ABSTRACT

With tighter design demands where factors of safety are lower and material specifications are stricter, knowledge of the stress state in materials and structures is increasingly important. This is true both on the manufacturing floor as well as for items in service. Ultrasonic techniques offer the unique capability of establishing macro stresses along a finite path length, as well as gradients near to the surface. The principles of ultrasonic stress measurement and specific applications of surface skimming waves are described. Stress resolution capabilities are covered. Specifically applications discussed include rolled and welded steel and aluminum plate, pressure vessels, turbine rotors, discs and blades and railroad rail and wheels.

INTRODUCTION

Various techniques are available for the measurement of residual stress. Thompson, Lu and Clark(1996) and Schneider(1997) describe several ultrasonic methods. Stress measurement with critically refracted longitudinal (L_CR) waves is described in Bray and Stanley (1997). Residual stress is never measured directly, but indirectly through the strain induced by the residual stress. The nondestructive methods are based on the relationship between the physical or crystallographic parameters and the residual stress. Ultrasonic techniques rely on the variations in the time of flight difference of ultrasonic waves, which can be related to the residual stress state through third order elastic constants of the material.

Cost and portability are the two main factors limiting the extensive use of the non-destructive techniques. Ultrasonic instrumentation has an advantage in this aspect because it has the lowest cost among the previously cited methods. Boards with data acquisition rates of 100 MHz or more are normally used in conjunction with regular personal computers. These boards are able to sample data at every 10 nanoseconds. When acquiring data, the researcher usually looks for a single position in time, like the time that the wave crosses the null in the amplitude scale. Because of the linear interpolation between two consecutive data points, it is possible to get estimates of this time with resolution of arrival time of 0.01 ns or better. Stress resolution is a function of variables of the entire system. Improved probe coupling practices have achieved resolution of 20 MPa (3 ksi) (Bray and Chance, 1999). Commercial ultrasonic flaw detectors with less resolution may be used for applications where stress variations are large.

THEORY

Ultrasonic stress measurement techniques are based on the relationship of wave speed in various directions with stress. Figure 1 shows elements of a bar under tension where the wave propagates in three perpendicular directions. The first index in the velocities represents the propagation direction for the wave and the second represents the direction of the movement of the particles. In (a) the wave propagates parallel to the load and V₁₁ represents the velocity of the particles in the same direction (longitudinal wave), meanwhile V₁₂ and V₁₃ represents the velocity in a perpendicular plane (shear waves). In (b) and (c) the waves propagate in the other directions and the velocities are also shown. The sensitivity of these waves to the strain is shown in the Figure 2 for rail steel. The most significant variation in travel-time with the strain was found for longitudinal waves, followed by the shear waves when the particles vibrate in the direction of the load. The other waves do not show significant sensitivity to the deformation.

 

Fig 1: Velocity of plane waves and stress field in orthogonal coordinate system.

Fig 2: Relative changes of wave speed with strain.

 

An extensive literature review of acoustoelasticity is contained in Bray and Stanley (1997) and will be summarized here. The speeds of the plane waves traveling parallel to load can be related to the strain (a ) by the following expressions:

(1.a)

(1.b)

(1.c)

where:
a₁, a₂, a₃ = Components of the homogeneous triaxial principal strains
r_o = Initial density
l, m = Second order elastic constants (Lame's constants)
q = a₁ + a ₂ + a₃
l, m, n = Third order elastic constants
V₁₁, V₁₂, V₁₃ = Velocities of waves in the direction 1 with particle displacement in the directions 1, 2, 3 respectively
For a state of uniaxial stress, a₁= e, a₂ = a ₃= -n.e , where e is the strain in the direction 1 and n is the Poisson's ratio. Using these values, equation 1.a becomes:

 

(2)


The relative sensitivity is the variation of the velocity with the strain and can be calculated by equation 3. In this equation, L₁₁ is the acoustoelastic constant for L_CR waves.

 

(3)


The values for acoustoelastic constants for other directions can be obtained in the same way. Experimental values for the acoustoelastic coefficients for several waves and materials are given in Table 1. The variation in the V₁₁ velocity, controlled by the coefficient L₁₁, is much greater than the other ones, indicating that these waves are the best candidates to be used in the stress evaluation.

Stress can be calculated by the one-dimensional application of the stress-strain relations in elastic solids. Equation 3 can be rearranged to give the stress variation in terms time-of-flight (dt/t_o), as shown in the equation 4, where t_o is the time for the wave to go through a stress free path in the material being investigated.

 

(4)

where ds = stress variation (MPa)
           E = elasticity modulus (MPa)

The same equation can be used for the other directions of the waves, provided the value of the acoustoelastic coefficient L is changed.

