Stress-strain relationship in Terfenol-D ABSTRACT. Keywords: Terfenol-D, magnetostriction, transducer, elastic modulus, Delta-E effect

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1 Stress-strain relationship in Terfenol-D Rick Kellogg, Alison Flatau Aerospace Engineering and Engineering Mechanics, Iowa State University, Ames, IA ABSTRACT The variability of Young's modulus (the AE effect) in giant magnetostrictive Terfenol-D has a significant impact on the performance and modeling of Terfenol-D transducers. While elastic modulus variability introduces nonlinearities in the transducer input/output relationship that are often deemed undesirable, it also affords opportunities for achieving novel device performance attributes. In this investigation, Terfenol-D's modulus of elasticity is characterized under controlled thermal, magnetic, and mechanical loading conditions. Quasi-static cyclic compressive stress testing methods are used to quantify the variability in Young's modulus over a wide range of d.c. applied magnetic fields and stresses. Elastic modulus changes of four-fold or more are demonstrated through the variation of a d.c. applied magnetic field. The effect of decreasing cyclic stress amplitude giving rise to an increase in Terfenol-D's apparent elastic modulus is also examined. The thermally controlled transducer used throughout this investigation is described. This conference paper is a shortened version of the paper titled "Experimental Investigation of Terfenol-D's Elastic Modulus" that has been submitted for peer reviewed journal publication. Keywords: Terfenol-D, magnetostriction, transducer, elastic modulus, Delta-E effect 1. INTRODUCTION Effective use of a transducer requires knowledge of its displacement and force output for a given input as well as its response to variable applied loads. As transducers utilizing the magnetostrictive alloy Terfenol-D find increased use, a better understanding of their characteristics is needed to improve modeling and control. Many effects are associated with the magnetostrictive phenomenon (i.e. magnetically induced strains), one of which is an operational variability in the apparent Young' s modulus. While elastic modulus variability complicates conventional transducer uses for generating outputs or sensing, it also opens the door to new possibilities. The utility of a material with a modulus of elasticity that can be changed through the variation of one d.c. electrical input parameter is extensive. Suggested applications include variable frequency resonators, variable acoustic delay lines and parametric amplifiers. This investigation explores changes in Terfenol-D's (TbDy1Fe1.9s.2 where x = 0.3) elastic modulus under quasi-static stress and d.c. applied magnetic field conditions. The circumstances under which continuous modulus changes exceed 400% are presented in addition to observations of an unusually high apparent modulus for low cyclic stress conditions. Young's modulus of elasticity, E,reflects a material's stiffness normalized for geometry over the elastic regime of its stress versus strain response. Young's modulus of elasticity is considered a constant material property for conventional materials. However, for materials that transduce energy from one state to another such as magnetostrictive materials, the apparent E is not constant. In magnetostrictive materials E is also dependent on the material's magnetic state due to magnetomechanical coupling. Changes in modulus of elasticity with magnetization were recognized over one hundred years ago in materials such as iron and nickel. Quantifying the phenomena, the percent modulus change is known as the "Delta-E effect" (ze effect) [Lee 1955]. The AE effect is defined in Equation 1 as the material's Young's modulus at magnetic saturation (E3) minus Young's modulus at the magnetic unsaturated state (E0)divided by Young' s modulus at the magnetic unsaturated state. E E0 (1) The observed AE effect is small in iron and nickel, being on the order of 0.4 to 18%. However, AE effects as large as 680% have been reported by Clark et al. for giant magnetostrictive materials such as Terfenol-D (Th.67Dy23Fe2) at cryogenic temperatures [Clark 1993]. In an earlier study by Clark et al., te effects up to 150% were measured for Terfenol-D (Th.3Dy.7Fe2) using applied d.c. magnetic fields up to 342 ka/m at room temperature. In this case, the modulus ranged from Smart Structures and Materials 2001: Smart Structures and Integrated Systems, L. Porter Davis, Editor, Proceedings of SPIE Vol (2001) 2001 SPIE X/01/$

