Creep Expansion of Porous Ti-6Al-4V Sandwich Structures

DOUGLAS T. QUEHEILLALT, BILL W. CHOI, DANIEL S. SCHWARTZ,and HAYDN N.G. WADLEY

Low-density titanium alloy sandwich structures consisting of a porous core and fully dense face sheetscan be produced by consolidating argon gas charged powder compacts followed by hot rolling andannealing to expand the gas-filled pores. Little is known about the rate of pore expansion, its dependenceupon temperature, and the morphological evolution of the pore shape during expansion. In situ eddycurrent and laser ultrasonic sensors have been combined with metallographic and texture measurementsto measure the relative density, the elastic moduli, and the microstructural evolution of Ti-6Al-4Vsandwich panels during the annealing stage of low-density core (LDC) processing. The eddy currentdata indicated that expansion began during, the heating phase, reached a maximum expansion rate(D˙) of 2 3 1025 s21 at approximately 685 8C, and had almost ceased (D

˙, 1 3 1026 s21) after

annealing for 4 hours at 920 8C. The elastic moduli were found to decrease with increasing temperatureand volume fraction of porosity. The initial (as-rolled) microstructure consisted of a lamellar a 1 bmicrostructure with an a-phase lath thickness of 2.0 mm and contained a distribution of oblate-shapedpores with aspect ratios of up to 10. During the expansion process, it recrystallized into an equiaxeda 1 b structure with an a-phase grain diameter of 7.5 mm with spheroidal pores with aspect ratiosof up to 3. The combination of the two sensors was found to enable the in situ determination of boththe porous cores relative density and its elastic properties. These are the two material indices thatgovern the elastic response of a sandwich structure.

I. INTRODUCTION (“HIPing”) to create a 95 to 98 pct dense material withgas-filled porosity. These consolidated samples are then hotSTRUCTURALLY efficient sandwich panel structuresrolled to create a plate consisting of a core with a finelyare attracting renewed interest with the emergence ofdispersed pore distribution. The structure contains pores withultralight foamed or highly porous metals.[1] Numerous solid,high internal pressures and thin, fully dense face sheetsliquid, and vapor phase processes are being developed for(inherited from the original canister). The small pores arethe manufacture of light alloy foams[2–5] and highly porousthen re-expanded using a high-temperature annealing step.alloys.[6–11] By bonding these low-density materials betweenThis process results in a low-density cored sandwich struc-stiff, dense face sheets, light-weight structurally efficientture with the core containing 35 to more than 50 pct porosity.sandwich structures can be developed to replace costly hon-The mechanical performance of the panel is governed byeycomb structures. When stiff ultralow relative density coresthe core’s relative density and elastic properties and the face(2 to 5 pct of theoretical density) are used, recent calculationssheet’s thickness and, therefore, are dependent on the degreeindicate that these structures can be more structurally effi-of expansion of the porous core.[13,14]

cient than semimonocoque, stringer-stiffened plates andLittle is quantitatively known about the rate of pore expan-shells.[12] However, titanium panels, with porous core rela-

sion, its dependence upon initial pore pressure, the effecttive densities of 0.5 or more, are also attracting interest forof expansion temperature, the change in pore shape thatthe shaped wingskins of aircraft.accompanies expansion, the relationship between the spatialHere, we experimentally explore a powder metallurgydistribution and morphology of the pores and the resultanttechnique first proposed by Kearns et al.[8,9] and Martinelastic properties of LDC sandwich structures.[15] The expan-et al.[10,11] for the production of Ti-6Al-4V porous coredsion kinetics and elastic properties are measured in situsandwich structures. Figure 1 schematically illustrates howduring the final high-temperature annealing step of the pro-these low-density core (LDC) sandwich structures are pro-cess by using a combination of eddy current and laser ultra-duced. The process begins with the filling of a thick-walledsonic sensors. This pair of sensors enables both the creepTi-6Al-4V canister with Ti-6Al-4V powder and back-fillingexpansion (via eddy currents) and the time-of-flight (TOF)with between 1 and 7 atmospheres of argon gas. The canisterof longitudinal and shear elastic waves (via laser ultrasonics)is then sealed and consolidated by hot isostatic pressingto be measured simultaneously. The expansion and TOF datacan be combined to determine the evolving relative densityand elastic properties during the annealing cycle.

DOUGLAS T. QUEHEILLALT, Graduate Assistant, and HAYDN N.G.WADLEY, Associate Dean for Research and Research Professor in Materi-als Science, are with the Intelligent Processing of Materials Laboratory,

II. EXPERIMENTAL PROCEDURESMaterials Science Department, University of Virginia, Charlottesville, VA22903. BILL W. CHOI, formerly Research Associate with the Intelligent

A. Sample PreparationProcessing of Materials Laboratory, Materials Science Department, Univer-sity of Virginia, is Postdoctoral Research Staff Member, Lawrence Liv- The LDC Ti-6Al-4V sandwich panels were prepared byermore National Laboratory, Livermore, CA 94551. DANIEL S.

the Boeing Company (St. Louis, MO), Figure 1. A 10 3SCHWARTZ, Engineer, is with The Boeing Company, St. Louis, MO 63166.Manuscript submitted March 15, 1999. 10 3 2 cm3 Ti-6Al-4V HIP canister (1.5 mm wall thickness)

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 31A, JANUARY 2000—261

rolling, the face sheets were ,300 mm thick with a relativedensity of the as-rolled product of ,0.98 (a density of 4.32g/cm3).

Rectangular (3 3 5 cm2 in dimension) samples weremachined from the sandwich panels for expansion studies.The samples were lightly ground on both sides to removeany oxide. One surface was diamond polished to a 6 mmfinish to create a reflective surface for the laser interferome-ter. The samples were then ultrasonically cleaned, placed inthe sensor setup, and inserted into a vacuum furnace, whichwas evacuated to ,1026 torr. The samples were then heatedat a rate of 10 8C/min to 920 8C, held for up to approximately9.5 hours, and then slowly cooled to ambient temperature.To obtain the temperature dependence of the elastic modulifor fully dense Ti-6Al-4V (Appendix A), a 3 3 5 cm2 fullydense 2.7 mm thick sample of Ti-6A1-4V was prepared inthe same manner to the porous samples. The fully dense Ti-6Al-4V sample was then annealed at 788 8C for 45 minutes,furnace cooled to 482 8C, and air cooled to room temperature.This resulted in an untextured a 1 b structure with anaverage a-phase grain diameter of 4.5 mm.

