The lowest temperature at which creep can occur in a given material is generally
, where Tm is the melting temperature of the material in degrees Kelvin.
Total engineering creep strain can be expressed by the following formula:
Where ε is the theoretical stress,
is the change in the materials length and
is the materials original length.
The strain rate describes the rate of change in the strain of a material with respect to time.
Where
is the strain rate;
is the change in strain and
is the change in time.
The rate of deformation caused by creep is called the creep rate. The creep rate for a material with a constant stress and constant temperature can be calculated using the following formula:
Steady State Creep Rate:
Where Q is the activation energy; n is the stress exponent; A is a material constant; R it the universal gas constant and T is the temperature in degrees Kelvin.
The activation energy Q can be determined experimentally, by plotting the natural log of creep rate against the reciprocal of temperature. The gradient of the subsequent slope is equal to
.
Fig. 1 - Natural log of strain rate against reciprocal of temperature. [2]
For this experiment we are using a constant temperature for the three specimens. The Arrhenius equation can then be simplified to give a power law relationship:
Where A is a constant that depends on the given material.
Rearranging this equation the material constant A can be found:
The value of A can also be found by plotting the natural log of the strain rates against the natural log of the applied stress values. Here the value of A is equal to the exponential of the intercept of the line created by this plot.
The stress exponent n can be determined by plotting the natural log of the strain rate against the natural log of the applied stress. The gradient of this slope is equal to the stress exponent n.
Fig. 2 - Natural log of strain rate against natural of applied stress [2]
The stress component n is defined by the following equation:
Stages of Creep
Primary creep occurs at the initial stages of creep. In this stage the strain rate is relatively higher and then begins to gradually decrease.
Secondary creep is also called the steady state creep stage. This occurs after the primary creep stage and the creep rate changes to a constant. In this stage there is no increase or decrease in the creep rate.
Tertiary creep is the last stage of creep. The creep rate moves from the steady state of the secondary stage to a continuous increase. The creep rate progressively increases until the material reaches its breaking point and it ruptures.
Materials
Fig. 3 - Analogue Creep Testing Machine – Not used in experiment [3]
- Lever-arm creep testing machine.
- Various “dead-weight” masses. For this experiment there were 1.0, 1.2 and 1.4 kg masses.
- Various lead creep specimens compatible with the creep testing machine. Similar to that in Fig. 4.
- Linear Variable Displacement Transducer in contact with the lever.
- Analogue to Digital convertor in the form of a PCI card.
- Data logging computer program.
- Computer.
Because the creep testing machine uses a lever similar to that in Fig. 3, a mechanical advantage takes place. This needs to be taken into consideration when analysing the results. The lever in the creep testing machine in the experiment has an 8:1 mechanical advantage.
The machine pictured in Fig. 3 uses an analogue dial for recording displacement. The creep testing machine used in this experiment uses an LVDT transducer. This is in contact with the lever and sends displacement data to the A/D card in the form of electrical signals.
Fig. 4 – Lead Creep Specimen [4]
Method
- The three lead specimens are measured for their length and cross sectional area. For the first of the three tests, a 1kg load level is selected.
- The top end of the first specimen is installed in the top grip of the creep testing machine.
- The bottom end of the specimen is installed in the lower grip of the creep testing machine.
- The creep testing machine is zeroed. In this experiment zeroing wasn’t possible so the recorded displacement results were offset by 6.039. This was remedied by adding 6.039 to all recorded displacements.
- The data logger program is started while choosing an appropriate file name. For this experiment ‘data1.txt’ was chosen for the first specimen.
- The load is now applied to the specimen in the creep machine. The data logger will record the elapsing time and the deformation in the specimen.
- The specimen will eventually rupture due to the increasing creep and at this stage pressing stop in the program will end the logging.
- For the second specimen a load of 1.2kg is selected. A different filename is chosen in the data logger program. For this experiment ‘data2.txt’ was chosen for the second specimen.
- The process is repeated until the specimen fails.
- For the third and last specimen a 1.4 kg load is chosen. Again a different filename is selected in the data logger program. For this experiment ‘data3.txt’ was chosen for the third specimen.
- The process is repeated for the last time until the specimen fails.
- The results are then analysed as described below.
Results
Fig. 5 - Specimen 1 – Strain against Time with 1kg
Fig. 6 - Specimen 2 – Strain against Time with 1.2kg
Fig. 7 - Specimen 3 – Strain against Time with 1.4kg
Fig. 8- Specimen 1 – Strain Rate against Time with 1kg
Fig. 9 - Specimen 2 – Strain Rate against Time with 1.2kg
Fig. 10 - Specimen 3 – Strain Rate against Time with 1.4kg
Fig. 11 – Table of Values Calculated from Experimental Results
Fig. 12 - Natural log of strain rate against natural of applied stress – 3 specimens
(a)Estimationis made of the maximum applied stress that the material can withstand considering creep of less than 1% per year.
