Bradley A. Lerch and Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

Viscoelastic Response of the Titanium Alloy

Ti-6-4: Experimental Identifi cation of Timeand Rate-Dependent Reversible and Irreversible

Deformation Regions

NASA/TM—2014-216584

April 2014
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Bradley A. Lerch and Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

Viscoelastic Response of the Titanium Alloy

Ti-6-4: Experimental Identifi cation of Timeand Rate-Dependent Reversible and Irreversible

Deformation Regions

NASA/TM—2014-216584

April 2014

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

This work was supported by NASA programs IVHM (Integrated Vehicle and Health Management) and SSAT (Systemwide Safety 

and Assurance Technologies). The authors appreciate the efforts of William Brown of Sierra Lobo, Inc., in preparing the test 

rig and samples. The texture measurements were performed by Dr. Richard Rogers of NASA. We also acknowledge the review 

comments of Dr. Tim Gabb, NASA Glenn.

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Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2014-216584 1 

Viscoelastic Response of the Titanium Alloy Ti-6-4: Experimental 

Identification of Time- and Rate-Dependent Reversible and 

Irreversible Deformation Regions 

Bradley A. Lerch and Steven M. Arnold 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Summary 

In support of an effort on damage prognosis, the viscoelastic behavior of Ti-6Al-4V (Ti-6-4) was 

investigated. This report documents the experimental characterization of this titanium alloy. Various 

uniaxial tests were conducted to low load levels over the temperature range of 20 to 538 °C to define 

tensile, creep, and relaxation behavior. A range of strain rates (610–07 to 0.001 s–1) were used to 

document rate effects. All tests were designed to include an unloading portion, followed by a hold time at 

temperature to allow recovery to occur either at zero stress or strain. The titanium alloy was found to 

exhibit viscoelastic behavior below the “yield” point and over the entire range of temperatures (although 

at lower temperatures the magnitude is extremely small). These experimental data will be used for future 

characterization of a viscoelastic model. 

Introduction 

Under NASA’s Systemwide Safety and Assurance Technologies (SSAT) project within the Aviation 

Safety Program the development, implementation, and experimental verification of a lifing (prognosis) 

methodology for structural components (e.g., jet engines) operating at high temperatures is underway. An 

objective of this program is to develop techniques to monitor vehicle condition and adjust performance 

profiles and maintenance schedules to ensure safe flight and long vehicle life. A key aspect of such a 

methodology is to correctly determine the extent of component degradation, thus requiring the 

deformation and damage behavior of the involved material to be well understood. In general, components 

are designed to operate in a material’s elastic regime. However, there can be stressed locations in the 

component, loading and temperature anomalies, and other unusual events that cause permanent 

deformation to occur. With continued events the amount of deformation that the component experiences 

increases until local ductility is exhausted and a crack forms. Through a stress amplification process at the 

crack tip, the crack can then be easily driven through the component, causing a noticeable change in 

physical behavior, such as vibration, excess flexure, and acoustic events, to name a few, that should 

enable detection using a suitable diagnostic process. Given an accurate material model to assess the life of 

the component, the remaining structural capability of the component can be predicted and either repaired, 

removed from service, or have its service conditions changed. 

A prerequisite for meaningful assessment of component durability and life, and consequently design 

of structural components, is the ability to accurately predict stresses, strains, failure modes, and their 

subsequent interaction and evolution occurring within a loaded structure. Furthermore, since constitutive 

material models provide the required link between stress and strain, this by necessity demands an 

appropriate constitutive behavior model for any material (be it monolithic or composite) before that 

material can be certified for use by a designer. Historically, metallic constitutive models have assumed 

the reversible or “elastic” regime to be time independent and the irreversible or “inelastic” strains to be 

either time independent (“plastic”), or at elevated temperatures, time dependent (“viscoplastic”). Prior 

research efforts on the titanium alloy TIMETAL 21S (Titanium Metals Corporation) determined that 

strains in the reversible regime can be both time independent and time dependent (Refs. 1 and 2) 
depending on the temperature. Therefore concepts from viscoelasticity, which previously had not been 

typically applied to metals, actually need to be applied to the constitutive equations employed to analyze 

metals. Furthermore, in the regime of time-dependent strains, because of the wide spectrum of rate 

dependence of the material in both the reversible and irreversible domains, multiple mechanisms or 

relaxation spectra need to be included. The more mechanisms that are used, the more likely the 

characterized model is to appropriately predict the behavior across the range of service conditions. 

The GVIPS (generalized viscoplasticity with potential structure) model is a comprehensive 

viscoelastoplastic constitutive model that aims to describe a material’s behavior whether in the 

viscoelastic or viscoplastic region (Refs. 3 and 4). 

The GVIPS model was successfully characterized for the titanium alloy TIMETAL 21S in the above 

references. A logical extension to this would be to determine if this response is present in another, more 

widely used titanium alloy, Ti-6Al-4V (Ti-6-4). Ti-6-4 is an attractive candidate because it was estimated 

that 50 percent of all titanium used in aeroengine applications is Ti-6-4 (Ref. 5). For example, the alloy is 

commonly used for compressor blades and cases, fan blades and cases, fixtures, bolts, and tubing. The 

maximum use temperature of the material is generally assumed to be 427 °C (800 °F). However, to 

account for over-temperature excursions and local hot spots within components, testing is frequently 

performed at temperatures as high as 538 °C (1000 °F). Since this alloy was developed in the 1960s, one 

would assume there should be an extensive amount of data in the open literature available for model 

characterization. Unfortunately, although there have indeed been extensive studies performed on Ti-6-4, 

most (1) involve special cases (i.e., unusual heat treatments, varying microstructures, extreme load or 

temperature conditions, etc.), (2) provide limited data, or (3) provide inappropriate data for use in 

characterizing the model. Therefore, existing literature data did not help in restricting or reducing the 

amount of testing required for characterizing the GVIPS model and at best could only be used to provide 

limited guidance. 

Consequently, the motivation behind the current study was to provide a sufficiently complete 

experimental database of response curves to enable the determination and full characterization of the 

viscoelastic response of the material (and the GVIPS constitutive model) over the service temperature 

range of interest to aeronautical engines. This report provides the experimental data itself, which will be 

used to characterize the viscoelastic model, the subject of a future report, and details on how these 

experiments were conducted. Future reports will address the viscoplastic and damage portions of the 

model. Data included in the current report are monotonic test results from low-level tensile, creep, and 

stress relaxation experiments. At the end of each test, the samples were unloaded and held for extended 

time at temperature and either at a fixed strain or stress level to permit full recovery of the deformation. 

Various loading rates were also employed. Finally, some of the basic viscoelastic parameters used in the 

model are calculated and discussed. 

A list of symbols used in this report is given in Appendix A. 

Experimental 

For the given test program the Ti-6Al-4V material came from one lot of 16-mm-thick rolled plate, 

ordered to AMS specification 4911 (Ref. 6). The plate was received and used in the mill-annealed 

condition. The chemical composition was analyzed in house (Table I) and met the specification. 

The supplier provided tensile properties at room temperature. These are listed in Table II and also 

satisfy the AMS 4911 specification. 

Sections from the three primary directions of the plate were mounted, polished, and subsequently 

etched using Kroll’s reagent to document the microstructure. Texture measurements were performed 

using an x-ray diffractometer equipped with a sealed Cu tube and a graphite monochrometer set to K

radiation. Pole figures were established for the plate. 

NASA/TM—2014-216584 2 
TABLE I.—CHEMICAL COMPOSITION 

OF Ti-6Al-4V PLATE 

Element Weight percenta 

Al 6.28 

V 3.85

Fe 0.15 

O 0.14 

C 0.035 

N 0.01 

Ti Remainder 

H 0.0070 

Y 0.005 a

H and Y measurements were provided by 

 material supplier.

TABLE II.—Ti-6Al-4V ROOM-TEMPERATURE TENSILE PROPERTIESa 

Orientation Yield strength, 

MPa (ksi) 

Tensile strength, 

MPa (ksi) 

Elongation, 

percent 

Reduction in area, 

percent 

Longitudinal 926 (134.3) 1003 (145.5) 15.0 42.6 

Transverse 940 (136.3) 998 (144.7) 16.0 44.9 

a

Data were provided by material supplier. 

Dynamic modulus tests were conducted on samples sectioned from the plate in the longitudinal 

(rolling) and short and long transverse (perpendicular to the rolling) directions. The samples were 

rectangular parallelepipeds, 40 by 12 by 3 mm in size. The tests were conducted using the impact 

vibration method from ASTM E1876 (Ref. 7). Tests were conducted from 28 to 700 °C in argon with data 

collected every 5 °C. The heating rate was 200 °C/h. The resulting resonance frequencies were on the 

order of 10 kHz. Both modulus of elasticity and shear modulus were measured; however, only modulus of 

elasticity is presented in this paper. 

Static modulus was measured on every loaded sample. The modulus was measured at 20 °C using a 

manual cycle to stress levels of approximately 14 MPa. Modulus measurements were occasionally taken 

at elevated temperature using a smaller load level. High-resolution stress and strain measurements were 

used for these tests. The modulus was also calculated from each viscoelastic test (be it tensile, creep, or 

relaxation) by fitting a line to the initial portion of the stress-strain curve. 

Test samples were taken from the plate and machined to a 150-mm-long, cylindrical, dogbone sample 

having a 6.4-mm-diameter gauge section. The specimen design is given in Figure 1. The longitudinal axis 

of each sample was aligned parallel to the rolling direction of the plate. The samples had the final stock 

removed from the gauge using light passes to minimize residual stresses. The samples were not heat 

treated after machining. The gauge section had a final hand polish parallel to the longitudinal axis of the 

sample to ensure that the final machining marks were also parallel to the specimen’s axis. 

NASA/TM—2014-216584 3 
All specimens were tested on the same 50-kip (222-kN) hydraulically actuated load frame containing 

a 20-kip (89-kN) load cell. The 50-kip frame was used to provide good lateral stiffness. The specimens 

were held by water-cooled, hydraulically-actuated, collet-type grips. Axial alignment was checked using a 

strain-gaged sample and following the procedure used in ASTM E1012 (Ref. 8). The maximum bending 

strain at zero load was ke	

. 

