AIDJEX Bulletin #40 - Polar Science Center - University of Washington

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Since small changes in the yield surface may change the system of equations from hyperbolic to elliptic, it is of interest to know what the corresponding change in the actual stress state will be. Hodge (1950) has shown that a small change in a von Mises yield surface produces small changes in the stress field in the problem of an expanding circular hole in a plate, while for a notched bar in tension the stress state does not vary smoothly as the system goes from hyperbolic to elliptic. Problems in the plasticity of soils are solved by using a Coulomb yield function. They differ from metals problems where the materials are typically assumed incompressible. De Jong (1959) and Haithornwaite (1963) have argued that if this is to be the case, then a non-normal flow rule must be used. At the same time, Saint-Venant's hypothesis that the principal directions of the stress and strain tensors coincide is found to give unrealistic results. The alternative, formulated by Shield (1955) and Jenike (1961), is to allow the cohesion of the soil to be a function of density and to allow workhardening. Spencer (1964) obtains equations governing the velocity field for an incompressible material, but does not integrate along the characteristic directions. Likewise, Morrison and Richmond (1976) extend Spencer's result to include body forces and solve the problem of gravity flow through a restricting channel. This solution, as with all solutions in soils plasticity, is for a state of plane strain. When the Coulomb yield criterion is used the possibility of non-real characteristic roots does not occur, so the equations are always hyperbolic. The problem of repeated characteristic roots is not encountered, either. Recently, Sodhi (1977) has taken the plane strain solution of Morrison and Richmond (1976) and applied it to the flow of pack ice througharestricting channel. By correlating lead directions in the icepackwithcharacteristic directions, he predicts the conditions required for ice breakout to occur and, by measuring the relative orientation of leads through a point, deduces the cohesive strength of the ice cover. This model differs both in the yield surface and the flow rule from the AIDJEX plastic model and the results are not applicable to our work. 114
As can be seen from the references, the mathematical theory of characteristics is well established for many systems of equations, and much work has been done to define the characteristic curves for plasticity models. Some of the results that we present have been published previously. However, the lack of common notation and approach has made it difficult to understand whether or not a given result is applicable to our model. Therefore, we have introduced the model to be studied and have made the simpler classical analysis in a format that is consistent without analysis of the complete model. This approach has allowed us to understand the limitations and lack thereof in each simple case. Since the mathematical characteristics of the system of equations may depend on the particular form of the equations, we must discuss the particular form of the model to be considered. We choose to work with an ideal plastic form of the model so that the time variable does not enter the system of equations. It has been well documented that acceleration of the ice cover on temporal scales of one day are small enough that inertia may be neglected. With this assumption in a rigid plastic material model we have time entering the model only in the form of a hardening law. To eliminate this dependence, we have neglected hardening and assumed that hardening occurs at a slower rate. We consider both the existence of real characteristic curves and the form of the governing equations along these curves when the system is hyperbolic. The characteristic directions are discussed in detail and are related to solution variables of importance. Several examples of this relation are shown to help understand how the characteristic directions might be useful in interpreting the performance of the model. The analysis is performed for the most general system, which includes advection of the.materia1, but a non-normal flow rule is required to complete the analysis. In this system, stress and velocity variables are coupled. The flow rule is not normal, but it is defined from a potential function as an assumption that makes principal directions of stress and stretching coincide. The properties of this system are also examined when advection is neglected and when a normal flow rule is assumed. The problems that arise when a normal flow rule is assumed are discussed, but this analysis is not completed. 115
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DATA BUOYS AIDJEX BULLETIN No. 40 J
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NOTE TO FRIENDS, AND BYSTANDERS . .
