Dynamics of propagating phase boundaries in NiTi

ARTICLE IN PRESS

Journal of the Mechanics and Physics of Solids 54 (2006) 2136–2161 www.elsevier.com/locate/jmps

Dynamics of propagating phase boundaries in NiTi J. Niemczura, K. Ravi-Chandar Department of Aerospace Engineering and Engineering Mechanics, Center for Mechanics of Solids, Structures and Materials, The University of Texas at Austin, Austin, TX 78712-0235, USA Received 2 January 2006; received in revised form 30 March 2006; accepted 2 April 2006

Abstract Propagating boundaries of phase transformation have been generated in polycrystalline NiTi specimens under a tensile impact loading condition. Multiple strain gages were used to monitor the time evolution of the strain at different spatial locations in the specimen. Nucleation and propagation of multiple phase fronts were detected in these experiments; the phase front speed was found to be in the range between 37 and 370 m/s. The strain measurements were interpreted through the onedimensional analysis of Abeyaratne and Knowles [1997. On the kinetics of an austenite-martensite phase transformation induced by impact in Cu–Al–Ni shape-memory alloy. Acta Mater. 45, 1671–1683] and a model of partial phase transformation in the polycrystalline specimen. The driving force for the motion of the phase front was evaluated from the measurements in order to establish the kinetic relation. r 2006 Elsevier Ltd. All rights reserved. Keywords: Shape memory alloys; Stress-induced phase transformation; Kinetic relation

1. Introduction Shape memory alloys exhibit many interesting characteristics, the most important being their ability to return to the original shape after large deformations, either on unloading or after a modest increase in temperature. This characteristic arises due to the reversible, diffusionless phase transformation between the martensitic and austenitic phases of the material. Many metallic alloys, and some ceramics and polymers, exhibit such shape Corresponding author.

E-mail address: [email protected] (K. Ravi-Chandar). 0022-5096/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2006.04.003

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2137

memory effects. The discussion in this paper focuses on a nearly equiatomic NiTi alloy. An extensive investigation of the constitutive properties of this alloy is provided in the works of Leo et al. (1993) and Shaw and Kyriakides (1995) who investigated the mechanical response of NiTi specimens at very low loading rates that suggest an equilibrium analysis of the problem; references to earlier work may be found here as well. One way to cause a phase transformation in shape memory alloys is to change the temperature. At room temperature the specimen is in the martensitic phase, with a monoclinic lattice structure. After the temperature is raised past a critical point, As, (austenite start temperature), the specimen begins to transform to the austenitic structure, with a body-centered cubic (bcc) lattice structure. At a temperature Af (austenite finish temperature), the entire specimen is in the austenitic phase. This phase transition from martensite to austenite is an endothermic process. When the temperature is decreased from Af, the specimen undergoes a phase transformation from the austenitic state to the martensitic state, beginning at Ms and ending at Mf. This phase transition is an exothermic process. Stress-induced phase transformation is the other method that causes the shape memory alloy to undergo a phase transformation. In this case, a specimen held at a constant temperature in an austenitic state stretches or compresses elastically upon initial loading. However, at a critical stress level, the bcc lattice structure becomes unstable and the martensitic phase is nucleated at one or more sites in the specimen; this is called stressinduced martensitic transformation (SIM). Upon continued loading, the SIM phase transformation spreads across the specimen and envelops the entire specimen; the speed of propagation of the phase boundary that separates the austenite and martensite is dictated by the rate of loading and perhaps by the specimen geometry. Upon unloading, the reverse process occurs and the austenitic transformation nucleates and subsequently envelops the entire specimen at which point the specimen continues to unload elastically. If the recovery of shape appears after unloading, the material is said to be pseudoelastic and if the recovery is generated by a heat-treatment, then material is said to exhibit shape memory. Leo et al. (1993) and Shaw and Kyriakides (1995) examined the progression of the SIM under extremely slow loading conditions (strain rates in the range of 4
105–4
102 s1); Escobar and Clifton (1995) performed experiments to follow SIM under extremely high rates of loading (strain rates in the range 104 s1) in single crystal specimens of the CuAlNi shape memory alloy; Escobar et al. (1999) also examined NiTi at high strain rates. Results from these investigations are summarized in Section 2 in order to place the present work in the proper context; in this work, we examine the dynamics of phase transformations in NiTi shape memory alloy at strain rates that are in the range 1–100 s1, intermediate to the range considered in these earlier studies. Many investigators have explored the problem of dynamic propagation of phase transformation fronts analytically. Abeyaratne et al. (2001) present a comprehensive discussion of the work in this area; the main outcome of such analysis is that in order to complete the formulation of the problem, a nucleation criterion for the onset of phase transformation and a kinetic relation describing the relationship between the driving force and the speed of propagation of the phase front—both obtained either from micromechanical models or measurements—must be added to the momentum balance equations. The main objective of the present work is to present a direct experimental observation of a dynamically propagating phase front and to determine from such observations, the underlying kinetic relation for the materials.

ARTICLE IN PRESS 2138

J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

This paper is organized as follows: in Section 2, we describe some relevant previous experimental work on the propagation of phase fronts in NiTi alloys. This is followed by a quasi-static characterization of the NiTi used in the present study. The one-dimensional theory of propagation of phase fronts in phase transforming materials is described in Section 3; this is a straightforward summary of the work of Knowles (1999), presented simply for ease of interpretation of the experiments performed in this work. The design of a one-dimensional dynamic tensile loading experiment is described in Section 4. The experimental observations on the dynamic propagation of phase fronts and their interpretation are presented in this section as well. 2. Background on propagation of phase fronts in shape memory alloys Investigations of propagating phase transformations in polycrystalline shape memory alloys have traditionally been at extreme rates of loading. In the case of Leo et al. (1993) and Shaw and Kyriakides (1995), the strain rates were extremely low, in the range of 4
105–4
102 s1. At the other extreme, Clifton and Escobar (1995) and Escobar et al. (1999) performed experiments to follow SIM under extremely high rates of loading (strain rates in the range 104 s1) in single crystal specimens of the CuAlNi shape memory alloy and in NiTi foils where the phase front was left to its own kinetics. It is instructive to review these experiments since they bracket two extremes of loading conditions and provide the point of departure for our investigations. We use the work of Shaw and Kyriakides (1995) who used commercial grade polycrystalline NiTi wires, 1.07 mm in diameter to describe the low loading rate response of the material. The composition was 50.1 at% of Ni with Af ¼ 62 1C. The specimen was instrumented with four custom made extensometers along its 64 mm length. Each extensometer had a 2.5 mm gage length, and was held in place with knife edges and spring clamps. In addition, many of the specimens were also instrumented with a number of thermocouples (with sensing elements 76 mm in diameter). The instrumented specimens were then mounted on an electromechanical test machine and pulled in uniaxial extension at nominal strain rates in the range 4
105–4
102 s1, with the load and the extension monitored along with the strains in the extensometers and the temperature in the thermocouples. The experiments were performed in air and in water in order to vary the thermal conditions; the temperature was varied in the range from 17.6 1C to 100 1C. The result from one typical experiment of Shaw and Kyriakides (1995), performed at 70 1C with a strain rate of 4
104 s1, is shown in Fig. 1. The measured values of the stress and the strain as monitored by the different extensometers during the experiment are shown in Fig. 1a. An xt diagram indicating the progression of the transformation is shown in Fig. 1b. The sequence of events observed is described completely by Shaw and Kyriakides; here we give a summary in order to set the stage for our own investigations. The stress increases monotonically until about 20 s; corresponding to this, the strain in all the gages 1–4 increases monotonically as well. At about 20 s, the stress exhibits a plateau, and all the extensometers indicate a corresponding plateau. This is indicative of nucleation of a localized transformation from austenite to martensite; nucleation of this phase transformation occurs near the top grip due to a local stress concentration there. With increasing applied displacement, the nucleated phase change propagates as a front along the length of the specimen, similar to the propagation of Lu¨der’s bands (Hall, 1970). At