Heating was accomplished using direct induction by means of a pair of three-wind copper induction 

coils. These coils provided a uniform temperature gradient along the sample gauge and maintained it to 

1 percent of the desired test temperature over the entire range of test temperatures. Temperature was 

measured using type-K thermocouples welded onto the sample. The temperature was measured and 

controlled at the bottom intersection between the specimen radius and the 12.7-mm grip end. This 

temperature was calibrated to a sample containing a total of five thermocouples welded both at this 

location and at four other points within the gauge length. This eliminated the welding of thermocouples 

onto actual samples (within the gauge length) and influencing the test results. It was discovered that the 

employed induction coil configuration not only provided an excellent thermal gradient, but it was very 

robust in that even larger-than-expected variations in thermocouple placement did not affect either the 

gradient or the absolute temperature. The test rig was enclosed by plastic shielding to minimize thermal 

drifts during the tests. 

Axial strain was measured using a commercially available extensometer with Al2O3 probes for use at 

elevated temperatures. The gage length of the extensometer was 12.7 mm. The extensometer was water 

cooled for stability at high temperatures. The resolution of the extensometer at 538 °C was 2 
 


10–6 
). Diametral strain was also measured on every sample using an optical micrometer. This allowed a 

diameter measurement to be taken at one point around the circumference of the sample and at a location 

just below the top radius of the sample. This device had a resolution at 538 °C of 0.002 mm, which 

equates to a strain of 300 
 The resolution of both extensometers improves as the temperature decreases. 

Depending on the test type, tests were conducted in air either in load or strain control. The test 

procedure was 

(1) Conduct a room-temperature modulus test in load control. 

(2) Record the axial and diametral strain at zero load. 

(3) Heat up the sample and hold for 15 to 30 min to achieve thermal equilibrium. 

(4) Record the axial and diametral strain at zero load at the test temperature. 

(5) Conduct a modulus test at the test temperature. 

(6) Zero the axial extensometer, switch to strain control if warranted, and start the test. 

Creep tests were conducted by loading in load control to the desired load value and holding for a 

period of 24 h. All tests were constant load (not stress) tests. Upon attaining the 24-h hold period, the load 

was removed and the specimen held at zero load for another 24 h and allowed to recover. The rate of 

loading was 0.001 s–1 (or its load equivalent Eewhere E is the static modulus at the test temperature and 

strain rate e) unless otherwise indicated. The loading rate and unloading rate were always identical. 

Electronic noise increased with increasing temperature with the worst case being at 538 °C. The stress 

variation at 538 °C was approximately 1 MPa during creep and 0.35 MPa during recovery, indicating 

excellent control of the stress. 

Stress relaxation tests were conducted in strain control to a predetermined strain level and held at that 

strain for 24 h to relax out the stress. After the 24-h period the specimen was unloaded to zero load, and 

the control mode was switched to load control. The sample was then held at zero load to recover (creep) 

for 24 h. Alternatively, a few tests were unloaded to zero strain and held for 24 h to recover through stress 

relaxation. The loading and unloading rates were again identical. The variation in strain due to noise was 

greatest at 538 °C and was approximately 10 
 

NASA/TM—2014-216584 4 
Tensile tests were not conducted to failure for this report. Instead the loading portion of the creep and 

stress relaxation tests were used for tensile data. This also provided a larger number of replicates for 

assessing the specimen-to-specimen scatter. Alternate loading rates were occasionally employed, which 

ranged from 610–07 to 0.001 s–1 (or their equivalent elastic load rates). These rates represent the slowest 

rate that the test system could provide, and the fastest rate that the machine could accommodate without 

greatly overshooting the desired stress or strain limits. As will be shown later in this report, the fastest 

rate of 0.001 s–1 was sufficiently fast to lock in all time dependency of the material. 

During the tests, computer acquisition of time, load, axial strain, and diametral strain was conducted. 

Data acquisition times varied from a point every 0.01 s to capture transient events, to every 120 s for 

capturing data over times of minimal changes (e.g., toward the end of a given creep or relaxation test). 

Additionally, signals were fed to both an x-y recorder and a strip chart recorder to verify data quality. The 

strip chart also had the advantage of capturing longer times (e.g., over the weekend) when the test had 

already terminated, but the load conditions were still being maintained, thus increasing the information 

over longer-than-planned test times. The x-y recorder provided an extra check on the modulus. At low 

loads, the analog recorder was able to filter out noise (which was contained in the digital signal) and often 

provided a better representation of the modulus. 

Test Results 

Microstructure 

A three-	-	Figure 2. ! grains (light phase) 

show elongation in all three directions, with the largest elongation noted along the rolling direction. The 

grain structure is inhomogeneous, making measurements of the average grain size unpractical. A rough 

-grain diameter is 6 μm, and the length in the rolling direction can be as much as 100 μm. 

Hardness was measured on the samples and was found to be HRC 30 and is consistent with the hardness 

of annealed Ti-6-4 in the literature (Refs. 9 and 10). 

NASA/TM—2014-216584 5 
Texture 

Figure 3(a) shows the calculated full pole figures for the (0002) and (10 1 0) poles of the  Ti phase 

for the plate. The strong overlap of the  Ti (110) pole with both the  Ti (0002) and (10 1 1) poles 

prevented a successful extraction of -phase pole figures. Values for selected maxima in the pole figures 

are indicated in units of multiples of a random distribution (m.r.d.). For a perfectly isotropic sample, 

ll points on a pole figure. The sample was found to have a transverse (T) texture, where the 

basal planes align perpendicular to the rolling plane with the c-axis parallel to the transverse direction 

(TD) as illustrated in Figure 3(b). Note from the (10 1 0) pole figure that the (10 1 0) pole (prismatic 

planes) tends to be parallel to the rolling direction (RD). To a lesser extent, this pole is also aligned with 

the thickness direction. A quantitative measure of texture sharpness is the texture index, which is 

computed from the orientation distribution function. Its value lies between 1 (for a texture-free sample) 

and infinity (for a single crystal). The texture index for this material was 1.55 indicating a moderate 

rolling texture. 

NASA/TM—2014-216584 6 
Modulus 

Dynamic modulus ED measured in both transverse and longitudinal directions is shown in Figure 4. 

The solid lines represent the dynamic modulus for samples taken in all three planes both parallel 

(longitudinal) and perpendicular (transverse) to the rolling direction. Repeats were conducted for each 

orientation. Since the reproducibility is excellent, although it should be noted that the repeated samples 

were both taken out of the same area of the plate, only selected tests were plotted. The data show that the 

modulus in the transverse direction is noticeably higher than that in the longitudinal direction. At room 

temperature the modulus is 112 GPa for the longitudinal direction and 126 GPa for the transverse, 

representing a 12 percent difference in modulus. The dynamic modulus decreases linearly with increasing 

temperature until it is 77 and 91 GPa, for the longitudinal and transverse, respectively, at 700 °C, an 

approximate 30 percent drop over room temperature. 

Viscoelastic Testing 

Given that previous work on TIMETAL 21S showed the material to be viscoelastic and rate 

dependent (Ref. 2), it is assumed that Ti-6-4 will exhibit similar viscoelastic behavior at elevated 

temperatures. Consequently, it is important to establish the key viscoelastic parameters that define the 

viscoelastic model put forth in Saleeb and Arnold (Ref. 1) and Arnold, Saleeb, and Castelli (Ref. 2). The 

"#"	$	ble 

and irreversible domains. This can be determined exactly using the viscoelastic subtraction method put 

forth in Arnold, Saleeb, and Castelli (Ref. 2), 

IR  
 	  ES  (1) 

where 	 is the applied stress level (just below the value of the proportional limit, PL, but above the 

threshold stress) where the creep test was held; ES is the infinitely slow modulus, the stiffness that would 

be present if the material were loaded at an infinitely slow speed; 
IR is the irreversible strain 

determined after unloading and allowing sufficient time for all recovery to take place. Note that  is the 

threshold stress in shear. For this work and since all of the tests conducted were uniaxial, future reference 

to  will refer to the uniaxial stress value, Y. The two are related with the following relationship: 

3

Y

 
 (2) 

NASA/TM—2014-216584 7 
Note the time-independent elastic modulus ES needs to be found as does the full relaxation spectrum of 

the material; that is, for each of the viscoelastic mechanisms (i.e., Maxwell elements) introduced, the 

spring stiffness Em and dashpot viscosities % (or relaxation time 
 
 Em ) need to be computed. 

Furthermore, since the dynamic modulus of the material (ED) is equal to the sum of the other stiffnesses, 

 
 
 

N

i

i ED ES Em 1

( ) (3) 

it is apparent that given ED and ES one can easily solve for the sum of the nonequilibrium stiffnesses. 

Figure 5 graphically depicts ES and Y and indicates that they can be obtained at a given temperature by 

theoretically conducting as few as three tests: two creep or relaxation tests below the threshold stress Y

(reversible) and a creep test above the threshold stress (irreversible). Consequently, brief descriptions of 

the tensile (T), creep (C), and relaxation (R) tests performed to obtain the necessary data follow. A list of 

all tests conducted for determining the viscoelasticity response of Ti-6-4 is given in Table III. 

Tensile 

Specimens were loaded in tension at a strain rate of 0.001 s–1 to a specified strain limit corresponding 

to a stress generally above the proportional limit (or apparent yield point) at a given temperature. For the 

purposes of this report, only the modulus and the yield points will be reported. A modulus at 20 °C was 

recorded before each test, and the average value was 116 GPa with a standard deviation of 2.8 GPa. The 

modulus and yield points are given in Table IV as a function of temperature, both decreasing as 

temperature increases. The modulus decreased 34 percent from 20 to 538 °C. The PL associated with a 

total strain rate of 0.001 s–1 decreased from 737 MPa at 20 °C to 114 MPa at 538 °C, a decrease of

85 percent. These values agree with those given in the Aerospace Structural Metals Handbook (Ref. 11) 

for annealed Ti-6-4. 

