Dynamic mechanical properties of straight titanium alloy arch wires R.P. Kusyl" T.W. Wilsont tProfessor of Orthodontics/Dental Research Center and Biomedical Engineering University of North Carolina Building #210-H, Room 313 Chapel Hill, NO 27599-7455 SFormerly NIH Post-doctoral Fellow (DE05487) at the Dental Research Center; presently employed at Camille Dreyfus Laboratories, Research Triangle Institute, Research Triangle Park, NC Received June 19, 1989 Accepted June 13, 1990 Dent Mater 6:228-236, October, 1990 Abstract- Eight straight-wire materials were studied: an orthodontic titaniummolybdenum (li-Mo) product, TMA'"; three orthodontic nickel-titanium (Ni-Ti) products, Nitinol", Titanal TM, and Orthonol'"; three prototype alloys, a martensitic, an austenitic, and a biphasic alloy; and a hybrid shape-memory-effect product, Biometal TM. Each wire was prepared with a length-to-cross-sectional area of at least 3600 cm -1. With an Autovibron Model DDV-II-C used in the tensile mode, each sample was scanned from -120 to + 200°C at 2°C/rain. From the data base, plots of the log storage modulus, log tan delta, and percent change in length vs. temperature were generated. Results showed that the dynamic mechanical properties of the alloys within this Ti system are quite different. The TiMo alloy, TMA", was invariant with temperature, having a modulus of 7.30 X 1011 dyne/cm 2 (10.6 X 106 psi). The three cold-worked alloys-Nitinol TM, Titanal", and Orthonol'"-appeared to be similar, having a modulus of 5.74 X 101~ dyne/cm 2 (8.32 X 108 psi). The biphasic shape-memory alloy displayed a phase transformation near ambient temperature; whereas the hybrid shapememory product, Biometal'", underwent a 3-5% change in length during its transformation between 95 and 125°C. Among the Ni-Ti wires tested, several different types of alloys were represented by this intermetallic material.
ithin the last ten years, titanmm alloys have had a significant impact on the treatment of orthodontic patients. Prior to this time, the force-to-activation ratios of stainless steel or cobalt-chromium •systems were not so very different, since both had substantially the same elastic modulus (Thurow, 1981). Although multi-stranded wires modified the stiffness p a r a m e t e r by changing not only the number of wire strands but also the helix angles (Kusy and Dilley, 1984; Kusy and Stevens, 1987), it was not until the addition of two basic titanium systems-titanium-molybdenum (Ti-Mo) and nickel-titanium (Ni-Ti)-that the most profound change in orthodontic mechanics occurred (Goldberg and Burstone, 1979; Burstone and Goldberg, 1980; Andreasen and Hileman, 1971; Andreasen, 1980; Burstone et al., 1985; Miura et al., 1986). Where before only variable cross-section orthodontics was practiced, now variable modulus orthodontics was possible (Burstone, 1981). The characteristics of Ni-Ti alloys are not all alike. Heat treatment, deformation history, and changes in
Wl
composition have profound effects on the performance of alloys (Special Metals, 1984; Eckelmeyer, 1976; Kousbroek, 1984). The two phases that are commonly found in these alloys, martensite (M) and austenite (A), represent the low- and hightemperature phases, respectively. The names of these phases are the same as those used in the steel industry and describe a diffusionless shear transformation that occurs during non-equilibrium cooling (Rostoker and Dvorak, 1965; Dautovich et al., 1966). Because the critical phase-transformation temperature range (TTR) of conventional Ni-Ti alloys can be varied from -200°C to +150°C (ITI, undated literature), martensite or austenite or even both phases can co-exist at oral temperature. Whether the prevailing phase is "active" (so-called when one phase will transform to another phase by thermal or mechanical activation) or "stable" depends upon the deformation history that the alloy has undergone (Buehler and Wiley, 1962; Everhart, 1971). In the present work, the mechanical properties of eight representa-
TABLE 1 STRAIGHT LENGTHS OF TITANIUM ALLOYS STUDIED Product "Stable" Alloys TMA'"* Nitinol'M. Titanal TM* Orthonol'"s "Active" Alloys SMC (M)** SMC (A)** SMC (M + A)** Biometa] . . . .
