Acoustoelastic Constants

As shown in §4 the stress in the roll material can be measured through the speed of sound. This changes with stress due to the acoustoelastic effect. In order to relate the change in the speed of sound to stress, the acoustoelastic properties of the material must be understood. Testing was completed to measure these properties in the material used for the press fit plugs installed as part of the industrial testing.

7.2.1 Experimental Setup

A load was applied to a plug which was identical to the plugs installed in the work roll. This plug was instrumented with 10 MHz shear and longitudinal sensors. As the load was increased the time of flight was calculated be measuring the time difference between two successive ultrasonic reflections.

Figure 7.9: Schematic and Photo of Plug Loading Test Setup.

It was important that the plug was loaded evenly to ensure a uniform compression. In order to achieve this, the load had to be applied normal to the surface of the plug. Any misalignment in

the loading would result in shear or non-uniform stresses. This was achieved by loading the plug via a ball as shown in Figure 7.9. This accounted for any misalignment in the rig and ensured that no shear loading was transmitted to the instrumented plug. A tool steel plate was placed between the ball and plug to ensure that the plug was not loaded just at the point of the ball contact. The load was applied by a hydraulic ram and measured using a load cell. This test equipment was provided by Tribosonics Ltd., Sheffield, UK.

On the opposite side of the plug to the loading ball the annular face surrounding the sensors was supported on a mounting plate. This was the same manner that the plug was supported when it was mounted in the roll. By using a plug and loading conditions representative of those seen in the final application the results gained should emulate those seen in the rolling mill implementation.

7.2.2 Results

The measured change in Time-of-Flight (ToF) for both the shear and longitudinal waves as the plug was loaded is shown in Figure 7.11. The compressive stress plotted was calculated using the cross sectional area of the plug and the load measured from the rig load cell. The negative change in time-of-flight corresponds to a reduction, and the negative stress represents compression.

Figure 7.10: Measured change in Time-of-Flight against Compressive Stress. The ultrasonic path length is 70±0.1mm and the absolute unstressed Time-of-Flights are Longitudinal =

11.66µs and Shear = 21.54µs.

It was expected that both plots would be linear, however the figure shows that both datasets appear to consist of two linear sections. The plots transition between these at around ~-45 MPa.

The plots show a very similar change in the time-of-flight.

From this the apparent speed of sound was calculated by dividing the ultrasonic path length (twice the plug length) by the time-of-flight. This speed of sound calculation doesn’t account for the plug deformation. In this case the deformation will result in a slight shortening of the plug, and therefore an over-prediction in the value of the speed of sound. The values gained for the data in Figure 7.10 are plotted in Figure 7.11.

Again a slight increase in the data gradient is seen at ~-45 MPa. However unlike in Figure 7.10 there is a difference in magnitude between the longitudinal and shear results. This is due to the differing speed of sound of the two wave types.

Figure 7.11: Percentage change in the apparent Speed of Sound against Compressive Stress.

Figure 7.12: 10.137mm diameter indentation left by loading ball in the Tool Steel plate.

When the test was disassembled it was observed that the loading ball had deformed the tool steel plate that was used to transmit load to the plug. This indentation is shown in Figure 7.12.

It is suspected that this plastic deformation happened at higher loads although this is not expected to have affected the acoustoelastic measurement.

The gradient above ~-45 MPa for both Figure 7.10 and Figure 7.11 appears to have a linear trend, but with a non-zero intercept. Non-uniform stress within the plug has been identified as the most likely source for this unexpected behaviour. The calibration assumes an ideal plane stress being applied through the plug. The whole of the top face of the plug is loaded, however the bottom face of the plug is only supported at its shoulders and not directly behind the sensors, due to the gap in the plug mounting plate required to extract the sensor cables as shown in Figure 7.11. Therefore, the stress does not transmit from the bottom to top faces of the plug in direct alignment with the ultrasonic path. Additionally, the rig will only have a finite stiffness, there will be some deformation of the rig and sample holder which will increase with higher loading. This means at higher loads the plane stress assumption is likely to be invalid. Because of this only data for the initial linear section of the graph was used to calculate the relationship between stress and acoustic velocity.

