4.4 Curve fitting with three different methods
The analysis was carried out
The analysis was carried out considering three different methods.considering three different methods.
In method 1 we assume the creep model to have the formula of
We proceed to determine the material parameters for the creep model from We proceed to determine the material parameters for the creep model from the experimental data. Also we assume the parameter m to be equal to one.
the experimental data. Also we assume the parameter m to be equal to one.
We tried to estimate the creep strain rate and the total strain with the We tried to estimate the creep strain rate and the total strain with the calculated parameters. The results obtained are compared with the calculated parameters. The results obtained are compared with the experimental results. Analysis of the above comparison is done
experimental results. Analysis of the above comparison is done to study theto study the creep behavior in this particular method.
creep behavior in this particular method.
In method 2 we have considered the
In method 2 we have considered the same creep model similar to same creep model similar to method 1.method 1.
Here we have assumed the parameter ‘m’ to be constant for each material Here we have assumed the parameter ‘m’ to be constant for each material independent of stress and with varying ‘A’. We also determine the material independent of stress and with varying ‘A’. We also determine the material parameters
parameters in in this this method method and and also also analysis analysis was was carried carried out out similar similar toto method 1.From the comparison of the experimental and analytical results method 1.From the comparison of the experimental and analytical results conclusions were drawn.
conclusions were drawn.
The method 3 was carried out assuming the same creep model similar to t The method 3 was carried out assuming the same creep model similar to thehe above two methods. Here we have assumed that the material parameters above two methods. Here we have assumed that the material parameters
‘A’ and ‘m’ vary at each stress for different materials. Analysis was done
‘A’ and ‘m’ vary at each stress for different materials. Analysis was done for the creep model using the same approach similar to method 1 and
We know the creep formula as, We know the creep formula as,
m
‘m=1’ in the in the above expression.above expression.
The values of ‘A’ can
The values of ‘A’ can be calculated for four different stresses by comparingbe calculated for four different stresses by comparing to the experimental results and the average value is considered, and the to the experimental results and the average value is considered, and the same is substituted in the above formula. We try to estimate the parameter same is substituted in the above formula. We try to estimate the parameter
‘A’
‘A’ assuming the other parameter assuming the other parameter ‘m’=1‘m’=1.The values of ‘A’ are estimated.The values of ‘A’ are estimated from the above logarithmic Creep rate
from the above logarithmic Creep rate Vs. logarithmic stress graphs.Vs. logarithmic stress graphs.
From fig.14 it can be found that there is a huge deviation in the results From fig.14 it can be found that there is a huge deviation in the results between experimen
between experimental and theoretical when we use Average ‘A’ value. Onetal and theoretical when we use Average ‘A’ value. One of the possible reasons could be the error in the calculation of ‘A’ value of the possible reasons could be the error in the calculation of ‘A’ value from the experimental data. Now we tried to approach the ‘A ‘values by from the experimental data. Now we tried to approach the ‘A ‘values by applying suitable numerical methods. The new values are substituted once applying suitable numerical methods. The new values are substituted once again. The calculations are performed by Matlab. The value of ‘A’ is again. The calculations are performed by Matlab. The value of ‘A’ is obtained by iterative calculations. The results from theoretical formula are obtained by iterative calculations. The results from theoretical formula are verified with the experimental results. Results are plotted in Matlab as verified with the experimental results. Results are plotted in Matlab as shown below,
shown below,
The results are tabulated as shown below:
The results are tabulated as shown below:
PP
PP PPC-0.75 PPC-0.75 PPC-1.5 PPC-1.5 PPC-2.25PPC-2.25
n 10.28
n 10.28 8.71 8.71 11.76 11.76 12.2012.20 m
m 1 1 1 1 1 1 11
Average Average
‘A’
‘A’
3.338
3.338××1010−−1919 19.74619.746
18
10 18
10−−
×
×
6.993
6.993××1010−−2121 30.7230.72××1010−−2222
Approached Approached
‘A’
‘A’
0.8011
0.8011××1010−−1919 0.9380.938××1010−−1717 1.17851.1785 ××1010−−2121 4.3624.362××1010−−2222 Table2: Results from method 1
Table2: Results from method 1
0
Figure10: Creep strain rate Vs Stress for PP Figure10: Creep strain rate Vs Stress for PP
0
0
Figure12: Creep strain rate Vs Stress for PPC1.5 Figure12: Creep strain rate Vs Stress for PPC1.5
0
For PP C2.25 BC2.25 B efefore Iteraore Iterativtive Ae A pproximation of Approximation of A Ap
Ap proacproac hedhed Experimental Experimental Average Average
Figure13:
Figure13: Creep strain Creep strain rate Vs rate Vs stress for stress for PPC2.25PPC2.25
From the results above it can be
From the results above it can be understoounderstood that the parameter ‘m’ = 1 d that the parameter ‘m’ = 1 doesdoes not give desired experimental results, and we
not give desired experimental results, and we proceed to method 2.proceed to method 2.