Alloy
Phase(s) Present
Sizes Tested (")
Usage
Ti-Mo Ni-Ti Ni-Ti Ni-Ti
Beta Martensite Martensite Martensite
0.016 0.016 0.016; 0.018 0.016 x 0.022
Orthodontics Orthodontics Orthodontics Orthodontics
Ni-Ti Ni-Ti Ni-Ti Ni-Ti
Martensite Austenite Biphase Hybrid Martensite
0.010; 0.016; 0.018 0.016 0.016 0.006
Experimental Experimental Experimental Robotics and Biomedical Applications
*American Ormco, Glendora, CA, USA. + Unitek Corporation, Monrovia, CA, USA. #Lancer Orthodontics, Carlsbad, CA, USA. {}Rocky Mountain Orthodontics, Denver, CO, USA. **Special Metals Corporation, New Hartford, NY, USA. + +Toki Corporation, Irvine, CA, USA.
228 KUSY & WILSON~DYNAMIC MECHANICAL PROPERTIES OF ARCH WIRES
tive titanium alloys were evaluated by dynamic mechanical analysis (DMA). This technique was chosen over others [e.g., differential scanning calorimetry (Lee et al., 1988), electrical resistivity (Nishida and Wayman, 1988), thermal mechanical analysis (Wendlandt, 1986), and static mechanical tests (Lee et al., 1988)] because DMA can measure the presence of phase transitions, thermal expansion coefficients, and elastic moduli simultaneously.
loRvl,
*Because of similarnotation used in dynamic mechanicalmeasurements and in metallurgy, the symbol "A" refers to crosssectional area as well as to austenite. When used within the propercontext, however, this duplicity should not present any confusion.
]
I
l I
['M PX I
I
I ADC I
¢1
EXPERIMENTAL
Materials.-Among the four "stable" alloys that were evaluated as straight lengths (Table 1), one was a metastable single phase Ti-Mo alloy of composition Ti-ll.5Mo-6Zr-4.5Sn by weight (Goldberg and Shastry, 1984). The other three stable alloys were martensitic Ni-Ti alloys of nominal composition Ni-45Ti-3Co in which the amount of cold work (--8-10% strain) had suppressed any thermo-elastic behavior (Cross et al., 1969; Wayman, 1977; Collings, 1984). Because of processing variations (Unitek, 1987), the force-deflection plots of the straight Nitinol TM wire vs. the preformed Nitinol TM wire displayed pseudo-elastic behavior vs. conventional linear elasticity (cf. Figs. 7 and 31 of Stush, 1985, and Fig. 1 of Miura et al., 1986). Among the four "active" alloys that were evaluated as straight lengths (Table 1), the proprietary SMC (A) exhibited a stress-induced martensitic transformation (SIM) (Otsuka and Wayman, 1975; Wayman, 1977) that should be comparable with a select group of the pre-formed arch wires-Nitinol SE TM, SentalloyTM, and NiTU" (Burstone et al., 1985; Miura et aI., 1986). Unfortunately, those orthodontic products were not available in straight lengths. The proprietary martensitic active alloy, SMC (M), and the proprietary biphasic alloy that contained both martensite (M) and austenite (A) phases*, SMC (M+A), exhibited the thermo-elastic martensitic transfor-
I COMPUTER J
Fig. 1. Overall schematic of the Autovibron Model DDV-II-C. In addition to the VIBRON and its immediate support equipment (cf. Fig. 2) are shown the COMPUTER, the muitiprogrammer (MULT.), the a-d converter (ADC), the signal generater (SG), the multiplexer (MPX), and the temperature controller (T), ~ee
eel
leeee
ee
eBeeee
e ee
t e
t
X m n m
mmeeoee
× P DRV
) VIBRON i
L
SAMPLE FURNACE STRAIN GAUGE STRESS GAUGE DRIVER
Fig. 2. Physical layout of the VIBRON's sample (---) and furnace (...) relative to its strain gauge (X), stress gauge (P), and driver (DRV).