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2139

Fig. 1. (a) Stress and strain histories at four axial positions from an experiment conducted at a displacement rate of d/L ¼ 4
104 s1, (b) x– t diagram of significant events (reproduced with permission from Shaw and Kyriakides, 1995).

around 40 s, a second transformation front is nucleated at the bottom grip again due to local stress concentrations there and begins to propagate upwards. These two fronts of phase transformation propagate steadily towards the center of the specimen as the specimen is stretched; during this propagation all extensometers indicate a constant strain until the front reaches the extensometer location, at which point, the strain jumps through the transformation strain gT and again remains constant. Sequential arrival of the front from gage 4 to 3 and from 1 to 2 can be observed from the strain signals in Fig. 1a and is represented better in the x–t diagram in Fig. 1b. Upon unloading at about 160 s, two propagating phase fronts—of transformation from martensite back to austenite—are formed near the center of the specimen and these propagate steadily towards the top

ARTICLE IN PRESS 2140

J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

and bottom grips. Once again, this propagation occurs at constant stress and each extensometer indicates a constant strain until arrival of the front at the extensometer. At about 300 s, the bands reach the lower and upper grips and the specimen is completely transformed back to austenite. From the x–t diagrams, it is readily seen that the speed of propagation of the phase transformation front is 0.25 mm/s for the austenite-tomartensite transformation and 0.32 mm/s for the martensite-to-austenite reverse transformation. These speeds are dictated by the kinematics of the problem as we will show in _   gþ Þ, where n the number of fronts and s_avg is the average phase Section 3: n_savg ¼ d=ðg front speed. The transformation kinetics cannot be determined from these experiments directly since the sample is in a state of equilibrium. Escobar et al. (1999) performed experiments to follow SIM under extremely high rates of loading (strain rates in the range 104 s1) in NiTi specimens. In these experiments, a NiTi plate (called the flyer) was launched from a compressed gas gun at speeds in the range 200 m/s and made to impinge on a stationary NiTi plate (called the target). A Ti–55.72 wt% Ni polycrystalline alloy with Af ¼ 17 1C was used as the flyer and target. The contacting surfaces of the flyer and target were at an angle with respect to the direction of motion of the flyer, thus imparting a normal stress and shear stress along the contact surface. Compression and shear waves generated upon impact propagate both into the flyer and the target. If a phase front is generated at the impact face, it travels into the flyer and the target at some speed Cp. The normal and transverse velocities are measured on the back surface of the target with interferometers and interpreted in terms of the conditions at the impact interface and the wave propagation in the target. At the back surface, the initial compressive wave arrives at time t1, followed by the arrival of the compressive wave reflected by the propagating phase front, t2, and finally by the shear wave from the initial impact at time, t3. The velocity signals measured on the back surface were used to determine the characteristics of the propagating phase boundary. In particular, imposing the kinematic jump condition, discussed in detail in Section 3, s_ ¼ 1vU=1vU, where 1vU is the jump in the particle velocity across the phase front and 1gU is the jump in shear strain across the phase boundary, the phase velocity can be determined; Escobar et al. (1999) estimated this to be about 327 m/s with a corresponding shear stress ahead of the front of 46.5 MPa. In a variation of the above test, Escobar et al. (1999) used a sandwich configuration in which a 300 mm Ti—56.5 wt% Ni polycrystalline foil (Af ¼ 3 1C) was attached to a steel flyer. Upon impact of the flyer plate on the target, the NiTi foil attains equilibrium conditions rapidly. Once again, measurements of the normal and transverse particle velocities at the back surface of the target were used to extract the response of the NiTi foil. The gradual increase in the transverse velocity that was measured on the target back surface was interpreted to indicate propagation of a diffuse boundary of transformation. A time-dependent Ginzburg–Landau equation was used to model the transformation in the polycrystalline specimen and with the kinetic coefficient in the model obtained by comparison to the experiment. These high-strain rate experiments suffer from two main shortcomings; first, the extremely short duration of the loading pulse allows the phase front to propagate very short distances; for example, at a speed of 300 m/s, the phase front moves through about 300 mm if the loading pulse duration is about 1 ms. Second, direct measurement of the phase front is not possible, and one has to rely on interpreting the velocity measurements on the back surface and infer the conditions at the phase front through interpretations of a model. Thus, it would be more expedient to develop an

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2141

experimental scheme where direct measurements of the motion of the phase front may be obtained. There are a few other high-strain rate tests performed on shape memory alloys. Liu et al. (1999) conducted experiments in the split-Hopkinson pressure bar arrangement; small cylindrical specimens of nearly equiatomic NiTi in the martensitic state were subjected to compressive strains in the range 20% at a strain rate of about 3E3 s1. From these tests, they were able to measure the stress–strain response that was controlled initially by detwinning followed by elastic and plastic deformation of the detwinned martensite. Comparison to the stress–strain response obtained at low strain rates indicated very little influence of the rate of loading except beyond the onset of plastic deformation in the martensite. Lagoudas et al. (2003) attempted to investigate phase front propagation in long rod specimens in a Hopkinson bar arrangement. In this experiment, a 100 ms compressive pulse was generated by impacting a cylindrical steel rod on another long steel rod; this corresponds to the typical input bar in a split-Hopkinson pressure bar apparatus. A longNiTi specimen rod placed in contact with the input bar to transmit this compression was instrumented with a number of strain gages to monitor the passage of the loading wave and possible phase fronts. However, due to the short duration of the loading pulse and the low energy levels available in their experiments, stress-induced martenistic transformation could not be generated and hence phase front propagation was not observed. Instead, only detwinning deformation of martensite could be observed and this was shown to be a dispersive wave just like plastic waves. For analysis of the phase transformations from our experiments, the constitutive model for the material must first be established experimentally; we establish this from a slow strain rate tensile test since the modulus of both phases should be strain rate insensitive. The shape memory alloy used was polycrystalline NiTi with a composition of 55.84 wt% Ni and 44.07 wt% Ti. The specimens were flat annealed with an oxide surface and had an Af around 80 1C. Each specimen had a thickness of 0.254 mm and was cut to have a width of 5.08 mm. The specimen (Test # 4-22-05) was instrumented with a strain gage for monitoring the local response; this specimen has a gage length of 65.71 mm and the strain gage (with a gage length of 3 mm) was fixed at a distance of 22.21 mm from the top grip. Once the specimen was in place in the test machine (Instron Model 4482), the test chamber was brought up to a temperature of 95 1C. The specimen was pulled at a crosshead rate 0.762 mm/min corresponding to a nominal strain rate of 1.9
104 s1; clearly equilibrium conditions could be assumed for this test. The variation of the nominal stress and strain monitored by the strain gage as a function of the nominal strain (cross-head position divided by the gage length) is shown in Fig. 2a. The initial linear variation of stress and strain corresponds to the austenitic phase. At about 523 MPa, stress-induced martensitic transformation occurs at some location in the specimen, possibly at one of the grips. This is marked with a dotted vertical line that crosses the strain axis at a nominal strain of 0.0149. With further extension of the specimen, the phase transformation propagates as a front with very small increment in the stress; the strain gage, located in the austenitic region, indicates a strain that is proportional to the load. Clearly, many barriers to front motion can be identified by the gradual increase and abrupt drops in the stress. The stress–strain data for this test verifies that as the phase transformation passes a barrier, the stress drops and as a result the material at the strain gage unloads elastically and then reloads elastically. Finally, when the phase front arrives at the strain gage location, the gage registers a large increase in strain. At a nominal

