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:
“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)
“But the surface of the Earth was meant for man. He wasn’t meant to live in a hole in the ground.”
—Edward L. Bernds (b. 1911)
“F.R. Leavis’s “eat up your broccoli” approach to fiction emphasises this junkfood/wholefood dichotomy. If reading a novel—for the eighteenth century reader, the most frivolous of diversions—did not, by the middle of the twentieth century, make you a better person in some way, then you might as well flush the offending volume down the toilet, which was by far the best place for the undigested excreta of dubious nourishment.”
—Angela Carter (1940–1992)
“One can describe a landscape in many different words and sentences, but one would not normally cut up a picture of a landscape and rearrange it in different patterns in order to describe it in different ways. Because a photograph is not composed of discrete units strung out in a linear row of meaningful pieces, we do not understand it by looking at one element after another in a set sequence. The photograph is understood in one act of seeing; it is perceived in a gestalt.”
—Joshua Meyrowitz, U.S. educator, media critic. “The Blurring of Public and Private Behaviors,” No Sense of Place: The Impact of Electronic Media on Social Behavior, Oxford University Press (1985)