NASA/TM—2014-216584 8 
TABLE III.—TEST MATRIX FOR VISCOELASTIC CHARACTERIZATION 

Specimena Test type Strain rate, 

s

–1

Loadup 

mode 

Strain levelb Stress level,b

MPa 

Comments 

(a) 20 C 

17 Creep 0.001 Stress 0.0025 279 Below threshold stress Y

17C2 Creep 0.001 Stress 0.0050 558 

17C3 Creep 0.001 Stress 0.0056 629 

17C4 Creep 0.001 Stress 0.0063 698 

17R Relaxation 0.001 Strain 0.0064 718 

17R2 Relaxation 0.001 Strain 0.0067 754 Above Y

17R3 Relaxation 0.001 Strain 0.0070 785 Above proportional limit PL

17R4 Relaxation 0.001 Strain 0.0073 814 

17R5 Relaxation 0.001 Strain 0.0076 844 

17C5 Creep 0.001 Stress 0.0083 891 

(b) 316 C 

18 Creep 0.001 Stress 0.0028 275 Below Y

18R Relaxation 0.001 Strain 0.0029 276 

18C Creep 0.001 Stress 0.0045 441 

18C2 Creep 0.001 Stress 0.0045 441 

18C3 Creep 0.001 Stress 0.0049 479 Above Y

18C4 Creep 0.001 Stress 0.0059 547 

18T Tensile 0.001 Strain 0.0180 650 

18TC Creep 0.001 Stress 0.0087 274 

16R Relaxation 0.001 Strain 0.0002 14 Below Y

16R2 Relaxation 0.001 Strain 0.0004 38 

16R3 Relaxation 0.001 Strain 0.0007 72 

16C Creep 0.001 Stress 0.0010 103 

16R4 Relaxation 0.001 Strain 0.0014 141 

16R5 Relaxation 0.001 Strain 0.0021 197 

16C2 Creep 0.001 Stress 0.0024 236 

16C3 Creep 0.001 Stress 0.0027 270 

16C4 Creep 0.001 Stress 0.0031 304 

16R6 Relaxation 0.001 Strain 0.0034 344 

16R7 Relaxation 0.001 Strain 0.0167 636 Above Y

54C Creep 0.001 Stress 0.0025 237 Below Y

54C2 Creep 0.001 Stress 0.0032 309 

54C3 Creep 0.001 Stress 0.0039 377 

54C4 Creep 0.001 Stress 0.0045 447 

54C5 Creep 0.001 Stress 0.0049 482 

54C6 Creep 0.001 Stress 0.0053 515 

54C7 Creep 0.001 Stress 0.0057 549 Above Y

54C8 Creep 0.001 Stress 0.0063 596 

54R Relaxation 0.001 Strain 0.0180 650 

(c) 371 C 

59R Relaxation 0.001 Strain 0.0014 132 Below Y

59C Creep 0.001 Stress 0.0015 138 Below Y

59C2 Creep 0.001 Stress 0.0037 344 Above Y

59R2 Relaxation 0.001 Strain 0.0180 599 Unload to zero strain (2 cycles) 

 

NASA/TM—2014-216584 9 
TABLE III.—Concluded. 

(d) 427 C 

25C Creep 0.001 Stress 0.0004 34 Below Y

25R Relaxation 0.001 Strain 0.0004 39 

25R2 Relaxation 0.001 Strain 0.0008 74 

25C2 Creep 0.001 Stress 0.0023 206 Above Y

25R3 Relaxation 0.001 Strain 0.0180 585 Above Y

26R1 Relaxation 0.001 Strain 0.0004 34 

26R2 Relaxation 0.0000008 Strain 0.0004 34 Below Y, 

below slow-rate PL

26R3 Relaxation 0.001 Strain 0.0004 35 Repeat of R1 

26C1 Creep 0.0000008 Stress 0.0008 68 Below Y, 

above slow-rate PL

26C2 Creep 0.001 Stress 0.0026 239 Above Y, 

below fast-rate PL

26C3 Creep 0.0024 Stress 0.0048 445 Above Y, 

below fast-rate PL

8R Relaxation 0.001 Strain 0.0011 103 Below Y

7C1 Creep 0.001 Stress 0.0040 36 Had prior test at 552 MPa 

7C2 Creep 0.001 Stress 0.0006 55 Below Y

7C3 Creep 0.001 Stress 0.0009 82 

7C4 Creep 0.001 Stress 0.0011 96 

7C5 Creep 0.001 Stress 0.0014 123 

7C6 Creep 0.001 Stress 0.0017 151 

7C7 Creep 0.001 Stress 0.0020 178 

7C8 Creep 0.001 Stress 0.0028 247 Above Y

7C9 Creep 0.001 Stress 0.0035 316 Above Y

83C Creep 0.001 Stress 0.0008 70 Below Y

83C1 Creep 0.001 Stress 0.0010 97 Below Y

83C2 Creep 0.001 Stress 0.0026 243 Above Y

83C4 Creep 0.001 Stress 0.0122 577 Above Y

83C5 Creep 0.001 Stress 0.0008 70 Below Y

83C6 Creep 0.001 Stress 0.0011 97 Below Y

83C7 Creep 0.001 Stress 0.0027 243 Above Y

(e) 482 C 

47R Relaxation 0.001 Strain 0.0002 15 Below Y

47C Creep 0.001 Stress 0.0001 14 Below Y

47C2 Creep 0.001 Stress 0.0008 69 Above Y

47R2 Relaxation 0.001 Strain 0.0179 553 Unload to zero strain (2 cycles) 

(f) 538 C 

28 Creep 0.001 Stress 0.0001 7 Below Y

28R Relaxation 0.001 Strain 0.0002 16 Below Y

28C Creep 0.001 Stress 0.0004 33 Above Y

28C2 Creep 0.001 Stress 0.0004 33 

28T Tensile 0.001 Strain 0.0038 294 

28T2 Tensile 0.001 Strain 0.0178 502 

28TR Relaxation 0.001 Strain 0.0002 14 Below Y

28TC Creep 0.001 Stress 0.0002 14 Below Y

28TC5 Creep 0.001 Stress 0.0004 33 Above Y

92R Relaxation 0.001 Strain 0.0001 8 Below Y

92C Creep 0.001 Stress 0.0001 7 Below Y

92C2 Creep 0.001 Stress 0.0004 34 Above Y

92R2 Relaxation 0.001 Strain 0.0179 508 Unload to zero strain 

60R Relaxation 0.001 Strain 0.0001 7 Below Y

60C1 Creep 0.00000063 Stress 0.0020 104 Above Y a

Test specimens are designated by sample number, test type (C is creep; R, relaxation; and T, tensile), and test number. b

Values for response signal taken at the end of the loadup. 

NASA/TM—2014-216584 10 
TABLE IV.—TENSILE PROPERTIES AND KEY VISCOELASTIC PARAMETERS FOR VISCOELASTIC TESTS 

Test 

temperature, 

C 

Sample 

number 

 

Tensile propertiesa Viscoelastic parameters 

Elastic

modulus, 

E, 

GPa 

Proportional 

limit, 

PL, 

MPa 

0.02% yield 

point, 

MPa 

0.2% yield

point, 

MPa 

Infinitely 

slow 

modulus, 

ES, 

GPa 

Threshold 

stress, 

Y, 

MPa 

Y/PL

538 28 84.19 105 261 397 21.3 16.5 0.16 

538 92 86.39 122 215 471 30.5 15.6 0.13 

538 60 82.46 20 59 --- 6.6 --- --- 

482 47 84.81 303 431 513 52.6 37.8 0.12

482 14 87.29 178 354 484 --- --- 0.22 

427 25 89.98 476 506 533 68.8 170.9 0.36 

427 83 93.15 473 518 552 73.9 217.1 0.46 

427 63 90.53 --- --- --- 65.5 90.5 --- 

427 26 91.70 --- --- --- 75.4 199.7 --- 

427 8 91.63 --- --- --- 55.6 --- --- 

371 59 94.19 502 529 553 80.6 316.3 0.63 

316 18 97.50 593 616 600 95.7 461.9 0.78 

316 16 98.46 525 547 575 97.0 >344.8c 0.66 

316 54 96.19 603 651 632 95.5 517.8 0.86 

200 50 103.36 587 640 677 102.3 516.6 0.88 

100b --- --- --- --- --- 107.3 634.3 0.91 

20 10 112.60 846 883 926 112.6 795.4 0.94 

20 17 111.70 737 880 --- 110.8 732.8 0.99 a

At strain rate of 0.001 s–1. b

Values at this temperature were interpolated.

c

This was the highest value measured that was still reversible. The next test in this series was taken at the 0.2% yield point. 

Modulus is plotted in Figure 6 as a function of temperature and strain rate. The solid line is the 

dynamic modulus data taken from Figure 4 for the longitudinal direction. The solid points in Figure 6 

represent the static modulus calculated from individual test samples in the overall program and cover the 

entire range of strain rates. In general, the modulus is shown to decrease with increasing temperature, for 

both the dynamic and static tests. At low temperatures, the modulus for the cylindrical samples (static 

modulus) coincides with the data for the dynamic tests and this is to be expected since (1) the modulus 

NASA/TM—2014-216584 11 
was measured in the same direction (along the rolling direction) and (2) at low temperatures rate 

dependence is not anticipated. However, at approximately 316 °C, strain rate becomes important, and the 

static modulus deviates from those of the dynamic data. This drop becomes more drastic as the applied 

strain rate decreases. Dotted lines are drawn in to aid in the discrimination of the various strain rates. 

Error bars representing one standard deviation on the static modulus are shown where there is sufficient 

data. At 538 °C the modulus for the slowest strain rate (610–07 s–1) is only 63 GPa, which is 20 GPa 

lower than the dynamic modulus at the same temperature. Note, however, that for strain rates equal to or 

faster than 110–05 s–1, the modulus is still approximately the same as the dynamic tests, which has a 

strain rate on the order of 1.0 s–1. Hence there is a strain rate effect on modulus starting at approximately 

316 °C, with a slower strain rate leading to a lower modulus. The rate effect increases with increasing 

temperature as one would expect. The remaining curve (dashed line) is the calculated stiffness ES

representing the modulus at an infinitely slow strain rate (to be explained later). 

The PL values, threshold, and 0.02 and 0.2 percent yield points are shown in Figure 7 as a function of 

temperature. These values were taken from individual tests loaded at a strain rate of 0.001 s–1. Error bars 

for 1 standard deviation are again given where sufficient data were available. Smooth curves were drawn 

through all points associated with the same yield definition to facilitate the observation of a decreasing 

yield with increasing temperature. The final set of data represents the threshold values calculated from 

Equation (1). It is noteworthy that the threshold values fall significantly below those of the fast-rate 

proportional limit. 