2 43 GPa at zero applied field to GPa at a saturating field of 342 kaim [Clark For the results presented in this paper, modulus changes not referenced to the modulus at magnetic saturation, Es, are reported using the notation AEH2H1. The AEH2HI effect quantifies modulus changes due to different d.c. applied magnetic fields, Hi and H2, as follows: EH2 H1 H2H1 = whereeh2eh]. (2) EH2 The AEH2H1 notation will be appropriate for quantifying Terfenol-D's modulus response for the measurement and application ofthe AE effect in which magnetic field strengths are limited (i.e. under non-saturating applied magnetic field conditions). 2. TRANSDUCER DESIGN With the objective of testing Terfenol-D under controlled quasi-static conditions, thermal, magnetic and mechanical operating regimes were considered in the design of the water-cooled transducer used in this investigation. A schematic of the water-cooled transducer is shown below in Figure 1 where the 6.35 mm diameter, 50.8 mm long Terfenol-D rod under study resides at transducer's center. Surrounding the Terfenol-D rod is a sense coil (not depicted) for measuring the sample's magnetic induction while a Hall Effect chip at the sample's midpoint measures the applied magnetic field. Exterior to the nonmagnetic alignment structure is a cooling tube encased solenoid producing the applied magnetic fields. Completing the transducer's magnetic circuit, a steel casing encloses the entire assembly. Loads were transmitted to the Terfenol-D rod through a load cell and an output shaft. For this study, thermal control maintained steady state operation at 21 1 C (70 F 2 F) while d.c. magnetic fields ofup to 258 ka/m were generated. Mechanical measurements included the output shaft displacement using an LVDT (linear variable differential transformer) as well as longitudinal strain measurements of the Terfenol-D rod using strain gauges. Four strain gauges were mounted directly under the sense coil in the center % portion of the rod for the best comparison of mechanical and magnetic data. Two sets of two gauges each were positioned on opposing sides of the Terfenol-D rod with all gauges wired in series to cancel out any induced bending moments. With each gauge having active regions of 4.0 by 12.7 mm, the relatively large gauge area provided an averaging effect since magnetostriction in Terfenol-D is not necessarily uniform. A MTS machine was used to generate the variable force loads applied to the Terfenol-D sample. A force transducer located in series with the transducer mechanical output shaft transmitted these loads while acquiring load data. Tests conducted with low cyclic load amplitudes of 89 N superimposed on d.c. loads as high as 3.3 kn required the use of an output shaft guided with Teflon-lined linear bushings to minimize frictional effects. Teflon bushings are also used to maintain alignment of the Terfenol-D rod under study. The output-shaft and structure around it are made of magnetic steel to provide a magnetic flux path from the case to the Terfenol-D sample. This continuous flux path served to reduce demagnetization effects in the Terfenol-D thereby improving transducer efficiency. Data acquisition and amplifier output control for the solenoid were accomplished using a DAQ board along with software running on a PC computer. Data was collected at a rate of 1 00 samples 4 165mm per second through 20 Hz low pass filters. The solenoid current and Hall Effect chip were sampled to monitor the applied magnetic field while a sense coil and flux meter provided sample magnetization data. Figure 1 Water Cooled Transducer. 118m 542 Proc. SPIE Vol. 4327

3 3. MAJOR LOOP COMPRESSION TESTING AND RESULTS Major and minor ioop cyclic compression tests were conducted to characterize Terfenol-D's apparent Young's modulus and magnetization changes with quasi-static stress variation at various d.c. applied magnetic fields. Cyclic test methods were used to reveal mechanical and magnetic hysteretic effects given Terfenol-D's nonlinear behavior. The major loop testing procedure commences with demagnetization of the Terfenol-D sample under zero load. Demagnetization was accomplished by driving the transducer solenoid with a 1.0 Hz exponentially decaying sinusoidal current. The resulting magnetic field decayed from an amplitude of 258 ka/m down to 2.5 AIm over 240 cycles and then finally was reduced to zero. This demagnetization process was used to bring the Terfenol-D sample to a reproducible demagnetized state prior to the start of each cyclic compression test (although a small remnant magnetization persisted). Following demagnetization, a d.c. applied magnetic field, H, was established by linearly increasing H from zero to the desired level at 3.2 ka/m per second. Completing the testing procedure, a MTS machine using feedback load-control compressed the sample from zero to 3.34 kn and back to zero load at a compression rate of 3.56 kni mm. This imposed a maximum compressive stress of MPa on the 6.35 mm diameter Terfenol-D sample. Repeating the