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161 800

0.08

Stress - (MPa)

600

0.06

σ

400

0.04

Strain

2142

ε 200

0.02

0 0

0.01

0.02

0.03

(a)

0.04 0.05 0.06 Nominal Strain -
/L

0.07

0.08

0.09

0 0.1

700 600

Stress (MPa)

500 400 300 200 100 0 0

(b)

0.01

0.02

0.03

0.04 Strain

0.05

0.06

0.07

0.08

Fig. 2. (a) Quasi-static test of NiTi with a strain gage signal from 22.21 mm below the top grip (Test # 4-22-05), (b) stress–strain curve for quasi-static test of NiTi using a strain gage signal from 22.21 mm from the top grip (Test # 4-22-05).

strain of 0.058, the strain gage signal rapidly increases from 0.011 to 0.07 as can be seen in Fig. 2a. We view this as a demonstration that strain gages are capable of monitoring the large strains associated with the arrival of the phase front. From correlating the measured stress and strain at the gage location, we show the stress–strain path followed by a material point in Fig. 2b; in particular during propagation of the transformation through the strain gage area, the stress is at a constant level, sp ¼ 523 MPa. At a nominal strain of 0.085, the material began to load elastically in the martensitic phase. Upon unloading, the material exhibits an elastic response with some small hysteresis in the martensitic phase. At a nominal strain of 0.062, the reverse phase transformation arrives at the strain gage and the strain changes from 0.051 to 0.0144 gradually. A nucleation peak is not observed since the transformation from martensite to austenite usually occurs easily at the center of the specimen where the austenite-to-martensite

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2143

transformation bands merged (see Shaw and Kyriakides, 1995). The strain at the gage then remains constant until the phase front has enveloped the entire specimen, at which point the material unloaded elastically until the specimen is completely unloaded. It can be seen that there is a residual strain in the strain gage once the material is completely unloaded. Once completely unloaded, the specimen was then cycled multiple times at the same strain rate, always unloading just as the stress in the martensitic state reached about 750 MPa. It should be noted that on the first cycle the phase transformation front has to overcome a few barriers as it grows from the nucleation site throughout the bar. In subsequent cycles, these barriers to the phase transformation disappear; on subsequent cycling, the stress level does not reach values as large as the previous loading cycle during phase transformation. However, the material appears to approach a stabilized loading-unloading curve with some small permanent deformation with each given cycle. The dynamic experiments reported in this paper were performed mostly on virgin specimens and hence we expect that they exhibit the behavior corresponding to the one displayed in Fig. 2; the cycled specimens do indicate significant differences in the propagation of the phase fronts, and these will be analyzed in a future contribution. 3. Dynamics of phase transformations We now turn our attention to the problem of one-dimensional tensile wave propagation in materials capable of phase transformations. The formulation described here is a summary of the relevant aspects of the development presented by Knowles (1999). Consider a one-dimensional rod occupying xX0; the governing equations of motion are, s0 ðgÞgx ¼ rvt , vx ¼ gt ,

ð1Þ

where r is the mass density, gðx; tÞ ¼ qu=qx  ux is the Lagrangian strain, uðx; tÞ is the displacement, vðx; tÞ ¼ qu=qt  ut is the particle velocity. The first equation represents balance of momentum and the second equation represents compatibility conditions. sðgÞ is the nonlinear stress–strain relationship, and must be specified for the material. This form of the equations of motion for one-dimensional waves in a nonlinear material was derived independently by Rakhmatulin (1945), Taylor (1958) and von Karman and Duwez (1950). In the present problem, the NiTi shape memory material is assumed to be a two-phase elastic material that exhibits an idealized trilinear stress–strain curve as shown in Fig. 3. The material begins to deform elastically in the austenite phase with an elastic modulus of mA until it reaches a peak stress value sM. At this point, the material undergoes a stressinduced phase transformation to martensite. The stress value at which the material is completely transformed to martensite is sm. The strains that correspond to these two stress levels are gM and gm, respectively. In addition, the thermal transformation from austenite to the martensite phase at zero stress results in a transformation strain of gT. In the fully transformed state, the martensite is again assumed to be linearly elastic with an elastic modulus of mM. The branch of the stress–strain curve that lies between gM and gm, is unstable since the strain increases with decreasing stress; the stress–strain law can

ARTICLE IN PRESS 2144

J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

 M

* µA

µM

m

M

T

m



Fig. 3. The trilinear constitutive model for two-phase material (Knowles, 1999).

then be written as: 8 mA g; > > hn > o > mA mM
> > < gM gm gM  mA gm  gT g   i sðgÞ ¼ mM mM >  1 g g  g g þ > M m mA mA M T > > > > : m
g  g ; M T

0ogogM ; !

gM ogogm ; (2) g4gm :

In terms of experimental calibration of such a trilinear model, we note that the elastic modulus of the austenitic phase, mA, the martensitic phase, mM, and the transformation strain, gT, are readily measured from the experimental results shown in Fig. 2. Using the strain gage data from Test # 4-22-05, the Young’s modulus of the austinitic phase was found to be mA ¼ 64.8 GPa and the Young’s modulus for the martensitic phase was found to be mM ¼ 41.4 GPa. The transformation strain gT is found by extrapolating the straight line fit for the martensitic region until it intersects the strain axis: gT ¼ 0.058. On the other hand, the critical stress and strain levels (gM, sM) and (gm, sm) that correspond to the nucleation levels for austenite-to-martensite transformation and martensite-to-austenite transformation, respectively, cannot be determined from such quasi-static tests since nucleation levels measured in such tests are influenced significantly by defects, barriers and stress concentrations. The plateau stress measured in the experiment corresponds to the steady-state propagation stress and is influenced by the specific geometry in which the localization occurs. Nevertheless, the construction of this trilinear constitutive model helps in understanding the dynamics of phase transformations. If the entire specimen is either in the austenitic or martensitic state, the equations of motion reduce to the wave equation distinct wave speeds corresponding pstandard ffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiwith ffi to each phase: cA ¼ mA =r and cM ¼ mM =r. If, however, the loads are imposed dynamically, it is possible to form separate regions of different phase; furthermore, such

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2145

boundaries, called phase boundaries, may move at appropriate speeds dictated by the energetics of the problem. These moving phase boundaries represent jump discontinuities in the field quantities that must satisfy appropriate jump conditions: 1sU þ r_s1vU ¼ 0, 1gU_s þ 1vU ¼ 0,

ð3Þ

where 1U denotes a jump in the quantity across the phase front and s_ is the speed of the phase boundary. Eliminating the particle velocity between the two conditions in Eq. (3), we obtain r_s2 ¼

1sU . 1gU

(4)

Clearly, the speed of the phase front is left undetermined in the above formulation since there is no restriction on the state on either side of the phase boundary, other than that they be located in the different phases. The energy dissipation in the process of moving the phase boundary may be examined to determine the restrictions posed by this: the dissipation rate in any interval ðx1 ; x2 Þ can be written as the difference between the rate of external work and the rate of change of kinetic energy and potential energy, Z o d x2 n r 2 v þ W ðgÞ dx, (5) DðtÞ ¼ sðx; tÞvðx; tÞxx21  dt x1 2 where W is the strain energy per unit reference volume. The jump relations are used to write down the dissipation across the phase boundary: 1 D ¼ 1svU þ 1W U_s þ 1 rv2 U_s. (6) 2 This can be simplified further by eliminating the particle velocity vðx; tÞ; the dissipation rate is then written as 1 f  W ðgþ Þ  W ðg Þ  ðsþ þ s Þðgþ  g Þ. (7) 2 where f is the driving force for the phase transformation. Note that the superscripts + and  denote the state ahead and behind the moving phase boundary. Eq. (7) represents the driving force for a phase front that goes from ðgþ ; sþ Þ to ðg ; s Þ. For the trilinear material model used here, f can be evaluated explicitly in terms of the state ahead and behind the front. However, the speed of the phase boundary is still left undetermined since f40 may be satisfied for an infinite combination of + and  states. For a given state ðgþ ; sþ Þ, the end state ðg ; s Þ and the propagation speed s_ must still be determined. Abeyaratne and Knowles (1991, 1997) indicate that this determination will come from an additional equation called the kinetic relation of the form: D ¼ f  s_40;

f ¼ jðs_Þ.