Creep 

Several creep tests were performed at various temperatures below the corresponding apparent yield 

points. Specimens were loaded at a rapid load rate (the elastic equivalent of a strain rate of 0.001 s–1) in an 

attempt to lock in all time dependency. Initial levels for a given specimen were maintained in the 

reversible regime (below Y) whereas final stress levels for a given specimen were in the irreversible 

regime (above Y). Representative response curves from a given specimen are shown in Figure 8, where 

both creep and recovery (where the time at which the load was removed was reset to zero) portions of the 

curves are shown for loads below Y (14 MPa) and above Y (69 MPa). The salient features for samples 

loaded in the reversible regime are that at a constant load, an initial primary creep region is observed 

followed by a shutdown of creep, thereafter yielding a constant strain with respect to time. 

NASA/TM—2014-216584 12 
Consequently, no secondary creep regime is experienced, which is consistent with full strain 

reversibility.1

 The reversibility is confirmed by unloading to zero load, and holding this load at 

temperature for 24 h. This is in contrast to behavior in the irreversible regime, during which primary 

creep transitions to nonzero secondary creep (see the solid blue line). Upon unloading the specimen 

experiences some recovery but does not return to zero strain as it did in the reversible regime, and 

maintains a permanent strain offset (e.g., in Fig. 8 the 69 MPa load case recovers to approximately 600 ). 

Relaxation 

Relaxation tests were also performed in both the reversible and irreversible regimes. Tests were 

conducted with a loading-unloading total strain rate of 0.001 s–1, again to lock in all time dependency. 

A representative response curve for a sample tested in the reversible regime at 371 °C and held at a total 

strain of 0.0014 is shown in Figure 9. The stress in the sample relaxed 22 percent during the 24-h hold 

with saturation (relaxation shutdown) 	sat near the end of the hold period. Similar behavior is also 

observed at 427 °C for relaxation at 0.0011 total strain, wherein the stress in the sample relaxed 

38 percent. Upon unloading to zero stress and switching to stress control, the strain recovered to zero. A 

relaxation test at 427 °C that was unloaded to zero strain and then recovered is shown in Figure 9 (see the 

dashed line). Note that at zero strain (the time at which this occurred has been reset to zero in the figure) 

the specimen had a compressive stress of 40 MPa, which almost completely recovered to near zero after 

40 h. Relaxation in the irreversible regime neither shuts down, nor does the specimen recover to zero 

upon unloading. 

1

A secondary creep rate of zero is a definite indication of reversibility only at elevated temperatures where a 

thermal recovery exists. At low temperatures, thermal recovery is very small and a zero creep rate does not 

necessarily imply reversibility. To document reversibility in this case the sample has to be unloaded and fully 

recover. 

NASA/TM—2014-216584 13 

 
Both creep (C) and relaxation (R) data at each test temperature (often for multiple specimens) are 

given in Figures 10 to 21. For each temperature a plot of stress-strain and strain-time is shown for both 

loading and recovery portions of the test. In these plots, the response variables advance continuously, and 

time is not reset to zero upon unloading. Also shown in these plots is the delineation between reversible 

and irreversible deformation as indicated by Y. For stress versus time and/or strain plots, Y is a 

straightforward limit value and is shown by a box representing the range of values for the various 

samples. However, in strain-time space one needs to develop a corresponding range. Consequently, the 

upper reversible strain bound (i.e., the accumulated strain after “infinite” amount of time) is represented 

by Rmax Y/ES, where the minimum and maximum values for ES were used to calculate this upper strain

limit. The values of ES used either represent the within-sample scatter when there is only one sample per 

temperature or the within-temperature values for multiple specimens. Similarly, the maximum achievable, 

instantaneous reversible (i.e., elastic) loadup strain is determined using the expression Y/ED. Comparing 

this value with the strain value after loadup of a given test, one can immediately observe if the test is 

above or below the threshold. In Figure 11 both ED and ES were used to calculate these ranges. 

Consequently, the instantaneous strain equivalent for Y (using ED) is the best indicator of the boundary 

between reversible and irreversible deformation in the strain-time space. This maximum elastic strain 

(Y/ED ) is shown in Figure 11, whereas on almost all other strain-time plots only the upper reversible 

strain limit Y/ES is shown. For both these calculations a range is usually given rather than a single value. 

This is because the values of ED and ES are typically from multiple samples. If only one sample was tested 

per temperature, within-sample values of ES were used to calculate the range. It should also be noted that 

all of the stress-strain curves from Figures 10 to 21 show linear behavior during loading of the sample. 

NASA/TM—2014-216584 14 
NASA/TM—2014-216584 15 
 

NASA/TM—2014-216584 16 
 

NASA/TM—2014-216584 17 
NASA/TM—2014-216584 18 
NASA/TM—2014-216584 19 
NASA/TM—2014-216584 20 
Room-Temperature Testing 

A viscoelastic test series was conducted on sample 17 at 20 C to examine if viscoelasticity is active 

at such low temperatures. Time-dependent deformation both under creep and relaxation have been 

observed in Ti-6-4 at room temperature (Refs. 9 and 12 to 18) and other titanium alloys (Refs. 15 and 19 

to 21). Here the objective will be to determine if this behavior is limited to the irreversible regime alone. 

The test series is shown in Figures 20 and 21. Four creep tests were conducted at low loads, followed by 

five relaxation tests that are situated both below and above the proportional limit, and finally ending with 

a high-stress creep test. Figure 20 shows that PL is high at 20 C, and the corresponding amount of timedependent behavior is exceedingly small. This is also shown in Figures 21 and 22 where an identifiable 

amount (100 ) of creep can be observed starting at a stress of 690 MPa. A better view of the time 

dependence can be seen by removing the loading strains and plotting only creep strain (i.e., strain 

accumulation subsequent to loadup) as in Figure 22. Here, time dependency can be observed at stresses as 

small as 552 MPa. At a still higher stress of 690 MPa (test 17C4) almost 100  of creep (approximately 

1.6 percent of the total strain) can be seen with an apparent shutdown at about 60 000 s. Upon unloading, 

the tests recover to near zero strain (Fig. 23) up through 690 MPa. 

The next five tests were relaxation tests and their stress responses are shown in Figure 24. Only the 

test at the lowest strain (0.0064) lies below Y. Relaxation at stresses near Y shows a stress drop of 6 MPa 

(1 percent) for a strain level of 0.0064 (below Y) and a stress drop of 14 MPa (2 percent) for a strain level 

of 0.0067 (above Y), indicating measurable time dependency. These tests were subsequently unloaded to 

zero strain and held to recover through stress relaxation as shown in Figure 25 (with the time reset to 

zero). Although none of the samples relaxed back to zero stress, the two tests at the lowest strain levels 

could be considered to have fully reversed within experimental error. Calculation of two of the 

viscoelastic parameters for this sample yielded ES '*+Y  732.8 MPa. The fast-rate modulus 

for this sample was 111.7 GPa and indicates that ES  ED by 0.9 GPa. Likewise Y  PL by 4.2 MPa. 

Hence, viscoelasticity provides a very small part (2 percent) of the reversible deformation at room 

temperature, with 98 percent being time-independent elastic response. 

NASA/TM—2014-216584 21 
NASA/TM—2014-216584 22 
Determination of Key Viscoelastic Parameters at 427 C 

In this section, the procedure for determining the key viscoelastic properties of an alloy will be 

elucidated using actual results from sample 25, tested at 427 °C. As stated previously, the first step is to 

estimate the value of the threshold between reversible and irreversible behavior. Since Y is initially 

unknown, a good starting point is to keep the stress value of either creep or relaxation below 1/3 to 1/2 of 

the PL obtained from the slowest strain rate tensile test at the desired temperature. These low stress values 

should ensure that the test remains in the reversible regime. However, resolution is often poor in this 

range, and the linear region is small. If one remains within the reversible domain, multiple tests can be 

conducted without altering the material state, thus enabling efficient utilization of specimens in any 

viscoelastic test program. This of course is predicated on the assumption that all recovery times (in 

between tests) are sufficiently long to completely reverse any prior viscoelastic deformation. 

Consequently, we begin with the lowest stress level and progressively increase the stress level on 

subsequent tests to yield better data resolution. 

The first test conducted under load control on sample 25 was a creep test (loaded at an equivalent 

strain rate of 0.001 s–1) at 34 MPa, which should be clearly in the reversible regime since the PL for this 

sample at 427 C is 476 MPa. It is observed in Figure 26 that creep behavior was exhibited during the 

24-h hold at 34 MPa in the axial direction. Clearly, the specimen crept <$
=


shutting down at approximately 40 000 s. The creep rate at the end of the hold period was 3.910–10 s–1, 

while at the beginning it was 1.010–7, a decrease of nearly 3 orders of magnitude. After unloading, the 

24-h recovery period resulted in a return to near-zero strain. During the recovery phase, the strain 

recovers to a value of 50 =
? 000 s, approximately 26 percent of its maximum. Note that in this and 

other figures, the time at the initiation of recovery (beginning of the unload) is typically reset to zero to 

facilitate display of the data. The recovery rate at the end of this test was –3.010–9 s–1. 

The next test on this sample was a stress relaxation experiment (loaded at a strain rate of 0.001 s–1) 

@J=
"QQ	
Pa and PL  

476 MPa, or 36 percent of the proportional limit). The strain versus time response is plotted in Figure 27 

and clearly illustrates the excellent control of strain during the 24-h relaxation period. The recovery 

period (conducted under stress control subsequent to the stress being unloaded to zero) is also shown in 

this figure, indicating the strain recovers beyond zero and into the compression strain regime (i.e., 

NASA/TM—2014-216584 23 
–
[=
\!$	–4.810–9 s–1. The stress versus time response 

for this test is observed in Figure 28. The stress relaxes from a maximum value of 39 MPa and decreases 

until shutdown at 32.7 MPa, a 16 percent relaxation. The stress rate at the end of the relaxation period was 

–8.310–5 MPa/s. Note the curve denoted as recovery lies on the abscissae and illustrates the excellent 

ability to control stress at zero load during the 24-h recovery period. Clearly, Figures 27 and 28 are 

conjugates of one another (i.e., strain vs. time and stress vs. time, respectively). 