We have the total strain given by We have the total strain given by
t
Now strain vs. time results are plotted as shs. time results are plotted as shown in Fig.14own in Fig.14-18,-18,
0
CASE1:Strain Vs Time for PP under 12.33Mpa CASE1:Strain Vs Time for PP under 12.33Mpa
Experimental Experimental Theoritical Theoritical
Figure14:
Figure14: Strain Vs time for PP under 12.33MpaStrain Vs time for PP under 12.33Mpa
0
CASE1:Strain Vs Time for PP under 17.33Mpa CASE1:Strain Vs Time for PP under 17.33Mpa
Experimental Experimental Theoritical Theoritical
Figure15:
Figure15: Strain Vs time for PP under 17.33MpaStrain Vs time for PP under 17.33Mpa
0
CASE1:Strain Vs Time for PPC0.75 under 17.33Mpa CASE1:Strain Vs Time for PPC0.75 under 17.33Mpa
Experimental Experimental Theoritical Theoritical
Figure16:
Figure16: Strain Vs time for PPC0.75 under 17.33MpaStrain Vs time for PPC0.75 under 17.33Mpa
0
CASE1:Strain Vs Time for PPC1.5 under 17.33Mpa CASE1:Strain Vs Time for PPC1.5 under 17.33Mpa
Experimental Experimental Theoritical Theoritical
Figure17: Strain Vs time for PPC01.5 under 17.33Mpa Figure17: Strain Vs time for PPC01.5 under 17.33Mpa
0
CASE1:Strain Vs Time for PPC2.25 under 17.33Mpa CASE1:Strain Vs Time for PPC2.25 under 17.33Mpa
Experimental Experimental Theoritical Theoritical
Figure18: Strain Vs time for PPC2.25 under 17.33Mpa Figure18: Strain Vs time for PPC2.25 under 17.33Mpa
From the fig.14 to 18 above we can see the strain vs. time is linear which is From the fig.14 to 18 above we can see the strain vs. time is linear which is not correct according to the experimental results. This may be due to the not correct according to the experimental results. This may be due to the assumptions made, and also the creep model we assumed in our case may assumptions made, and also the creep model we assumed in our case may not be appropriate. So we
not be appropriate. So we proceed to method 2.proceed to method 2.
4.4.2 Method 2 creep stage, with varying ‘A’
creep stage, with varying ‘A’
Since the secondary creep rate has much significance in the design fields, Since the secondary creep rate has much significance in the design fields, we consider secondary creep here. From the Norton-Bialy’s creep laws:
we consider secondary creep here. From the Norton-Bialy’s creep laws:
m derivation of such functions is as
derivation of such functions is as follows.follows.
The time derivative gives The time derivative gives
1
This derivation results in the known Time-Hardening rule, where creep This derivation results in the known Time-Hardening rule, where creep strain rate is expressed as a function of the stress
strain rate is expressed as a function of the stress σ σ and time t. and time t.
We solve them to find out the value for ‘m’. The values of ‘A’ are We solve them to find out the value for ‘m’. The values of ‘A’ are calculated by substituting the values of ‘m’ in above equations. The same is calculated by substituting the values of ‘m’ in above equations. The same is tried at four different stresses. We arrive at four different ‘A’ values.
tried at four different stresses. We arrive at four different ‘A’ values.
Similar approach as the method 1 is
Similar approach as the method 1 is done here for ‘A’ value. The results aredone here for ‘A’ value. The results are plotted for exp
plotted for experimental vs. theorerimental vs. theoretical.etical.
