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:

    “Whoever undertakes to create soon finds himself engaged in creating himself. Self-transformation and the transformation of others have constituted the radical interest of our century, whether in painting, psychiatry, or political action.”
    —Harold Rosenberg (1906–1978)

    “A society which allows an abominable event to burgeon from its dungheap and grow on its surface is like a man who lets a fly crawl unheeded across his face or saliva dribble unstemmed from his mouth—either epileptic or dead.”
    —Jean Baudrillard (b. 1929)

    “Measured by any standard known to science—by horse-power, calories, volts, mass in any shape,—the tension and vibration and volume and so-called progression of society were full a thousand times greater in 1900 than in 1800;Mthe force had doubled ten times over, and the speed, when measured by electrical standards as in telegraphy, approached infinity, and had annihilated both space and time. No law of material movement applied to it.”
    —Henry Brooks Adams (1838–1918)

    “An element of exaggeration clings to the popular judgment: great vices are made greater, great virtues greater also; interesting incidents are made more interesting, softer legends more soft.”
    —Walter Bagehot (1826–1877)