Stress-strain equations for viscoelastic materials with emphasis on two dimensional flow

Bogue, D. C.

English

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R sI: 
FINAL REPORT ON 
"CONSTITUTIVE EQUATIONS IN TWO-DIMENSIONAL FLOW" 
National Aeronautics and Space Administration 
RA IV lt-
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September 1964 - August 1969 
Submitted by 
D. C. Bogue 
Department 'of Chemical and Metallurgical Engineering 
University of Tennessee 
March 1969 
https://ntrs.nasa.gov/search.jsp?R=19690016027 2020-03-12T04:09:43+00:00Z
3 •1
FINAL REPORT ON 
"CONSTITUTIVE EQUATIONS IN TWO-DIMENSIONAL FLOW" 
CONTRACT NUMBER: NsG 671 
PERIOD OF CONTRACT: September 1964 - August 1969 
CONTRACTOR: Department of Chemical and Metallurgical Engineering 
University of Tennessee, Knoxville, Tennessee 
INVESTIGATORS: 
Principal: 
D.C.Bogue, Professor of Chemical Engineering 
Associates:
approx. support period: 
(1) E. B. Adams 3 years 
(2) R. L. Boles 2 years 
(3) H. L. LaNieve 1 year 
(4) J. 0. Casey 1 1/2 years 
(5) I. J. Chen 1 1/2 years
OBJECTIVES: 
To develop and test constitutive equations for viscoelasticu1uid, 
especially in two-dimensional flows. 
2 
SUMMARY OF RESEARCH ACCOMPLISHED 
In a general way the research has been concerned with developing 
constitutive (stress-strain) equations for viscoelastic materials, 
with particular emphasis on accelerative ("two-dimensional") flows. 
Such flows are. of very general, practical significance in that all 
extrusion processes involve the squeezing down of materials from 
large cross-sections to small ones. The research is summarized below 
in the order in which the projects developed; the overall objectives 
and results are noted very briefly, with reference being made to the 
published articles for the details. Actually, due to the step-funding 
arrangements, Mr.' Chen and Mr. Casey are presently being .supported, 
in part, from the contract and will continue to be through August 1969. 
In terms of the experimental work and overall scope, however, the 
research has been substantially completed and will be reported at this 
time. The finishing details will be available with the publication 
of their work. 
1. Birefringent Studies in Entrance Flows; Analysis of Entrance Flows 
(Adams, Fields *) 
In work started about seven years ago, we have measured stress-
birefringent patterns in viscoelastic flows, at sharp-edged and tapered 
channel ("two-dimensional") entrances. Polarized light is passed 
* Mr. T.R. Fields' work contributed to the project although he was supported 
by a NASA Traineeship and not by the contract.
I 
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through the flowing test material to produce optical patterns which, 
with certain assumptions, can be quantitatively converted to stresses. 
The present work has centered on interpreting these stresses in terms 
of viscoelastic theory (Adams, Whitehead and Bogue). As an outgrowth 
of this work, more general theoretical analysis of converging-diverging 
flow was carried out using full integral viscoelastic theory (Adams and 
Bogue). The most recent work has been concerned with birefringent 
measurements of much more elastic fluids at entrances. A very striking 
stress pattern, resembling a shock wave, has been observed and is not 
fully understood at the present time (Fields and Bogue). 
2. Pressure Drops in Entrance Flows (LaNieve, Boles) 
In two projects, we dealt directly with an important although very 
complicated practical problem: that of predicting pressure drops in 
tube entrances. We obtained some help from the theoretical analyses 
noted. above in establishing the parameters and correlating groups, but 
the final correlations are best described as empirical. In the first 
work we correlated pressure drops at sharp-edged entrances with an 
independent rheological measurement, normal stresses (LaNieve and Bogue). 