B. Eddy Current Measurements

Eddy current sensors consist of metal coils through whichelectric currents are passed in order to induce weak electriccurrents in nearby conducting materials.[16,17] These currentscan be monitored (by observing the electric fields with anearby antenna coil) and used for quantitative measurementof materials properties such as electrical conductivity ormagnetic permeability, the dimensions of conducting sam-ples, and the detection of cracklike discontinuities in met-als.[16,17] Since they can be constructed from high-temperature materials, they are beginning to be used as insitu sensors during materials processing for such applicationsas probing internal temperature variations during the castingof metals and alloys,[18,19] monitoring the dimensionalchanges accompanying densification during HIPing of met-als powders and metal matrix composites (MMCs),[20,21]

measuring solidification conditions and temperature profilesduring the Czochralski growth of GaAs and Si,[22–25] anddetecting the liquid-solid interface during vertical Bridgmangrowth of CdTe and (Cd,Zn)Te solid solution alloys.[26–30]

A multifrequency eddy current approach was used to mea-sure sample thickness during the expansion process. It wasbased upon the sensor’s sensitivity to its separation distancefrom the sample. A cross-sectional sketch of the eddy currentprobe sensor is shown in Figure 2. Two identical sensors

Fig. 1—(a) through (d ) Schematic of the entrapped gas expansion processfor the production of LDC Ti-6Al-4V sandwich panels.

was filled with Ti-6Al-4V PREP powder. The HIP canisterwas hot vacuum outgassed and pressurized with three atmos-pheres (,0.3 MPa) of argon gas. The gas-charged powdercompact was HIPed at 1040 8C for 6 hours at a pressure of103 MPa. After HIPing, the sample heated in air to 900 8C,held for 30 minutes, and rolled (5 pct reduction per pass),with a 3-minute reheat at 900 8C between rolls, to a finalthickness of about 3.3 mm. Forty cross-rolled passes wereused for this rolling step. During deformation, strong diffu-sion bonds formed between the HIP canister and the powder, Fig. 2—Schematic of the cross-sectional view of the two-coil probe-type

eddy current and laser ultrasonic sensor.resulting in an integrally bonded sandwich structure. After

262—VOLUME 31A, JANUARY 2000 METALLURGICAL AND MATERIALS TRANSACTIONS A

were used to monitor the position of each surface of the method.[31] Since no physical contact is made with the sam-sample. A pair of 4 turn spirally wound (0.254 mm diameter) ple, the method can be utilized at elevated temperaturesplatinum primary (current inducing) and secondary and hostile environments, where conventional piezoelectric(antenna) coils were used to excite and detect eddy currents. transduction methods fail.[32] Laser ultrasonic approachesThe primary coils inner turn diameter was 11.2 mm and the have been used for determining the elastic moduli,[33–37] theouter coil had a diameter of 26.4 mm. The secondary coil grain size,[34,38,39] and texture of materials.[40,41,42] They canhad an inner diameter of 9 mm and an outer diameter of 24 also be used for thickness gaging of thin films providedmm. Both coils were embedded in ceramic (alumina) disks to the film’s elastic moduli and density are known.[43–46] Thisminimize changes in the coil diameter (and thus the sensor’s method is now beginning to be used as process controlresponse) upon heating and cooling. The eddy current sensor sensors for in situ measurements of internal temperaturecomponents were assembled in an alumina housing and gradients,[47,48] monitoring solid-liquid solidificationretained securely in place with an alumina threaded locking fronts,[49,50] and solid-solid phase changes.[48,51–53]

ring. The two sensors were positioned on either side of the The laser ultrasonic approach emerged after the develop-sample during the annealing process. Small (3 mm) holes ment of Q-switched lasers, which allow the rapid (,10 ns)were drilled through the center of each sensor to allow laser deposition of significant electromagnetic energy (10 to 100access to each face of the sample (Figure 2). mJ) on a surface. This causes local transient heating,

Each probe measured the separation distance between the resulting in the generation of energetic elastic waves (ultra-secondary coil position and the position of the sample’s sound). At low laser power densities (,106 W/cm2), local-surface adjacent to the probe. The separation distance ized heating in the region of irradiation produces an elasticbetween the pair of probes increased slightly as the sensor pulse only by thermal expansion of the heated volume.[31]

was heated due to thermal expansion of the ceramic. A This expansion results in the generation of both longitudinalthermal expansion correction accounted for this. Detailed and shear waves. At higher power densities (.108 W/cm2),analysis of the two-coil eddy current sensor approach can the incident laser pulse causes material ablation. Duringbe found elsewhere.[29,30] The sensors exploit the observation ablation, atoms and small liquid droplets are ejected fromthat at very high frequencies, skin effects limit the depth of the surface at high velocities creating a transient force normalflux penetration in a metallic sample. In the limit, the eddy to the surface and a large amplitude elastic impulse directedcurrent losses and the real part of the complex impedance normal to the sample. This generates a strong longitudinalgo to zero. The imaginary component of the impedance, pulse normal to the surface.[31]

when normalized by the sensor’s response with no sample Optical detection of ultrasound can be accomplished inter-present, is determined by the ratio of the flux linking the ferometrically by collecting the light reflected (or scattered)sample and the secondary coil to the total flux. As the sample from a surface as it is subjected to an ultrasonic disturbance.expands and its surface moves toward the sensor, more flux Surface displacement interferometers interfere the reflected/lines link the sample and the normalized high-frequency scattered light with a reference beam; the interference resultsimpedance decreases. A calibration curve is used to correct in a measurement of the optical phase, which is directlythe normalized impedance to the distance between the related to the instantaneous surface displacement.[54] Inprobe’s face and the sample surface. velocity interferometry, changes in the frequency of the

Multifrequency impedance measurements were per- reflected or scattered light are monitored and the dopplerformed using a HEWLETT-PACKARD* 4194A imped- shift is used to measure velocity. Surface displacement inter-

ferometers (Michelson or Mach-Zehnder) are widely used*HEWLETT-PACKARD is a trademark of Hewlett-Packard Company,Colorado Springs, CO. at low frequencies with highly reflective surfaces, whereas

velocity interferometers offer higher sensitivities at highance/gain-phase analyzer. A RF power amplifier was used frequencies and do not require highly reflective surfaces.[31]

to increase the primary coil current and thus to enhance the The ultrasonic TOFs between precisely positioned sourcevoltage signal monitored across a 1 V precision resistor and receiver points were measured (Figure 3). An approxi-in the primary test circuit. This was monitored with the

mately 10-ns duration Q-switched Nd:YAG laser pulse of“reference” channel of the analyzer. An attenuator was used1.064 mm wavelength was used as the ultrasonic source.in the line to the analyzer’s “test” channel to prevent over-The energy per pulse was ,10 mJ. The roughly Gaussianloading of the secondary coil signal. The multifrequencyspatial beam profile of the multimode pulse was focused tomeasurement was automated using a computer algorithman approximate circular spot 1 mm in diameter. Thus, thein conjunction with software installed on the impedanceincident source power density on the sample surface wasanalyzer. The impedance analyzer’s software recorded the,125 MW/cm2. Although this type of source generates ultra-test and reference channel voltages for 51 logarithmicallysonic frequencies in the 1.0 to 3.0 MHz range, the wave-spaced frequencies between 50 kHz and 5 MHz. It thenlength of the ultrasound was therefore several orders ofcalculated the gain and phase angle difference and the nor-magnitude larger than the pore size.malized impedance components at each frequency. A pro-