NASA/TM—2014-216584 24 
The third test of this series is a creep test at a stress of 206 MPa (equal to 43 percent of the PL), which 

will be shown subsequently to be above the threshold and in the irreversible regime. The test exhibits 

substantial creep (Fig. 29) during a 24-h hold (i.e., 1100 =
\. Both a primary and secondary creep regime 

are exhibited with no indication of shutdown over the 24-h creep period. The strain rate at the end of the 

creep period is 6.010–9 s–1. It is important to remember that the hold period for either creep or relaxation 

above Y needs to equal or exceed the hold period used in the recovery regime. This ensures that all 

reversible time dependence will be equally dissipated before the test is stopped. The unloaded sample 

recovers (at zero stress) to 500 =

@-h period, and reached a strain rate of –2.310–9 s–1 at the end 

of the recovery period. The strain does not appear to extrapolate to zero in any reasonable timeframe 

greater than 24 h. Note the amount of signal noise is significantly reduced at this load level in comparison 

NASA/TM—2014-216584 25 
to the previous load levels. The conjugate curves illustrating the transverse strain behavior are described 

in Appendix B. Transverse strain was measured during most tests, and Poisson’s ratio was usually 

calculated from it. These are both discriminating variables in comparing models because they are often 

more sensitive to minute changes in deformation. 

Viscoelastic Parameters 

The ES can be established from calculating the secant modulus at saturation (i.e., once the response 

variable—be it strain or stress—has shut down) from a single reversible creep or relaxation test. 

However, it is always better to have a series of creep and stress relaxation tests that were conducted at 

various levels in the reversible regime. Both types of tests should be held for at least 24 h (this maximum 

time may be material dependent) to allow the controlling viscous mechanism time to dissipate. Note as 

temperature increases or for different materials the hold time period may have to be increased to ensure 

that sufficient dissipation occurs. Multiple tests enable one to account for associated uncertainties. 

Figure 30 depicts two relaxation tests and one creep test (two discussed previously) at 427 °C. The 

loading rate in each case was either 0.001 s–1 or its load equivalent during the creep test. The points at 

which the viscous mechanisms dissipate are denoted on the figure. A best-fit line through those points 

yields a secant stiffness of 68.8 GPa. This represents the time-independent stiffness ES of this sample 

under the given test conditions. There is a small amount of scatter in the dissipation points, and this is 

typical when executing these types of tests. In fact, it was observed that whichever type of test was 

conducted first (for this sample it was creep) gave the most time-dependent deformation, resulting in the 

lowest single point value for ES. Six out of seven samples showed this behavior. For the purposes of this 

report, all dissipation points were used, and an average ES was calculated. Therefore, the use of more 

points (more tests) would improve the confidence of the estimation of ES, but this would be accomplished 

at the cost of additional days of testing time (typically 2 days per test including the recovery). The 

difference in the calculated ES using individual dissipation points rather than an average of all is what led 

to the range of upper strain values Rmax given in Figures 13 and 17. 

NASA/TM—2014-216584 26 
Utilizing the concept of viscoelastic subtraction (Ref. 2), the threshold value Y, delineating reversible 

and irreversible regimes, can be explicitly calculated using Equation (1). This is shown graphically in 

Figure 31, where the reversible and irreversible tests from sample 25 are now combined. The value of 	

is 206 MPa, and 

IR is 0.0005. Using ES  68.8 GPa, Equation (1) gives a value of 171 MPa for Y. Note 

the viscoelastic subtraction concept was derived using the idea of similar triangles; thus tests conducted in 

the irreversible region must be above Y and below PL (i.e., is applicable only in the linear region of stressstrain curves). The PL for this specimen (no. 25) is given in Table IV as 476 MPa. This indicates that 

there can be extensive irreversible deformation at stresses below the apparent “yield” point. 

This above procedure was repeated for each specimen over the full temperature range of interest to 

yield ES and Y as a function of temperature. These values are listed in Table IV. Both ES and Y are plotted 

as a function of temperature T in Figure 32; the homologous temperature T/Tm is also shown (where Tm is 

the melting point), taken to be 1704 C (1977 K). The longitudinal dynamic modulus ED defined in 

Equation (3) and plotted in Figures 4 and 6 is also plotted and represents the highest modulus achievable 

at a given temperature. Note that ES is equivalent to ED up to temperatures of 316 °C; consequently this 

temperature delineates the temperature below which rate-independent elasticity applies. Above this 

temperature rate dependency becomes active and the infinitely slow modulus ES decreases at a faster rate 

than ED. This can be more easily viewed in Figure 6, which also shows that ES is much lower at any given 

temperature than the slowest practical strain rate static modulus. For example at a temperature of 538 °C 

and a strain rate of 610–07 s–1, the static modulus is 63 GPa, whereas ES, is only 30 GPa. In Figure 32 

both the values for Y and PL for strain rates of 0.001 s–1 are shown to decrease with increasing 

temperature. At room temperature, Y and PL are almost identical (Y is approximately 95 percent of PL), 

whereas at 538 °C they are significantly different: Y is approximately 10 percent of PL (see last column in 

Table IV), and near zero stress. At temperatures above 538 °C, it is assumed that Y is either zero or so 

small as to be unmeasureable. The trend line for Y represents the delineation between the reversible and 

irreversible deformation regimes over the temperature range of interest. Figure 32 constitutes the 

deformation modeling map for the Ti-6-4 alloy and provides insight into the required modeling 

assumptions as a function of stress-temperature regimes. It is interesting to note that significant time 

dependence is observed at homologous temperatures (T/Tm) as low as 0.29 in this alloy, which is consistent 

with the range of values (0.25 to 0.30) normally cited for initial creep activity in most structural alloys. 

NASA/TM—2014-216584 27 
Rate Effects 

Ideally Y could be determined by obtaining the proportional limit of an infinitely slow strain rate 

tensile test. The question is what constitutes an infinitely slow tensile test? Based on the creep rate at 

complete dissipation for the test depicted in Figure 26, the imposed loading rate would need to be on the 

order of 3.910–10 s–1 at the temperature of 427 C. This is significantly slower than a typical 

servohydraulic control limit of 6.010–7 s–1 and could not be achieved here in a tensile test. Figure 33 

provides the proportional limits for a number of measurements made for very fast (0.001 s–1) and very 

slow (6.010–7 s–1) strain rates over the temperature range of interest. The trend line for Y determined 

using the viscoelastic subtraction method plotted in Figure 32 is included in this figure for comparison. 

Obviously, the fast-strain-rate proportional limits agree with the extracted values for Y up to 

approximately 316 °C, whereas they are significantly higher than the calculated Y values at higher 

temperatures. As noted before, as temperatures rise and stress levels fall, the noise level in the 

measurements become more problematic and make accurate determination of both PL and Y (since the 

determination of ES is problematic as well) more difficult. It is noted in Figure 33 that the PL values at 

very slow strain rates are an excellent estimate for Y. Furthermore, these results strongly suggest a PL

fraction rule that is 50 percent of the PL obtained from a very slow strain rate tensile test would be a good 

starting stress for the viscoelastic series. This estimate should ensure that a given test is within the 

reversible regime when conducting viscoelastic exploratory/characterization testing, regardless of the 

temperature. 

NASA/TM—2014-216584 28 
Since viscoelastic subtraction uses a test above Y, but below the PL, and since the PL is a function 

of strain rate, one could ask what strain rate needs to be employed to calculate Y. In an attempt to 

ascertain this, Y was calculated on a sample (i.e., sample 26) using both very fast and very slow strain 

rates. The test sequence is shown in Figure 34. Tests R2 and C1 were conducted at a very slow strain rate 

(810–7 s–1). All other tests were conducted at a very fast strain rate (0.001 s–1). Moreover, the first three 

tests (relaxation) were conducted below the slow-rate PL of 55 MPa. The fourth test (creep) was 

conducted above the slow-rate PL, but still under the suspected Y value. The remaining two tests were 

conducted above Y, but below the fast-rate PL and appear to be “fully elastic” during loadup. The final 

creep value of 445 MPa is within the region of fast-rate PL values, which ranges from 375 to 476 MPa. 

The strain versus time behavior for these tests is depicted in Figure 34 along with the corresponding 

R

and Rmax associated with the deduced Y value of 199 MPa. Figure 34(a) shows that only the two highest 

stress creep tests fall above Y since the initial loadup strains exceed Y/ED and the maximum accumulated 

strain after 24 h exceed Rmax . The slower rate creep test (68 MPa), while still under the threshold value, 

does not show evidence of shutting down during creep and appears to exhibit a constant rate, which is in 

contrast to previous experience. Although this test depicts a final recovery strain of 110  after the 

standard 24-h recovery period, continued testing showed that the recovery strain decreased further to 

58  after another 18 h and was still decreasing. Consequently, this suggests that for slow-strain-rate 

loading more than 24 h may be needed to reach saturation of the time-dependent processes, particularly at 

lower temperatures. Figure 35 documents the relaxation behavior of the first three tests, all taken to a 

strain limit of 0.0004. The relaxation behavior is essentially identical regardless of the loading rate. All 

three tests also saturate to approximately the same equilibrium stress level of 27.5 MPa, which provides a 

68.75-GPa estimate for the infinitely slow modulus (i.e., ES 	sat / e ). This indicates that deformation 

response under the threshold value is rate independent, as long as the test is maintained below the ratedependent proportional limit. 

NASA/TM—2014-216584 29 
NASA/TM—2014-216584 30 
History Dependence of the Threshold 

In order to investigate the effect of prior deformation on the threshold value, a test (sample 83) was 

conducted in which Y was first established, a large viscoplastic deformation was then applied, and the 

threshold was again determined. Figure 36 displays the stress-strain response at 427 C for this test series. 

The dotted lines represent the data from probes on samples with no permanent deformation, whereas the 

solid lines show data from the probes after large sample deformation. All of the tests conducted in this 

series were creep with two tests below the threshold and one slightly above to determine Y. The largescale deformation test was crept at 577 MPa, which is both above the threshold and also above the 

0.2 percent yield point (552 MPa) of this sample. This creep test achieved a total strain of 4.8 percent 

before it was unloaded and recovered at zero load. 

