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
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
-
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:
The reference and current volumes of an element are
where .
Therefore,
or,
so,
So we get
or,
Read more about this topic: Finite Strain Theory
Famous quotes containing the words transformation of, surface, volume and/or element:
“If you say, Im for equal pay, thats a reform. But if you say. Im a feminist, thats ... a transformation of society.”
—Gloria Steinem (b. 1934)
“All beauties contain, like all possible phenomena, something eternal and something transitory,something absolute and something particular. Absolute and eternal beauty does not exist, or rather it is only an abstraction skimmed from the common surface of different sorts of beauty. The particular element of each beauty comes from the emotions, and as we each have our own particular emotions, so we have our beauty.”
—Charles Baudelaire (18211867)
“And all the great traditions of the Past
They saw reflected in the coming time.
And thus forever with reverted look
The mystic volume of the world they read,
Spelling it backward, like a Hebrew book,
Till life became a Legend of the Dead.”
—Henry Wadsworth Longfellow (18091882)
“Justice has its anger, my lord Bishop, and the wrath of justice is an element of progress. Whatever else may be said of it, the French Revolution was the greatest step forward by mankind since the coming of Christ. It was unfinished, I agree, but still it was sublime. It released the untapped springs of society; it softened hearts, appeased, tranquilized, enlightened, and set flowing through the world the tides of civilization. It was good. The French Revolution was the anointing of humanity.”
—Victor Hugo (18021885)