The results are tabulated as
The results are tabulated as shown below,shown below,
PP
PP PPC-0.75 PPC-0.75 PPC-1.5 PPC-1.5 PPC-2.25PPC-2.25 n
Table3: Results from method 2 Results from method 2
0
CASE2:For PP With average A and Constant m values CASE2:For PP With average A and Constant m values App
Approroacheachedd Experimental Experimental
Figure19:
Figure19: Creep strain rate Vs Stress for method 2Creep strain rate Vs Stress for method 2
The results were drawn in the same figures together with the Experimental The results were drawn in the same figures together with the Experimental results from Fig.20 to 23 for
results from Fig.20 to 23 for the different materials.the different materials.
0
CASE2:Strain Vs Time for PP under 17.33Mpa CASE2:Strain Vs Time for PP under 17.33Mpa
Experimental Experimental Theoritical Theoritical
Figure20: Strain Vs time for PP under 17.33Mpa Figure20: Strain Vs time for PP under 17.33Mpa
0
CASE2:Strain Vs Time for PPC0.75 under 17.33Mpa CASE2:Strain Vs Time for PPC0.75 under 17.33Mpa
Experimental Experimental Theoritical Theoritical
Figure21: Strain Vs time for PPC0.75 under 17.33Mpa Figure21: Strain Vs time for PPC0.75 under 17.33Mpa
0
0 5500000 0 110000000 0 1155000000 0.01
0.01 0.015 0.015 0.02 0.02 0.025 0.025 0.03 0.03 0.035 0.035
Time(Seconds) Time(Seconds)
S S t t r r a a i i n n
CASE2:Strain Vs Time for PPC1.5 under 17.33Mpa CASE2:Strain Vs Time for PPC1.5 under 17.33Mpa
Experimental Experimental Theoritical Theoritical
Figure22:
Figure22: Strain Vs time for PPC1.5 under 17.33MpaStrain Vs time for PPC1.5 under 17.33Mpa
0
0 2200000 0 4400000 0 6600000 0 8800000 0 110000000 0 112200000 0 114400000 0 1166000000 0.01
0.01 0.015 0.015 0.02 0.02 0.025 0.025 0.03 0.03 0.035 0.035
Time(Seconds) Time(Seconds)
S S t t r r a a i i n n
CASE2:St
CASE2:St rain Vs Time for PPC2.25 under 17rain Vs Time for PPC2.25 under 17.33Mpa.33Mpa
Experimental Experimental Theoritical Theoritical
Figure23: Strain Vs time for PPC0.75 under 17.33Mpa Figure23: Strain Vs time for PPC0.75 under 17.33Mpa
The total strain in this
The total strain in this method is given bymethod is given by
m
The Relative error in % for method 2 is
The Relative error in % for method 2 is as shown below,as shown below, PP
PP PPC0.75 PPC0.75 PPC1.5 PPC1.5 PPC2.25PPC2.25 12.33(Mpa)
Table4: Relative error in % of results when compared to experimental Relative error in % of results when compared to experimental results. experimental results, and therefore method 3
experimental results, and therefore method 3 was introduced.was introduced.
4.4.3 Method 3
parameters from the creep equationm the creep equation..
1 different stresses. And by performing necessary calculations we get the different stresses. And by performing necessary calculations we get the values of the parameters can be obtained. Now we plot the curves for time values of the parameters can be obtained. Now we plot the curves for time vs. strain using the obtained parameters, and they are verified with the vs. strain using the obtained parameters, and they are verified with the experimental values.
experimental values.