The ultrasonic receiver was a Mach-Zehnder heterodynegram activated the measurement periodically during thelaser interferometer, which responded to the out-of-planeprocess and stored the impedance data together with thesurface displacement associated with wavefront arrivals. Itsensors temperature.was powered by a 5 mW HeNe laser, which produced acontinuous Gaussian beam of 632.8 nm wavelength focused

C. Laser Ultrasonic Measurements to a circular spot ,100 mm in diameter. The time-integrated(displacement) signal from the interferometer was bandpassLaser-based ultrasonic techniques use a laser pulse to

generate ultrasound and an interferometric detection filtered between 10 kHz and 10 MHz and recorded with a

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 31A, JANUARY 2000—263

Fig. 3—Schematic diagram of the high-temperature laser ultrasonic sensing facility with integrated vacuum furnace.

precision digital oscilloscope using a 2-ns sampling interval The expansion rate then dropped with further heating to theplateau temperature of 920 8C. After about 6 hours at 920and 8-bit analog-to-digital conversion. To improve the sig-

nal-to-noise ratio, each waveform was an average of ten 8C, the expansion rate had dropped to ,1026 s21 (a 20-folddecrease from the maximum expansion rate). Two expansionpulses collected at a pulse repetition rate of 20 Hz. A fast

photodiode identified the origination time for the ultra- tests were conducted and both sets of data were in closeagreement.sonic signals.

The propagation of high-frequency elastic waves (ultra- Since mass can assumed to be conserved during the proc-ess cycle, the density can be readily calculated from thesound) in isotropic polycrystalline bodies is directly related

to the (dynamic) elastic moduli of the body. In the long expansion data. Figure 4(b) shows the density of both thesandwich structure and the core as a function of the processwavelength limit, the longitudinal and shear wave velocities

(evaluated from the TOF data) can be expressed in terms cycle time. Again, the data of Figure 4(b) has been correctedto remove contributions from the temperature-dependent lin-of the low-frequency limit (static) elastic constants and the

density (measured with the eddy current sensor).[55] The ear thermal expansion of the sample. Figure 4(b) indicatesthat the relative density of the sandwich structure decreasedlinear elastic behavior of these bodies can be fully described

by two independent elastic stiffness constants.[56] from 0.98 to 0.82, while that of the porous core decreasedfrom an initial value of 0.95 to 0.77.

The expanding porous core of the sandwich panels isIV. RESULTS AND ANALYSIS constrained by the integrally bonded face sheets, which

results in expansion strain anisotropy. The eddy current sen-A. Density Sensingsor measured strains in the easy expansion direction, i.e.,

Figure 4 shows the sample’s expansion and the accompa- normal to the rolling plane. The strain anisotropy was deter-nying density decrease as a function of process cycle time mined by measuring the total macroscopic strain normal tomeasured with the eddy current sensor. The data of Figure («33) and within («11 5 «22) the rolling plane after the anneal-4(a) has been corrected for the temperature-dependent ther- ing cycle. The ratio of the true strain normal to and withinmal expansion contributions[57,58,59] to the sensed expansion the rolling plane was found to be 20:1. Therefore, the actualand therefore corresponds to only the expansion caused by core density (D*) waspore growth.* Expansion began during the heating phase of

D* 5 D0(1 2 [«11 1 «22 1 «33]) 5 D0(1 2 1.1«33) [1]*An analysis of the thermal expansion for Ti-6Al-4V is shown in Appen-

where D0 is the initial core density.dix A.

the process cycle and continued into the annealing phase. TheB. Elastic Moduli Sensingexpansion rate is also shown in Figure 4(a). The maximum

expansion rate (D˙

5 2 3 1025 s21) occurred ,1.1 hours Figure 5 shows the TOF for the longitudinal (Figure 5(a))and the shear (Figure 5(b)) waves propagating through theinto the process cycle as the temperature reached ,685 8C.

264—VOLUME 31A, JANUARY 2000 METALLURGICAL AND MATERIALS TRANSACTIONS A

(a)

(a)

(b)

(b)

Fig. 5—Laser ultrasonic response for (a) the longitudinal and (b) the shearwave TOF evolution during annealing.

vl 5 !Er

1 2 v(1 1 v)(1 2 2v)

[2](c)

Fig. 4—Eddy current response for (a) the macroscopic expansion and (b)the relative density evolution during annealing incurred by (c) the tempera- vs 5 !E

r1

2(1 1 v)5 !G

r[3]

ture waveform.

where E is Young’s modulus, G is the shear modulus, r isthe density, and the Poisson’s ratio (v) can be expressed as

sandwich structure as a function of process cycle time. Thev 5

0.5(vl /vs)2 2 1(vl /vs)2 2 1

[4]data indicate that the longitudinal and shear wave TOFsincreased during the heating phase, went through a maximum

Since the elastic properties of fully dense Ti-6Al-4V arerate of change near the end of the heating phase, and thenknown a priori,* the ultrasonic velocities of both the longitu-slowed during the annealing phase. The TOF data for both

experiments are in close agreement. The increase in the*The elastic moduli for untextured polycrystalline Ti-6Al-4V were evalu-

sandwich structure’s longitudinal and shear wave TOFs ated independently, the analysis of which is shown in detail in Appendix A.resulted from the combined effect of temperature-dependentsoftening of the elastic moduli (during the heating portion dinal and shear waves propagating in the face sheets and

the porous core can be decoupled. Figures 6(a) and (b) showof the annealing cycle) and the increase in the volume frac-tion of porosity, which also caused a reduction in the elastic the evolution of Young’s and the shear moduli for the fully

dense face sheets and evolving porous core as a function ofmoduli of the porous core.The longitudinal (vl) and shear (vs) wave velocities were process cycle time, respectively. It is clear that the tempera-

ture-dependent softening of the elastic moduli contributedcomputed by dividing the sample’s instantaneous thicknessby the corresponding TOFs. Once the two ultrasonic veloci- a significant portion to the decrease in the elastic moduli of

the evolving porous core. The additional decrease in theties and the density were known, the two independent elasticmoduli were evaluated using[55,56] elastic moduli of the porous core was due to the volume

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 31A, JANUARY 2000—265

(a)

Fig. 7—Scanning electron micrographs showing the cross section (facesheets plus porous core) of the Ti-6Al-4V low density core sandwichstructure for (a) the as-rolled and (b) post annealed conditions.

compact, resulting in the formation of the macroscopic sand-wich structure (i.e., a porous core sandwiched between fullydense face sheets). Slight variations in face sheet thickness(nominal thicknesses ranged from 250 to 350 mm) wereincurred during the HIPing and hot rolling consolidationprocess steps. The as-rolled macrostructure (Figure 7(a))(b)shows a relatively homogenous distribution of pores that

Fig. 6—The (a) Young’s modulus (E ) and (b) shear modulus (G) evolution tended to be elongated in the rolling plane. The post annealedevaluated from the eddy current measured expansion and the laser ultrasoni- macrostructure (Figure 7(b)) shows a homogeneous distribu-cally measured longitudinal and shear wave velocity data.