demagnetization, magnetic field application and cyclic compression steps for various levels of d.c. applied magnetic field produced the combined results shown in Figure 2. The figure depicts strain plotted against stress for each continuous stress cycle from zero to MPa and back to zero. The d.c. applied magnetic field levels tested ranged from 0 to ka/m at ka/m intervals. A pair of arrows indicating the progression of data collection with sample compressive loading and unloading. is shown for the zero applied field case. Several features of Figure 2 deserve examination. First of all, the strain versus stress loops are clearly hysteretic with ioop closure nearly occurring for all but the zero applied field case. The hysteresis is indicative of energy loss occurring during the compression and decompression cycle. Compared to conventional metals, the overall hysteresis is large due to energy losses developed through the coupling of mechanical and magnetic energy Figure 2 Major strain vs stress loop for d.c. applied fields states. Notably, the hysteresis is reduced where either levels, H, ( ka/m at 16.1 ka/m steps). mechanical or magnetic processes dominate the other one, such as at high compressive stresses with lower applied magnetic fields and for high applied fields with lower stresses. Secondly, examining the strain at zero stress, note that strain increases at a decreasing rate with higher applied magnetic fields as saturation magnetostriction is approached. Finally, the primary feature of interest in Figure 2 is how the slope of the strain versus stress data varies with stress and applied magnetic field. The ratio of change in strain per change in stress gives the material's compliance, 5", with the inverse ratio providing the material's modulus of elasticity El'. Use of the superscript "H" implies compliance and modulus values evaluated at a constant applied magnetic field. As the slope of the strain versus stress data increases, the compliance increases and the modulus of elasticity decreases. The strain plotted against stress format will be useful for later comparison of mechanical and magnetic observations with respect to stress. Terfenol-D's modulus for a given applied magnetic field and stress state is a result of the interplay between the mechanical and magnetic energy regimes. In the right hand portion of Figure 2, at low stresses, magnetic effects dominate the overall strain response due to a dependence of the available magnetostriction on applied magnetic field. In the left-hand portion of the figure, where applied stress is high, mechanical effects dominate the strain response. For all applied fields, as the compressive load increases, the traces asymptotically approach the zero applied field case (bottom left) giving rise to a modulus of elasticity that is dominated by mechanical effects. Having examined the coarse features of the strain versus stress and magnetization responses of Terfenol-D, additional insight into the material's elastic modulus and magnetomechanical coupling may be gained through further analysis of the data. Considering the first leg of the strain versus stress plot of Figure 2 where the sample undergoes compression only, the slope of the traces may be calculated to determine the apparent Young' s modulus, Figure 3 shows Terfenol-D's elastic modulus as a function of compressive stress for d.c. applied magnetic fields, H, ranging from 0 to ka/m in steps of 16.1 ka/m. Evident from the plot, modulus changes with applied magnetic field (the AEH2H1 effect) are stress dependent. Proc. SPIE Vol

4 Considering compressive stress magnitudes below 8 MPa, an increasing H produces increasing modulus values. In contrast, above an 80 MPa compressive stress magnitude, increasing H produces decreasing modulus values for the data shown. This apparent dichotomy in modulus response to applied field may be explained by examining the balance between stress anisotropy and applied magnetic field energies as well as by noting which easy axes are predominately populated by the magnetic domains for a particular compressive stress and applied magnetic field combination. Recall that compressive loading of the Terfenol-D sample favors magnetic domain orientation with the <1 1 1> easy axes perpendicular to the sample's longitudinal axis and that the applied magnetic field favors domain orientation with the <1 1 1> easy axes nearer the rod's longitudinal axis. -60 Addressing the modulus response for stress stress, MPa magnitudes less than 8 MPa, relatively modest axially applied Figure 3 Young's Modulus for d.c. applied field levels H (0 magnetic fields overcome the stress anisotropy and favor vm at 16.1 ka!m steps). magnetic domain orientation with the <1 1 1> easy axes within 19.5 of the sample's longitudinal axis. Low levels of compressive