The strain versus time diagram is given in Figure 37 with dotted lines again depicting the initial 

probes. This chart shows that the before and after creep responses are similar, with the initial response 

being slightly higher than the final response at the lowest load, the same at the medium load, and slightly 

below at the highest load above Y. It should also be noted that the strain during recovery for both sets of 

below-Y tests ended near zero strain. In both load cases the post-yield tests recovered to slightly lower 

strains than the pre-yield tests, but this is assumed to be due to experimental scatter. However, the tests 

for the above-Y loads exhibited similar behavior in that the post-yield test also recovered to slightly lower 

values. The viscoelastic parameters were calculated from this test series and are shown in Table V. 

Although both the infinitely slow modulus ES and Y are similar for both pre- and post-yield tests, the 

values after yielding are slightly higher. However, both sets fall in the range of the viscoelastic parameters 

given in Table IV. The last row under the 427 C test in Table V lists a second Y value calculated by 

using the original value of ES, 74 GPa. Using this value for the stiffness makes little difference in the 

resulting threshold values. 

An additional test series (7C) was conducted at 427 C on a sample that had prior creep deformation 

at 552 MPa (past the 0.2 percent yield stress), followed by recovery. Several creep tests were 

subsequently performed to determine the post-yield viscoelastic parameters. The modulus ES was 

calculated to be 78 GPa and threshold Y, 235 MPa. These values are again just slightly outside 

(5 percent) the range of those given in Table IV. 

NASA/TM—2014-216584 31 
Finally, a test was conducted at 538 C with pre- and post-yield viscoelastic probes. A tensile test to 

502 MPa (1.8 percent strain) with subsequent unloading and recovery was performed between the probes 

to induce large-scale deformation. The viscoelastic parameters are given in Table V and show that the 

tensile deformation has no effect on the stiffness ES. There was a decrease in the value of the threshold. 

Again all values fall in the range of those listed in Table IV for tests at 538 C. Hence, it is reasonable to 

assume that the viscoelastic parameters are minimally influenced by inelastic flow. Further given that the 

threshold value, Y, is unaffected by inelastic flow (i.e., no isotropic hardening) this material can also be 

idealized by a kinematic hardening formulation consistent with Saleeb and Arnold (Ref. 1). 

NASA/TM—2014-216584 32 
TABLE V.—PRE- AND POST-YIELD EFFECT ON 

VISCOELASTIC PARAMETERS FOR Ti-6Al-4V 

 Infinitely slow 

modulus, 

ES, 

GPa 

Difference, 

percent 

Threshold stress, 

Y, 

MPa 

Difference, 

percent 

427 C 

Pre-yield 74 217 

Post-yield 81 9.7 226 3.9 

 Using ES ?@ 227 4.6 

538 C 

Pre-yield 21 17 

Post-yield 21 0.0 14 –12.4 

Comparison With TIMETAL 21S 

Since it has been shown that Ti-6-4 displays viscoelastic behavior, it will be compared to what was 

observed for TIMETAL 21S (Refs. 1 and 2). The data from these past programs were reevaluated and 

plotted along with Ti-6-4 on the deformation modeling map shown in Figure 38. Both alloys show similar 

behavior with the TIMETAL 21S showing better properties at higher temperatures. This is to be expected 

since TIMETAL 21S was developed to be used at higher temperatures (Ref. 22) and to have creep 

resistance equivalent to Ti-6-4. TIMETAL 21S exhibits a higher threshold stress than Ti-6-4. Although 

they appear in Figure 38 to have identical thresholds at 500 C and above, this is just a resolution issue. 

TIMETAL 21S actually has a threshold at 500 C, which is 5 to 6 times that of Ti-6-4. Both the dynamic 

modulus (ED) and the time-independent modulus (ES) for TIMETAL 21S are also larger than Ti-6-4. 

Furthermore, the delineation between rate dependency and rate independency is over 100 C greater for 

TIMETAL 21S than for Ti-6-4. 

NASA/TM—2014-216584 33 
Discussion 

Whereas the basic information required for characterizing the viscoelastic model put forth by Saleeb 

and Arnold (Ref. 1) employs testing of a basic nature (e.g., tension, creep, and stress relaxation), 

acquiring the required data necessitates a more difficult test protocol involving unloads with long hold 

times. The stress levels at which most of the tests are performed are low, with both the stress and strains 

being near the noise floor of the equipment. This becomes more evident at higher test temperatures, which 

unfortunately is the regime of interest where viscoelastic deformation dominates. For example, at 538 °C 

there is a 75-percent decrease in the stress because of viscoelastic relaxation, whereas at 316 °C there is 

only a 3-percent decrease. Additionally, the stress levels at 538 °C (16 MPa) are obviously much lower 

than those attained at 316 °C (280 MPa). Hence, electronic and thermal noise, as well as test rig stability, 

is critical to the success of these tests. Similarly, small load offsets typical of loading and gripping 

samples must be minimized, or they could result in stresses occurring above the threshold levels. Also, 

when control-mode switches are utilized, they must occur smoothly (no bumps in stress or strain) in order 

to prevent overshooting of the desired value and landing in the irreversible range. These factors make the 

conducting of seemingly simple tests more difficult. 

It should be reminded that each step of the viscoelastic tests requires a minimum of 2 days to 

complete: the first 24 h in the loading stage and the second 24 h in the recovery phase. This timeframe is 

convenient for testing because changes do not occur in the middle of the night. Also, 24 h is sufficiently 

long to cover any time-dependent behavior that may be experienced in aeronautical applications. For 

materials used in power generation, deep-space probes, or other long time-at-temperature applications, 

this dwell time may have to be revisited. Obviously, the hold times in these phases can be reduced, but are 

done so at the risk of errors in the conclusions. In both phases the deformation mechanisms are required 

to fully dissipate, and this requires a certain timeframe. This study has shown that more time is required 

for the viscoelastic mechanisms to dissipate or completely recover at higher temperatures than at lower 

temperatures. Results suggest that 24 h is not quite sufficient at 538 °C for the creep process to 

completely dissipate. However, at 316 °C, the process can dissipate in 6 h. Hence, variable hold times 

could be employed to minimize overall test times. Unfortunately, the required hold times are not only 

temperature dependent, but also material dependent. Therefore, advanced knowledge of the material 

behavior is required to effectively minimize test times, and this is most likely resolved through a series of 

screening tests, possibly negating any time-saving advantage associated with modifying the dwell times. 

To calculate the viscoelastic constants ES and Y, there are theoretically only three tests required at any 

given temperature. Two of these are conducted below Y in the reversible regime. The last test is 

conducted above Y in the irreversible regime, but below PL. These three tests can all be performed on the 

same sample, since the deformation resulting during the first two experiments—conducted within the 

reversible domain—is fully recovered. In fact, it is advised that they all be performed on one sample to 

eliminate specimen-to-specimen scatter in the results. The basic test procedure is shown in Figures 5, 30, 

and 31. The tests below Y consist of a combination of creep and relaxation tests. They can be both the 

same type of test, or can be different (one creep and one relaxation test). The preferred test above Y is a 

creep experiment, since viscoelastic subtraction requires explicit measurement of the irreversible strain 

accumulated after all recovery is achieved. This is most easily obtained from a creep test, although a 

stress relaxation experiment could also be used. 

Although only three tests (see Fig. 5) are necessary for establishing the viscoelastic parameters, 

practice usually requires more testing. More tests conducted below Y provide additional points enabling 

better linear regression fits of ES. The results indicate that the first test conducted, and all others like it, 

seems to yield the lowest value of ES in a sample (see Fig. 30), suggesting it is prudent to get additional 

points from both types of tests. Moreover, it is often difficult to determine if dissipation or full recovery 

has occurred, and more tests can aid in this determination. Part of the difficulty of determining this is that 

the deformation behavior, particularly at low stress values, can be influenced by specimen-specific 

history, such as residual stresses (in part due to machining), local microstructure variations, surface 

roughness, and specific temperature variations, to name a few. These may lead to odd deformation 

NASA/TM—2014-216584 34 
behavior such as negative creep; recovery shutdown at nominal, nonzero values; recovery into the 

negative strain regimes; and recovery as increasing rather than decreasing values. Again these effects are 

more readily overcome at lower temperatures where the values of both Y and PL are high and are high 

enough that most of these effects become nonissues. However, the viscoelastic effects are once again 

more important at the higher temperatures. Nonetheless, a reasonable amount of testing to yield accurate 

results per temperature is four tests, resulting in a total of 8 test days per sample. 

Even though the final viscoelastic test is above Y and the specimen does incur some irreversible 

deformation, the stress levels are still below those of the proportional limit. Hence the specimen contains 

comparatively little permanent deformation and may be used for subsequent high-strain (viscoplastic 

(post-yield) or damage) testing. Given that the sample contains prior history, the researcher must 

determine whether this is important to subsequent testing or not. All of the samples used in this study 

were used for further viscoplastic tests. The determination of whether or not the prior viscoelastic history 

was important was made only after high-strain tests were conducted on virgin samples and were available 

for comparison. Experience would suggest that these viscoelastic specimens can at a minimum be used to 

obtain initial estimates of monotonic tensile behavior at a given strain rate to failure. 

Two key events determining the success of these tests is the dissipation of viscoelastic deformation 

and the complete reversibility of the deformation during unloading and recovery. Since these 

determinations are complicated by the aforementioned problems, some guidance will now be given. 

Complete dissipation is theoretically defined by an instantaneous slope of zero for the response variable 

(stress or strain). It was determined that creep rates of less than 6.010–09 s–1 were sufficient to declare 

that the mechanisms had dissipated. Similarly, relaxation rates of less than 3.310–04 MPa/s were 

required. Recovery was always based upon a strain proximity to zero rather than a rate. Experience 

showed that recovery values within 0100  were suitable to account for experimental error. If the 

recovered strain was larger than this, the test was declared to have been in the irreversible regime. 

Although this process was tempered with any subsequent experiments per the aforementioned discussion 

on repeat tests below Y. The observed strain rate at the end of recovery ranged between –5.010–9 and 

–3.810–10 s–1. 

