At 08:52 AM 2001/06/16 +0200, Falk Koenemann wrote:
>...
>There is a reason for using a cube as unit body. As long as you understand
>pressure and stress as force per unit area you need that unit plane; and
>thinking of a cube is then nearly inevitable. But this is an artifact of the
>concept.
>
>We have two definitions of pressure, P = f/A and P = dU/dV. The latter is by
>far the more fundamental one because pressure is a state function; and since
>it is one form of the most fundamental relation in physics - the ratio
>energy/mass - it is also the basis of all of potential theory. The two
>definitions are not identical; the mathematical relation between the two is
>the subject of my published paper. But if you manage to think of pressure
>only and exclusively in terms of energy density P = dU/dV, the shape of V
>is no longer predispositioned by other concepts.
I would argue that the two definitions are fully compatible and equally
valid. P = dU/dV may be theoretically more fundamental, but in practice
only P and V are directly measureable, and devices that measure P depend on
P = f/A. Only by measuring/monitoring/controlling P and V can we
experimentally keep track of changes in U (i.e., dU = PdV).
>
>By assuming a priori that your unit volume is a cube, you thereby constrain
>the shape so that you have identical r in the principal directions. In
>effect you give away one degree of freedom in continuum mechanics - which
>you do have in discrete mechanics; but why is this done? What is the
>theoretical justification?
>...
A thermodynamic system is any part of the universe we choose to consider.
Let our system be a one-unit-of-volume part of a large solid body in a
state of constant, uniform, homogeneous elastic strain. I would contend
that the shape of the system is arbitrary. It can be a cube, a sphere or
any other shape that encloses one unit of volume, provided only that the
forces acting on the outer surface of the system are such as to maintain
its state of constant, uniform, homogeneous elastic strain. Accordingly,
since our choice of system shape has no energetic consequences, we might as
well choose a shape which will make the mathematics comprehensible and
tractable, i.e. a cube with unit volume and unit area on each of its six
faces, so that the normal and shearing components of externally applied
force on each face will be constant, uniform, homogeneous, and numerically
identical to stress = f/A (or to P = f/A in the special case where the
normal forces are equal and the shearing forces are zero).
Dugald M Carmichael Phone/V-mail: 613-533-6182
Dept of Geological Sciences and Geological Engineering
Queen's University FAX: 613-533-6592
Kingston ON K7L3N6 E-mail: [log in to unmask]
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