stress have a relatively minor effect on the domain orientation in the presence of high-applied magnetic field energies. With increasing applied field energies, the domains are held more rigidly in place. Keeping in mind that strain accompanies magnetization changes in magnetostrictive materials, then limiting domain rotation limits strain. Furthermore, limiting the strain response for a given applied stress change results in a higher apparent modulus of elasticity. In summary, at low levels of compressive stress, an increase in the applied field correlates to an increase in the apparent elastic modulus. Looking at the modulus response in Figure 3 for stress magnitudes greater than 80 MPa, high compressive stresses and low applied field levels favor domain orientation with the <1 1 1> axes perpendicular to the sample's longitudinal axis. With domains aligned along these perpendicular <1 1 1> axes, further increases in stress have little effect on the sample's magnetization and magnetostriction. Consequently, the high stress, low applied field traces reflect the material's mechanical modulus of elasticity. Subsequently, the effect of an increasing applied field causes domain rotation into the applied field direction with an accompanying magnetostriction. Under the action of stress, the field-induced magnetostriction is available for reversal in addition to the mechanical-only strain thus allowing for larger changes in total strain with stress. As a result of the magnetostriction, the apparent modulus of elasticity decreases with increasing magnetic field levels. This magnetically induced softening of the apparent modulus occurs up to the point at which the magnetic energy is sufficiently high to dominate the stress anisotropy. Further increases in the applied field will then result in an increase in the apparent modulus where additional magnetic energy essentially induces magnetic hardening of the apparent modulus. With an explanation in hand for the moduli response to applied magnetic fields under low and high magnitude compressive stress conditions, the intermediate conditions producing the moduli minima observed in Figure 2 are now examined. Moduli minima occur when stress anisotropy and applied magnetic field energies balance one another. The energy balance allows domain rotation to occur easily between the <1 1 1> axes perpendicular to the sample's longitudinal axis and the < > axes within 19.5 of the longitudinal axis. As a consequence ofthe magnetostriction produced from domain rotation, substantial strains occur with stress giving rise to the modulus minima. 20 Secondary modulus minima are also observed to occur for each o applied field level of Figure 3. As the applied field increases, the secondary minima occur at larger compressive stress levels.. Although the exact mechanism for the secondary modulus minima. io is unclear, it is possibly related to magnetic moments jumping between energy wells. 5 A further consideration of the modulus data presented in Figure 3 is that the values of the moduli minima vary with applied field and their corresponding stress locations. The moduli minima values, in GPa, are plotted against the applied magnetic field level Applied field, kajm for which the moduli minima occurred in Figure 4. The trend is for Figure 4 Modulus minima values for d.c. applied fields, H, increasing moduli minima values with increasing applied field. ( ka/m at ka/m steps) Proc. SPIE Vol. 4327

5 With the interest of controlling Terfenol-D's Young's modulus with d.c. applied magnetic fields, or in other words exploiting the AEH2HI effect, it becomes useful to recast the modulus versus stress data of Figure 3. Figure 5 depicts Terfenol-D's modulus plotted as a function of applied magnetic field for various levels of constant compressive stress. The circles denote the discrete modulus values extracted from Figure 3 and are connected by lines marking the constant stress levels. Choosing a representative modulus response, the MPa case (highlighted trace) is now examined. As the applied field is increased from zero to 50 ka/m, the modulus decreases from 72 GPa to 14 GPa. Thereafter, as the applied field is increased to the maximum field of ka/m, the modulus is observed to increase to 69 GPa. The AEH2H1 effect in this case, from H2 0 ka/m and Hi 50 ka/m, is calculated as (72 GPa-14 GPa)/ MPa constant stress line.. 0 / loo 200 GPa giving AEH2H1 4 14%. Applied field, kajm Considering the overall effect of compressive stress on.... Figure 5 Modulus at d.c. compressive stress values (6.9 the modulus response to applied magnetic field, high MPa with 6.9 MPa steps) for d.c. applied fields H compression levels increase the modulus to as high as 112 GPa ( ka/m at 16.1 ka/m steps). Estimated data point with low levels of applied field, H, and