IMPOSED SINUSOIDAL STRAIN
INPUT:
Y
IN-PHASE
RESPONSE
OUTPUT: OUT-OF-PHASE
RESPONSE
/Z
~
"t'.~
E" E
t
Y
Fig. 3. Operating principle of the VIBRON. By imposition of a sinusoidal strain on the sample, an in-phase and out-of-phase stress response results that is characteristic of the operating temperature, angular frequency (~), and time (t). The real (E') and imaginary (E') components of the complex modulus (E*) may be defined in terms of the cosine and sine of the phase angle (8).
Dental Materials/October 1990 229
E
t
rE t,=
Tan Delta
_J
i
..-%-_ _ ee e
% Change in Length eee
°
o @ o 4)
lille•
e'•°
• ooo oeoo°
oooOe oee ~@@eloi
@e @°
Temperature Fig. 4. Representative sample output of the VIBRON showing the storage modulus (E',
), the loss tangent (tan 8, ---), and the percent change In length (...). As the temperature is changed and a phase transition is encountered, E' discretely changes magnitude, tan 8 passes through a maximum, and the percent change in length shifts its slope.
sultant stress gauges (X and P) were measured as the multiplexer (MPX) toggled between T, X, and P. The analogue-to-digital converter (ADC) transformed the signal currently selected by the MPX into the multip r o g r a m m e r (MULT.). F r o m the known sample geometry, the signal that was generated, and the three inputs of MPX, the dynamic mechanical properties were calculated by the COMPUTER. In actual operation, the sinusoidal strain (e) that is imposed on the sample gets out of synchronization with the stress (a) (cf. Figs. 2 and 3). This phase lag between the imposed • and the response ~ results from the time necessary for atomic or molecular rearrangements and their associated relaxation phenomena to occur. Given that the imposed frequency (f, cycles/ sec) may be expressed in terms of the angular frequency (~, radians/ sec),
T.M.A. ® - 0.016"
12-
o~ = 2~rf,
~-.
(1)
/
the values of • and ~ may be expressed as • = •o sin (cot) ~'~ E
11
I-3:;I ' / a / 3~
(3 e-
_J
~
0 -J
10 r-
0~
-120
I
I
I
'
-80
-40
0
40
!
80
I
120
160
Fig. 5. Dynamic mechanical spectra for the stable orthodontic Ti-Mo product, TMA".
Method$. - T h e dynamic mechanical properties were measured by means
= ~o sin (cot + 8),
(3)
in which 8 (radians) is the phase angle between them. The magnitude of may be re-cast into two components, one in-phase with strain (ao cos 8) and one 90° out-of-phase with strain (ao sin 8), that is, = ~o sin cot cos 8 + ao cos cot sin 8
(4)
200
Temperature (C)
marion commonly referred to as the shape memory effect ( S M E ) (CoilLugs, 1984; Otsuka et al., 1971). The last material, a hybrid martensitic alloy that consists of nickel and tit a n i u m in a p p r o x i m a t e l y e q u a l weights (Told, 1988), was one of two not tested in a 0.016" diameter and represented the latest generation of SME alloys.
and
-1
!