MATERIALS EFFECTS

Besides the internal strain condition, the texture in the material is another effect that could cause some measurable changes in the wave velocities. The non-uniformity of the microstructure usually is not significant, therefore the texture effect to the ultrasonic wave travel time will be very small ~ 0.02% (Tang and Bray, 1996). However, comparing to ~ 0.1% stress-related travel time change, it is still a countable error source in the practical stress evaluation and the effects need to be minimized. The common texture orientation for rolled steel is (100)[011], but other orientations may still exist. Bray and Stanley (1997) discuss the effects of texture and stress on wave velocities in rail steels. Tests of the potential the potential texture effect in a 4140 steel bar showed that if spatial position could be well controlled the texture effect of the wave propagation can be as low as 0.02% or 2 ns in the travel-time measurement

 

Material	Load	L21	L23	L22	L11	L12
Aluminum	Compressive	-2.0	+0.6
Aluminum (Tanala, et al, 1995)	Tension - RD		+0.46	+0.68	-2.7
Aluminum(Tanala, et al.,1995)	Tension - TD			+0.93	-3.1
Ductile Cast Iron As-cast Annealed Normalized Q & T	Compressive Compressive Compressive Compressive				-2.15 -3.89 -3.92 -2.98
Rail Steel(1080)	Tension Compressive	-1.5	+0.09	+0.27	-2.38 -2.45	-0.15
4140 Steel (Tang and Bray, 1996)	Tension (2.25 MHz) Tension (5 MHz)				-2.2 -2.36
316L Stainless Steel (Tanala, et al., 1995)	Tension	-1.5	-1.2	~0	-2.1
Clear acrylic, aircraft grade	Tension				-2.14
Table 1a: Acoustoelastic constants (Lij) for longitudinal and shear waves in engineering materials

 

Material	Load	LR13	LR23
Aluminium	Tension - RD	-1.1	+0.5
Aluminium	Tension - TD	-0.48	+0.5
316L Stainless Steel	Tension
Table 1b: Acoustoelastic constants (LRij) for Rayleigh waves in engineering materials (Tanala, et al, 1995)

CHARACTERISTICS OF THE L_CR TRANSDUCER

Ultrasonic techniques are regularly used for flaw detection in mechanical components and systems, and more recently for stress evaluation. Angle beam techniques direct the ultrasonic energy to regions not easily reachable with the normal beam approach. Snell's Law, as given by Equation (5),

 

(5)

where q₁ is the angle of the incident longitudinal wave
q₁^' is the angle of the refracted longitudinal wave
C₁ is the longitudinal wave speed in the incident material and
C₁^' is the longitudinal wave speed in the inspected material

describes the relationship of the incident and refracted beams in angle beam inspection. The L_CR technique is a special case of angle beam inspection, the L_CR being excited when the angle of incidence is slightly greater than the critically refracted angle. For the PMMA (Plexiglas) and steel combination shown in Figure 3, q₁ will be 28degrees. Typical speeds for bulk longitudinal waves in PMMA and steel are 2730 m/s and 5900 m/s, respectively.

 

Fig 3: LCR Probe for PMMA Steel Combination

Fig 4: LCR Beam Profile.

 


There are two proposed classifications for waves excited in the critically refracted mode, namely longitudinal surface creeping wave (LSCW) and subsurface longitudinal wave (SSLW). The LSCW has been described as one that decays rapidly in the first few centimeters after leaving the probe. Yet, for ultrasonic stress measurement work, longitudinal critically refracted waves (L_CR) waves are observed to travel long distances, over 300 mm (12 in.) and retain a suitably strong amplitude. Thus, the subsurface longitudinal waves generated by this type of probe are truly bulk waves propagating long distances at near bulk longitudinal wave speed, just underneath the surface. The longitudinal wave beam profile on Figure 4 shows that the L_CR wave is indeed generated at the surface of the steel plate. This energy is within the principal lobe of the refracted signal. The maximum energy of the principal lobe (SSLW) is at an angle of refraction of 84°. There is a stronger lobe at 67°, but this is only a side lobe. The L_CR is shown to be approximately 50 percent as string as the major lobe. Since the lobe shape is affected by frequency, there is reason to believe that stress gradients may be measured (Bray and Tang, 1997). Another important characteristic of this wave is that, in comparison to the shear waves, it is more sensitive to stress and, yet, less sensitive to localized material texture changes.