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SUMMARY OF TECHNICAL DEVELOPMENTS I
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milestones, kept the vendor working
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The scene during deployment of HF-N
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The initial AIDJEX scientific plan
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poizr-orbiting satellite, which is
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t@ the spacecraft during each orbit
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BUOY NAHE COMMUN. COMMANDS FIXES TA
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L( 0 I 2 3 4 3 Radial error, kilome
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The buoy will be subjected to tempe
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nics occupy about 12 feet of the 17
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Ill -- - KEY AIUJEX WIN WW AEBa 0 4
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event that a melt pond should form
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chute supplied by Paranetics Inc. i
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6'" IqORGANIC LITHIUM BP.TTEI?Y IMP
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OM (AD RAMS) UREMEN~ \ AIRDROP N m
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provides a grid from which geostrop
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words; a four-bit air temperature w
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PERFORMANCE OF MET/OCEAN BUOYS IN A
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Sensor samples were taken every thr
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strengths are known, were used with
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the ice (causing reduced shear). An
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(McPhee, Mangum, and Martin) , TABL
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70°N 75'N n 3 P 17OOW (D Y 170°W
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12OOW * U Station 25. Xeasurements
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(McPhee , Mangum, and Martin) 2701
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(McPhee, Mangum, and Martin) a 0 a
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"i BUOY 4 I 43 360- 309- 257 206 15
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a30gl 251 I 60 511 r 43 BUOY 4 I
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n 4, (McPhee, Mangum, and Martin) 8
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(McPhee, Mangum, and Martin) 33 W c
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Page 70 and 71: I 25 23 t h--:!li[ NCI PiHCEl4l 22
Page 72 and 73: .5 .4 .3 0 A R 2 R 0 W J - 0 B U -
Page 74 and 75: 9 1.5 1.0 c 8 w .5 R 930 $ 565 1000
Page 76 and 77: Clock corrections were applied rout
Page 78 and 79: Q) u o z z i 4- 4- Z + + z f n 9 Ef
Page 80 and 81: TABLE 1 DIFFERENCES IN DAILY TEMPER
Page 82 and 83: Laboratory calibration of the ADRAM
Page 84 and 85: 3.20- - 1 t W o? 2 3.051 ...:-.;..
Page 86 and 87: POSITION MEASUREMENTS OF AIDJEX MAN
Page 88 and 89: some redundancy. Four points were u
Page 90 and 91: 75.5 75.: - E75.1 LLI % y75.c - 1 1
Page 92 and 93: The translocation principle removes
Page 94 and 95: POSITION ACCURACY To evaluate the q
Page 96 and 97: TABLE 1 STAND-ALONE ACCURACY 68th P
Page 98 and 99: Az i mu t h s When both receivers a
Page 100 and 101: DATA PROCESSING Following the field
Page 102 and 103: model ice motion in the statistical
Page 104 and 105: FREC 1 INCY / DOPPLER / FREQUENCY /
Page 106 and 107: 35 30 25 v) > 0 2 20 LL 0 r= E m 15
Page 108 and 109: An example of a large-scale buoy ar
Page 110 and 111: observations are needed in remote a
Page 112 and 113: useful for determining how various
Page 114 and 115: as well as velocity fields, we must
Page 118 and 119: The transformation of the system of
Page 120 and 121: e able to judge how well such a mod
Page 122 and 123: then the flow rule becomes Q = +A -
Page 124 and 125: [-B' -F Finally, we eliminate Carte
Page 126 and 127: differential equation. 2. Discontin
Page 128 and 129: Substituting (36) and (37) into (34
Page 130 and 131: arbitrarily assign the values of 1
Page 132 and 133: The special case when advection can
Page 134 and 135: TABLE 1 STRETCHING, STRESS, ASP CHA
Page 136 and 137: ate of deformation, we have normali
Page 138 and 139: CHARACTERISTIC EQUATIONS The ordina
Page 140 and 141: One way to circumvent the problems
Page 142 and 143: The equations governing solutions a
Page 144 and 145: It is readily seen that when body f
Page 146 and 147: which may be substituted into equat
Page 148 and 149: a the stress derivative dCldS remai
Page 150 and 151: Although we have not studied in thi
Page 152 and 153: Courant, R., and D. Hilbert. 1962.
Page 154 and 155: A RESEARCH PLAN TO TEST THE AIDJEX
Page 156 and 157: the barges managed to reach their d
Page 158 and 159: APPROACH A baseline simulation 0-f
Page 160 and 161: SENSITIVITY OF ICE RESPONSE TO DIFF
Page 162 and 163: Since free drift of ice is computed
Page 164 and 165: and what is observed becomes less i
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stresses appear to be relatively un
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ehavior is directly related to accu
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TABLE 1 COMPARISON OF ESTIMATED ERR
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a) Base line b) 24-hour prediction
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\ / a) Base line b) 24- hour predi
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a) Base line b) 24- hour predi ct i
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\ / a) Base line b) 24-hour predict
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.. a) Base line b) 24- hour predict
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J2 a) Base line b) 24-hour predicti
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) 24-hour prediction c) 36-hour pre
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8 Figure 17. AIDJEX surface pressur
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APPENDIX 1 AIDJEX DATA FILES 1. Pos
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9A. Surface-1 eve1 ai r pressure (d
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19. Surface pressure (Val i dated)
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APPENDIX 2 AIDJEX DATA TRANSFERRED
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I -7-71327GEO A Cards (continued) 7
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B Cards (continued) 201
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AIDJEX Bulletin #40 - Polar Science Center - University of Washington

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