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)

    “A big leather-bound volume makes an ideal razorstrap. A thin book is useful to stick under a table with a broken caster to steady it. A large, flat atlas can be used to cover a window with a broken pane. And a thick, old-fashioned heavy book with a clasp is the finest thing in the world to throw at a noisy cat.”
    —Mark Twain [Samuel Langhorne Clemens] (1835–1910)

    “We must be very suspicious of the deceptions of the element of time. It takes a good deal of time to eat or to sleep, or to earn a hundred dollars, and a very little time to entertain a hope and an insight which becomes the light of our life.”
    —Ralph Waldo Emerson (1803–1882)