(8)

Such a relation must be obtained either from a thermodynamic modeling of the nucleation and growth of the phase transformation or from experiments aimed at determining such a relation; we follow the latter approach in this paper. In the case of an equilibrium problem, it is possible to obtain the speed of the phase front from kinematic considerations alone. Consider a bar of length L subjected to an _ let a phase boundary be located at a position x ¼ sðtÞ; extension of one end at a rate of d;

ARTICLE IN PRESS 2146

J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

the total elongation of the bar d is split between the austenitic and martensitic states as:   s s d ¼ g s þ gþ ðL  sÞ ¼ þ gT s þ ðL  sÞ. (9) mM mA At this point, there is no unique solution for the mixed-phase state of equilibrium since infinitely many pairs ðs; sÞ are possible for any given extension. The appropriate solution is found by using the minimum energy criterion. The energy of the bar is minimized when the stress in the bar is equal to the Maxwell stress, s . The Maxwell stress for the trilinear material is shown in Fig. 3 and breaks up the curve so that the triangle between s and sM and the triangle between s and sm are of equal area. There is still another possible source of nonuniqueness: the above considerations dictate that a segment of length s would have transformed to martensite and the remaining segment of length L–s would remain as austenite. There is no requirement, especially in a polycrystalline material, that there be only one nucleation site and that all regions of martensite be contiguous. Multiple bands of transformed material may exist with the total adding up to a segment of length s. Typically, however, in experiments at the slow strain rates (as in the Shaw and Kyriakides experiments) only one or two bands nucleate, mostly from the end-clamp regions and move into the main gage section of the specimen. Such slow rate tests may be considered as a sequence of equilibrium states and Eq. (9) may be used to determine the time variation of the position of the phase front. In this case, we can take s fixed at sn and show that
 _ s_ g  gþ ¼ d. (10) If there are n such fronts each moving at a uniform speed, then the speed of each front is simply s_=n. Shaw and Kyriakides (1995) observed this in their experimental measurements. Sylwestrowicz and Hall (1951) and Moon and Vreeland (1968) measured such kinematically constrained propagation of Lu¨der’s bands and made similar observations. 4. Experimental investigation of phase transformation in NiTi alloy We now describe the details of an experimental investigation of the dynamics of phase transformations in a NiTi alloy. A detailed description of the experimental arrangement— essential to the understanding and interpreting the observed results—is presented first. 4.1. Experimental design The experimental arrangement used to monitor propagating phase fronts in NiTi under conditions of dynamic loading is not too different from the original Hopkinson (1901) experiment. In the original experiment, Hopkinson hung a bar of the specimen material from the ceiling and loaded it dynamically by impacting a flange at the bottom end of the specimen with a falling weight. This generated a tensile pulse that propagated towards the top end of the specimen. von Karman and Duwez (1950) used a similar arrangement, but propelled a projectile to generate the tensile impact. Our experimental arrangement is similar to these experiments in that one end of the specimen is fixed while a velocity is imposed at the other end; a schematic diagram of this arrangement is shown in Fig. 4. The NiTi specimen is a strip 0.254 mm thick and 5.08 mm wide; the length of the specimen is 70 mm, with an additional 20 mm underneath the end clamps. At the right end, the

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2147

Fig. 4. Schematic diagram of the experimental arrangement for tensile impact on a shape memory alloy specimen.

specimen is clamped into a fixture that is attached to a 25 mm diameter, 2 m long steel bar. This bar runs through the projectile launching tube and is clamped just after exiting the closed end of the projectile tube. At the impact end of the specimen, the clamp holding the specimen is attached to a flange which serves as the target for the projectile. A steel tube is used as the barrel for launching the projectile and directs pressurized air onto a smaller tube that serves as the projectile. The projectile’s dimensions are as such that it slides freely between the launch-tube and the fixed bar that holds one end of the specimen. The hollow cylindrical projectile (inner and outer diameter of 25 and 34.62 mm) impinges on the solid flange at a speed V. The flange that holds the NiTi specimen is 50.62 mm in diameter and 15.48 mm thick. The projectile is just long enough (150 mm) so that when it hits the flange its back end is partially over the fixed bar. This arrangement guides the projectile over the specimen and onto the impact flange. This allows the projectile to hit the flange in the same place in a repeatable manner and also to load the specimen to a large enough strain level so that complete SIM phase transformation takes place along the entire specimen. In order to heat the material to the austenite phase, a heating tape was used. This heating tape was fixed to the inside of a glass tube that had a large enough diameter to allow free projectile motion. The glass tube was long enough that it was held in place by inserting one end of the glass tube over the launch tube. Thus the specimen can be heated to the appropriate temperature while the projectile is allowed to pass by without being deflected by the heating tape or the specimen. The temperature of the specimen was controlled with an Omega Model CN2110-R20 controller; all of the dynamic experiments reported here were performed when the specimen reached 951C. In order to measure the strains and the arrival of the phase front, a strip of strain gages and single gage were used as shown schematically in Fig. 5a. A photograph of the specimen near the impact end with the strip strain gages and its wiring is shown in Fig. 5b. The strip of gages used was Vishay Micro-Measurements EA-06-031MF-120 with option SE. Fig. 5a shows the grid pattern for this strain gage strip. The strain gages in the strain gage strip each had a gage length of 0.79 mm and were spaced 2.03 mm apart from the centerlines of the gage pattern and were electrically independent. Thermal compensation was not very important in this experiment since the test does not last longer than 2 ms. The single gage used at the other end of the specimen was the Kyowa Strain Gage KFG-3-120-C1-11L3M2R which has the lead wires already attached to the strain gage. The single gage had a gage length of 3 mm. Both types of strain gages were chosen due to their very short gage lengths. This is important since the gage effectively averages

ARTICLE IN PRESS 2148

J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

Fig. 5. (a) Schematic diagram of placement of strain gages on the specimen, (b) photograph of a specimen with gages in tensile impact setup.

the strain experienced under the entire gage length. If a jump in strain travels across the strain gage, then the strain gage will smooth out the jump in its output signal over a finite time which is based on its gage length. The high temperature adhesive M-Bond 600 from Vishay Micro-Measurements was used since the temperature during the experiment is 95 1C. One interesting point to note about the adhesive and the strain gages is that the manufacturer stated a strain limit of 3% while the strain gages routinely experienced strains of 7% and sometimes as large as 9.5% during the dynamic experiments without peeling. The gages were applied as close as possible to the clamped ends while leaving enough room for the wires. Holes were drilled completely through the central portion of the impact flange and the clamp on the impact flange so that the wires could leave the setup without interfering with the projectile or the specimen. A copper sheet was attached to the bottom of the specimen and had bondable terminals, which allowed interconnection between the strain gages and the Wheatstone bridge amplifier. Thin wires were used to connect the strain gages to the bonding terminal due to the close spacing of the gages and the need to bend the wires over without breaking the solder joint. The thicker wires were used to connect to the amplifier circuit since they would not break as easily as the thin wires during the process of loading the specimen. The projectile impacting the specimen is assumed to be in constant contact with the flange being impacted and both travel at a constant velocity. No direct measurement was made to determine the end condition of the specimen. An accelerometer was attached to the impact end; however, it did not yield data that could be analyzed quantitatively in terms of the speed of the flange end of the specimen; but the signal produced was used to trigger the oscilloscopes recording the data from the strain gages. The strain gages were connected to a Wheatstone bridge amplifier (Vishay Measurements Group Model 2210A) whose output was connected to three high-speed digital oscilloscopes. Three oscilloscopes stacked on top of each other had to be used since each oscilloscope had only four input