(2)
of a programmable mechanical testing device and computer work station, the Autovibron Model DDV-IIC [IMASS, Inc., Accord (Hingham), MA] (Figs. 1 and 2). In this apparatus, a signal generator (SG) actuated the driver (DRV) to cycle a pre-loaded sample at a specified frequency within the VIBRON's clamps. As the temperature (T) of the furnace was controlled to -+ I°C by a monitor on the platinum thermocouple, an imposed strain and the re-
230 KUSY & WILSON/DYNAMIC MECHANICAL PROPERTIES OF ARCH WIRES
or equivalently, a/Co = (~o/•o) sin cot cos 8 + (Cro/•o) cos cot sin 8
(5)
so that = E' eo sin cot + E " • o cos cot
(6)
where E' = ~o/eo cos 8
(7)
E" = Cro/eo sin 8.
(8)
and
The parameter, E', is the real part of the modulus or the storage mod-
ulus, so-called because it is equivalent to the maximum potential energy that is elastically stored and released during the deformation cycle. On the other hand, E" is the imaginary part of the modulus or the loss modulus, so-called because it is associated with the energy dissipation as heat when a material is deformed. From the parameters, E', E", and 8, general relationships are obtained (Fig. 3): J E* I = (E' 2 + E" 2)zm (9)
and E* were collected at a rate of one datum per three min and stored on disk. These raw data were then converted to finished form by use of Eqs. 9 and 10 and plotted (Fig. 4). A curve-smoothing algorithm was applied to those "noisy" log E' and log tan ~ plots, by continuous averaging of the reciprocals of six contiguous data points. By biasing these transformed values toward the baseline of the curves, we eliminated most transient excursions.
and
RESULTS
Stable AIIoys,.-:Because stable alloys generally lack any phase transitions within the temperature range of in. terest (-120 to +200°C), t h e small offset shown for TMA TM was regarded as an artifact (Fig. 5). When a damping peak was observed, it rarely appeared in Nitinol" and Titanal TM (Figs. '6 and 7) but oceasionally appeared in Orthonol TM at about -10°C (Fig. 8). Because Nitinol TM,
NITINOL ® - 0.016"
tan 8 = E"/E'
(10)
Eq. 9 defines the complex modulus, I E*], as the vectorial sum of the real and imaginary components of the moduli. Eq. 10 defines the loss tangent, tan 8, which is a measure of the internal friction or damping. In effect, tan ~ is the dissipation factor that measures the lag between the imposed strain and the resulting stress or the ratio between the energy dissipated as heat and the maximum energy stored in the sample per cycle. Fig. 4 shows the sample output of E', tan ~, and the percent change in length for a hypothetical metal alloy undergoing a phase transition (Fig. 4) (Nielsen, 1974; Murayama, 1978). In the present study, 13 samples having acceptable length-to-cross-sectional area ratios (L/A I> 3,600 cm-z) were pre-loaded in uni-axial tension (20 g) and oscillated at a frequency of 11 Hz over a temperature range from -120 to + 200°C at a nominal heating rate of 2°C/min. Output voltages of the stress transducer (Vz) and the strain transducer (V~_)were converted to tan 8 V/a ]Vz - V2] = 2 sin(8/2) = tan ~(11)
12
-1