tion of pores that tended to be more spherical in nature. Thiswas expected since a majority of the expansion occurred inthe through plate direction.

fraction of porosity dependence of the elastic properties. It Figure 8 shows a three-dimensional composite of the as-is interesting to note that during the heating portion of the rolled and the post annealed microstructures. In the as-rolledannealing cycle, the rate of decrease of the elastic properties sample (Figure 8(a)), the porosity distribution consisted ofof the porous core is slightly greater than that of the fully pore sizes ranging from a few microns in diameter up todense face sheets. This is attributed to the small increase in ,150 mm, including morphologies that varied from spheri-volume fraction of porosity during the annealing phase. It cal- to oblate-shaped pores. It is clear that the rolling processis a commonly known fact that the elastic properties of aids in producing oblate-shaped pores, which are predomi-most materials decrease as the relative density of the body nantly aligned in the rolling plane. The inset of Figure 8(a)decreases. Although there have been limited studies on the shows a heavily deformed microstructure of the as-rolledeffects of volume fraction of porosity on the elastic properties plate consisting of elongated, interconnected a-phase grainswith increasing temperature, several thorough studies[61,62,63] with intergranular transformed b-phase. In the post annealedhave demonstrated that there is little or no variation in the sample (Figure 8(b)) the pore morphology showed a rela-porosity dependence of the elastic properties with increas- tively uniform distribution of semispherical pores ranginging temperature. in diameter up to ,150 mm in size. In the post annealed

sample, the microstructure had recrystallized to a fine equi-axed structure of primary a-phase and intergranular retained

C. Microstructural Analysis (metastable) b-phase grains. The size and shape distributionof the pores are examined further in Section 3.1. Porous core evolution

Figure 7 shows representative cross-sectional slices of the The inset micrographs of Figure 8 clearly show a lamellara 1 b structure for the as-rolled condition and a recrystal-Ti-6Al-4V LCD sandwich structure for the as-rolled (Figure

7(a)) and post annealed (Figure 7(b)) conditions. It was lized equiaxed a-phase with intergranular transformed b-phase structure in the post annealed condition. It shouldobserved that during consolidation, a uniform, fully coherent

diffusion bond formed between the HIP canister and powder be noted that the high-temperature microstructure, which

266—VOLUME 31A, JANUARY 2000 METALLURGICAL AND MATERIALS TRANSACTIONS A

(a)

Fig. 8—Three-dimensional composite micrographs of the cross section for(a) the as-rolled and (b) the post annealed microstructure showing the sizeand spacial distribution of porosity. High-magnification micrographs of thea 1 b phase microstructure are shown in the inserts.

governs the creep deformation behavior, cannot be deducedfrom the slow furnace-cooled microstructures shown in Fig-ure 8. It is well known that the volume fraction of the a-to b-phase (Va/Vb) is dependent on the temperature.[64,65]

During furnace cooling, b-phase transforms to secondary a (b)phase and some secondary a-phase is also precipitated on

Fig. 9—Evolution of the average grain diameter of (a) the entire annealingthe primary a-phase; therefore, the a-phase grain size aftercycle and (b) the normalized isothermal behavior of LDC Ti-6Al-4V at

furnace cooling to room temperature may not accurately 920 8C.indicate the primary a-phase grain size, volume fraction, ordistribution at the annealing temperature.[66]

found to be 2.0 mm. Figure 9(b) shows the normalized a-2. Grain growth kineticsphase grain diameter as a function of isothermal annealingIt is clear from Figure 8 that the initial LDC poroustime. The data were normalized such that the initial conditionmicrostructure recrystallized. During recrystallization, thecorresponded to the inflection point of the heating and iso-post annealed microstructure shows a propensity for equi-thermal annealing waveform. Ghosh and Hamilton[69] haveaxed a-phase grains to form. To investigate the recrystalliza-investigated the strain rate dependence of the grain growthtion and grain growth kinetics, a series of time-interruptedfor Ti-6Al-4V at 927 8C and found that the average grainannealing treatments were conducted on 18 1 3 1 cm2

diameter is well represented bysamples. The samples were encapsulated in evacuated quartzampoules (1026 torr), annealed with the same process sched- d 5 d0(t/10)n [5]ule and water quenched at various times to freeze in the

where d0 is the initial grain diameter, t is the time (inhigh-temperature microstructural features. It was found thatminutes), and n is a strain rate parameter n 5 1.8(«̇t 1the a-phase recrystallization began by polygonization and0.00005)0.237. Our data show good agreement with their datathe formation of subgrains. A dihedral angle was formed atnormalized to the initial condition (d0 5 4.0 mm and «̇t 5the a/b interface and the subgrain grew into adjacent a-0) (Figure 9(b)).phase laths during which the misorientation between newly

formed a-phase subgrains increased. During the a-phase 3. Pore size/shape evolutionFigure 10 shows both the cross-sectional areas and aspectsubgrain growth, the b-phase laths subdivided. This coin-

cided with models proposed by Margolin and Cohen[67] and ratio distributions for the as-rolled (Figure 10(a)) and thepost annealed (Figure 10(b)) conditions. The cross-sectionalGogia et al.[68] for the recrystallization behavior of Ti-

6Al-4V. areas were calculated assuming ellipsoidal shaped pores andthe aspect ratio was taken as the length of the major axisFigure 9(a) shows the average a-phase grain diameter as

a function of process cycle time. The initial grain diameter divided by the minor axis. The sampling population was500 randomly chosen pores. The as-rolled microstructurewas taken to be the average a-phase lath thickness and was

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 31A, JANUARY 2000—267

(a)

(b)

(c)

(d )

Fig. 10—Cross-sectional areas (a and c) and aspect ratio distributions (band d ) for the as-rolled and post annealed microstructures.

exhibited a tight distribution (,500 mm2) of cross-sectionalareas (Figure 10(a)) with a wide range of pore aspect ratios(up to 10) (Figure 10(b)). During the process cycle, the poresgrew up to 5000 mm2 (Figure 10(c)), while the distributionof pore aspect ratios decreased (less than 3) (Figure 10(d)).The a 1 b phase microstructures of both the as-rolled andpost annealed samples for the face sheets were similar innature to that of the porous core.

4. Texture analysisFigure 11 shows (0002) basal plane pole figures for (a)

a random oriented powder distribution, (b) the as-rolled and(c) the post annealed porous cores obtained on sectionsnormal to the rolling plane. Both of the pole figures shownin Figures 11(b) and (c) exhibit a higher a-phase basal (0002)intensity than expected for a random distribution of grain

Fig. 11—Basal plane (0002) pole figures for (a) a random oriented powderorientation (Figure 11(a)). This indicates that both the as-distribution and (b) the as-rolled and (c) post annealed porous cores.rolled and post annealed microstructures exhibited (0002)

texture.