Assuming 31,536,000 seconds in a year:
The slope of the line in Fig. 12 gives the value for n. The exponential of the intercept of the line in Fig. 12 gives the value for A. Subbing for A and n and rearranging:
(b)
Estimation is made for the maximum applied stress considering a total time to failure of more than 10 years.
Again an assumption of 31,536,000 seconds in a year is taken. For the strain at failure an average was taken from the data for specimens 1 and 2, giving 13.134.
Subbing in for A and n and rearranging:
Discussion
From looking at the strain against time graphs, Fig. 5, 6, & 7, the different stages of creep can clearly be seen. In the primary stage the strain rate is relatively high and this can be seen visually by the steeper slope at this section on the graph. The slope in the primary stage then begins to decline indicating a decrease in the strain rate. This is despite the applied stress and temperature remaining constant. This can be explained by strain hardening occurring in the lead due to dislocations in the crystalline structure.
Looking at these graphs it can be seen that their slopes reduce further to a minimum and for a time stay nearly constant. This is a visual indication of the secondary stage in the creep process where the strain rate becomes nearly constant. Here there is a recovery process in the lead due to thermal softening. The recovery balances the effect of the strain hardening causing the strain to reach its steady state.
At the right hand side of the same graphs it can be seen that the slope increases. In Fig. 6 and Fig. 7 this is shown more clearly where the slope increases exponentially. This increase in slope after the steady state is a visual indication of the tertiary stage in creep. The increased strain rate, as visualised by the increasing slope, is caused by necking. The necking begins due to local variations in stress concentrations in the specimen due to microscopic differences, defects or impurities. After the necking the cross-sectional area of the specimen decreases resulting in rapidly increasing stress concentrations. This increases the strain rate exponentially leading to fracture.
In figures 8, 9 and 10 where the strain rate is graphed against time, the secondary creep stage can be seen more clearly. Here the steady state creep rate is visualised by a straight line with a value of y = 0. In the same graphs the secondary stage is bordered by two spikes in the strain rate. The left hand side has a smaller spike due to no work hardening having occurred and the specimens reacting to the applied load. The strain rate then decreases as discussed earlier. The right hand side shows a much larger spike due to the exponentially increasing strain rate caused by the necking.
The stress component n is defined by the following equation:
The stress component is then found by calculating the slope of against as seen in Fig.12. The material constant A can be found on the same graph by calculating the exponential of the intercept. Alternatively A can be found rearranging the power law equation:
Fig. 14 – Theoretical values for A against the experimental value.
In Fig. 14 it can be seen the values for A when using the power law equation compared against the value of found from Fig. 12. The differences are negligible and can be explained by errors as discussed below. The results of the experiment then confirms the steady state creep law.
Errors
If the masses are applied suddenly to the machine it will have a higher resulting stress on the specimen compared to a mass applied more gently. This is due to impact loading and will cause a higher deformation and creep in the specimen.
The precision of the machine used in the experiment will have a result on the error. Also over time a machine needs to be calibrated. In this experiment it was not possible to calibrate the machine so this needed to be compensated in calculation later.
Any vibrations on the machine or the LVDT will impact on the readings. This can occur through impact loading resulting in cyclical loading vibrations or it might be outside forces such as a table being moved.
As discussed earlier, the creep rate is impacted by temperature. Changes in temperature due to draft or other influences could result in a change in the creep rate.
No two lead specimens are exactly the same. There will be minor differences due to impurities in the metal or small defects such as notches caused by wear. Due to the manufacturing of the specimens there could be minor differences in their shape and area. All of these differences will have an impact on the results.
Friction in the creep testing machine will resist the stresses caused by the “dead-weight” masses. Ideally this friction will be at a minimum, however some friction will always still remain and this will be a source of error. Most of this friction will be concentrated at the fulcrum of the lever arm on the creep testing machine.
Electromagnetic interference in the electrical circuitry can impact on the recordings from the LVDT. Also any stray components in the system such as parasitic capacitances will also cause some interference.
Rounding errors in the software or algorithm or later by the user will result in cumulative errors.
References
[1]Jones, D. R. H. 2011. Engineering Materials 1, Fourth Edition: An Introduction to Properties, Applications and Design. 4Edition. Butterworth-Heinemann.
[2] DoITPoMS. 2006. Creep Rate Equation. University of Cambridge. Available at: http://www.doitpoms.ac.uk/tlplib/creep/rate_equation.php (Accessed: 01/04/2014)
[3] HSM34 Creep Testing Machine. 2004. Strength of Materials Brochure. P-A-Hilton Ltd. Available at: http://www.p-a-hilton.co.uk/files/17403%20Strength%20of%20Materials_19_04_2013.pdf (Accessed: 01/04/2014)
[4] DB/0908 Datasheet. Creep Test Specimens. TecQuipment Ltd. Available at: http://www.tecquipment.com/Datasheets/CPSpecimens_0908.pdf (Accessed: 01/04/2014)