ithin the last ten years, titanmm alloys have had a significant impact on the treatment of orthodontic patients. Prior to this time, the force-to-activation ratios of stainless steel or cobalt-chromium •systems were not so very different, since both had substantially the same elastic modulus (Thurow, 1981). Although multi-stranded wires modified the stiffness p a r a m e t e r by changing not only the number of wire strands but also the helix angles (Kusy and Dilley, 1984; Kusy and Stevens, 1987), it was not until the addition of two basic titanium systems-titanium-molybdenum (Ti-Mo) and nickel-titanium (Ni-Ti)-that the most profound change in orthodontic mechanics occurred (Goldberg and Burstone, 1979; Burstone and Goldberg, 1980; Andreasen and Hileman, 1971; Andreasen, 1980; Burstone et al., 1985; Miura et al., 1986). Where before only variable cross-section orthodontics was practiced, now variable modulus orthodontics was possible (Burstone, 1981). The characteristics of Ni-Ti alloys are not all alike. Heat treatment, deformation history, and changes in
Wl
composition have profound effects on the performance of alloys (Special Metals, 1984; Eckelmeyer, 1976; Kousbroek, 1984). The two phases that are commonly found in these alloys, martensite (M) and austenite (A), represent the low- and hightemperature phases, respectively. The names of these phases are the same as those used in the steel industry and describe a diffusionless shear transformation that occurs during non-equilibrium cooling (Rostoker and Dvorak, 1965; Dautovich et al., 1966). Because the critical phase-transformation temperature range (TTR) of conventional Ni-Ti alloys can be varied from -200°C to +150°C (ITI, undated literature), martensite or austenite or even both phases can co-exist at oral temperature. Whether the prevailing phase is "active" (so-called when one phase will transform to another phase by thermal or mechanical activation) or "stable" depends upon the deformation history that the alloy has undergone (Buehler and Wiley, 1962; Everhart, 1971). In the present work, the mechanical properties of eight representa-
TABLE 1 STRAIGHT LENGTHS OF TITANIUM ALLOYS STUDIED Product "Stable" Alloys TMA'"* Nitinol'M. Titanal TM* Orthonol'"s "Active" Alloys SMC (M)** SMC (A)** SMC (M + A)** Biometa] . . . .
Alloy
Phase(s) Present
Sizes Tested (")
Usage
Ti-Mo Ni-Ti Ni-Ti Ni-Ti
Beta Martensite Martensite Martensite
0.016 0.016 0.016; 0.018 0.016 x 0.022
Orthodontics Orthodontics Orthodontics Orthodontics
Ni-Ti Ni-Ti Ni-Ti Ni-Ti
Martensite Austenite Biphase Hybrid Martensite
0.010; 0.016; 0.018 0.016 0.016 0.006
Experimental Experimental Experimental Robotics and Biomedical Applications
*American Ormco, Glendora, CA, USA. + Unitek Corporation, Monrovia, CA, USA. #Lancer Orthodontics, Carlsbad, CA, USA. {}Rocky Mountain Orthodontics, Denver, CO, USA. **Special Metals Corporation, New Hartford, NY, USA. + +Toki Corporation, Irvine, CA, USA.
228 KUSY & WILSON~DYNAMIC MECHANICAL PROPERTIES OF ARCH WIRES
tive titanium alloys were evaluated by dynamic mechanical analysis (DMA). This technique was chosen over others [e.g., differential scanning calorimetry (Lee et al., 1988), electrical resistivity (Nishida and Wayman, 1988), thermal mechanical analysis (Wendlandt, 1986), and static mechanical tests (Lee et al., 1988)] because DMA can measure the presence of phase transitions, thermal expansion coefficients, and elastic moduli simultaneously.
loRvl,
*Because of similarnotation used in dynamic mechanicalmeasurements and in metallurgy, the symbol "A" refers to crosssectional area as well as to austenite. When used within the propercontext, however, this duplicity should not present any confusion.