The biggest factor in identifying reversibility of a test is whether there is a secondary creep rate 

(nonzero rate) or not. This poses the question at what creep rate can the test be considered to be zero? The 

strain rates during dissipation (i.e., loading portion of test) were found to be between 1.010–10 and 

6.010–9 s–1, whereas the strain rates at the end of the recovery portion of the test in the reversible regime 

were between –3.810–10 and –5.010–9 s–1. It is interesting that the strain rates during dissipation in the 

reversible regime were very similar (ignoring the sign) to the final recovery rates. Above Y, the final 

creep rate was always larger than the final recovery rate. There were few tests conducted around the 

proportional limit to draw conclusions about rates in this regime. Given these observations, one could 

conclude that a creep rate smaller than 1.010–9 s–1 indicates fully dissipated mechanisms (i.e., zero 

secondary creep rate), and would seem to indicate reversibility. 

Stress relaxation tests tended to dissipate with stress rates between –3.810–5 and –3.310–4 MPa/s in 

the reversible regime. There were too few tests conducted above Y and below the proportional limit to 

compare stress rates in the irreversible regime. These results show that judging dissipation is subjective, 

particularly without a series of like tests centered around Y. The best judge of whether or not the test fell 

into the reversible regime is to ensure that the axial recovery strain returned to a near-zero value. Even 

here, multiple tests make this decision easier. 

Traditional metallic behavior assumes that inelastic behavior (flow) at room temperature does not 

occur until the applied stress exceeds the yield point. Time-dependent deformation behavior at room 

temperature below the yield point in titanium alloys, especially in Ti-6-4, is not new and has been 

documented previously (Refs. 12 to 17). In the majority of these cases, the authors used a 0.2-percent 

offset as their yield point definition. However, the current work shows that time-dependent deformation is 

possible at values far below this point. For example, PL at 20 C for the sample represented in Figure 20 

(no. 17) is given in Table IV as 737 MPa. Yet this sample has significant time-dependent behavior (see 

NASA/TM—2014-216584 35 
NASA/TM—2014-216584 36 

Figs. 21 and 22) below the proportional limit at a stress of 690 MPa. Wapniarsky, Rotem, and Rosen 

(Ref. 13) performed creep tests at room temperature on annealed Ti-6-4 at stresses as low as 85 percent of 

the 0.2-percent yield (904 MPa) and showed evidence of time dependency. Although this stress level is in 

the range of Y values observed here, it is unknown whether or not Wapniarsky’s levels were above the 

threshold of those particular samples. Wapniarsky does state that the observed creep is transient at all the 

stress levels investigated, and they exhibited only a primary creep regime followed by eventual shutdown. 

At least some of their applied stress levels were above Y, and still complete dissipation was observed, 

implying that a few of these tests were in the irreversible regime. The authors did not unload and recover 

the samples, so it cannot be definitively known which of the tests, if any, were reversible, since full 

dissipation alone is not a sufficient criterion at low temperatures to define reversibility. It should also be 

noted that because of the very long time periods (2000 h) necessary to achieve full dissipation, a very 

long clock would be required in the GVIPS model. 

Lastly, the reader is reminded that when dealing with elevated-temperature response, understanding 

rate dependence is essential. Figure 6 shows the modulus as a function of strain rate and temperature. For 

specimens conducted at fast rates (e.g., 110–4 to 110–3 s–1) the moduli are nearly identical to those 

conducted under dynamic conditions. Thus the loading rates of 0.001 s–1 used in most of the viscoelastic 

testing was sufficient to lock in practically all material-time dependency. Only at temperatures of 482 C 

and above did the fast-rate static modulus deviate from the dynamic modulus, where the strain rate from 

the dynamic modulus tests is approximately 3 orders of magnitude larger than the fastest static strain rate. 

Although this implies that the applied static rates still should have been faster, we feel that the difference 

between the curves was negligible. Moreover, faster rates would have decreased the ability to properly 

control the tests. Clearly, lower loading rates, certainly less than 110–4 s–1, would have allowed some of 

the time-dependent behavior to be released as the sample was loaded. Obviously, the desired loading rate 

is both temperature and material dependent. There may be some materials for which no realistically 

achievable rate can totally lock in all time dependency. Additionally, there may be other materials where 

any realistic load rate is sufficient to lock in time dependency. 

Sinha (Ref. 23) investigated the relaxation behavior of Ti-6Al-2Sn-4Zr-6Mo to describe its viscous 

strain behavior. For fast loading rates between 110–3 and 110–2 s–1, he found that the static modulus also 

agreed with the values from dynamic tests. Sinha and Kearsey (Ref. 24) documented the same finding on 

a nickel-base superalloy, IN–738LC. They suggested that because of the fast loading rates that “inelastic 

strain was not introduced during loading.” Actually, inelastic strain is indeed being incurred during this 

time in spite of the observed linear stress-strain response (apparent elastic behavior). It is this “locked-in 

strain” that manifests itself as viscoelastic behavior during subsequent creep or relaxation. During Sinha 

and Kearsey’s tests, samples were unloaded to zero load and allowed to recover for up to 1000 s. 

Viscoelastic recovery (termed by Sinha and Kearsey as “delayed elastic strain”) was observed in each 

case, but did not return to zero strain. Relaxation and creep tests showed the absence of a full dissipation 

of mechanisms in stress and strain, respectively, during the loading portion of the tests. This indicates that 

either all of the tests had been loaded into the irreversible regime and none of them were at loads below 

the threshold or (most likely) that the hold times were far too short. The current work indicates that 

significantly longer times are required to allow all time-dependent behavior to cease. Sinha (Ref. 23) did 

indicate that the recoverable strain had been recognized to be a significant portion of the total strain at 

high temperatures. He also implied (Ref. 25) that time dependency may occur below a threshold, where 

threshold was defined as an internal or back stress. Therefore, the time-dependent reversible and 

irreversible response and modeling approach used for Ti-6Al-4V could be applicable to many nickel-base 

superalloys. 
Summary of Results 

The time and rate dependence of deformation in both the reversible and irreversible regimes were 

investigated and documented for annealed Ti-6Al-4V. A series of tensile, creep, and stress relaxation tests 

was conducted from 20 to 538 C. Unloading followed by a recovery period was performed at the end of 

each test. These tests were in support of characterization of the GVIPS (generalized viscoplasticity with 

potential structure) viscoelastoplastic constitutive model, which will be used in damage prognosis for 

aircraft engines. The salient features of this work are 

(1) The modulus was found to be rate dependent, particularly above 316 C. It was bounded by stiffness 

limits: the dynamic modulus ED at very fast rates (upper limit) and the modulus ES at infinitely slow 

rates (lower limit). 

(2) This material was found to be viscoelastic at low stresses over the entire range of test temperatures 

based on the following observations: 

(a) At stresses much below the value of the proportional limit (PL) the material exhibited time 

dependency that completely dissipated given sufficient time. 

(b) Upon unloading to zero load, the strains recovered to zero in a viscous fashion. 

(c) There was an experimentally derived stress below which the material deformed by reversible 

mechanisms and above which it deformed by irreversible mechanisms. This stress threshold is 

referred to in the model as Y. 

(d) Viscoelastic strain was shown to be a significant portion of total strain at elevated temperatures. 

(e) The value for Y was low and a small proportion (10 percent) of the fast-rate PL at high 

temperatures and a large fraction of PL (99 percent) at low temperatures. 

(3) Below the threshold, primary creep eventually dissipated, with final creep rates being less than 

610–9 s–1. 

(4) Above the threshold, creep tests transitioned from primary to secondary creep regimes. 

(5) Time-dependent deformation, although extremely small, was observed at temperatures as low as 

20 C and existed both below PL and Y. 

(6) The experimental results substantiate the fundamental assumptions of the viscoelastic portion of the 

GVIPS model. 

Conclusions 

In conclusion, there can be substantial amounts of time-dependent reversible and irreversible 

deformation below the proportional limit (or traditionally defined yield points), of this material and over 

the entire range of temperatures studied. The irreversible or permanent deformation occurs above a 

threshold stress, which can be found experimentally. Below this threshold, all deformation is reversible. 

This threshold was found to be invariant of the loading rate. The deformation types can be displayed in a 

deformation-modeling map, which shows the active mechanisms in each regime and the key viscoelastic 

partitioning parameters—the dynamic modulus (ED), the modulus at infinitely slow rate (ES), and the 

uniaxial threshold stress (Y)—as a function of temperature. These parameters were found to decrease with 

increasing temperature. Moreover, the modulus decreased with strain rate at high temperatures, reaching a 

lower limit at the calculated value ES, which represents the infinitely slow strain rate. The viscoelastic 

parameters ES and Y do not appear to be affected by prior deformation in this material, thus suggesting 

that this material can be idealized using a kinematic hardening formulation. 

Finally, the data presented herein will be used for characterizing the viscoelastic portion of the 

GVIPS (generalized viscoplasticity with potential structure) model and documenting the ability of the 

model to describe the viscoelastic behavior of Ti-6Al-4V. Similar experimental and modeling 

publications will also follow that characterize the viscoplastic and damage behavior of Ti-6Al-4V and 

include predictions of complex deformation behavior. 

NASA/TM—2014-216584 37 

Appendix A—Symbols 

e total axial strain 

ec axial creep strain 

esat saturation strain 

et transverse strain 

e relaxation strain (applied strain level) 

eloading strain rate 

eccreep rate 

E static modulus 

ED dynamic modulus 

Em Maxwell spring stiffness 

ES infinitely slow modulus 

PL proportional limit 

T temperature 

Tm melting point 

Y uniaxial threshold stress 




IR irreversible strain after achieving full recovery 



R

 reversible strain 

Rmax maximum reversible strain 

% dashpot viscosity 

 threshold stress in shear 


 relaxation time 

 Poisson’s ratio 

	 stress 

	sat saturation stress 

	0.02% 0.02 percent yield point 

	0.2% 0.2 percent yield point 

	 applied stress level 

NASA/TM—2014-216584 39 

Appendix B—Transverse Strain Behavior 

The viscoelastic model put forward by Saleeb and Arnold (Ref. 1) assumes that all moduli (i.e., Es, 

Em, and n) are coincident, which implies time independency of Poisson’s ratio. Consequently, in this 

study an attempt to experimentally assess the time dependency of Poisson’s ratio was made. Previous 

attempts at this were inconclusive because the contacting diametral extensometer was found to creep into 

the sample over time yielding erroneous values (Ref. 2). For this reason, a noncontacting, optical 

extensometer was chosen specifically for this test program on Ti-6Al-4V (Ti-6-4). Transverse (or 

diametral) strain data were collected on nearly every test. 