In cold rolling compressive stresses of up to several hundred MPa are expected. Only using the data for compressive stresses up to 45MPa to determine the acoustoelastic relationship means that the results potentially need to be extrapolated to stresses up to an order of magnitude higher. This could be a source of error, however, it was demonstrated in §4.5 and shown in Figure 4.4, that the stress to Time-of-Flight change is expected to remain linear up to at least 1GPa therefore the extrapolation is expected to hold.

The change in time-of-flight and speed of sound, along with linear trend lines and equations, are plotted in Figure 7.13 and Figure 7.14.

Figure 7.13: Measured change in Time-of-Flight against Compressive Stress, -5 to -40 MPa.

Figure 7.14: Percentage change in the Speed of Sound against Compressive Stress, -5 to -40 MPa.

While it would be expected that the trends pass through the origin, their equations show that they have a slight offset. This is likely to be due to the system settling as the load is initially applied, the results below -5 MPa have therefore been ignored. The offsets have been discarded and only the slope of the trend line has been used to quantify the relationship between the load and the speed of sound.

Figure 7.15: Contributions to the change in Time-of-Flight for a Longitudinal Wave.

Knowing the stress and cross sectional area it was possible to calculate the strain and therefore the deflection of the plug. This allowed the contributions of both the density and plug deflection to the change in ToF to be calculated using the equations in §4. The difference between measured change in ToF and the combined density and deflection effects was assumed to be solely down to the acoustoelastic effect. The results of these calculations for both the shear and longitudinal results and are plotted in Figure 7.15 and Figure 7.16.

Figure 7.16: Contributions to the change in Time-of-Flight for a Shear Wave.

Table 7.3 shows the contribution of each effect as calculated from the equations in Figure 7.15 and Figure 7.16. Also stated is the back calculated acoustoelastic constant calculated from Equation 4.40.

Density Deflection Elasticity Total

ns/MPa % ns/MPa % ns/MPa % ns/MPa % α

Long. 0.0303 11.95 0.0606 23.91 0.1627 64.18 0.2535 100 -2.685 Shear 0.0547 20.21 0.1095 40.45 0.1064 39.31 0.2707 100 -1.757 Table 7.3: Contributions of Density, Deflection and Elasticity to the change in Time-of-Flight.

7.2.3 Acoustoelastic Testing Discussion and Conclusions

The value for the longitudinal acoustoelastic constant compares well with those found in literature, as shown in Table 4.1. Also the ratio of the contributions from the deflection and elasticity effects compare favourably with those found in previous studies (Chen, Mills, & Dwyer-Joyce, 2015). No equivalent values for the shear waves were found, and no previous study appears to have accounted for the effect of density.

The data used (Figure 7.13) to calculate the contributions to the change in time-of-flight and acoustoelastic constants demonstrated excellent linearity. The square of Pearson’s correlation coefficient is 0.9971 and 0.9984 for the longitudinal and shear data respectively. The

combination of good data correlation, and consistency with literature values, gives confidence that the results reported are accurate.

A number of potential sources of error were identified in this testing. Some plastic deformation of a loading plate was observed, although it is not thought that this will have affected the acoustoelastic results. The inability to support the entire bottom face of the plug, due to the presence of the sensors and the need to extract cabling, may have resulted in some bending of the plug. This meant that only the results for compressive stress up to -40 MPa were used for the calculation of the acoustoelastic constants. The peak stresses seen in cold rolling are likely to be multiple times this magnitude, and so the acoustoelastic relationship will have to be extrapolated beyond that measured. Because the relationship is linear this extrapolation should be valid, however the error is likely to increase with greater extrapolation. The ultrasonic waves were digitised with a 10 ns time-step. This is large in comparison to the measured time-of-flight changes. The resolution was increased beyond that of the digitiser using the time-of-flight interpolation methods discussed in §4.6, these represent another potential source of error.