delay the onset of the maximum error I.4 GPa. moduli minima to higher levels of H. Conversely, lower compressive stress levels result in lower initial modulus values and moduli minima occurring at lower applied field levels. With the modulus response to applied field and stress of Figure 5 in mind, the approach to designing and implementing a transducer utilizing the AEH2H1 effect becomes apparent. First of all, the largest changes in modulus may be achieved with the least change in applied magnetic field by using the appropriate prestress. Minimizing the magnitude of applied field change is warranted since a transducer solenoid producing larger applied fields requires more power and heat dissipation capability. Note that the required change in applied field is of interest here since magnets may be used to provide constant levels of applied magnetic field. The second important factor to successfully using the AEH2HI effect is to maintain the desired stress level. The prestress mechanism should not impart significant stress changes to the sample as the sample undergoes magnetostriction with changes in applied magnetic field. Although instructive, the modulus response with applied field in Figure 5 will provide only an approximation of what to expect in a transducer used for the L\EH2H1 effect. The reason is that magnetostrictive materials are sensitive to the amplitude of imposed cyclic stresses as well as the sequence of stress and magnetic field application [Moffet 1991]. Cyclic stress amplitude effects will be addressed in the following section where minor loop stress testing is covered. Concerning the effects of applied field and stress sequence, the data just presented are the result of application of the magnetic field prior to the compressive stress. However, transducers employed to use the AEH2H1 effect would most likely utilize springs to provide a prestress and consequently the Terfenol-D would be subjected to applied magnetic fields only after the prestress application. This idea and related issues are elaborated in the accompanying manuscript, "Investigation of a Terfenol-D tunable mechanical resonator." 4. MINOR LOOP COMPRESSION TEST RESULTS Minor loop compression testing was undertaken to more closely approximate the conditions Terfenol-D would experience in a mechanical resonator. Successful implementation of the AEH2H1 effect in Terfenol-D to produce a variable frequency mechanical resonator would require an understanding of the effects of cyclic stress superimposed on various levels of d.c. compressive stress. The test procedure permits prestress to be introduced prior to the applied magnetic field as well as providing a measure of the mechanical and magnetic responses to varying cyclic stress conditions and d.c. applied magnetic fields. The testing procedure begins with demagnetization of the Terfenol-D sample under zero load. As done in the major loop stress tests, demagnetization was accomplished with a 1.0 Hz sinusoidal applied magnetic field decaying exponentially from an amplitude of 258 ka/m. The applied field decayed to 2.5 A/m over 240 cycles and then was finally reduced to zero. Following demagnetization, a d.c. compressive stress along with a superimposed sinusoidal minor ioop stress was established Proc. SPIE Vol

6 using a MTS machine in feedback load-control mode. Mechanical and magnetic measurements were recorded while the applied magnetic field was varied for different levels of d.c. compressive stress. Starting with test conditions where the d.c. compressive stress was established along with a 4.0 Hz 28 MPa cyclic stress, the applied field was then increased from zero to ka/m in ka/m increments. Data were recorded at each d.c. applied magnetic field level. This process was repeated for d.c. stress levels ranging from 6.9 to MPa in 6.9 MPa intervals. Figure 6 shows the combined results of the strain versus stress response for the various levels of applied magnetic field and d.c. compressive stress. Reading each column of minor loops from bottom to top, as the applied field, H, is increased, the average slope and enclosed area of each minor loop changes. Increased average loop slopes correspond to a relative increase in compliance (or reduction in modulus). Fatter ioops are a result of hysteresis and indicate greater energy loss than their more linear counter-parts. Examining the fourth column from the right, it corresponds to a d.c. compressive stress of MPa and is marked with the overlying arrow indicating the direction of increasing applied magnetic field. Starting at the bottom with zero applied field, the ioops are relatively flat and not hysteretic. As applied field is increased to 80.6 ka/m (4th loop from bottom), the loops progressively rotate counterclockwise, indicating decreasing modulus, as