Torsion of Cylindrical and Prismatic Bars in the Presence of Steady Creep

1958 ◽  
Vol 25 (2) ◽  
pp. 214-218
Author(s):  
S. A. Patel ◽  
B. Venkatraman ◽  
P. G. Hodge
Keyword(s):  
Closed Form ◽  
Lower Bounds ◽  
Cross Sections ◽  
Creep Behavior ◽  
Upper And Lower Bounds ◽  
Elastic Analysis ◽  
Steady Creep ◽  
Pure Torsion ◽  
Closed Form Solutions ◽  
Creep Problem

Abstract This paper is concerned with the steady creep behavior of cylindrical and prismatic bars in which the deformations are caused by pure torsion. The creep problem is first reduced to one in nonlinear elasticity by means of the elastic analog. The elastic analysis is then carried out by means of the principles of minimum energies. These principles yield upper and lower bounds on the angle of twist. Closed-form solutions also are presented for some cross sections.

scholarly journals Creep-Velocity Bounds and Glacier-Flow Problems

1967 ◽  
Vol 6 (46) ◽  
pp. 479-488 ◽  
Author(s):  
Andrew C. Palmer
Keyword(s):  
Cross Sections ◽  
Upper And Lower Bounds ◽  
Steady Creep ◽  
Mean Velocity ◽  
Flow Problems ◽  
The Mean ◽  
Hand Calculation ◽  
Glacier Flow ◽  
Rate Relation ◽  
Creep Velocity

Abstract A general result due to Martin can be used to find upper and lower bounds on velocities in steady-creep problems. This method can be applied to glacier flow if ice can be assumed to satisfy a powerlaw stress–strain-rate relation. Bounds on the mean velocity over the glacier cross-section and on the mean velocity on the surface are determined for a particular example (a uniform parabolic channel, with powerlaw exponent 3) and they are shown to bound quite closely the exact solutions due to Nye. Bounds can be found rapidly by hand calculation. The method can be applied to real glacier cross-sections measured in the field.

scholarly journals Expected Logarithm and Negative Integer Moments of a Noncentral χ2-Distributed Random Variable

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 1048
Author(s):  
Stefan Moser
Keyword(s):  
Closed Form ◽  
Lower Bounds ◽  
Degrees Of Freedom ◽  
Random Variable ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Basic Properties

Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral χ2-distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed.

scholarly journals On closed-form tight bounds and approximations for the median of a gamma distribution

PLoS ONE ◽  
2021 ◽  
Vol 16 (5) ◽  
pp. e0251626
Author(s):  
Richard F. Lyon
Keyword(s):  
Lower Bound ◽  
Closed Form ◽  
Lower Bounds ◽  
Gamma Distribution ◽  
Upper Bound ◽  
Entire Range ◽  
Shape Parameter ◽  
Upper And Lower Bounds ◽  
Elementary Functions ◽  
Asymptotic Analyses

The median of a gamma distribution, as a function of its shape parameter k, has no known representation in terms of elementary functions. In this work we use numerical simulations and asymptotic analyses to bound the median, finding bounds of the form 2−1/k(A + Bk), including an upper bound that is tight for low k and a lower bound that is tight for high k. These bounds have closed-form expressions for the constant parameters A and B, and are valid over the entire range of k > 0, staying between 48 and 55 percentile. Furthermore, an interpolation between these bounds yields closed-form expressions that more tightly bound the median, with absolute and relative margins to both upper and lower bounds approaching zero at both low and high values of k. These bound results are not supported with analytical proofs, and hence should be regarded as conjectures. Simple approximation expressions between the bounds are also found, including one in closed form that is exact at k = 1 and stays between 49.97 and 50.03 percentile.

scholarly journals Creep-Velocity Bounds and Glacier-Flow Problems

1967 ◽  
Vol 6 (46) ◽  
pp. 479-488
Author(s):  
Andrew C. Palmer
Keyword(s):  
Cross Sections ◽  
Upper And Lower Bounds ◽  
Steady Creep ◽  
Mean Velocity ◽  
Flow Problems ◽  
The Mean ◽  
Hand Calculation ◽  
Glacier Flow ◽  
Rate Relation ◽  
Creep Velocity

AbstractA general result due to Martin can be used to find upper and lower bounds on velocities in steady-creep problems. This method can be applied to glacier flow if ice can be assumed to satisfy a powerlaw stress–strain-rate relation. Bounds on the mean velocity over the glacier cross-section and on the mean velocity on the surface are determined for a particular example (a uniform parabolic channel, with powerlaw exponent 3) and they are shown to bound quite closely the exact solutions due to Nye. Bounds can be found rapidly by hand calculation. The method can be applied to real glacier cross-sections measured in the field.

Upper and lower bounds for the torsional stiffness of a prismatic bar

1972 ◽  
Vol 328 (1573) ◽  
pp. 295-299 ◽  
Keyword(s):  
Lower Bound ◽  
Cross Section ◽  
Lower Bounds ◽  
Upper Bound ◽  
Variational Principles ◽  
Circular Cross Section ◽  
Torsional Stiffness ◽  
Upper And Lower Bounds ◽  
Steady Creep

Upper and lower bounds for the torsional stiffness of a prismatic bar in steady creep are derived in a unified manner from the theory of complementary variational principles. The lower bound is known in the literature, but the upper bound appears to be new. The results are illustrated with calculations for a bar with circular cross-section.

Upper and Lower Bounds of Frequencies for Cantilever Bars of Variable Cross Sections

1966 ◽  
Vol 33 (4) ◽  
pp. 948-950 ◽  
Author(s):  
J. H. Gaines ◽  
Enrico Volterra
Keyword(s):  
Lower Bounds ◽  
Cross Sections ◽  
Transverse Shear ◽  
Natural Frequencies ◽  
Numerical Results ◽  
Upper And Lower Bounds ◽  
Variable Cross ◽  
Tabular Form ◽  
Rotatory Inertia ◽  
Transverse Vibrations

Upper and lower bounds of frequencies of transverse vibrations of cantilever bars of variable cross sections are presented, taking into account the effects of transverse shear and of rotatory inertia. Numerical results for the first four natural frequencies are presented in tabular form for different inertia characteristics of the bars.

Viscous Flow Through Tubes of Multiply Connected Cross Sections

1959 ◽  
Vol 26 (4) ◽  
pp. 573-576
Author(s):  
F. A. Gaydon ◽  
H. Nuttall
Keyword(s):  
Lower Bounds ◽  
Cross Sections ◽  
Essential Feature ◽  
Ritz Method ◽  
Cylindrical Tube ◽  
Volume Flow ◽  
Upper And Lower Bounds ◽  
Internal Boundary ◽  
Relaxation Methods ◽  
Multiply Connected

Abstract A new method is proposed for estimating the volume flow of a viscous incompressible fluid through a cylindrical tube of multiply connected cross section. The method brackets the magnitude of the volume flow between upper and lower bounds. The essential feature of the method is that the calculation of both upper and lower bounds is based upon the same approximating function for the velocity distribution, thus avoiding the usual approach to a lower bound via the Rayleigh-Ritz method. For multiply connected cross sections of the form discussed, a Rayleigh-Ritz solution of sufficient accuracy becomes extremely laborious. Efforts to solve the problem by relaxation methods are also rendered difficult by the presence of high-velocity gradients in the vicinity of an internal boundary, particularly when this is a small circle. In contrast to these methods the one presented achieves the result with considerably less labor; moreover, the method is directly applicable to simply connected cross sections with many-sided boundaries.

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