268—VOLUME 31A, JANUARY 2000 METALLURGICAL AND MATERIALS TRANSACTIONS A

This is important because texture strengthening of themechanical properties could be modified through the controlof the crystallographic texture. Changes in texture have beencorrelated with changes in tensile strength and Young’s mod-ulus, and it has been shown that Young’s modulus variessignificantly with the relation between the stress axis andthe basal pole.[70] It has also been shown that regardless ofthe slip plane and in the absence of twinning, multidirectionalcross rolling of Ti-6Al-4V will lead to a basal plane tex-ture.[71] A basal plane texture would be desirable for suchapplications as pressure vessels, etc., which are biaxiallystressed. This texture is expected because hot cross-rollingof Ti-6Al-4V tends to align the normal of the (0002) basalplane of the a-phase normal to the rolling plane.[72] Thisresults in an elastically anisotropic structure due to the factthat Young’s modulus normal to the basal plane (E33 5 ,145

(a)GPa) is significantly larger than Young’s modulus withinthe basal plane (E11 5 ,100 GPa) at room temperature.[73]

D. Elastic Moduli Analysis

This elastic anisotropy of the LDC sandwich structuresis clearly observed by comparison of both Young’s (Figure6(a)) and the shear (Figure 6(b)) moduli for the textured,porous sandwich structure. During the heating portion ofthe annealing cycle, it is believed that the decreased modulidue to temperature-dependent softening of the elastic modulidominate over the decreased moduli caused by the volumefraction porosity dependence of the evolving porous core.Once the temperature reached ,800 8C to 900 8C, thedecreased elastic moduli due to the evolving porous corebegan to dominate and the elastic moduli of the porous corewas lower than the fully dense face sheets. Their are threedominant microstructural attributes that dictate the elastic (b)behavior of these sandwich structures: the temperature-

Fig. 12—The temperature and texture corrected (a) Young’s modulus anddependent softening of the elastic moduli, the volume frac- (b) shear modulus measured via laser ultrasonics and that predicted bytion of porosity-dependent elastic moduli, and the crystallo- Budiansky’s model in conjunction with the eddy current data.graphic texture–dependent elastic moduli. In order to fullyunderstand the macroscopic elastic behavior, it is necessaryto analyze each of these mechanisms individually. It has been where v 5 v0(1 2 VP); K0 and G0 are the bulk and shearreported[74] that the volume fraction of porosity-dependent moduli, respectively; and v0 is Poisson’s ratio for the fullyelastic moduli of Ti-6Al-4V porous cores, produced by the dense material. The Young’s modulus was determined fromlow density core processing method (nontextured with spher- the well-known analytical relationshipical pores), is well predicted by Budiansky’s model[75] for thepore volume fraction dependence of the elastic properties. E 5

9KG3K 1 G

[8]Although a number of analytical models for the depen-

dence upon relative density (r*/r0 5 1 2 VP) of the elastic Typical properties for fully dense isotropic polycrystallinemoduli have been developed, where (VP) is the volume frac- Ti-6Al-4V used were: K0 5 120 GPa, E0 5 114 GPa, G0tion of porosity,[76] many of these models were originally 5 42.5 GPa, v0 5 0.342, and r0 5 4.42 g/cm3.[77]

developed for two-solid phase systems and adapted to porous Figure 12 shows the Young’s modulus (Figure 12(a)) andmaterials by assuming the density and elastic stiffness of the shear modulus (Figure 12(b)), which have been correctedthe second phase tends toward zero. Budiansky[75] used a for the temperature-dependent moduli. Figure 12 also showsself-consistent approach to analyze a body containing ran- the simulated moduli evolution for an untextured polycrys-domly distributed spherical pores. In his model, the porosity talline Ti-6Al-4V porous core predicted from the densitydependence of the bulk modulus (K ) and the shear modulus data of Figure 4(b) and Eqs. [6] through [8]. It can be seen(G) are represented by that the experimental data follow the same functional form

as the moduli evolution predicted by Budiansky’s model andthe eddy current data; however, the absolute values of bothK 5 K0F1 2

3VP (1 2 v)2(1 2 v) G [6]

Young’s and the shear moduli are greater for the experimen-tal data. It is believed that this increase in the absolute valuesof the moduli can be attributed to texture strengtheningG 5 G011 2 15VP1 1 2 v

7 2 5v2G [7]incurred during processing. The variability observed during

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 31A, JANUARY 2000—269

the initial stages of heating is attributed to the increase inmodulus caused by the oblate shape of the pores during theinitial stages of the annealing treatment. Budiansky’s modeldoes not account for nonspherical pores.

V. DISCUSSION

The driving force for the expansion of these structurallyporous sandwich structures is governed by an increase inpore gas pressure due to increased temperature and the con-comitant decrease in creep-dependent microstructural vari-ables (e.g., elastic moduli, flow stress, yield strength, creepconstants, etc.). During the initial stages of heating, as tem-perature and time were increased, the expansion rateincreased (Figure 4(a)). However, when the temperaturereached ,685 8C, the expansion rate reached a maximum

Fig. 13—Relative pore pressure evolution predicted from the density evolu-and began to decrease with increasing temperature and aftertion during annealing, P0 5 2 MPa.about 6 hours at 920 8C dropped to ,1026 s21. Several

mechanisms are believed to contribute to this behavior,including the change in internal gas pressure caused by thetemperature increase and the macroscopic expansion and the isothermal stage of the annealing cycle. This behaviorthe retarded expansion rate caused by the increased creep is expected because, during the initial stages of heating, theresistance due to grain growth. gas pore pressure increases due to the increase in temperature

It is believed that the extensive deformation incurred dur- and begins to retard due to the competing densificationing the hot rolling in the a 1 b phase field of these sandwich effects. The sharp decrease in gas pore pressure at 90 minutespanels enhances expansion characteristics of these materials is a direct result of isothermal soaking at 920 8C.due to superplastic behavior. Fine-grained Ti-6Al-4V exhib- In order to fully understand how the rate of change ofits superplastic behavior between 750 8C and 950 8C, and pore pressure along with the temperature dependence of thegrain boundary sliding is assumed to be the dominant mecha- critical materials’ parameters and mechanisms, predictivenism for the superplastic behavior.[78–82] It has been shown models can be used to predict and optimize the process.that partitioning of the substitutional alloying elements The analysis of expansion of these porous cored sandwichbetween the a and b phases (Al to a, V to b) retards grain structures depends on a micromechanical approach in whichgrowth and increases the superplastic temperature range to the constitutive behavior of a representative volume element900 8C to 950 8C.[79] At temperatures above 950 8C, the a- of material is modeled and then the overall (macroscopic)phase volume fraction decreases, rapid coarsening of the b- response is obtained via a stochastic geometry model.[84]

phase occurs, and superplastic behavior is lost. At tempera- The analysis of the rate of pore expansion can be treatedtures below 750 8C, the superplastic response is retarded by the application of variational principles of continuumdue to the volume fraction of the b-phase, and its diffusivities mechanics to a single, noninteracting, spherical void in anare insufficient to accommodate the grain boundary slid- infinite creeping matrix in which a unit cell model, baseding mechanism.[80,81] on that of Elzey and Wadley,[85] has been developed incorpo-

Therefore, it is reasonable to assume the rate of change rating time-dependent power law creep and diffusional creepof pore pressure along with the temperature dependence of to predict the relative density of LDC Ti-6Al-4V sandwichthese critical materials’ parameters and the evolving sys- structures.[86] The analysis of this model used microstructur-tem’s geometry govern the relative magnitude of each mech- ally sensitive constitutive laws for power law creep and theanism. If it is assumed that the argon gas behaves ideally diffusion accommodated grain boundary sliding responseand that the permeability of argon in these structures is for a Ti-6Al-4V matrix containing a variable distribution ofnegligible, then the normalized pore pressure (with respect pore shapes and sizes to predict the macroscopic responseto the initial pore pressure prior to expansion) evolved during to the two primary input conditions (heating rate and initialannealing can be represented by gas pore pressure).