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2149

channels. Strain gages 1–4 were connected to the four channels of the bottom oscilloscope. The signal from the fourth gage was also connected to the first channel of the middle oscilloscope; this served as a way to get the multiple oscilloscope recordings synchronized with respect to each other. Strain gage signals from channel 5 to 7 were then connected to the three remaining inputs for the middle oscilloscope. The seventh gage was also connected to the first channel of the top oscilloscope so that the top oscilloscope could then be synchronized with the middle and bottom oscilloscopes. The eighth and ninth gages were then connected to two of the remaining input channels of the top oscilloscope. The last channel of the top oscilloscope was reserved for the single strain gage signal if the single strain gage was being used; if not then the 10th gage of the strain gage strip was connected to this input channel. Fifteen thousand samples were taken at a rate of five mega samples per second. The voltage output of the Wheatstone bridge amplifier is then converted into strain values by using the calibration measurements taken prior to the test. In order to keep the specimen from breaking before a phase transformation passed through the entire specimen, a cylindrical tube was clamped behind the back end of the impact flange to limit the maximum strain experienced by the specimen. A cylindrical tube was used so that the accelerometer attached on the back end of the flange is not damaged. The cable for the accelerometer and the strain gage wires pass through the tube so that they are not pinched between the impact flange and the clamped tube. It was found that if the cylindrical tube is clamped at a distance from the back end of the flange so as to allow an average strain of the specimen to reach 10% then the specimen would not break and the specimen would be allowed to undergo a full phase transformation from the austenitic phase to the martensitic phase. 4.2. Experimental results and interpretation Table 1 lists the specimen conditions and gage locations for each of the quasi-dynamic tests. The specimen condition refers to history of the specimen either as being a virgin material or that it has been cycled through SIM as described in Section 2. The pressure level refers to the pressure used to launch the projectile that impinges the impact flange. The location for both the strip of gages and singles gages is the distance between the impact end clamp and the middle of the first gage. All the tests, except Test # 1-7-2005, have the strain gage strip near the impact end. The strain gage strip in Test # 1-7-2005 is near the fixed end with Gage # 1 as the strain gage closest to the fixed end. A few other preliminary trials were also made to debug the experiment; data from these were not used in interpreting the propagation of phase fronts. Raw experimental measurements of gðx; tÞ for each of the tests listed in Table 1 are shown in Figs. 6–9. Table 1 Specimen condition and strain gage locations Test #

Specimen condition

Pressure level (psi)

Distance of Gage # 1 from impact end (mm)

Distance of single gage from impact end (mm)

11-8-2004 12-30-2004 1-4-2005 1-7-2005

Virgin Virgin Cycled Virgin

60 100 100 100

6.16 9.46 8.06 63.26

— 61.91 63.08 7.49

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2150 0.06

0.05

Strain

0.04

Ch 1

0.03

Ch 3

Ch 2

4

5 10 6 9

Ch7&8

0.02

0.01

0 0

0.0001

0.0002

0.0003 Time (s)

0.0004

0.0005

0.0006

Fig. 6. Strain history of impacted shape memory alloy, NiTi, during Test # 11-8-04.

0.1 0.09 0.08 0.07

Strain

0.06 0.05

Extra Gage

Ch 1

Ch 2

0.04

Ch 3 4,7,9

0.03

Ch 5

0.02 0.01 0 0

0.0001

0.0002

0.0003 Time (s)

0.0004

0.0005

0.0006

Fig. 7. Strain history of impacted shape memory alloy, NiTi, during Test # 12-30-04, nearly vertical increase in strain signal is associated with breaking of lead wires.

We now turn to a description and interpretation of the results from the tensile impact experiments. A typical experimental result shown in Fig. 7 is used as an example. This figure shows the strain output of the different strain gages as a function of time for Test #

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2151

0.1 0.09 0.08 0.07 Ch 5

Strain

0.06 0.05

Ch 9 Extra Gage

0.04

Ch 3 Ch 1

Ch 2

2 4

Ch 4

0.03 Ch 8 & 9

Ch 1

0.02 0.01 0 0

0.0001

0.0002

0.0003

0.0004

0.0005 Time (s)

0.0006

0.0007

0.0008

0.0009

0.001

Fig. 8. Strain history of impacted shape memory alloy, NiTi, which has been cycled, during Test # 1-4-05.

0.1 0.09 0.08 0.07

Strain

0.06 0.05 Extra Gage

0.04

3

4

5

6

9

Ch 7 & 8

0.03 Ch 2

0.02 Ch 1

0.01 0 0

0.0001

0.0002

0.0003 Time (s)

0.0004

0.0005

0.0006

Fig. 9. Strain history of impacted shape memory alloy, NiTi, during Test # 1-7-05, Ch 1 and 2 indicate peeling of the strain gage. Gage # 1 is the closest gage to the fixed end.

12-30-04. The time reference is based on the signal from the accelerometer that was used to trigger the oscilloscope. The specimen was initially at 95 1C, in the austenitic state with a pffiffiffiffiffiffiffiffiffiffiffi one-dimensional bar wave speed C A ¼ mA =r ¼ 3566 m/s.

ARTICLE IN PRESS 2152

J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

The stress amplitude and pulse shape of the initial elastic wave from the impact depends on the impedance mismatch and the relative length of the projectile and the flange. This is, in fact, the main advantage of this loading scheme over the split-Hopkinson bar apparatus; the loading on the specimen is spread over a time greater than about 500 ms. The initial elastic wave front propagates the length of the specimen in about 19.5 ms and hence reflects many times between the two ends of the specimen. Thus, this particular test should not be interpreted as a wave propagation test, but as a quasi-dynamic test, based on the strain signals monitored on the specimen. As can be seen in Fig. 7, all the strain gages on the specimen, separated spatially by 2.03 mm from each other, indicate an identical increase in strain with time (until about 200 ms) suggesting that the strain evolves homogeneously in the specimen (at least along the 2 cm length over which the strain gages are affixed); thus we may write gðx; tÞ ¼ g_ 0 t. Using strain compatibility, we get an estimate of R Lthe imposed speed at the flange end of the specimen: vx ¼ gt ¼ g_ 0 ; integrating vð0; tÞ ¼ 0 gt dx ¼ g_ 0 L, since vðL; tÞ ¼ 0. Thus, the impact loading, typically in the range 2.8–6.3 m/s, provides a uniform strain along the specimen with a constant strain rate g_ 0 40  90 s1 that is significantly larger than that obtained in the tests of Shaw and Kyriakides (1995), but also significantly smaller than in the plate impact tests of Escobar et al. (1999). This large strain rate, sustained for times on the order of 500 ms, enables this experimental setup to provide an appropriate condition for evaluation of the propagation of phase boundaries. With this interpretation of the dynamic experiment as a constant strain rate test, there are a number of key observations that can be made from the experimental results displayed in Fig. 7.