In later work we also dealt with tapered entrances using a similar 
correlation scheme (Boles and Bogue). 
3. Rheological Properties of Polymer Melts (Chen) 
Basic studies of the rheological properties of polymer melts
( 
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-	 (polyethylenes, polystyrene, polypropylene) have been carried Out on 
the Weissenberg Rheogoniometer. The emphasis has been on strongly 
non-linear elastic effects, by starting and stopping the applied strain 
rate as a function of time. These data have been analyzed with integral-
type viscoelastic theories. The objective of this work has been to 
find distinctive rheological differences between polymer melts and to 
use the parameters from such measurements in correlating complex flow 
effects, such as melt fracture. 
4. Macroscopic Energy Balances and Stability Analyses (Casey) 
In contrast to the detailed differential analysis of entrance flows 
carried out by Adams, we have also set down and solved overall, macroscopic 
balances. Tke.hope was to be able to associate the large entrance 
pressure drops with the elastic energy stored by the fluid. The 
elastic energy can be calculated, given certain constitutive theory, 
before and after the entrance.without knowledge of . he details;ofthe 
flow at the entrance itself. The conclusion was that this approach is 
not generally satisfactory: that is, that the large entrance pressure 
drops are due to irreversible losses and not to stored elastic energy. 
It was possible, however, to state the overall balances in a systematic 
manner such that they may be used in other problems. Casey's work has 
now turned to a rather different area: that of finding well-defined 
rheological experiments (such as extensional flow) which can be associated 
directly with entrance flows, as a route to characterizing melt fracture.
5 
PAPERS RESULTING FROM THE CONTRACT 
Papers published or accepted for publication or presentation; 
1. Adams, E.B., J.C. Whitehead and D.C. Bogue, "Stresses in a 
Viscoelastic Fluid in Converging and Diverging Flow", AIChE 
Journal, 11, 1026-32 (1965). (partly supported by the contract) 
2. LaNieve, H.L. and D.C. Bogue, "Correlation of Capillary Entrance 
Pressure Drops with Normal Stresses", J. Applied Polymer Science, 
12, 353 (1968). (partly) 
3. Fields, T.R., Jr., and D.C. Bogue, "Stress Birefringent Patterns 
at a Sharp-Edged Entrance", Trans. Soc. Rheo, 12:1, 39-55 
(1969). (partly) 
4. Adams, E.B., and D.C. Bogue, "Viscoelasticity of Accelerative 
Flows: A Simplified Integral Theory", AIChE Journal, in press. 
(fully) 
5. Boles, R.L., and D.C. Bogue, "Entrance Flows of Polymeric Materials, 
Part I. Pressure Drop", to be presented and preprinted for the 
Society of Plastics Engineers Meeting, Mayv'1969; submitted to 
Polymer Engineering and Science. (partly) 
Papers proposed: 
1. Chen, I.J. and D.C. Bogue, "Rheological Properties of Polymer 
Melts; and Development of Constitutive Equations for Polymer 
Melts". (partly) 
2. Casey, J.0. and D.C. Bogue, "Overall Energy Balances in 
Viscoelastic Fluids and Analysis of the Tube Entrance Problem". 
(fully) 
THESES COMPLETED OR IN PROGRESS IN CONNECTION WITH THE ABOVE WORK 
1. Adams, E.B., "A Simplified Constitutive Equation for Two-
Dimensional Viscoelastic Flow", PhD thesis, August 1967. 
2. LaNieve, H.L., "Correlation of Capillary Entrance Pressure 
Drops with Normal Stress Data", PhD thesis, March 1966. 
3. Fields, T.R., Jr., "Stress-Birefringent Patterns at a Sharp-
Edged Entrance", MS thesis, March 1967. 
4. Boles, R.L., "An Experimental Study of Viscoelastiô Flow in 
a Conical Entrance Region", PhD thesis, December 1967. 
5. Casey, J.O., PhD thesis, in progress. 
6. Chen, I.J., PhD thesis, in progress.


Constitutive equations in two-dimensional flow Final report, Sep. 1964 - Aug. 1969

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