PP0

5 1TT021

DD021

1 2 D0

1 2 D 2 [9]VI. CONCLUSIONS

The expansion kinetics and elastic properties of Ti-6Al-where D0, T0, and P0 are the initial core relative density,temperature, and gas pore pressure, respectively; and D, T, 4V sandwich panels have been measured during the high-

temperature annealing step of the LDC process sequence byand P are the instantaneous core relative density, tempera-ture, and gas pore pressure, respectively.[83] Figure 13 shows using a combination of eddy current and laser ultrasonic

sensors. By using these sensors, the creep expansion (viathe relative gas pore pressure predicted from Eq. [9] usingthe densification and temperature rates shown in Figures eddy currents) and the TOF of longitudinal and shear waves

(via laser ultrasonics) were simultaneously measured. The4(b) and (c), respectively. It can be seen from Figure 13 thatthe driving force for core expansion (gas pore pressure) expansion and TOF data were analyzed to determine the

porous cores’ evolving density and concomitant decrease inincreases rapidly during the initial stage of heating, beginsto retard with core expansion, and decreases rapidly during elastic properties of the sandwich structure. The use of these

270—VOLUME 31A, JANUARY 2000 METALLURGICAL AND MATERIALS TRANSACTIONS A

two sensors enables control of the two most important mate-rial parameters for efficient sandwich structures.

The eddy current data indicated that expansion beganduring the heating phase of the process cycle and continuedinto the annealing phase. The maximum expansion rate (d

˙5

2 3 1025 s21) occurred ,1.1 hours into the process cycleas the temperature reached ,685 8C. The expansion ratethen dropped with further heating to the plateau temperatureof 920 8C. After about 6 hours at 920 8C, the expansion ratehad dropped to ,1026 s21 (a 20-fold decrease from themaximum expansion rate). The laser ultrasonic data indi-cated that the elastic moduli exhibited a concomitantdecrease from initial values of Young’s and the shear moduliof 120 and 45 MPa to final values of 42 and 15 MPa,respectively.

The as-rolled microstructure exhibited a relatively homog-enous distribution of pores, which tended to be elongated

(a)in the rolling direction (aspect ratios of up to 10), where thepost annealed microstructure exhibited a more homogeneousdistribution of pores, which tended to be spherical in nature(aspect ratios of up to 3). This was accompanied by therecrystallization and grain growth of the initial lamellar a 1b structure in the as-rolled condition to an equiaxed a-phase with intergranular transformed b-phase structure inthe postannealed condition. It is concluded that in additionto the microstructural characteristics, the texture strengthen-ing of the (0002) basal plane of the a-phase influenced theelastic properties of the LDC Ti-6Al-4V sandwich structures.

APPENDIX A

The temperature dependence of the fractional change inlength, Dl /l0, of fully dense Ti-6Al-4V between 20 8C and1110 8C has been measured by Williams[57] and McGee andMatthews.[58,59] Figure A1 shows their data for the fractional (b)change in length as a function of temperature. The data in

Fig. A2—Experimentally measured temperature dependence for (a)Figure A1 is well fitted by the cubic relationship Young’s (E ), the bulk (K ), and the shear (G) moduli; and (b) Poisson’s

ratio (v) of fully dense equiaxed polycrystalline Ti-6Al-4V.Dl/l0 (pct) 5 21.7679 3 1022 1 9.3989

3 1024 T 1 6.6987 3 1028 T 2 [A1]

1 1.2464 3 10210 T 3 the coefficient of linear thermal expansion, a, of Ti-6Al-4Vcan be evaluated by differentiation of Eq. [A1] with respectwhere T is the temperature in degrees Celsius. Subsequently,to temperature and is obtained in the quadratic form:

a(C21) 5 9.3989 3 1026 1 1.3397 3 1029 T[A2]

1 3.7392 3 10212 T 2

where T is the temperature degrees Celsius.If it is assumed that Ti-6Al-4V is elastically isotropic,

then the coefficient of volumetric thermal expansion or coef-ficient of thermal expansion, b, is equal to 3 times thecoefficient of linear thermal expansion and is represented by

3a 5 b 5 21r1­r

­T2P

[A3]

Equation [A3] can be integrated with respect to temperatureto determine the temperature dependence of the density:

1n rT 5 1n r20 2 3 eT

20

a(T )dT [A4]

Fig. A1—Temperature dependence of the thermal linear expansion, Dlt /l0 ,for Ti-6Al-4V; data taken from Williams[57] and McGee and Matthews.[58,59] where rT is the density at a given temperature and r20 is the

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 31A, JANUARY 2000—271

Table AI. Experimentally Determined Temperature-Dependent Polynomial Fits of the Elastic Properties for Fully DenseTi-6Al-4V

Elastic Properties Temperature-Dependent Moduli (GPa), T in 8C

Bulk modulus (K ) 120.58 2 7.795 3 1023 T 2 1.483 3 1025 T 2

Young’s modulus (E ) 114.07 2 2.787 3 1022 T 2 2.721 3 1025 T 2

Shear modulus (G) 42.55 2 1.181 3 1022 T 2 9.625 3 1026 T 2

Poisson’s ratio (v) 0.344 1 1.815 3 1025 T 1 3.773 3 1028 T 2

2. S. Akiyama, H. Ueno, K. Imagawa, A. Akira, S. Nagata, K. Morimoto,density at 20 8C (r20 5 4.42 g/cm3). By integrating Eq. [A4]T. Nishikawa, and M. Itoh: U.S. Patent No. 4,713,277, Dec. 15, 1987.and substituting a room-temperature density of 4.42 g/cm3

3. I. Jin, L.D. Kenny, and H. Sang: U.S. Patent No. 4,973,358, Nov.into Eq. [A4], the temperature-dependent density can be 27, 1990.represented by 4. J. Baumeister: DE Patent 40 18 360, 1991.

5. J. Baumeister and H. Schrader: U.S. Patent No. 5,151,246, Sept.r (g/cm3) 5 4.4226 2 1.2264 3 1024 T

[A5] 29, 1992.6. V.I. Shapovalov: U.S. Patent No. 5,181,549, Jan. 26, 1993.