At about 193 ms, Gage # 1 (Ch 1) indicates a significant departure from the uniform strain response shown by all of the other gages. In fact, the strain increases rapidly from about 0.0142 to 0.0529 (in about 96.6 ms) suggesting that a stress induced martensitic transformation has occurred. It should be noted that averaging of the strain signal over the gage length (790 mm) occurs as the phase transformation spreads across this gage element; we will estimate the effect of this averaging in Section 4.2.1. At 266 ms, Gage # 2 (Ch 2) exhibits a rapid strain increase from about 0.01755 to 0.0501, spread over a time interval of 95 ms. Subsequently, Gages # 3, # 4, etc. indicate the sequential onset of SIM at each gage location. This behavior is very similar to that observed by Shaw and Kyriakides (1995) and is a clear indication of the propagation of the phase front from the impact end towards the fixed end of the specimen. It is essential to determine that the strain increase observed in these experiments corresponds to phase transformation from austenite to martensite and not to plastic deformation in the austenite. This was explored by monitoring the strain in the specimen at longer times, when unloading of the specimen occurs. As shown in Fig. 8 at 380 ms, Gage # 1 (Ch1) shows a rapid decrease in strain from 0.04645 to 0.02052, spread over a time interval of 20 ms. At 453 ms, Gage # 2 is the next gage to exhibit a rapid decrease in strain. Subsequently, other gages exhibit a sequential drop in the strain to levels that are about 2%. This rapid decrease in strain is the result of reverse phase transformation from martensite back to austenite upon unloading, just as observed in the quasi-static tests. However, in the dynamic unloading, this transformation propagates as a front with characteristics similar to the forward transformation front. We take this strain measurement to be an indication that the observed changes in strain during loading are due to stress-induced martensitic transformation. However, we have

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161









2153

not analyzed the reverse transformation in detail to explore its kinetics in detail and we will not consider this aspect further in this paper. By tracing the time of arrival of the strain jump at each strain gage, we can estimate the propagation speed of the phase transformation front. For the data shown in Figs. 6–9, this speed was estimated by monitoring the arrival time of a strain level of 3.5% at each gage. The resulting variation of the phase front speed in the range of 37–369 m/s is significantly larger than those observed in the kinematically controlled experiments of Shaw and Kyriakides (1995) (about 0.25 mm/s), but in the range of speeds estimated by Escobar et al. (1999) in the plate impact experiments. The speed of the phase front is discussed further in Section 4.2.2. A rather striking feature of the experiment is that local strains along the length of the specimen may reach about 3%, before arrival of the propagating phase front; such strain levels have never been observed in the austenite in equilibrium or quasi-static experiments. We explore this more carefully in Section 4.2.3. A careful examination of the strain signals in Fig. 6 indicates that gages 7–10 undergo SIM in reverse order–i.e., a phase transformation front is propagating from Gage # 10 towards Gage # 7, independently of the front moving from Gage # 1 towards Gage # 6; eventually the two fronts merge. This points to the nucleation of multiple phase fronts, a point already noted by Shaw and Kyriakides at strain rates of about 102 s1. We explore nucleation of multiple phase fronts in Section 4.2.4. Finally, the propagating phase front exhibits dispersion. The broadening of the strain signal observed in Fig. 7 suggests that the phase front is smeared spatially over a much longer length as it propagates through the polycrystalline specimen. Such broadening was not observed in quasi-static experiments (see Fig. 1) and is attributed to the polycrystallinity and the adiabatic nature of the dynamic experiments. This is discussed further in Section 4.2.5.

In a dynamic test of a shape memory alloy, it is assumed that the specimen is in an adiabatic state during the propagation of a phase front. This assumption can be justified by estimating the diffusion length in one-dimensional heat conduction. The characteristic  diffusion length is l th k=V , where V is the velocity of the phase boundary and k ¼ k rcp is the diffusivity; k is the thermal conductivity, r is the density, and cp is the specific heat of a material. The characteristic diffusion length can be estimated by taking V to be the phase boundary propagation speed. From the material properties listed in Table 2, the diffusivity for NiTi is found to be 3.33
106 m2/s. In the slow rate test performed in an Instron, the band moves at about V ¼ 0.25 mm/s and hence the diffusion length is about 1.3
102 m; clearly the temperature ahead of a moving front will increase significantly, thereby raising the stress required to nucleate and propagate the band. This results in the suppression of further nucleation and stabilization of the propagating front in the slow rate tests as

Table 2 Thermodynamic properties of NiTi (source: [15]) k (W/(m1C))

r (kg/m3)

cp (J/(kg 1C))

18.0

6450

837.36

ARTICLE IN PRESS 2154

J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

reported by Shaw and Kyriakides (1995). On the other hand, the phase boundary front may travel at about 100 m/s during dynamic loading so that the characteristic length is 3.33
108 m which allows the assumption of adiabatic conditions to be made. 4.2.1. Strain averaging over the gage length The signals measured with the strain gages should be interpreted appropriately in order to account for time averaging due to the finite gage length of the sensor. A simple estimate of this effect may be obtained by considering a step wave of strain traveling at a speed Cp through the strain gage of lengthlG. The signal monitored by the strain gage will have a rise time of approximately t ¼ l G C p . For the strip strain gage elements used in this study, lG ¼ 0.79 mm; thus for a typical speed of propagation of the phase front of C p ¼ 50 m/s, the rise time is about 15.8 ms. A scale mark with this time is indicated in Fig. 7 on Gage # 1; this estimate suggests that while strain averaging definitely plays a role in dictating the steepness of the observed strain signal, the broadening of the strain pulses as it moves down the length of the specimen is not entirely due to this averaging. Some inherent dispersion in the phase front propagation is also evident. 4.2.2. Propagation speed of the phase front The time and location at which the phase transformation begins are not known. However, by following the progression of the strain in the various gages, it is possible to reconstruct the motion of the phase front. A graphical representation that is extremely useful in interpreting the measured strain data can be obtained by plotting an (x, t) diagram with the strain at each point in this diagram indicated by a color map. Such a map is shown in Fig. 10 for one of the tests; the propagation of the phase fronts is identified very easily in this diagram as the motion of the high-strain amplitude points in the (x, t) plane. It is clear from Fig. 10 that two bands propagated within the strain gage strip region, one moving from left to right and the other moving from right to left. Furthermore, the speed of propagation of these two fronts appears to be quite different, even though the strain jumps appear to be about the same magnitude. While the distance between strain gages is known, the exact time of arrival of a phase front cannot be identified, particularly due to the signal averaging and dispersion discussed above. In order to obtain a simple and consistent quantitative estimate of the phase front speed, the time at which a strain level of 3.5% was reached, denoted t3.5, was used as a measure of the time at which the phase transformation arrived at any gage location; t3.5 at each of the nine strain gages, corresponding to three different tests is shown in the x–t plot in Fig. 11. For Test # 12-30-04, the speed was in the range 37–370 m/s. 4.2.3. Nucleation and growth of phase fronts One rather startling result of these experiments is that prior to the arrival of the phase transformation front, the NiTi specimen appears to be strained to significantly higher strain levels than that at which phase transformation is observed in the quasi-static tests. For example, in Test # 12-30-04, Gage # 7 indicates that the strain prior to the arrival of the phase front at 420 ms is about 3.11% as marked in Fig. 7. If the austenite is considered to be a linearly elastic material up to all strain levels measured prior to the arrival of the phase front, then at the time of arrival of the phase front at the Ith strain gage in Test # A 12-30-04 the strain–stress state may be identified as a pair ðgA I ; mA gI Þ. At strain levels in the range 2–3%, the stress works out to be in the range 1.2–2 GPa; if this is true, it would

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2155

x 10-4 0.06 1 0.05

Time (s)

2

0.04 0.03

3

0.02 4 0.01 5 0 2

3

4

5 6 Strain Gage #

7

8

9

Fig. 10. x–t color map for strain gages 2–9 for Test # 1-07-04. The color scale bar denotes the level of strain. Gage # 2 is the closest of these gages to the fixed end.

