ASM Metals HandBook Vol. 8 - Mechanical Testing and Evaluation

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Table 3 Resonant specimen lengths for several pure metals

Resonant specimen length (λ/2) at: Young's modulus

20 kHz 2 MHz

Metal

GPa psi × 10

mm in. mm in.

Al 62 9.0 120 4.7 1.20 0.047

Cu 110 15.9 88 3.5 0.88 0.035

Ti 116 16.8 127 5.0 1.27 0.050

Fe 196 28.4 124 4.9 1.24 0.049

Ni 207 30.0 121 4.7 1.21 0.047

The resonance frequency of a bar of arbitrary length, as determined in the previously described experiment,

may not be the driving frequency of the converter that will be used for testing. The appropriate wavelength for

testing frequency is calculated by assuming that the speed of sound through the material remains constant for a

given material and temperature.

The dynamic modulus of elasticity, or dynamic Young's modulus, E, can be determined by combining Eq 1 and

6 to obtain:

E = ρ(λf)

(Eq 7)

The dynamic modulus differs from the static modulus (relaxed modulus) obtained from a tensile test. The static

modulus is inadequate for converting the strain amplitude to stress amplitude because it will include anelastic

contributions to strain that are absent at ultrasonic frequency. Use of the static modulus gives stress estimates

that are too low because the static modulus is typically less than the dynamic modulus.

Stress-Life Specimen Design. Several specimen designs for ultrasonic fatigue stress-life testing are shown in

Fig. 15. Although the uniform bar is the most easily produced specimen, it generally is not employed in testing.

The stress concentration due to the screw threads used for gripping causes the local stress in the thread to

exceed the maximum stress produced at the center of the specimen. Failure in the screw threads rather than in

the center of the specimen would result.

Fig. 15 Profiles of specimen designs for ultrasonic fatigue. References cited provide mathematical

analyses required to compute stress and strains.

Strain amplification is desirable at the center of the specimen. This is accomplished by reducing the cross-

sectional area of the gage section to produce a dumbbell-shaped specimen. By reducing the cross-sectional area,

the total power needed to drive the specimen into resonance is reduced. Consequently, the amount of heat

produced in the specimen is reduced, which also reduces the cooling requirements.

The choice of specimen design for a particular test depends on many factors, including the amount of material

available, maximum amplitude of the wave train, amount and type of cooling available or allowable, minimum

diameter to ensure stiffness and eliminate flexural modes, and desired strain level in the gage section. The

calculation of the appropriate resonant specimen geometry is the primary task in designing an ultrasonic fatigue

specimen. Other considerations for ultrasonic fatigue specimens are the same as those encountered in

conventional fatigue testing, including surface finish, minimization of residual stresses, capabilities of the

machine shop, and costs of producing the specimen.

Complex Specimen Geometries. More complicated expressions are needed to determine the dimensions and

strain amplitude at the point of maximum strain for specimens with nonuniform geometries. A dumbbell-type

specimen is resonant at frequency f if Eq 7 and the following equation are satisfied simultaneously (Ref 15):

(Eq 8)

The variable dimensions L

, L

, d

, and d

are shown in Fig. 16. Equations 7 and 8 are the basic design

equations for an ideal dumbbell specimen. Using this representation, any three dimensions can be selected; the

fourth can be calculated for a given wavelength. The maximum elastic strain on the gage length of the dumbbell

specimen shown in Fig. 16 is given by (Ref 15):

(Eq 9)

where k is 2π/λ and A

is displacement amplitude at the end of the dumbbell. Comparing Eq 9 with the

maximum strain obtained in a uniform bar shows that the term in square brackets is the magnitude of the

amplification of strain amplitude produced by the dumbbell shape. The term in square brackets is often referred

to as the strain amplification factor.

Fig. 16 Ideal dumbbell specimen dimensions

Direct measurement of the strain profile from a test specimen has been reported (Ref 41). A contacting probe

aids in measuring the displacement and strain distribution for specimens with complex geometries. The

displacement and strain amplitudes along the length of a circular, tapered dumbbell specimen are shown in Fig.

17. The amplification of the strain amplitude due to the dumbbell shape is clearly indicated.

Fig. 17 Distribution of displacement and strain amplitudes obtained from a dumbbell-shaped Ti-6Al-4V

specimen at 14.2 kHz. Source: Ref 41

If the ideal dumbbell calculations are used, small adjustments to the overall length of the specimen may be

necessary during specimen design to achieve the desired test frequency. For example, if a hole is tapped in one

end of the specimen to attach the specimen to the horn, the equivalent mass of material removed for the hole

must be replaced in the form of extra length of that dumbbell head. If the attachment stud is the same density as

the specimen, then no adjustments are required. However, if a steel stud is used to hold an aluminum specimen,

a mass adjustment to the length of the dumbbell head will be needed. Similar tuning considerations should be

made when placing fillets between the dumbbell heads and the gage section.