STRESS MEASUREMENT WITH THE L_CR TECHNIQUE

For stress measurement work, the L_CR probes are arranged in a tandem fashion (Figure 5), with one probe acting as the transmitter and the other as a receiver. In some cases, dual receivers are used. Distance between the probes (d) is kept constant by the rigid space bar, assuring that any change in travel-time between the two probes is due to stress or material variations, and not a change in probe spacing. The effects of stress, texture and temperature on the wave speed all contribute to the aggregate changes in data collected. The relationship caused by such parameters may be expressed as:

(6)

where t is measured travel-time, t^* is travel-time for a homogenous, isotropic, stress free member at a standard temperature, Dt_RS is the travel-time effect of the residual stress, Dt_T is travel-time effect of the temperature difference at the time of measurement from the standard measurement, Dt_F is travel-time effect of an applied (active) force, and Dt_TX is travel-time effect of the material texture. For residual stress measurement, Dt_Fcan usually be assumed to be zero.

The relationship of measured L_CR wave travel-time change and the corresponding uniaxial stress is given by:

 

(7)

where Ds is change in stress, E is Young's modulus, and L is the acoustoelastic constant for longitudinal waves propagating in the direction of the applied stress field. For assumed constant values of t^* and Dt_TX then, travel-time values (t) obtained at different locations on the rotor will indicate stress changes. Note that temperature variations may be established in the data set.

APPLICATIONS TO ENGINEERING COMPONENTS

Results of ultrasonic stress measurements in a number of engineering components will be described.

The stress fields in rolled and welded plate are amenable to ultrasonic stress measurement. Date were obtained from two 19 mm (0.75 in.) rolled steel plates that were butt welded before and after the stress relief. By comparing travel-times away from the weld, assumed zero stress, and at the weld, tensile stresses of 570 MPa (83 ksi) were shown to exist at the weld before the stress relief. In another test, the technique was able to distinguish between stress relieved and non-stress relieved patch welded 12 mm (0.5 inch) thick steel plates. Later, three aluminum plates were furnished for investigation by Kaiser Aluminum. One plate was as rolled, another stress relieved and another fully (O temper) annealed (Bray, Kim and Fernandes, 1999). Here, the L_CR technique clearly distinguished between the as rolled and the stress relieved plates.

Also, the results showed some unusual stress patterns in the full O temper plate, which were later confirmed. Applications for ductile cast iron are described in Bray and Stanley (1997).

Experimental work also has been performed on welded pressure vessels (Bray, 2000). Here, the 305 mm (12 in) diameter steel vessel was pressurized with water, and a pressure range of 0 - 69.9 MPa (0 - 10 ksi) was observed with both L_CR travel-time change and strain gauges. Additionally, stresses observed with L_CR near to the weld were comparable to values obtained by others using strain gauges in a similar vessel.

 

Fig 5: Typical tandem LCR probe.

Fig 6: Typical LCR arrival.

 

There have been several applications for rotating equipment. First, data collected in a steam turbine disk showed uneven stresses on the inlet and outlet sides. These results were confirmed by x-ray diffraction. Also, tests on a compressor rotor correctly identified the region causing a bow in the rotor (Bray, Tang and Grewal, 1997, Bray and Dietrich, 1997). Finally, the method has been used to characterize the effects of shock peening in titanium turbine blades for aircraft engines (Bray, 2000a). The holding quality of the shrink fit of generator retaining rings is dependent on the hoop residual stress in the ring since the operating, centrifugal induced stresses will tend to release the retention force. Leon-Salamanca, Reinhart, Bray and Golis (1989) describe an application of critically refracted longitudinal waves in evaluating this stress, before and after mounting of the ring.

Railroad applications have well demonstrated the usefulness of the technique (Bray and Stanley, 1997). Szelazek (1998) has reported success in using the results for observing thermally induced stress changes in rail. Railway wheel stresses have been observed with ultrasonic shear waves by Schneider (1998). Santos and Bray (2000) have used both shear waves and L_CR waves for wheel stress measurement.

For applications where the stress variations are high, commercial ultrasonic flaw detectors may be used for collecting L_CR data. Using a load frame designed for demonstration of the L_CR technique, data were collected on a strain gauged steel bar using both a PC containing a high speed digitizing card and an Epoch III digital ultrasonic flaw detector (Santos and Bray, 2000). The flaw detector was set so that a 7 ns resolution could be obtained, and load variations near to 4 percent over a stress range of 0 to 100 MPa. In comparison, the PC based data were in error by less than 1 percent over the load rang.

SUMMARY AND CONCLUSIONS

Ultrasonic stress measurement has its origin in the basic acousto elastic relationships. Other material variations such as texture and temperature may affect the results obtained from these measurements. Considering these limitations, however, there is ample data indicating that repeatable and reliable results are possible from many engineering components. One source of error is travel-time variations caused by couplant thickness changes. These can be minimized by applying pressure to the interface. Also, precise probe location control is needed. Good knowledge of the expected stress field and material variations is needed. More data on material properties is needed. Given these qualifications, it is quite likely that ultrasonic stress measurement will see growth in the next few years.