600

500

Time (µs)

400

300

200 Test# 11-8-04 Test# 12-30-04 Test# 1-7-05

100

0 0

10

20

30 40 Position (mm)

50

60

70

Fig. 11. Time of arrival for phase transformation (e ¼ 3.5%) as a function of position for each gage for each of the quasi-dynamic tests.

correspond to the highest stress levels ever observed in the austenite state at that point in the specimen. This estimate of the stress levels (denoted as ‘‘Model-A’’ for further reference) is extremely high and brings to question whether it is appropriate to assume that the material remains in the austenite state and strains homogeneously. This motivates us to consider a second possibility; if homogeneous nucleation of phase transformation occurs

ARTICLE IN PRESS 2156

J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

at multiple locations, the 790 mm region beneath the strain sensing element could be in a mixed state of austenite and martensite, thus accumulating strain to the levels observed, but without a distinctly propagating phase front. The moving phase front that was nucleated elsewhere then envelops this region and transforms most of the remaining austenitic material to martensite. In this model, we cannot determine the stress at the gage locations from the measured strain, but we will simply consider that the stress remains P constant at the propagation level, sP ; thus, the pair ðgA I ; s Þ represents the strain–stress state at that point (denote this as ‘‘Model-B’’). The available experimental observations do not yet permit us to determine which of these two models is appropriate. Comparing estimates of the nucleation stress and propagation stress for Lu¨der’s bands in steels, one cannot discount either possibility. The difference between the quasi-static and dynamic propagation of the phase front can be understood by considering the time sequence of events in each case. In the quasi-static test, the phase front nucleates at a stress concentration near the grips; thus, as pointed out earlier, the true nucleation state ðgM ; sM Þ cannot be measured in this quasi-static test setup. The peak stress observed in the quasi-static test is the nominal stress when nucleation of phase transformation occurs near one of the specimen grips ðgN ; sP Þ. Upon onset of this band, further progression of the phase transformation occurs not by additional nucleation at other sites, but by the propagation of the phase front. The stress for propagation dictated by the inhomogeneous deformation near the front end is typically lower than the stress required to nucleate phase transformation homogeneously. Hence the stress in the entire specimen never exceeds the propagation stress. Furthermore, since nucleation occurred at the weakest location (and/or the location with the largest local stress concentration), during propagation of the phase front, the austenitic regions remain at a constant stress and strain state, well below the nucleation threshold at any other point in the specimen. In contrast, in the dynamic test reported here, the strain rate is uniform everywhere and the nucleation of a band at any location does not significantly unload any other region; thus regions away from the band continue to strain either homogenously in the austenitic phase (Model-A) or heterogeneously with local transformations to martensite in individual grains (Model-B). M A similar attempt at determining the stress–strain state for each gage, ðgM I ; mM ðgI  gT ÞÞ M where gI is the strain in the martensitic state after passage of the phase front leads to unrealistic estimates of the stress because gM I ogT for all the gages. We have used the isothermal, quasi-static stress–strain behavior in making this estimate even though the region behind the phase front experiences a small temperature change (of about 10–12 K assuming no losses); this is because this temperature increase does not change gT significantly (see Shaw and Kyriakides, 1995) and hence the dilemma of gM I ogT persists. These measurements point to the need for additional experiments with more diagnostic instrumentation, but for now, we will analyze the present experiments through a model incorporating incomplete transformation to martensite in these polycrystalline specimens after passage of the phase front. 4.2.4. Multiple nucleation of propagating phase fronts Regardless of how the homogeneous straining to above 2% is interpreted, the experimental observations clearly indicate the propagation of a front of phase transformation. In fact, nucleation and propagation of multiple phase fronts can be confirmed from the experiments performed. The arrival time t3.5 from three tests, Test #

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2157

11-08-04, Test # 12-30-04 and Test # 1-7-05 is shown in Fig. 11 with respect to position on the specimen. In Test # 12-30-04, the 10-gage strip was attached with the center of the first gage located at 9.46 mm from the impact end and single strain gage was placed with its center at 8.09 mm from the fixed end of the specimen. The single gage experienced a strain jump 56.6 ms before the first gage near the impact end. In Test # 1-7-05, one strain gage was mounted at 7.49 mm from the impact end of the specimen and the 10-gage strip was attached with the middle of the first gage at 6.74 mm the fixed end. In this case, the gage in the strip closest to the fixed end experienced a jump in strain 75.7 ms before the single gage near the impact end. From Fig. 11, it is clear that a band is nucleated at both the impact end and the fixed end; furthermore, the first nucleation of the propagating phase front appears to occur near the fixed end. That additional nucleation sites occur is evident from the fact that t35 was always smaller for the gage farthest from the specimen end than the nearest gage, regardless of which end of the specimen. From Fig. 11, it is clear that the strip gage recorded a phase front moving through the strip from either ends, regardless of whether the strip was placed near the impact end or the fixed end. The speeds determined from their motion indicate that there must have been at least one more moving phase front nucleated at some location in the interior of the specimen. 4.2.5. Dispersion of phase front As indicated earlier, we have used t3.5 to identify the location of the propagating phase front and characterize its speed; this was partly due to the signal averaging over the gage length of the strain gage element, but also partly due to the observed dispersion of the phase front. As the phase transformation traveled away from the nucleation site, each strain gage recorded the transformation over a longer duration indicating an equivalent spatial broadening of the phase transformation front. In the first gage or two, the phase transformation front is indicated by a very steep rise in the strain level seen in the strain gages (with a duration t ¼ 95 ms); for subsequent gages t increases significantly. This dispersion of the phase transformation front may be the result of the effects of a polycrystalline material as illustrated in Fig. 12. The individual grains experience a phase transformation front that travels in a direction along a crystal plane of favorable orientation with respect to the loading axis, but the ensemble ‘‘phase front’’ in the polycrystalline specimen is then distributed spatially over a much large region. An upper bound for the spatial extent of the phase front can be estimated by taking the product of the transformation time recorded at any gage and the corresponding phase front speed, thus, we estimate that Cpt4.25 mm. Experiments with higher spatial resolution than that provided by the strain gages are necessary to explore this issue further. 4.2.6. Kinetic relation Finally, we come to a quantitative evaluation of the kinetic relation—the relationship between the driving force and the propagation speed of the phase front. For regions ahead of the propagating front, the temperatures are close to the initial test temperature, with a possibly small increase caused by the partial transformation discussed above. For the regions behind the phase front, the temperature increases from the latent heat of transformation can be about 10–12 K, but unlike the quasi-static problem, these temperatures only influence the deformations in the martensitic region, and therefore influence only the magnitude of the calculated driving force. In the absence of

ARTICLE IN PRESS 2158

J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

Fig. 12. Dispersion of phase front based on polycrystalline structure.

characterization of the temperature-dependent properties of the martensite appropriate to these strain rates, we use the quasi-static isothermal results. From the measurements, the strains g¯ þ and g¯  are known on either side of the propagating phase front at every gage; we denote them as g¯ þ ¯ I at the Ith gage. The bars indicate that these are the strains I and g averaged over the gage length. Then, the driving force f can be determined from Eq. (7) and correlated with the speed of the phase front estimated from t3.5 as indicated earlier. It is clear that a large variation in the propagation speed of the phase front is observed (see Fig. 11). This, of course, embeds the assumption that the stress–strain response determined from quasi-static experiments is suitable for use under the higher strain rates encountered in the propagating phase transformation front. However, due to the possible mixed state straining to levels of g¯ þ I and the possible incomplete transformation as discussed in Section 4.2.3, direct computation of the driving force from Eq. (7) is not possible. Therefore, in order to interpret the experimental results in terms of the driving force, we make the following assumptions: First, at the arrival of the phase front at the Ith strain gage, P ð¯gþ gþ ¯þ ¯ þ accordingly as we choose I ; mA g I Þ or ð¯ I ; s Þ represents the state corresponding to g  Model-A or -B discussed in Section 4.2.3. Second, since g¯ I ogT for all strain gages, the corresponding points after passage of the phase front fall on ð¯g g I ; mM ð¯ I  gT ÞÞ which places the martensitic region in compression. To circumvent this dilemma, we assume that there is incomplete transformation upon passage of the phase front and construct a stressstrain diagram for the mixed state simply based on the rule of mixtures. Consider that ahead of the propagating phase front, a volume fraction vþ M of the material has transformed to martensite as illustrated in Fig. 13; likewise, v M is the volume fraction of the material in the martensitic state behind the propagating phase front. Thus, for the states g¯ I the modulus is obtained through the rule of mixtures: m n ¼ mM vM þ mA ð1  vM Þ.