2 2.3351 3 1028 T 2 2 4.3354 3 10212 T 37. V.I. Shapovalov and A.G. Timchenko: Fiz. Metall., 1993, vol. 76,

p. 335.where T is the temperature in degrees Celsius.8. M.W. Kearns: U.S. Patent No. 4,659,546, Apr. 21, 1987.The temperature dependence of the ultrasonic velocity for 9. M.W. Kearns, P.A. Blenkinsop, A.C. Barber, and T.W. Farthing: Int.

a fully dense Ti-6Al-4V sample was evaluated from the J. Powder Metall., 1988, vol. 24 (1), p. 59.10. R.L. Martin and R.J. Lederich: Advances in Powder Metallurgy: Proc.measured TOF data for both longitudinal and shear waves.

1991 Powder Metallurgy Conf. Exp., Powder Metallurgy IndustriesThe samples’ instantaneous length was calculated from Eq.Federation, Princeton, NJ, 1991, p. 361.[A1] at each measurement temperature. Both the longitudinal

11. R.L. Martin: U.S. Patent No. 5,564,064, Oct. 8, 1996.and shear wave velocities exhibited a strong monotonically 12. B. Budiansky: Harvard University, MA, private communication, 1998.decreasing function of temperature. The temperature-depen- 13. M.F. Ashby: Materials Selection in Mechanical Design, Pergamon,

Oxford, United Kingdom, 1992.dent polycrystalline elastic moduli for Ti-6Al-4V were14. F.R. Shanley: Weight-Strength Analysis of Aircraft Structures, Dover,deduced from the ultrasonic velocities, and the temperature-

New York, NY, 1960.dependent density, Eq. [A5]. The temperature dependencies 15. Workshop on Ultralight Metal Structures, University of Virginia, Char-of the elastic moduli (E, K, and G) are shown in Figure lottesville, VA, Sept. 1997.A2(a). Figure A2(b) shows the temperature dependence of 16. H.L. Libby: Introduction to Electromagnetic Nondestructive Test Meth-

ods, Wiley, New York, NY, 1971.Poisson’s ratio (v), calculated from Young’s and the shear17. F. Foster: Nondestructive Testing Handbook, 2nd ed., Columbus, OH,moduli of Figure A2(a). Similar to the ultrasonic velocity

1986, vol. 4.data, the elastic moduli exhibit a monotonically decreasing 18. M.L. Mester, A.H. Kahn, and H.N.G. Wadley: Proc. 4th Int. Aluminumfunction of temperature. The elastic moduli and Poisson’s Extrusion Technology Seminar, The Aluminum Association, Washing-

ton, DC, 1988, p. 259.ratio data for Ti-6Al-4V was well fitted by polynomial rela-19. A.H. Kahn, M.L. Mester, and H.N.G. Wadley: in Intelligent Processingtionships. Table AI shows the third-order polynomial fits

of Materials, H.N.G. Wadley and W.E. Eckart, eds., TMS, Warrendale,for the temperature dependence of the elastic moduli andPA, 1990, p. 293.

Poisson’s ratio. Our data are in close agreement with that 20. B.W. Choi, J. Marshall, Y.K. Deng, C. McCullough, B. Paden, andof Fukuhara and Sanpei[60] for temperature dependence of R. Mehrabian: Acta Metall. Mater., 1990, vol. 38 (11), p. 2245.

21. H.N.G. Wadley, R.J. Schaefer, A.H. Kahn, M.F. Ashby, R.B. Clough,the elastic properties of Ti-6Al-4V. Therefore, the polycrys-Y. Geffen, and J.J. Wlasich: Acta Metall. Mater., 1991, vol. 39 (5),talline elastic moduli for Ti-6Al-4V have been fully charac-p. 979.terized from 20 8C to 1000 8C, covering the broad spectrum

22. K.S. Choe, J.A. Stefani, J.K. Tien, and J.P. Wallace: J. Cryst. Growth,of temperatures encountered during annealing of LDC sand- 1989, vol. 97, p. 622.wich structures. 23. J.A. Stefani, J.K. Tien, K.S. Choe, and J.P. Wallace: J. Cryst. Growth,

1990, vol. 106, p. 611.24. J.P. Wallace, J.K. Tien, J.A. Stefani, and K.S. Choe: J. Appl. Phys.,

ACKNOWLEDGMENTS 1991, vol. 68, p. 550.25. K.S. Choe, J.A. Stefani, T.B. Dettling, J.K. Tien, and J.P. Wallace: J.

We are grateful to Ravi Vancheeswaren for many helpful Cryst. Growth, 1991, vol. 108, p. 262.discussions concerning predictive microstructural modeling. 26. G.J. Rosen, F.M. Carlson, J.E. Thompson, W.R. Wilcox, and J.P Wal-

lace: J. Electronic Mater., 1995, vol. 24 (5), p. 491.This work has been performed as part of the research of27. K.P. Dharmasena and H.N.G. Wadley: J. Cryst. Growth, 1997, vol.the Multidisciplinary University Research Initiative (MURI)

172, p. 337.program on Ultralight Metal Structures supported by 28. B.W. Choi and H.N.G. Wadley: J. Cryst. Growth, 1999, vol. 208, p. 219.DARPA and ONR under contract No. N00014-96-I-1028, 29. B.W. Choi: Ph.D. Dissertation, University of Virginia, Charlottesville,monitored by Steve Wax (DARPA) and Steve Fishman VA, 1996.

30. K.P. Dharmasena: Ph.D. Dissertation, University of Virginia, Char-(ONR).lottesville, VA, 1996.

31. C.B. Scruby and L.E. Drain: Laser Ultrasonics: Techniques and Appli-cations, Hilger, New York, NY, 1990.REFERENCES

32. Report on the Workshop on Industrial Applications of Laser Ultrason-ics, Johns Hopkins University, Baltimore, MD, 1997.1. L.J. Gibson and M.F. Ashby: Cellular Solids: Structure and Properties,

2nd ed., Cambridge University Press, Cambridge, United Kingdom 33. C.A. Calder, E.C. Draney, and W.W. Wilcox: J. Nucl. Mater., 1981,vol. 97 (1–2), p. 126.1997.

272—VOLUME 31A, JANUARY 2000 METALLURGICAL AND MATERIALS TRANSACTIONS A

60. M. Fukuhara and A. Sanpei: J. Mater. Sci. Lett., 1993, vol. 12, p. 1122.34. J.D. Aussel and J.P. Monchalin: Ultrasonics, 1989, vol. 27, p. 165.35. M. Paul, B. Haberer, and W. Arnold: Mater. Sci. Eng. A, 1993, vol. 61. R.E. Fryell and B.A. Chandler: J. Am. Ceram. Soc., 1964, vol. 47 (6),

p. 283.A168, p. 87.36. A. Moreau and F. Taheri: Mater. Sci. Forum, 1996, vols. 210–213, 62. W.R. Manning and O. Hunter, Jr.: J. Am. Ceram. Soc., 1969, vol. 52

(9), p. 492.p. 235.37. D.T. Queheillalt and H.N.G. Wadley: J. Appl. Phys., 1998, vol. 83 63. O. Hunter, Jr., H.J. Korklan, and R.R. Suchomel: J. Am. Ceram. Soc.,

1974, vol. 57 (6), p. 267.(8), p. 4124.38. A. Le Brun, J.L. Lesne, O. Cassier, F. Goncalves, and D. Ferriere: 64. J. Pilling, D.W. Livesey, J.B. Hawyard, and N. Ridley: Met. Sci., 1984,

vol. 18, p. 117.in Nondestructive Characterization of Materials III, P. Holler, V. Haul,G. Dobmann, C. Ruud, and R. Green, eds., Springer-Verlag, Berlin, 65. M.T. Salehi, J. Pilling, N. Ridley, and D.L. Hamilton: Mater. Sci. Eng.