well as becoming more hysteretic. Increasing applied fields above 80.6 ka/m result in the loops rotating progressively clockwise, reflecting higher modulus. Looking at the plot as a whole, the d.c. stress and applied field combinations generating low moduli occur where intra-column loop spacing is largest. In other words, these are the regions where significant magnetostriction is occurring. Additionally, low modulus loops are accompanied by increased hysteresis, which will be discussed subsequently. Viewing the left-hand side of the figure, the effects of increasing d.c. compressive stress begin to overcome magnetostriction and suppress modulus changes with various levels of applied magnetic field. The minor loop's vertical spacing is significantly reduced, and the variation between minor ioop average slopes is limited. Overlaying the major strain versus stress loops of Figure 2 with the minor ioops from the previous figure for the 16. 1, 80.6, and ka/m applied magnetic field levels provides a comparison of the strain responses as shown in Figure 7. The minor loops are expected to fall within the major loops, however it is noteworthy that the upper right tips of the minor ioops are generally coincident with the return path (decompression) of the major loop. (Minor loop tips do not fall on the major loop return path in the lower left portion of the plot due to hysteresis of the major loop with the reversal of stress application. Had the major loop test been taken to sufficiently high compressive stress levels, the minor loop tips shown would have been coincident with the major loop return path.) The effect of the minor loop cyclic stress is to settle the magnetomechanical state ofthe Terfenol-D without going to the stress extremes ofthe major stress loop. Another significant feature offigure 7 is that the average slope of the minor stress loops are consistently flatter than their enclosing major loop. Although unconfirmed, this flatter slope (or limited strain change with stress) is most likely due to the coupling between mechanical and magnetic states. Under small cyclic stress amplitudes, domain wall bending predominates over domain wall motion. The magnetic moments' orientations are trapped energetically thus limiting magnetostriction. The limited magnetostriction then translates into a larger effective modulus. Further discussion on the minor loop stiffening effect will come later in this section. Performing a least squares regression fit on each minor strain-versus-stress loop of Figure 6 allows Terfenol-D' s modulus to be calculated for the various combinations of d.c. stress and d.c. applied magnetic field. The modulus values as a E Co. S ':1io Stress, MPa Figure 6 Minor ioop strains with 4.0 Hz 2.8 MPa cyclic stress superimposed on d.c. compressive stresses ( MPa with 6.9 MPa steps) and d.c. applied field, H, ( ka/m at 16.1 ka/m steps). E S -:-iio -ioo -so -oo Stress, MPa Figure 7 Major and minor ioop strains with 4.0 Hz 2.8 MPa cyclic stress superimposed on d.c. compressive stresses ( MPa with 6.9 MPa steps) and d.c. applied fields, H, (16.1, 80.6, and 161lAIm). S 546 Proc. SPIE Vol. 4327

7 function of applied magnetic field, H, are plotted in Figure 8 where continuous lines connect constant d.c. compressive stress conditions. Modulus value uncertainty, for compressive stress levels above 48.8 MPa, approach 10 % due to larger frictional effects occurring in the testing fixture under higher d.c. load levels. Error estimates were established using an aluminum T-4 sample having the same dimensions as the Terfenol-D rod. Similar to the major loop modulus results in Figure 5, the modulus first decreases to a minimum and then increases again with increasing applied magnetic field. Additionally, as the d.c. compressive stress level is increased, the modulus starts out higher as well as the modulus minima being larger with their occurrence being delayed to higher applied magnetic field levels. In contrast to the major loop modulus results, the minor loop results display higher modulus values overall. Evaluating the AEH2H1 effect for the MPa case (highlighted trace) the modulus is observed to decrease from a peak of 98 GPa at H2 0 ka/m to 26 GPa at Hi 56.4 ka/m. The modulus change is calculated as (98 GPa - 26 GPa)/26 GPa giving AEH2H1 = 277%. Clearly the z\eh2h1 effect for the minor ioops is significantly less than the 414% observed in the major loop stress method. Given Terfenol-D's hysteretic and nonlinear nature, the modulus response with minor loop cyclic stress was also expected to be nonsingular over the course of major loop d.c. applied magnetic field changes. Figure 9 shows the modulus results produced by minor loop cyclic stress tests as the d.c. applied field, H, is varied from zero to ka/m and back down to zero. Examining the