Results From The Analysis.

6.1- Introduct ion.

6.2- Theoretical Percentage Reduction In Area.

6.3- Theoretical Coating Thickness.

6.4- Theoretical Yielding Position Of The Wire.

6.5- Theoretical Pressure.

6.6- Theoretical Shear Stress On The Wire.

6.7- Theoretical Stress In The Wire.

6.1- Introduction

A computer programme was written utilizing iteration techniques and finite difference method (based on backward difference) to solve the equations simultaneously.

Developments and listing of the computer programme is provided in Appendix 4. Since all the equations were

dependent upon speed, they were solved for speeds varying from 0,1 m/s to 4.0 m/s and at 0.1 m/s intervals. Also the equations governing the deformation zone were solved at steps of A x = 1mm for every speed increment from the position of yield up to the step in the unit where it is

assumed that the deformation of wire ceases.

The data below are the known parameters which were used to solve the equations and were varied to show their effect on the performance of the unit.

i)- Dimensions of the stepped bore reduction unit;

hi - 0.0002 m h2 = 0.00005 m 11 = 0.05 m 12 = 0.02 m hi /h2 = 4 . 0 I1/I2 = 2 . 5

ii)- Data for polymer;

T\ = 110 NS/m2

a = 120000 N/m2 b = 4 x 1011 m 2/N

5 2

Tea = 5 x 10 N/m

iii)- Data for copper wire;

Yo Ko ' n N T D,

Results from the analysis were obtained in tabulated forms and are represented in graphical forms for

convenience. _

6.2- Theoretical Percentage Reduction In Area

Figure 73 shows the variations of the percentage reduction in area with drawing speed, comparing the Newtonian and the non-Newtonian solutions. The non-

Newtonian solution (I) represents the effect of shear rate on viscosity alone and the non-Newtonian solution (II) represents the combined effect of shear rate and pressure on the viscosity.

The Newtonian solution appeared to estimate higher percentage reduction in area compared to those from the other two solutions. The inclusion of the strain rate sensitivity into the solution made little effect in the

= 0 . 5 x 108 N/m2 = 4.4 x 108 N/m2 = 0.52 = 55000 = 3.8 = 0.0016 m

final results.

The non-Newtonian solution (I) commenced from zero at about 0.1 m/s and the predicted percentage reduction in area increased as the drawing speed was increased. The theoretical results were significantly reduced when the strain rate sensitivity of the wire material was included into the analysis. This effect became higher as the drawing speed was increased. Note that the speed at which slip was predicted reduced when the strain rate sensitivity was included in the analysis.

The non-Newtonian solution (II) generally predicted higher percentage reduction in area than those of solution

(I) and also introduced slip at slower speeds. The

inclusion of the strain rate sensitivity into the analysis again reduced the theoretical results. This theory

produced results that showed closer trends to those of the experimental results.

Figures 74-to 86 show the theoretical percentage reduction in area when the input data were varied. These results are presented in three parts as follows;

i)- The effect of the dimensional changes of the unit.

The effect of the dimensional changes in the theoretical results are shown in figures 74 and 75.

Different values of gap ratios (hi /h.2 ) and length ratios were fed into the eauations to show their effect on the predicted percentage reduction in area,

Figure 74 shows the effect of the gap ratio on variations of the percentage reduction in area versus

drawing speed.In order to produce these results hi remained constant whilst h2 was varied. As the gap ratio was

increased, the predicted percentage reduction in area also increased and the critical shear stress occured at slower drawing speeds.

The effect of the length ratio on the predicted percentage reduction in area are shown in figure 75. It has been observed experimentally that, the deformation of wire takes place in the first part of the unit, therefore 1 2 was kept constant and li was changed to obtain the theoretical results for different length ratios. The results of percentage reduction in area showed similar trends to those of figure 74 ie the higher the length ratio, the higher the deformation of the wire.

ii)-The effect of the polymer material.

Theoretical effect of changing the parameters representing the polymer melt rheology are shown in figures 76 to 80. The effect of temperature in the viscosity of polymer melt was not included in the

analysis, therefore the initial viscosity was varied which is assumed to have the effect of temperature change.

Figure 76 shows the theoretical effect of changing the initial viscosity of polymer melt on the percentage reduction in area.An increase in the viscosity caused the percentage reduction in area to increase and also

introduced slip at slower drawing speeds.

constant MK" is shown in figure 77 where the percentage reduction in area is shown to increase as MK M was reduced. Also as the result of increase in shear stress constant, the crirical shear stress was predicted to occur at

slower drawing speeds. When K = 0 , deformation of wire commenced at about 0.1 m/s and rapidly reached to 25% reduction in area at-about 0.5 m/s.

Figure 78 shows the effect of the critical shear stress on the predicted percentage reduction in area. A

small increase in the magnitude of the limiting shear stress appeared to have a substantial effect on the results. The higher the critical shear stress, the higher the percentage reduction in area before slip takes place. Note that in this case the results followed the same curve before the critical shear stress was reached.

The effect of the viscosity constant on the percentage reduction in area is shown in figure 79. The increase in "a" predicted higher deformation in the wire and note that a large increase in "a” was necessary to introduce a

noticeable change in the theoretical results.

Figure 80 shows the effect of the coefficient of pressure on viscosity of polymer melts. Higher values of lfb" predicted higher percentage reduction in area. When

-11 o

b = 6 x 10 m /N, slip was shown to occur at very slow drawing speeds and higher results were predicted over the entire range of speed.

iii)- The effect of the wire material.