Page:ComstockInertia.djvu/4

 This can be readily shown by a consideration of the figure. When the velocity of the element is along (x) there is an amount of work $$(v_{x} \cdot X_{x}) dy \cdot dz$$ done per second on the element by the tension ($$X_{x}$$) applied at the surface ($$O'A'$$), and this energy is instantly available at the surface ($$OA$$), where it is given out. The distance over which the energy is transmitted being $$dx$$ (the thickness of the element), the rate of energy-flow is

$-v_{x}X_{x} dy dz dx=-v_{x}X_{x} d\tau\,$,

where ($$d\tau$$) is the element of volume.



In like manner the velocity ($$v_{y}$$) and the shearing stress ($$Y_{x} \cdot dy \cdot dz$$) cause energy to be taken up at the surface ($$O'A'$$) and given out at the surface ($$OA$$), and we have the rate of flow along the x-axis

$-v_{y}Y_{x} d\tau\,$;

and finally the velocity ($$v_{z}$$) and the shearing stress ($$Z_{x} \cdot dy \cdot dz$$) give

$-v_{z}Z_{x} d\tau\,$.

Hence adding we have, if we call ($$f'_{x}$$) the density of flow along $$x$$,

$f'_{x}d\tau =-(v_{x}X_{x}+v_{y}Y_{x}+v_{z}Z_{x})d\tau\,$.

Obtaining the corresponding equations in similar way we have finally for the three components of the density of energyflow along the constraints in any system