return path, as applied field is decreased, the modulus response lags the modulus response of the increasing applied field leg. To the right of the modulus minimum, the return path modulus is higher and to the left of the modulus minimum the return path modulus is lower. As is the case with Terfenol-D's magnetostrictive response, the modulus response is also dependent on the material's magnetic history. Implementing the AEH2H1 effect in a device such as a mechanical resonator, the applied magnetic field would likely be varied only from zero up to a level where the modulus minimum was reached where a greater AEH2HI effect occurs. Figure 10 shows the hysteretic modulus response with applied field excursions varying from zero to 64.4 ka/m and back. Again, the return path modulus lags and is lower than modulus values of the increasing applied field leg. Although not shown, subsequent applied field passes gave repeatable modulus responses following the same hysteretic path n.. '. e - (0 to 64.4 ka/m) I bc.. 90 ::N\. +Nok Appted magnetic field, ka/m Figure 8 Minor ioop generated Young's modulus from 4.0 Hz 2.8 MPa cyclic stress superimposed on d.c. compressive stresses ( MPa with 6.9 MPa steps) and d.c. applied fields, H, ( kaim at 16.1 ka/m steps). + (64.4toOkA/r) 0 0, >- S 40 > Applied magnetic field, knm Figure 9 Modulus hysteresis with applied field from minor loop strains under 4.0 Hz 2.8 MPa cyclic stress superimposed on d.c MPa compressive stress and d.c. applied fields, H, ( ka/m ). Estimated data point maximum error 1.4 GPa. CO _ Applied magnetic tield, ka/m Figure 10 Modulus hysteresis with applied field from minor loop strains under 4.0 Hz 2.8 MPa cyclic stress superimposed on d.c MPa compressive stress and d.c. applied field levels, H, ( ka!m ). Estimated data point maximum error I.4 GPa. 70 Proc. SPIE Vol

8 Returning to the issue of apparent modulus variation with cyclic stress amplitude, it will be shown that as the cyclic stress amplitude is decreased, the sample's apparent modulus increases. The minor strain versus stress loops of Figure 1 1 were collected for Terfenol-D under a d.c. compressive stress of MPa and a d.c ka/m applied magnetic field. The 4.0 Hz ac. cyclic stress was varied from an amplitude of 4.9 MPa down to 0.7 MPa in 1.4 MPa steps while holding the d.c. stress and applied field constant. As the stress is cycled, the strain response follows the counterclockwise path indicated by the arrows. Tracking a strain versus stress loop, immediately counterclockwise of the stress reversal or loop tip, the strain response lags behind. Considering magnetostrictive effects, this lag in strain response is indicative of constrained magnetic moment rotation that becomes increasingly significant for smaller stress amplitudes. The overall effect is an increasing apparent Young's modulus with decreasing cyclic stress amplitude. Verifying whether Terfenol-D's apparent modulus increase with decreasing cyclic stress amplitude was genuine, the minor loop test procedure was performed on aluminum 2024-T4 for comparison. The Terfenol-D and aluminum samples had the same dimensions and were strain gauged identically. The samples were tested under a MPa d.c. compressive stress and a ka/m d.c. applied magnetic field. A least squares fit of each minor strain versus stress loop was calculated where the fitted line's slope provided the sample's effective compliance and hence the Young's modulus. Figure 12 shows the modulus results for the Terfenol-D and aluminum samples where a 4.0 Hz cyclic stress increasing from 1.4 to 4.9 MPa in 0.7 MPa steps was superimposed on the d.c. stress. The aluminum's modulus begins at 77.5 GPa with the 1.4 MPa stress amplitude and levels off at 74.5 GPa for the 4.9 MPa stress amplitude case. Aluminum's aforementioned modulus change amounts to a 4.0% decrease with the 74.5 GPa modulus value being only 1.9% higher than the standard book value of 73.1 GPa [Riley 1989]. Turning to Terfenol-D's modulus response, the modulus starts at 98.4 GPa for the 1.4 MPa amplitude cyclic stress and decreases to 72.3 GPa for the 4.9 MPa amplitude condition. In this instance, Terfenol-D's modulus change is a significant 36% decrease over the range of cyclic stress amplitudes tested. Terfenol-D' s modulus dependence on cyclic stress amplitude has been observed previously [Moffet 1991], and the effect will continue to be a factor in the application of tunable mechanical resonators. The final investigative effort of this section is to evaluate Terfenol-D's AEH2H1 effect under various cyclic stress amplitudes for a MPa d.c. compressive stress and d.c. applied magnetic fields ranging from zero to 64.5 ka/m. These d.c. stress and applied magnetic field conditions are targeted for use in a mechanical resonator. Terfenol- D's modulus response