(11) P

We assume that the stress in both austenite and martensite remains at s (Model-B). The volume fraction vþ M is obtained from the equation for the average strain P þ þ g¯ þ ¼ sP =mþ n ¼ s =½mM vM þ mA ð1  vM Þ

(12)

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2159

Fig. 13. Material ahead of and behind the phase front have some grains that have already transformed into the martensitic state.

A similar consideration behind the phase front yields the corresponding expression for g¯  and v M  P    g¯  ¼ sP =m n þ gT vM ¼ s =½mM vM þ mA ð1  vM Þ þ gT vM .

(13)

In calculating the driving force, we should consider the volume fraction of martensite before and after the phase transformation, and hence we write f ¼ W ð¯gþ Þ  W ð¯g Þ  sP ð¯gþ  g¯  Þ.

(14)

The strain energy is calculated assuming Model-B where the material remains at a constant stress at the propagation level during the phase transformation so that the strain energies are   sP W ð¯g Þ ¼ ð1  vM ÞW ðgM Þ þ vM W gT þ . (15) mM Fig. 14 shows the estimated kinetic relation for phase transformations in the dynamic tensile experiments. The large scatter in driving force for the same phase boundary velocity is greatly influenced by the uncertainty in the strain levels g¯ þ and g¯  at the phase transformation front. Hence, while a slightly increasing driving force is observed with increasing phase front speed, additional experiments are required to confirm the trends observed here. In particular, our assumption of constant stress in Model B must be verified through direct measurements. 5. Conclusions An important accomplishment reported here is the development of a technique to investigate the dynamics of phase transformations under quasi-dynamic loading

ARTICLE IN PRESS 2160

J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161 5.00E+07 4.50E+07

Driving Force (Pa)

4.00E+07 3.50E+07 3.00E+07 2.50E+07 2.00E+07 1.50E+07 1.00E+07 5.00E+06 0.00E+00 0

50

100

150

200

250

300

350

400

Phase Boundary Velocity (m/s)

Fig. 14. Estimated kinetic relation for phase transformation in the dynamic tensile experiments estimated with Eq. (14).

conditions. The experimental scheme implemented here had one end of a NiTi specimen fixed while the other end was loaded at a constant velocity. In the thin specimens, a strip of gages were attached with a high-temperature adhesive and used to monitor the strain levels at a number of positions along the length of the specimen during the duration of loading. This allowed us to monitor the arrival of the phase transformation as well as the strains before and after the phase transformation. There are a few key observations from this investigation of phase transformation in NiTi during quasi-dynamic loading.



 

 

The strain measurements made with strain gages are capable of indicating the arrival of the phase front at the gage location. Propagation of the phase front from one gage location to another are readily identified and the speed of propagation shown to be in the range 37–370 m/s. Dispersion of the phase front with propagation along the length of the specimen is observed; this is attributed to the polycrystallinity of the specimen. A model that allows for partial phase transformation underneath the strain gage before the arrival of a phase transformation front is suggested. This mixed state allows the stress level in the material to remain low, yet allow the high-strain levels that are measured by the strain gages to appear. The use of multiple gages during the dynamic loading of NiTi allowed us to confirm clearly that multiple nucleation sites or phase transformation. Using the data from the quasi-dynamic experiments and the constitutive properties of NiTi found from quasi-static tensile loading, the driving force of the phase transformation was found at each gage and related to the speed of propagation of the phase front. Correlation of these measurements was used to identify the kinetic relation for the NiTi material. The increase in the driving force with the phase front speed was not very significant for the range of velocities generated. However, further work is needed in elucidating the role of temperature in the estimation of the driving force.

ARTICLE IN PRESS J. Niemczura, K. Ravi-Chandar / J. Mech. Phys. Solids 54 (2006) 2136–2161

2161

References Abeyaratne, R., Knowles, J.K., 1991. Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114, 119–154. Abeyaratne, R., Knowles, J.K., 1997. On the kinetics of an austenite - martensite phase transformation induced by impact in a Cu–Al–Ni shape-memory alloy. Acta Mater. 45, 1671–1683. Abeyaratne, R., Bhattacharya, K., Knowles, J.K., 2001. Strain-energy functions with multiple local minima: modeling phase transformations using finite thermoelasticity. In: Fu, Y. Ogden, R.W., (Eds.), Nonlinear Elasticity: Theory and Applications. Cambridge University Press, Cambridge, pp. 433–490. Escobar, J.C., Clifton, R.J., 1995. Pressure-shear-impact induced phase transformation in Cu-14.44Al-4.19Ni single crystals. SPIE Active Mater. Smart Struct. 2427, 186–197. Escobar, J.C., Clifton, R.J., Yang, S.-Y., 1999. Stress-wave-induced martensitic phase transformations in NiTi. Shock Compression Condensed Matter 267–270. Hall, E.O., 1970. Yield Point Phenomena in Metals and Alloys. New York, Plenum Press, p. 33. Hopkinson, J., 1901. Original Papers. Cambridge University Press, Cambridge. Knowles, J.K., 1999. Stress-induced phase transitions in elastic solids. Comput. Mech. 22, 429–436. Lagoudas, D.C., Ravi-Chandar, K., Sarh, K., Popov, P., 2003. Dynamic loading of polycrystalline shape memory alloy rods. Mech. Mater. 35, 689–716. Leo, P.H., Shield, T.W., Bruno, O.P., 1993. Transient heat transfer effects on the pseudoelastic behavior of shape memory wires. Acta Metall. Mater. 41, 2477–2485. Liu, Y., Li, Y., Ramesh, K.T., Van Humbeeck, J., 1999. High strain rate deformation of martensitic NiTi shape memory alloy. Scr. Mater. 41, 89–95. Moon, D.W., Vreeland Jr., T., 1968. Stress dependence of mobile dislocation density and dislocation velocity from Lu¨ders band front propagation velocity. Scr. Metall. 2, 35–40. Rakhmatulin, K.A., 1945. Propagation of an unloading wave (in Russian). Prik. Mat. Mekh. 9, 91–100. Shaw, J.A., Kyriakides, S., 1995. Thermomechanical aspects of NiTi. J. Mech. Phys. Solids 43, 1243–1281. sma-inc, 2001. http://www.sma-inc.com/NiTiProperties.html, accessed on 6/14/2001 Sylwestrowicz, W., Hall, E.O., 1951. The deformation and ageing of mild steel. Proc. Phys. Soc. B 64, 495–502. Taylor, G.I., 1958, The plastic wave in a wire extended by an impact load, The Scientific Papers of G.I. Taylor, volume I, Mechanics of Solids, University Press, 467–479. von Karman, T., Duwez, P., 1950. The propagation of plastic deformation in solids. J. Appl. Phy. 21, 987–994.