Shocks Near the Sonic Line: A Comparison between Steady and Unsteady Models for Change of Type

While congratulating F. H. Robertson on his technical note in the January 1957 JOURNAL (page 52) on “short cut” methods of estimating aircraft direct operating costs, which certainly give a very rapid approximate answer, I am somewhat uncertain as to its practical value. In the analysis of possible projects, having established the general range of operating costs, which can usually be done by a combination of experience and inspection, interest lies primarily in the effect on cost of operation of such factors as change of type of power plant, fuel used, what complexity of structural designs is permissible from a cost aspect, how much is desirable from a weight saving aspect and so on. Unfortunately, Mr. Robertson's formulae do not allow such comparisons to be made primarily because of the assumptions made.

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Change of type in three-phase flow: A simple analogue

Journal of Differential Equations ◽

10.1016/0022-0396(89)90085-5 ◽

1989 ◽

Vol 80 (2) ◽

pp. 280-305 ◽

Cited By ~ 14

Author(s):

Barbara Lee Keyfitz

Keyword(s):

Phase Flow ◽

Three Phase ◽

Three Phase Flow ◽

Change Of Type

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The influence of material non-uniformity preceding shear-band formation in a model for granular flow

European Journal of Applied Mathematics ◽

10.1017/s0956792597003070 ◽

1997 ◽

Vol 8 (5) ◽

pp. 457-483 ◽

Cited By ~ 3

Author(s):

DAVID G. SCHAEFFER ◽

MICHAEL SHEARER

Keyword(s):

Shear Band ◽

Material Properties ◽

Shear Banding ◽

Shear Bands ◽

Small Variation ◽

Band Formation ◽

Time Period ◽

Single Location ◽

Change Of Type ◽

Short Time

The onset of shear-banding in a deforming elastoplastic solid has been linked to change of type of the governing partial differential equations. If uniform material properties are assumed, then (i) deformations prior to shear-banding are uniform, and (ii) the onset of shear-banding occurs simultaneously at all points in the sample. In this paper we study, in the context of a model for anti-plane shearing of a granular material, the effect of a small variation in material properties (e.g. in yield strength) within the sample. Using matched asymptotic expansions, we find that (i) the deformation is extremely non-uniform in a short time period immediately preceding the formation of shear-bands; and (ii) generically, a shear-band forms at a single location in the sample.

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