to changes in applied magnetic field was measured for cyclic stress amplitudes ranging from 1.4 to 4.9 MPa in 0.7 MPa steps. The results of these tests, shown in Figure 13, demonstrate a trend in the AEH2H1 effect with cyclic stress amplitude. As cyclic stress amplitude is increased, the AEH2H1 effect becomes more pronounced. Comparing the I.4 and 4.9 MPa cases (top and bottom traces), AEH2H1 effects for H2 = 0 and Hi = 48.4 ka/m are calculated as (118 GPa - 34 GPa)/34 GPa 269% and (88 GPa - 21 GPa)/21GPa 319% respectively. In contrast, the net difference in modulus with applied field is greater for the smaller stress amplitude cases. The modulus difference may be calculated as (118 GPa - 34 GPa) 84 GPa Figure 1 1 Minor loop strain variation with cyclic stress amplitudes (0.7, 2.1, 3.5, and 4.9 MPa) superimposed on a MPa d.c. compressive stress and subject to a d.c kaim applied field. \ bc S\ e--- Terfeno-D B- Juminum6O61-T6 95.".. : : S Stress, MOe -e CycUc stress amplitude, MPa Figure 12 Modulus hysteresis with 4.0 Hz cyclic stress amplitudes ( MPa) superimposed on a MPa d.c. compressive stress and subject to a d.c. 16.lkAIm applied field Applied megrettc field, katm Figure 13 Modulus variation with d.c. applied fields ( ka/m) for 4.0 Hz cyclic stress amplitudes ( MPa) superimposed on a 28.1 MPa d.c. compressive stress. Estimated data point maximum error 1.4 GPa. 548 Proc. SPIE Vol. 4327

9 and (88 GPa - 21 GPa) = 67 GPa for the 1.4 and 4.9 MPa cyclic stress cases respectively. The larger modulus difference with applied field for the 1.4 MPa stress amplitude test compared to the 4.9 MPa test arises from the fact that small cyclic stress amplitudes have a greater stiffening effect for lower d.c. applied magnetic field conditions. 5. SUMMARY Major and minor loop cyclic compression tests clearly demonstrated the AEH2H1 effect in Terfenol-D. As d.c. applied magnetic fields are increased from zero, the modulus response first decreases to a minimum and then undergoes an increase. The overall modulus values are impacted by the level of compressive stress where increasing compressive stress increases the modulus at low applied field levels and conversely decreases the modulus at high applied field levels. Reflecting a balance in Terfenol-D's stress anisotropy and the applied magnetic energy, modulus minima occur at proportionally higher compressive stress levels with an increase in the applied magnetic field. Considering further the minor loop cyclic stress testing results, the d.c. compressive stress level used can be optimized to achieve the desired modulus response profile. Employing the appropriate d.c. compressive stress level will give the largest modulus change while minimizing the applied magnetic field change required. Additionally, the amplitude of cyclic stress will impact the magnitude of the AEH2H1 effect observed. In general, as cyclic stress amplitudes increase, the AEH2H1 effect measured will increase due to smaller moduli minima, however the net difference in modulus decreases. Accompanying the modulus minima, Terfenol-D experiences the largest changes in magnetization and exhibits the largest mechanical and magnetic hysteresis. Finally, decreasing cyclic stress amplitudes leads to a significant increase in Terfenol-D's apparent modulus possibly due to constrained magnetic moment rotation. 6. REFERENCES Calkins, FT., Design, Analysis, and Modeling of Giant Magnetostrictive Transducers, Ph.D. dissertation, Iowa State University, Ames, Iowa, Cedell, T.L., Magnetostrictive materials and selected applications, Magnetically induced vibrations in manufacturing processes, Ph.D dissertation, Lund University, Lund, Sweden, Clark, A.E., and Savage, H.T., "Giant magnetically induced changes in elastic the moduli in Tb3Dy7Fe2," IEEE Transactions on Sonics and Ultrasonics Su-22(1), pp , January, Clark, A.E., "Magnetostrictive rare earth-fe2 compounds," Ferromagnetic Materials, Volume 1, E. P. Whohlfarth, editor, North-Holland Publishing Company, Amsterdam, pp , Clark, A.E., Restorff, J.B., Wun-Fogle, M., and Lindberg, J.F., "Magnetoelastic coupling and the AE effect in TbxDyl-x single crystals," I Appl. Phys. 73(1 0), pp , May, Dapino, M.J., Nonlinear and hysteretic magnetomechanical mode/for magnetostrictive transducers, Ph.D. dissertation, Iowa State University, Jiles, D.C., "Theory ofthe magnetomechanical effect," J. Phys. D Appl. Phys. 28(8), pp , Lee, E.W., "Magnetostriction and Magnetomechanical Effects," Rep. Prog. in Phys. 18, pp , Moffet, M.B., Clark, A.E., Wun-Fogle, M., Linberg, J., Teter, J.P., and McLaughlin, E.A., "Characterization of Terfenol-D for magnetostrictive transducers,".1 Acoust. Soc. Am. 89(3), pp , March, Riley, W. F., and Zachary, LW., lntr. to Mechanics ofmaterials, John Wiley and Sons, New York Proc. SPIE Vol