]
I
l I
['M PX I
I
I ADC I
¢1
EXPERIMENTAL
Materials.-Among the four "stable" alloys that were evaluated as straight lengths (Table 1), one was a metastable single phase Ti-Mo alloy of composition Ti-ll.5Mo-6Zr-4.5Sn by weight (Goldberg and Shastry, 1984). The other three stable alloys were martensitic Ni-Ti alloys of nominal composition Ni-45Ti-3Co in which the amount of cold work (--8-10% strain) had suppressed any thermo-elastic behavior (Cross et al., 1969; Wayman, 1977; Collings, 1984). Because of processing variations (Unitek, 1987), the force-deflection plots of the straight Nitinol TM wire vs. the preformed Nitinol TM wire displayed pseudo-elastic behavior vs. conventional linear elasticity (cf. Figs. 7 and 31 of Stush, 1985, and Fig. 1 of Miura et al., 1986). Among the four "active" alloys that were evaluated as straight lengths (Table 1), the proprietary SMC (A) exhibited a stress-induced martensitic transformation (SIM) (Otsuka and Wayman, 1975; Wayman, 1977) that should be comparable with a select group of the pre-formed arch wires-Nitinol SE TM, SentalloyTM, and NiTU" (Burstone et al., 1985; Miura et aI., 1986). Unfortunately, those orthodontic products were not available in straight lengths. The proprietary martensitic active alloy, SMC (M), and the proprietary biphasic alloy that contained both martensite (M) and austenite (A) phases*, SMC (M+A), exhibited the thermo-elastic martensitic transfor-
I COMPUTER J
Fig. 1. Overall schematic of the Autovibron Model DDV-II-C. In addition to the VIBRON and its immediate support equipment (cf. Fig. 2) are shown the COMPUTER, the muitiprogrammer (MULT.), the a-d converter (ADC), the signal generater (SG), the multiplexer (MPX), and the temperature controller (T), ~ee
eel
leeee
ee
eBeeee
e ee
t e
t
X m n m
mmeeoee
× P DRV
) VIBRON i
L
SAMPLE FURNACE STRAIN GAUGE STRESS GAUGE DRIVER
Fig. 2. Physical layout of the VIBRON's sample (---) and furnace (...) relative to its strain gauge (X), stress gauge (P), and driver (DRV).
IMPOSED SINUSOIDAL STRAIN
INPUT:
Y
IN-PHASE
RESPONSE
OUTPUT: OUT-OF-PHASE
RESPONSE
/Z
~
"t'.~
E" E
t
Y
Fig. 3. Operating principle of the VIBRON. By imposition of a sinusoidal strain on the sample, an in-phase and out-of-phase stress response results that is characteristic of the operating temperature, angular frequency (~), and time (t). The real (E') and imaginary (E') components of the complex modulus (E*) may be defined in terms of the cosine and sine of the phase angle (8).
Dental Materials/October 1990 229
E
t
rE t,=
Tan Delta
_J
i
..-%-_ _ ee e
% Change in Length eee
°
o @ o 4)
lille•
e'•°
• ooo oeoo°
oooOe oee ~@@eloi
@e @°
Temperature Fig. 4. Representative sample output of the VIBRON showing the storage modulus (E',
), the loss tangent (tan 8, ---), and the percent change In length (...). As the temperature is changed and a phase transition is encountered, E' discretely changes magnitude, tan 8 passes through a maximum, and the percent change in length shifts its slope.
sultant stress gauges (X and P) were measured as the multiplexer (MPX) toggled between T, X, and P. The analogue-to-digital converter (ADC) transformed the signal currently selected by the MPX into the multip r o g r a m m e r (MULT.). F r o m the known sample geometry, the signal that was generated, and the three inputs of MPX, the dynamic mechanical properties were calculated by the COMPUTER. In actual operation, the sinusoidal strain (e) that is imposed on the sample gets out of synchronization with the stress (a) (cf. Figs. 2 and 3). This phase lag between the imposed • and the response ~ results from the time necessary for atomic or molecular rearrangements and their associated relaxation phenomena to occur. Given that the imposed frequency (f, cycles/ sec) may be expressed in terms of the angular frequency (~, radians/ sec),
T.M.A. ® - 0.016"
12-
o~ = 2~rf,
~-.
(1)
/
the values of • and ~ may be expressed as • = •o sin (cot) ~'~ E
11
I-3:;I ' / a / 3~
(3 e-
_J
~
0 -J
10 r-
0~
-120
I
I
I
'
-80
-40
0
40
!
80
I
120
160
Fig. 5. Dynamic mechanical spectra for the stable orthodontic Ti-Mo product, TMA".