Fillets and Radii. Efficient propagation of acoustic energy along the length of a dumbbell specimen requires

that a smooth transition be provided between the heads and the gage section of the specimen. For some

specimen designs such as a circular or exponential tapered dumbbell, this smooth transition is the major design

element. Specimen designs that use the Neppiras formula (Eq 7 and 8) for an ideal dumbbell must provide a

fillet at the transition from head to gage section. Constant-radius fillets are easily machined and can be used if

other alternatives are not feasible.

Depending on the choice of radius, some heat will be generated at the fillet because a constant radius fillet does

not provide optimal transition in particle velocity. The recommended fillet design is the baud streamline fillet

(Ref 42). This design has a continuously varying fillet radius. Its shape is like that of a nonturbulent stream of

water as it drains out of a tank with a circular hole. It is highly efficient in providing a smooth transition of

particle velocity. This is quite useful for testing high-damping or precipitation-hardened materials. One

disadvantage of the baud streamline design is that it requires a tape- or computer-controlled lathe to produce the

desired profile.

Notched Bar Specimens. A notched specimen is a special condition of a dumbbell specimen, where L

is very

small and L

is close to λ/4 to maintain resonance. For a bar containing a narrow notch, the maximum strain in

the notch (exclusive of stress concentration factors) is the product of the maximum strain at λ/4 of a uniform

bar without the notch multiplied by the area ratio (d

)

(Ref 39). The L and d dimensions are defined in the

ideal dumbbell specimen shown in Fig. 16.

Finite element analysis of the notched bar specimen design shows that, despite differences in the stress

distribution along the specimen length between static- and dynamic-loaded specimens, the stress distribution

and configuration in the notch region are the same. The complete equation for the maximum stress, σ

max

, in a

notched resonant member can be calculated by multiplying the maximum strain, kA

· (d

)

, by the dynamic

modulus, E, and the stress concentration factor, K′

, as:

(Eq 10)

where K′

is the von Mises stress concentration factor. The von Mises stress concentration factor is used instead

of K

for analyzing high-cycle fatigue results (Ref 43). This is based on the findings that high-cycle fatigue

failure is dictated by the alternating von Mises stress, where K′

is approximately 10% less than K

. Definition of

additional parameters relating to notch fatigue, including notch bar fatigue strength and notch sensitivity, are

found in Ref 39.

Design of resonant fatigue crack growth rate specimens is based on physical concepts similar to stress-life

specimen design. The major difference is that a sharp crack is introduced into the design at the point of

maximum strain. Therefore, the relationships developed for purely elastic deformation are not exactly fulfilled

when appreciable plastic deformation occurs at the crack tip, when changes occur in Young's modulus due to

localized plastic deformation, or when the specimen is detuned by the growing crack. Discussion of these issues

can be found in Ref 44.

The first ultrasonic fatigue crack growth test specimen was a simple resonant bar with an electrodischarge

machined slot cut into one side of the bar (Ref 45). Typical geometries of fatigue crack growth specimens are

shown in Fig. 18, including single-edge-cracked specimens (Ref 19), double-edge-cracked specimens (Ref 46)

with axial loading, center-cracked specimens with axial loading (Ref 47, 48), and single-edge-cracked

specimens with transverse loading (Ref 49). Crack length, a, is shown.

Fig. 18 Specimen geometries for crack growth measurements under high-frequency resonance

excitation. (a) Center-cracked specimen. (b) Single-edge-cracked specimen. (c) Double-edge-cracked

specimen. (d) Single-edge-cracked specimen. (e) Center-cracked specimen. R, fatigue stress ratio; a,

crack length; b, specimen width; t, specimen thickness. Source: Ref 44

Fatigue crack growth test specimen length is controlled by the test frequency, as in the stress-life case.

However, the cross-sectional dimensions selected can vary considerably. The current trend in specimen design

is to incorporate the relevant criteria of conventional-frequency fatigue crack growth and fracture mechanics

into the design. One factor pertains to specimen thickness, d, which should be large in comparison to the plastic

zone size at the applied stress intensity range. This is consistent with the pertinent ASTM recommendation:

(Eq 11)

ΔK

max

is the maximum stress intensity range, and σ

is the yield strength of the material. The exception to this

rule arises when materials have high damping and produce large quantities of heat. In this case, the specimen

must be thin enough to ensure adequate cooling. The specimen also must be thick enough to suppress other

modes of vibration, such as flexural oscillations.

