = J~d\mathbf{A}^{T}\cdot d\mathbf{L} \,\!

so,

d\mathbf{a}^{T} \cdot \mathbf{F} = J~d\mathbf{A}^{T} \,\!

So we get

d\mathbf{a} = J~\mathbf{F}^{-T} \cdot d\mathbf{A} \,\!

or,

da~\mathbf{n} = J~dA~\mathbf{F}^{-T}\cdot \mathbf{N}\qquad \qquad \square \,\!

Read more about this topic:  Finite Strain Theory

Famous quotes containing the words transformation of, surface, volume and/or element:

    “The transformation of the impossible into reality is always the mark of a demonic will. The only way to recognize a military genius is by the fact that, during the war, he will mock the rules of warfare and will employ creative improvisation instead of tested methods and he will do so at the right moment.”
    —Stefan Zweig (18811942)

    “We’ve forgotten what it’s like not to be able to reach the light switch. We’ve forgotten a lot of the monsters that seemed to live in our room at night. Nevertheless, those memories are still there, somewhere inside us, and can sometimes be brought to the surface by events, sights, sounds, or smells. Children, though, can never have grown-up feelings until they’ve been allowed to do the growing.”
    —Fred Rogers (20th century)

    “A German immersed in any civilization different from his own loses a weight equivalent in volume to the amount of intelligence he displaces.”
    —José Bergamín (1895–1983)

    “Beauty is composed of an eternal, invariable element whose quantity is extremely difficult to determine, and a relative element which might be, either by turns or all at once, period, fashion, moral, passion.”
    —Jean-Luc Godard (b. 1930)