Method$. - T h e dynamic mechanical properties were measured by means
= ~o sin (cot + 8),
(3)
in which 8 (radians) is the phase angle between them. The magnitude of may be re-cast into two components, one in-phase with strain (ao cos 8) and one 90° out-of-phase with strain (ao sin 8), that is, = ~o sin cot cos 8 + ao cos cot sin 8
(4)
200
Temperature (C)
marion commonly referred to as the shape memory effect ( S M E ) (CoilLugs, 1984; Otsuka et al., 1971). The last material, a hybrid martensitic alloy that consists of nickel and tit a n i u m in a p p r o x i m a t e l y e q u a l weights (Told, 1988), was one of two not tested in a 0.016" diameter and represented the latest generation of SME alloys.
and
-1
!
(2)
of a programmable mechanical testing device and computer work station, the Autovibron Model DDV-IIC [IMASS, Inc., Accord (Hingham), MA] (Figs. 1 and 2). In this apparatus, a signal generator (SG) actuated the driver (DRV) to cycle a pre-loaded sample at a specified frequency within the VIBRON's clamps. As the temperature (T) of the furnace was controlled to -+ I°C by a monitor on the platinum thermocouple, an imposed strain and the re-
230 KUSY & WILSON/DYNAMIC MECHANICAL PROPERTIES OF ARCH WIRES
or equivalently, a/Co = (~o/•o) sin cot cos 8 + (Cro/•o) cos cot sin 8
(5)
so that = E' eo sin cot + E " • o cos cot
(6)
where E' = ~o/eo cos 8
(7)
E" = Cro/eo sin 8.
(8)
and
The parameter, E', is the real part of the modulus or the storage mod-
ulus, so-called because it is equivalent to the maximum potential energy that is elastically stored and released during the deformation cycle. On the other hand, E" is the imaginary part of the modulus or the loss modulus, so-called because it is associated with the energy dissipation as heat when a material is deformed. From the parameters, E', E", and 8, general relationships are obtained (Fig. 3): J E* I = (E' 2 + E" 2)zm (9)
and E* were collected at a rate of one datum per three min and stored on disk. These raw data were then converted to finished form by use of Eqs. 9 and 10 and plotted (Fig. 4). A curve-smoothing algorithm was applied to those "noisy" log E' and log tan ~ plots, by continuous averaging of the reciprocals of six contiguous data points. By biasing these transformed values toward the baseline of the curves, we eliminated most transient excursions.
and
RESULTS
Stable AIIoys,.-:Because stable alloys generally lack any phase transitions within the temperature range of in. terest (-120 to +200°C), t h e small offset shown for TMA TM was regarded as an artifact (Fig. 5). When a damping peak was observed, it rarely appeared in Nitinol" and Titanal TM (Figs. '6 and 7) but oceasionally appeared in Orthonol TM at about -10°C (Fig. 8). Because Nitinol TM,
NITINOL ® - 0.016"
tan 8 = E"/E'
(10)
Eq. 9 defines the complex modulus, I E*], as the vectorial sum of the real and imaginary components of the moduli. Eq. 10 defines the loss tangent, tan 8, which is a measure of the internal friction or damping. In effect, tan ~ is the dissipation factor that measures the lag between the imposed strain and the resulting stress or the ratio between the energy dissipated as heat and the maximum energy stored in the sample per cycle. Fig. 4 shows the sample output of E', tan ~, and the percent change in length for a hypothetical metal alloy undergoing a phase transition (Fig. 4) (Nielsen, 1974; Murayama, 1978). In the present study, 13 samples having acceptable length-to-cross-sectional area ratios (L/A I> 3,600 cm-z) were pre-loaded in uni-axial tension (20 g) and oscillated at a frequency of 11 Hz over a temperature range from -120 to + 200°C at a nominal heating rate of 2°C/min. Output voltages of the stress transducer (Vz) and the strain transducer (V~_)were converted to tan 8 V/a ]Vz - V2] = 2 sin(8/2) = tan ~(11)
12
-1