The dimension of the starting notch and the permissible fractional length of fatigue crack extension also

influences the design of the specimen width, b. For center-cracked specimens, the crack advance should be

limited to a/b ratios of less than 0.4 (Ref 48). For specimens that are wide in comparison to the crack length,

flexural contributions due to lateral displacement become significant.

Specimens tested under superimposed static loads (higher R ratios) require regions with cross sections larger

than the gage section in order to transmit the required tensile load. The specimen shown in Fig. 18(e) is similar

to that specified for conventional fatigue crack growth tests (Ref 50). Currently, specimen choice is equivocal.

Standardization of a test specimen will be determined by the ability of a specimen to provide the necessary

da/dN and ΔK data. Research is focusing on providing accurate stress intensity values under resonant

conditions.

Stress Intensity Concepts. Design of fatigue crack growth rate specimens is useful only if the stress intensity at

the crack tip can be calculated. In the simplest method of computing stress intensity range, the nominal stress

amplitude is obtained from the displacement amplitude, and the ΔK value is calculated as:

(Eq 12)

where F

(a/b) is a correction factor to account for the presence of a crack in the finite width of the specimen.

This method does not account for the effects of a growing crack on the resonance behavior of the system. It

assumes that the relationship between displacement and strain amplitude remains constant with detuning of the

specimen due to crack growth. This problem can be avoided by measuring the strain directly with strain gages

placed in the crack plane. Additional correction factors to the stress intensity formulism mentioned previously

(Eq 12) have been added to account for the growth of the crack by normalizing the measured effective stress to

the initial cross section (Ref 51).

A dynamic correction factor for ΔK also has been determined with finite element analysis. This results in a

factor that is a function of the ratio of crack length (a) to specimen width (w), specimen width to length (W/L),

and the instantaneous frequency (ν).

A summary of frequently used correction formulas to calculate ΔK is given in Table 4. Further study must be

undertaken to develop a standardized computation procedure for ΔK in resonant specimens.

Table 4 Computation of stress intensity ranges

Specimen geometry (loading mode) Equation

Ref

Side notch in octagonal bar (longitudinal)

Center-notch flat bar (longitudinal)

(a)

Center-notch flat bar (longitudinal), a/w < 0.5

Double-side-notch flat bar (longitudinal), a/w < 0.4

Single-edge-notch thin, flat strip (longitudinal), a/w < 0.6

Single-edge-notch rectangular bar (transverse)

(a)

Note: E, elastic modulus, U

displacement in the load line.

(a) By finite element analysis.

Source: Ref 14

Specimen Grips. The requirements for gripping an ultrasonic fatigue specimen are minimal. The only gripping

requirements in ultrasonic fatigue are maintenance of intimate contact between the specimen and horn to allow

good acoustic coupling and the absence of external forces that would disturb the resonance of the remainder of

the wave train. Consequently, a fatigue specimen with a 5 mm (0.2 in.) gage can be held in place by a single 6.4

mm (0.25 in.) stud while being fatigued to failure at a stress amplitude of 485 MPa (70 ksi) or more.

Generally, gripping is accomplished by an internal thread arrangement. This arrangement is adequate, even

with a specimen that is difficult to grip, such as thin-walled tubing. Gripping can be accomplished by an

external screw-down collar, which grips the specimen to an extension horn. This is particularly useful for tests

at high mean loads. Gripping can also be accomplished by brazing or welding the specimen to a threaded

adapter or directly to an acoustic horn. A holder for wire specimens has been developed that allows multiple

specimens to be tested in a batch mode (Ref 53). An interference fit of the specimen with the horn also is

satisfactory, as long as good acoustic coupling with the horn is obtained. Attachment of the specimen to the

acoustic horn may be accomplished using an adhesive. This is appropriate for fatigue testing of brittle materials,

such as glass and ceramics.

Most ultrasonic converter cases are grounded. If metal-to-metal contact is maintained throughout the wave

train, the specimen will be grounded also. An electrically floating specimen is needed if electrochemical

potential or corrosion current is to be measured during testing. The specimen can be isolated from the horn by

using an extension horn made of a nonconducting material, such as Lucite, to grip the specimen. Such grips

limit the magnitude of the stress amplitude that can be transmitted to the specimen because of high dissipation

of energy or low fatigue strength of these materials.

