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)

    “The surface of the earth is soft and impressible by the feet of men; and so with the paths which the mind travels. How worn and dusty, then, must be the highways of the world, how deep the ruts of tradition and conformity!”
    —Henry David Thoreau (1817–1862)

    “I dare say I am compelled, unconsciously compelled, now to write volume after volume, as in past years I was compelled to go to sea, voyage after voyage. Leaves must follow upon each other as leagues used to follow in the days gone by, on and on to the appointed end, which, being Truth itself, is One—one for all men and for all occupations.”
    —Joseph Conrad (1857–1924)

    “Cranks live by theory, not by pure desire. They want votes, peace, nuts, liberty, and spinning-looms not because they love these things, as a child loves jam, but because they think they ought to have them. That is one element which makes the crank.”
    —Rose Macaulay (1881–1958)