The results from the transverse strains were less than ideal because of a number of factors. First, the 

practical resolution of the optical micrometer was not adequate for the low strain levels observed in the 

viscoelastic tests. At 538 C a resolution of 0.002 mm could be resolved, which equates to a transverse 

J
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total change in diametral strain anticipated based on the known maximum axial strain for these test 

conditions. However, the optical measurement system does give good data for tests conducted at higher 

strains as will be shown in a future report on the viscoplastic and damage results. 

Second, the material was shown to have a moderate transverse texture (Fig. 3), and also displayed a 

directional dependence of the dynamic modulus (Fig. 4). Moreover, the material was not strictly 

transversely isotropic since there were slight differences in the degree of planar orientation between the 

transverse and thickness direction of the original rolled plate. The test samples were cylindrical with the 

load axis parallel to the rolled direction of the plate and the location of the other two plate directions were 

not maintained during machining. Hence the diametral extensometer measured the displacement of the 

diameter in some random and unknown transverse direction on the circumference of the samples. 

Correctly treating the material as orthotropic would have resulted in three independent Poisson’s 

ratios and would have required a great deal more testing to determine these values. Since the two 

transverse directions were not greatly different from one another, we considered treating the material as 

transversely isotropic, but still lacked sufficient data to determine all Poisson’s ratios. The simplifying 

assumption of isotropy was believed to be somewhat justified since the texture is moderate in nature; 

therefore, to calculate Poisson’s ratio  we utilized 

e

e  t  
 (4) 

Where et is transverse strain and e is axial strain. 

In spite of the above-mentioned complications, we felt that there was still value in reporting selected 

data on transverse strains. Even though the signal-to-noise ratio is poor, trends can still be discerned in the 

data. Likewise, trends can also be observed in Poisson’s ratio (assuming isotropy) even if the absolute 

values may be questionable. Finally, there are very few transverse stress-strain curves presented in the 

literature, and we thought that showing such curves would be of general interest. 

An example of the transverse strain response measured is described via sample 25, tested at 427 C. 

Its general behavior is typical of the tests at the other temperatures. The first test of this series is shown in 

Figure 39 for a creep test at a stress of 34 MPa and then in Figure 40 for its respective recovery at zero 

load. The axial strain response has already been shown in Figure 26. Figure 39 indicates that the total 

diametral strain initially increases (becomes more negative, or the diameter shrinks) during the loading 

and the first 30 000 s of creep, thereafter reaching an apparent constant average value of –258 . The 

calculated isotropic value of Poisson’s ratio for the loading portion of this test was 0.37. Upon unloading 

(Fig. 40) the diametral strain decreases (diameter expands) and reaches a constant value of –200 

almost immediately during the recovery phase. Although the transverse strain did not return to zero, it did 

shut down. Moreover, the axial strain at the end of recovery (Fig. 26) also failed to return precisely to 

zero. We suspect that both of these nonzero values fall within experimental scatter. 

NASA/TM—2014-216584 41 
The next step in this test series was a relaxation test (25R) at an axial strain of 430 . The 

corresponding axial strain and the stress response were shown in Figures 27 and 28. The diametral strains 

for the relaxation and recovery phases of this test are plotted in Figures 41 and 42, respectively. After a 

slight increase during the loading period, the diametral strain remains constant at a strain of –120  during 

the relaxation period. During the recovery, no discernible decrease in transverse strain is observed after the 

specimen is unloaded. The diametral values remain constant at approximately –70 . Again the diametral 

strain did not return completely to zero, but returned to a value less than half that observed in the previous 

test (25C). It should also be noted that unlike the previous load-controlled creep test, which showed primary 

NASA/TM—2014-216584 42 
creep strains in both the axial and transverse directions, this strain-controlled relaxation test exhibited no 

change (as expected if Poisson’s ratio is constant in time) from a constant value in either the axial or the 

transverse strain during the 24-h hold time. 

The third test in this series was a creep test, above the threshold, at 206 MPa (see Fig. 29). The 

corresponding diametral strain during this creep test is shown in Figure 43. Consistent with the axial strain, 

the diametral strain exhibits a primary creep regime followed by a secondary creep regime, with no 

appearance of ever reaching a saturation point. During the recovery period (Fig. 44) the diametral strain 

decreases, showing again a primary region followed by saturation at a transverse strain of approximately 

–97 . There is no indication that with continued time the diametral strain (or the axial strain) would return 

to zero, suggesting that the deformation was indeed irreversible and permanent. 

As stated previously, the viscoelastic model put forward by Saleeb and Arnold (Ref. 1) assumes 

coincidence of moduli, which implies time independency of Poisson’s ratio, although this assumption could 

not be definitively confirmed or refuted by the present test program given the experimental limitations 

stated. In nearly every test conducted in this study, diametral strain was measured using an optical 

micrometer. As observed in Figures 39 and 40, the amount of noise in the diametral signal was quite high. 

However, in most cases a trend in the diametral strain could still be observed, and thus a calculation of 

Poisson’s ratio could still be made at various times. It was shown that during viscoelastic creep tests 

(Fig. 39), the transverse strain eventually shut down during creep and recovered slightly after unloading 

(Fig. 40). During stress relaxation, the transverse strain remained unchanged during the relaxation period 

(Fig. 41). This is expected since the axial strain is being held constant during the test. Concurrently, the 

diametral strain decreases during unloading and remains constant during recovery (Fig. 42). During 

recovery in the reversible regime the diametral strains appear to have shut down, although not always at 

zero. A small diametral strain value (–610–5) often remained (Fig. 42), but this is believed to be within the 

acceptable range of scatter and given the experimental setup, assumed to be zero. 

NASA/TM—2014-216584 43 
NASA/TM—2014-216584 44 
A plot of Poisson’s ratio over time for various creep samples tested at different temperatures is shown 

in Figure 45. There are two groups of data representing stress levels above and below the threshold Y. For 

those below the threshold, the calculated Poisson’s ratio is approximately constant after the initial loading 

and yields an isotropic value of approximately 0.38. This is consistent with a range of values of 0.34 to 

0.38 given by Fukuhara and Sanpei (Ref. 26) over the temperature range of 20 to 538 C for Ti-6-4 using 

a dynamic method. 

NASA/TM—2014-216584 45 
At stress levels above the threshold, Poisson’s ratio continues to increase throughout the creep test. 

Three of the four tests exceed the isotropic plastic limit of 0.5. This could be due to two reasons: One, the 

material is orthotropic, permitting Poisson’s ratio to reach higher values, and two, at least two of the 

samples entered tertiary creep and failed. Also localized necking was observed on these specimens, 

allowing more transverse strain to be measured than calculated through the Poisson’s effect. Moreover, 

the orthotropic properties resulted in asymmetric necking forming an elliptical cross section at failure. 

Nonetheless, it is clear that the tests above the threshold had a significantly different Poisson’s response 

than those beneath the threshold. Also, the behavior in the reversible regime appears to be constant and 

consistent with that idealized by the model. 

A final note is made about the transverse measurements and the optical micrometer. The residual 

strain $$[
!_Q

diameter of 0.4 !		$Q"-grain 

thickness for this material is 16 times larger. A speck of dust on the sample surface can be in the range of 

4 to 50 m, and the surface roughness of the machined sample could be as high as 1 m. The possible 

errors introduced by any of these elements can be interpreted as transverse strains and could complicate 

the interpretation of the diametral strain values. Moreover, these errors are unique to the employed, 

noncontacting micrometer in which the micrometer is fixed on the load frame and the sample translates 

during loading within a fixed beam of light from the micrometer. Any variation in diameter along the 

gage length will be recorded as a change in the apparent diametral strain. At high temperatures surface 

oxidation can also increase the diameter and could be interpreted as a possible drift in transverse strain, 

although this aspect was thought to be insignificant in most of these tests. If future examination of 

Poisson’s ratio is desired in the viscoelastic regime with the goal of documenting Poisson’s ratio 

constancy throughout the creep tests, a specific and improved test setup would be required. A different 

sample geometry, such as a parallelepiped with a large specimen width to increase transverse strains, 

could be used. Otherwise a special high-resolution extensometer would need to be developed. 

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J"
"{|[@ 

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@"
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NASA/TM—2014-216584 46 
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?@"
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15. "!}$Q"['"|
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16. Chiu, S.S.; Eftis, J.; and Jones, D.L.: Prediction of Fatigue Life With and Without Hold Times Using 

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""''|? 

17. Imam, M.A.; and Gilmore, C.M.: Room Temperature Creep of Ti-6Al-4V. Metall. Trans., vol. 10A, 

?"@|@
{ 

18. Evans, W.J.: Stress Relaxation and Notch Fatigue in Ti-6Al-4V. Scrip. Metal., vol. 21, 1987, 



J|

? 

19. Hatch, A.J.; Partridge, J.M.; and Broadwell, R.G.: Room-Temperature Creep and Fatigue Properties 

of Titanium Alloys"
""[?"| 

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of Ti-5Al-
{~!"@"?J"'|' 

21. Savage, M.F.; Neeraj, T.; and Mills, M.J.: Observations of Room-Temperature Creep Recovery in 

!}$!"JJ}"

"'|'' 

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23. Sinha, N.K.: Limitations of Stress Relaxation Tests for Determining Stress Dependence of Strain 

Q!~"@'"
J"?J|?J[ 

24. Sinha, N.K.; and Kearsey, R.: Optimizing Deformation Path for Stress-Relaxation Tests on 

Supera$Q!}~
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"
 

25. Sinha, N.K.: Short Strain Relaxation/Recovery Tests for Evaluating Creep Response of Nickel-Base 

~$|?J'~"
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"{|{J 

26. Fukuhara, M.; and Sanpei, A.: Elastic Moduli and Internal Frictions of Inconel 718 and Ti-6Al-4V 

!~"
"J"

|24. 

NASA/TM—2014-216584 47