A, 1992, vol. A150, p. 1.1989.39. S. Faßbender, M. Kulakov, B. Hoffman, M. Paul, H. Peukert, and W. 66. D.S. McDarmaid, A.W. Bowen, and P.G. Partridge: J. Mater. Sci.,

1985, vol. 20, p. 1976.Arnold: in Nondestructive Characterization of Materials III, P. Holler,V. Haul, G. Dobmann, C. Ruud, and R. Green, eds., Springer-Verlag, 67. H. Margolin and P. Cohen: in Titanium ‘80, Science and Technology,

H. Kimura and O. Izumi, eds., TMS-AIME, Warrendale, PA, 1980,Berlin, 1989.40. A.C. Tam and W.P. Leung: Appl. Phys. Lett., 1984, vol. 45, p. 1040. p. 1555.

68. A. Gogia, D. Banerjee, and N.C. Birla: Trans. Ind. Inst. Metals, 1983,41. J.W. Wagner and J.B. Spicer: Acousto-Optics and Acoustic Microscopy,1992, vol. 140, p. 121. vol. 36 (3), p. 200.

69. A.K. Ghosh and C.H. Hamilton: Metall. Trans. A, 1979, vol. 10A,42. H.N.G. Wadley, Y. Lu, and J.A. Goldman; J. Nondestr. Eval., 1996,vol. 14 (1), p. 31. pp. 699-706.

70. A.W. Bowen: Mater. Sci. Eng., 1977, vol. 29, p. 19.43. A.C. Tam: Appl. Phys. Lett., 1984, vol. 45 (5), p. 510.44. R.J. Dewhurst, C. Edwards, A.D.W. McKie, and S.B. Palmer: Appl. 71. S.F. Fredrick and G.A. Lanning: Metall. Trans. B, 1975, vol. 6B, p. 601.

72. E.W. Collings: The Physical Metallurgy of Titanium Alloys, ASM,Phys. Lett., 1987, vol. 51 (14), p. 1066.45. D.A. Hutchins, K. Lundgren, and S.B. Palmer: J. Acoust. Soc. Am., Metals Park, OH, 1984.

73. High Performance Materials in Aerospace, H.M. Flower, ed. Chap-1989, vol. 85 (4), p. 1441.46. S.E. Bobbin, J.W. Wagner, and R. Cammarata: Appl. Phys. Lett., 1991, man & Hall, London, 1995.

74. D.T. Queheillalt, H.N.G. Wadley, and D.S. Schwartz: Porous andvol. 59 (13), p. 1544.47. S.J. Norton, L.R. Testardi, and H.N.G. Wadley: J. Res. Nat. Bur. Cellular Materials for Structural Applications, Materials Research

Society Symposia Proceedings, D.S. Schwartz, D.S. Shih, H.N.G.Standards, 1984, vol. 89 (1), p. 65.48. H.N.G. Wadley, S.J. Norton, F.A. Maur, and B. Droney: Phil. Trans. Wadley, and A.G. Evans, eds., Materials Research Society, Warrendale,

PA, 1998, vol. 521, p. 237.R. Soc. London A, 1986, vol. 320, p. 341.49. S.J. Norton, A.H. Kahn, F.A. Maur, and H.N.G. Wadley: Intelligent 75. B. Budiansky: J. Mech. Phys. Solids, 1965, vol. 13, p. 223.

76. R.W. Rice: in Treastie on Materials Science and Technology, R.K.Processing of Materials, H.N.G. Wadley and W.E. Eckart, eds., TMS,Warrendale, 1990, p. 293. MacCrone, ed., Academic Press, New York, NY, 1977, vol. 11.

77. Metals Handbook, [vol. 2], Properties and Selection: Nonferrous50. F.A. Mauer, S.J. Norton, Y. Grinberg, D. Pitchure, and H.N.G. Wadley:Metall. Trans. B, 1991, vol. 22B, p. 467. Alloys and Special-Purpose Materials, ASM INTERNATIONAL,

Metals Park, OH, 1990.51. M. Rosen, H.N.G. Wadley, and R. Mehrabian: Scripta Metall., 1981,vol. 15, p. 1231. 78. A. Arieli and A. Rosen: Metall. Trans. A, 1977, vol. 8A, pp. 1591-56.

79. J.A. Wert and N.E. Paton: Metall. Trans. A, 1983, vol. 14A, pp.52. R.D. Greenhough, R.J. Dewhurst, and C. Edwards: J. Appl. Phys.,1987, vol. 62, p. 4728. 2535-44.

80. M.L. Meier, D.R. Lesuer, and A.K. Mukherjee: Mater. Sci. Eng., 1991,53. M. Dubois, A. Moreau, M. Militzer, and J.F. Busiere: Scripta Metall.,1998, vol. 39 (6), p. 735. vol. 136, p. 71.

81. M.L. Meier, D.R. Lesuer, and A.K. Mukherjee: Mater. Sci. Eng., 1992,54. J.P. Monchalin: IEEE Trans. Ultrasonic Ferroelectric Frequency Con-trol, 1986, vol. UFFC-33 (5), p. 485. vol. 154, p. 165.

82. O.D. Sherby and J. Wadsworth: Progr. Mater. Sci., 1989, vol. 33, p. 169.55. R.L. Blisplinghoff, J.W. Mar, and T.H.H. Pian: Statics of DeformableSolids, Dover, New York, NY, 1965. 83. R. Vancheeswaren: University of Virginia, VA, private communica-

tion, 1998.56. B.A. Auld: Acoustic Fields and Waves in Solids, Kreiger, Malabar,FL, 1990. 84. H.N.G. Wadley and R. Vancheeswaren: JOM, 1998, vol. 50 (1), p. 19.

85. D.M. Elzey and H.N.G. Wadley: Acta Metall. Mater., 1993, vol. 4157. D.N. Williams: Trans. TMS-AIME, 1961, vol. 221, p. 411.58. W.M. McGee and B.R. Matthews: U.S. Air Force Report ASD-TDR- (8), p. 2297.

86. R. Vancheeswaren, D.M. Elzey, D.T. Queheillalt, and H.N.G. Wadley:62-335, 1962, vol. 2A, p. 413.59. W.M. McGee and B.R. Matthews: U.S. Air Force Report ASD-TDR- University of Virginia, Charlottesville, VA, unpublished research,

1999.62-335, 1962, vol. 3, p. 326.

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