REFERENCES

1. Bray, D. Report to Koch Industries, 1999, unpublished.
2. Bray, D., Report to General Electric Aircraft Engines, 1999a, unpublished.
3. Bray, D. E. and B. Chance, in Proceedings of the 6^th NDE Topical Conference, edited by C. Darvennes (ASME Int., San Antonio, Texas, 1999).
4. Bray, Don E., and Dietrich, M., "Stress Evaluation in High Speed Rotating Machinery with the L_CR Ultrasonic Technique," Proceedings of the 26^th Turbo Machinery Symposium, Bailey, Jean C., Tech. Ed., Texas A&M University, Houston, Texas, 16-18 September 1997, pp. 143 -149
5. Bray, Don E., Kim, S-J., and Fernandes, M., "Ultrasonic Evaluation of Residual Stresses in Rolled Aluminum Plates," Proceedings Ninth International Symposium on Nondestructive Characterization of Materials, Robert E. Green, Ed., Sydney Australia, June 28-July 2, American Institute of Physics, Melville, NY, 1999, pp. 443-448.
6. Bray, D. E and Stanley, R. K. Nondestructive Evaluation, Revised Edition, CRC Press, Boca Raton FL, (1997).
7. Bray, Don E., and Tang, W., "Evaluating Stress Gradients in Steel Plates and Bars with the L_CR Ultrasonic Wave," Approximate Methods in the Design and Analysis of Pressure Vessels and Piping Components, Proceedings 1997 ASME Pressure Vessels and Piping Conference, W. J. Bees, Ed., Orlando, FL, July 1997, PVP-Vol. 347, pp. 157-164.
8. Bray, D. E., Tang, W. and Grewal, D., "Ultrasonic Stress Evaluation in a Turbine/Compressor Rotor," Journal of Testing and Evaluation, Vol. 25, No. 5, September 1997, pp. 503-509.
9. Leon-Salamanca, T., Reinhart, E., Bray, D.E., and Golis, M., "Field Applications of an Ultrasonic Method for Stress Measurement in Structure," in Boogaard, J. and Van Dijk, G., eds., Nondestructive Testing (Proceedings 12th World Conference), Amsterdam, The Netherlands, pp. 1484-1489, April (1989).
10. Santos, Auteliano and Bray, Don E., "Ultrasonic Stress Measurement Using PC Based and Commercial Flaw Detectors," Review of Scientific Instruments, to appear, Sept. 2000.
11. Santos, A. A., and Bray, Don E., "Application of Longitudinal Critically Refracted Waves to Evaluate Stresses in Railroad Wheels," to appear, Topics on Nondestructive Testing, vol. 5, The American Society for Nondestructive Testing, 2000.
12. Schneider, E. and R. Herder, "Ultrasonic Evaluation of Stresses in the Rims of Railroad Wheels," ECNDT '98, Copenhagen 26-29 May 1998, Vol. 3, No. 6, NDTNet, June 1998.
13. Schneider, E., Structural and Residual Stress Analysis by Nondestructive Methods, edited by V. Hauk (Elsevier, Amsterdam, 1997), Chap. 4, pp. 522-563.
14. Szelazek, J., "Monitoring of Thermal Stresses in Continuously Welded Rails with Ultrasonic Technique," ECNDT '98, Copenhagen 26-29 May 1998, Vol. 3, No. 6, NDTNet, June 1998.
15. Tanala, E., G. Bourse, M. Fremiot and J. F. De Belleval, "Determination of Near Surface Residual Stress on Welded Joints using Ultrasonic Methods," NDT&E International, Vol. 28, No. 2, pp. 83-88, 1995.
16. Tang, W., and Bray, D. E., "Stress and Yielding Studies Using Critically Refracted Longitudinal Waves," NDE Engineering Codes and Standards and Material Characterization, Proceedings 1996 ASME Pressure Vessels Piping Conference, Montreal, PQ, July 1996. PVP-Vol. 322, NDE-Vol. 15, J. F. Cook, Sr., C. D. Cowfer, and C. C. Monahan, Eds. The American Society of Mechanical Engineers, New York, pp. 41-48.
17. Thompson, R. B. ,W. Y. Lu, and A. V. Clark Jr., Handbook of Measurement of Residual Stress, edited by J. Lu, M. James and G. Roy (Society for Experimental Stress Analysis, Bethel, Connecticut, 1996), Chap. 7, pp. 149-178.

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