Bounds and limit theorems for secondary creep in symmetric pressure vessels Page: 4 of 26
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denoted by a, rather than to an associated quantity such as the y defined
by Einarsson. Thus the bounds which result relate immediately to all of the
physical quantities of the problem without the need for further transformation.
For comparison purposes we record the equation central to developing
Einarsson's bounds. The cylinder equation, given by (51) of , and the
sphere equation, (94) of , are given respectively by the cases j - 2
and j - 3 of the equation
(x,t) _ -1 + 1 + x -j+j/n r Y d - y(x.,t
(j - 2,3) , (1.1)
where we have neglected the effects of strain-hardening included in (51)
and (94) of ; i.e., we have set m = 0. For the cylinder case (j 2),
the quantity y is related to the radial stress ar through equations (37)
and (55) of . In (3.12) of the present paper we show the relation,
again for the cylinder, between y and .the effective stress. The derivation
of (1.1) is given in ; however, the analysis leading to error bounds
for numerical solutions in the case of secondary creep, which is of major
interest to us, is contained in . In section 2 we obtain equations
for the sphere (2.28) and the cylinder (2.32). These are then unified
in equation (2.36). This latter equation plays the role analogous to
(1.1) and is the equation on which the analysis of the paper is based.
Our first results in section 3, which have no counterpart in  and
, state that o< 0 and r (ra) > 0 for all t > 0, where j - 2 for
cylinders and j - 3 for spheres. These results, together with some others,
are used to obtain bounds (3.8) for o(rt). They, in turn, may be used
to furnish upper and lower bounds for the radial displacement u(r,t) at
any point in the body, although only bounds for the displacement u(bt)
at the outer surface [(3.9) and (3.10)] are stated. These bounds reduce
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Edelstein, W.S. & Valentin, R.A. Bounds and limit theorems for secondary creep in symmetric pressure vessels, report, August 1, 1974; Illinois. (digital.library.unt.edu/ark:/67531/metadc1023364/m1/4/: accessed March 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.