References cited in this section

14. P. Bajons, in Ultrasonic Fatigue, J.M. Wells, O. Buck, L.D. Roth, and J.K. Tien, Ed., TMS-AIME,

Warrendale, PA, 1982, p 15

15. E.A. Neppiras, Proc. ASTM, Vol 59, 1959, p 691

19. S. Stanzl and E. Tschegg, Met. Sci., April 1980, p 137

39. L.E. Willertz and L. Patterson, in Ultrasonic Fatigue, J.M. Wells, O. Buck, L.D. Roth, and J.K. Tien,

Ed., TMS-AIME, Warrendale, PA, 1982, p 119

41. C.R. Sirian, A.F. Conn, R.B. Mignogna, and R.E. Green, Jr., in Ultrasonic Fatigue, J.M. Wells, O.

Buck, L.D. Roth, and J.K. Tien, Ed., TMS-AIME, Warrendale, PA, 1982, p 87

42. R.E. Petersen, Stress Concentration Factors, John Wiley & Sons, New York, 1973

43. R.E. Petersen, Trans. ASME (Appl. Mech. Sect.), Vol 58, 1936, p A-149

44. R. Stickler and B. Weiss, in Ultrasonic Fatigue, J.M. Wells, O. Buck, L.D. Roth, and J.K. Tien, Ed.,

TMS-AIME, Warrendale, PA, 1982, p 135

45. S. Purushothoman, J.P. Wallace, and J.K. Tien, Ultrasonics International 1973 Conference

Proceedings, IPC Science and Technology Publications, Guildford, Surrey, U.K., 1973, p 244

46. W. Hessler, H. Mullner, and B. Weiss, Met. Sci., May 1981, p 225

47. B. Weiss, R. Stickler, J. Fembock, and K. Pffafinger, Fatigue Eng. Mater. Struct., Vol 2, 1979, p 73

48. W. Hoffelner and P. Gudmundson, Eng. Fract. Mech., Vol 16, 1982, p 365

49. W. Hoffelner, J. Phys. E, Sci. Instrum., Vol 13, 1980, p 617

50. N. Dowling, Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue Growth, STP 637, ASTM,

Philadelphia, 1976, p 97

51. B. Weiss, Determination of The Threshold Stress Intensity Value of Mo and Mo-Alloys using a 20 kHz

method (German), Metall, Vol 34, 1980, p 636

52. S. Purushothoman and J.K. Tien, Metall. Trans. A, Vol 9, 1975, p 367

53. J. Babouk, K. Kromp, W. Kromp, and P. Bajons, in Ultrasonic Fatigue, J.M. Wells, O. Buck, L.D.

Roth, and J.K. Tien, Ed., TMS-AIME, Warrendale, PA, 1982, p 51

Ultrasonic Fatigue Testing

Applications

Ultrasonic fatigue testing has been applied successfully to many situations that require fatigue initiation and

crack growth data. Testing has been performed under a variety of loading conditions, specimen geometries, and

environmental constraints. With perhaps the exception of plastic strain-controlled testing and single-cycle

hysteresis testing, ultrasonic fatigue techniques can be readily applied to the problems of fatigue that

traditionally have been confronted at lower frequencies.

Plastic strain rate is a function of strain amplitude and waveform as well as cyclic frequency. In ultrasonic

fatigue testing, there is some question as to the effect of increased strain rate from testing at ultrasonic

frequencies. Experimentation generally is required, but it is also clear that body-centered cubic (bcc) materials

exhibit a much larger strain-rate dependence than face-centered cubic (fcc) materials.

The cyclic deformation of bcc materials is quite different from that of fcc materials and can be quite complex.

The athermal portion of the flow stress is due primarily to dislocation interactions leading to work hardening.

The effective stress is dominated by the lattice friction stress. Thus, strain rate and effect of impurities are

extremely important in the deformation behavior of bcc materials.

In general, the strain-rate sensitivity of engineering alloys and materials often requires experimental verification

because of the many possible combinations of alloy compositions, properties, and operating environments.

Changes in alloy composition affect the activation barriers to deformation, and the high-frequency behavior

might be moderated between the paradigms of bcc and fcc materials. Testing in aggressive environments can

offset the expected behavior. Additional information on strain-rate-dependent fatigue behavior can be found in

Ref 54, 55, 56.

References cited in this section

54. C. Laird and P. Charlsey, in Ultrasonic Fatigue, J.M. Wells, O. Buck, L.D. Roth, and J.K. Tien, Ed.,

TMS-AIME, Warrendale, PA, 1982, p 183

55. J.K. Tien, in Ultrasonic Fatigue, J.M. Wells, O. Buck, L.D. Roth, and J.K. Tien, Ed., TMS-AIME,

Warrendale, PA, 1982, p 1

56. L.E. Coffin, in Ultrasonic Fatigue, J.M. Wells, O. Buck, L.D. Roth, and J.K. Tien, Ed., TMS-AIME,

Warrendale, PA, 1982, p 423

Ultrasonic Fatigue Testing

Acknowledgments

Adapted from the article “Ultrasonic Fatigue Testing” by L.D. Roth (with contributions from participants of the

First International Conference on Ultrasonic Fatigue Testing) published in Mechanical Testing, Volume 8 of

the 9th Edition Metals Handbook.

Ultrasonic Fatigue Testing

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