The values of A and m for each material at different stress values are The values of A and m for each material at different stress values are shown in the table
shown in the table below,below,
Material
Material PP PP PPC0.75PPC0.75 Stress(Mpa)
Stress(Mpa) A A m m A A mm
12.33
12.33 3.8833.883××1010−−1616 0.520.52 6.976.97××1010−−1414 0.350.35
17.33
17.33 2.582.58××1010−−1717 0.520.52 1.381.38××1010−−1414 0.310.31
20.67
20.67 2.792.79××1010−−1717 0.410.41 7.357.35××1010−−1515 0.270.27
24
24 4.304.30××1010−−1818 0.570.57 3.543.54××1010−−1616 0.600.60
Table5:
Table5: Results from method 3 with assumed model Results from method 3 with assumed model
Material
Material PPC1.5 PPC1.5 PPC2.25PPC2.25 Stress(Mpa)
Stress(Mpa) A A m m A A mm
12.33
12.33 5.835.83××1010−−1717 0.300.30 3.513.51××1010−−1717 0.270.27
17.33
17.33 6.776.77××1010−−1818 0.220.22 1.931.93××1010−−1818 0.230.23
20.67
20.67 2.052.05××1010−−1919 0.43 0.43 1.8761.876
20
10 20
10−−
×
×
0.60 0.60
24
24 6.706.70××1010−−2121 0.81 0.81 1.2611.261
21
10 21
10−−
×
×
0.90 0.90
Table6:
Table6: Results from method 3 with assumed model Results from method 3 with assumed model
0
CASE3:For PP With average A and m values CASE3:For PP With average A and m values
Appro
Approachedached Experimental Experimental
Figure24: Creep strain rate Vs Stress for method 3 Figure24: Creep strain rate Vs Stress for method 3 The total strain is given by,
The total strain is given by,
m
The results were drawn in
The results were drawn in the same figures together with the the same figures together with the ExperimentalExperimental results, as shown from Fig.25 to 28 for the different materials.
results, as shown from Fig.25 to 28 for the different materials.
0
CASE3:St rain Vs rain Vs TTime for PP ime for PP undeunder 17.33Mpar 17.33Mpa
Experimental Experimental Theoritical Theoritical
Figure25: Strain Vs time for PP under 17.33Mpa Figure25: Strain Vs time for PP under 17.33Mpa
0
CREEP3:Strain Vs Time for PPC0.75 under 17.33Mpa CREEP3:Strain Vs Time for PPC0.75 under 17.33Mpa
Experimental Experimental Theoritical Theoritical
0
CASE3:Strain Vs Time for PPC1.5 under 17.33Mpa CASE3:Strain Vs Time for PPC1.5 under 17.33Mpa
Experimental Experimental Theoritical Theoritical
Figure27: Strain Vs time for PPC1.5 under 17.33Mpa Figure27: Strain Vs time for PPC1.5 under 17.33Mpa
0
CASE3:Strain V s s Time foTime for PPC2.25 under 17.33Mpar PPC2.25 under 17.33Mpa
Experimental Experimental Theoritical Theoritical
Figure28: Strain Vs time for PP2.25 under 17.33Mpa Figure28: Strain Vs time for PP2.25 under 17.33Mpa
From the calculations it is evident that there exists one set of ‘A’ and ‘m’
From the calculations it is evident that there exists one set of ‘A’ and ‘m’
values for each material at different stresses. This is clear from the plots values for each material at different stresses. This is clear from the plots above.
above.
Plots between A and stress values and also between m and stress values are Plots between A and stress values and also between m and stress values are as shown
as shown
1
122 1144 1166 1188 2200 2222 2244
10 101414 10 101515 10 101616 10 101717 10 101818 10 101919 10 102020 10 102121 10 102222
Stress(Mpa) Stress(Mpa)
A A
A
A VsVs.S.Sigmaigma
*PP
*PP
vPPC0.75 vPPC0.75 .PPC1.5 .PPC1.5 +PPC2.25 +PPC2.25
Figure29: Plot of A Vs Stress for method 3 Figure29: Plot of A Vs Stress for method 3
1 Relative error in % for case3 is
Relative error in % for case3 is as shown below,as shown below, PP
PP PPC0.75 PPC0.75 PPC1.5 PPC1.5 PPC2.25PPC2.25 12.33(Mpa)
Table7: Relative error in % of results when compared to experimental Relative error in % of results when compared to experimental results for method 3
results for method 3 4.4.1.1 The results show that:
4.4.1.1 The results show that:
From the figure.29 it is evident that for the nanocomposites PPC1.5 and
Nanocomposites osites PP PP and and PP0.75 PP0.75 have have similar similar behavior behavior from from the the figure.29.figure.29.
For nanocomposites PP there is a small increase in the ‘A' value with For nanocomposites PP there is a small increase in the ‘A' value with increase in stress at a particular instant, this may be due to the variation in increase in stress at a particular instant, this may be due to the variation in the material model
the material model