mechanical behavior of an ni-ti shape memory alloy under axial-torsional proportional and...

T. Jesse Lim

David L. McDowell Fellow ASME.

GWW School of Mechanical Engineering, Georgia Institute of Technology,

Atlanta, GA 30332-0405

Mechanical Behavior of an Ni-Ti Shape IVIemory Alloy Under Axial-Torsional Proportional and Nonproportional Loading Several biaxial proportional and nonproportional loading experiments are reported for thin-wall tubes of a pseudoelastic Ni-Ti shape memory alloy (SMA). In addition to the mechanical behavior, temperature was measured during the experiments. It is shown that the phase transformation exhibits asymmetrical behavior in the case of tension-compression cycling. The transformation strain rate is determined for selected histories by numerical differentiation of data. Under nonproportional loading, the rate of phase transformation does not follow a generalized J2-J3 criteria based on results of micromechanical simulations for proportional loading. The role of simultaneous forward and reverse transformations on the nonproportional transformation response is examined using a simple micromechanical model, and the direction of the inelastic strain rate is adequately predicted. Load- and strain-controlled experiments at different strain rates, with and without hold times, are reported and coupled thermomechanical effects are studied.

1 Introduction

Ni-Ti shape memory alloys (SMA) exhibit pseudoelasticity above the austenite finish temperature, Af. Pseudoelasticity is due to stress induced martensitic (SIM) transformation in which austenite (A) phase is transformed to the martensite (M) phase under stress. When the stress is reduced, the M phase transforms back to the A phase. Although most constitutive equations describing SMA phase transformations have included temperature dependence (cf. Huo and Mtiller, 1993; Raniecki et al , 1992; Liang and Rogers, 1991), few of these previous works considered fully coupled thermomechanical behavior. Recently, Leo et al. (1993) and Shaw and Kyriakides (1997) demonstrated that under mechanical cycling, the temperature in the SMA changes due to latent heat generation/absorption during phase transformation. The change of temperature during loading alters the stress-strain response and induces rate dependence. Our experiments further confirm this under somewhat more general loading conditions.

SMA constitutive models have been proposed at the meso-scale (cf. Graesser and Cozzarelli, 1994; Liang and Rogers, 1991; Raniecki et al , 1992) based on J2 type transformation theory as a generalization to the multiaxial case. However, our experimental work shows that the direction of transformation strain rate is not colinear with the deviatoric stress, in general. Furthermore, we show that a combined second and third invariant {J2-J3) flow potential based on a self-consistent micromech-anics solution (Patoor et al., 1995, 1996) for proportional loading does not accurately describe transformation under nonproportional loading conditions. Accordingly, generalizations must reflect the role of the orientation distribution of transformed martensite variants, variant coalescence, and related phenomena. The heterogeneity of transformation among grains effectively induces strong curvature of the macroscale transformation strain rate potential in the vicinity of the loading point.

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials Division February 17, 1998; revised manuscript received May 22, 1998. Guest Editors: H. Sehitoglu and Y. Chumlyakov.

2 Experimental Procedures and Specimens

Both proportional and nonproportional loading experiments were performed on biaxial thin-walled tube specimens shown in Fig. 1. All specimens were made of a Ni-Ti alloy with the near equiatomic composition (49.20 at % Ti-50.80 at % Ni), and balance trace elements including Al, C, O, Si, etc. After machining from bar stock, specimens were heat treated at 500°C for 20 minutes in atmospheric air, followed by air cooling. The austenite finish temperature as found by the differential scanning calorimetry method was A/ !« 0°C. The grain size in the gage section was about 40 //m, with about 50 grains across the wall thickness of the thin walled tube specimen. Four T-type thermocouples were attached by tape to all specimens in the positions shown in Fig. 1. The thermocouples were used to measure the change of temperature due to latent heat generation/ absorption during phase transformation, serving as an indication of the extent and nonuniformity of phase transformation. Strain was measured using an axial-torsional extensometer directly in contact with the specimen gage section.

3 Proportional Loading Histories

3.1 Uniaxial Loading. Specimen 1 was subjected to the uniaxial strain history outlined in Table 1. Axial strain is denoted by e. In Steps 3 and 4, the strain path was also cycled between ±3%. In Step 3, the strain was held in strain control for 300 s whenever the strain level reached - 3 % , - 2 % , 0, 2% and 3%. In Step 4, the control was switched to load control, so stress was occasionally held instead of strain. Figure 2 shows the stress-strain response of cycles 1 - 5 of Step 1 and the corresponding average temperature response among the thermocouples. The temperature measured by the three thermocouples differed at most by about 2°C and exhibited very similar variation; the presence of potentially distinct transformation fronts could therefore not be discerned. Assuming the transformation is homogeneous throughout the gage section, the temperature shown in the figures is the average of the measurements of the three thermocouples attached to the gage section.

The stress-strain response is pseudoelastic; as the A phase to M phase transformation proceeds during straining from 0% to

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JANUARY 1999, Vol. 121 / 9

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i.d.=16.5

o.d.=20.4

encl=25.4

Dimensions in mm

T.C.=Position of thermocouple

Fig. 1 Biaxial tliin-walied tube specimen

3%, latent heat is released, resulting in specimen heating. For M to A transformation during unloading from 3% to 0%, latent heat is absorbed. When the specimen is compressed from 0% to - 3 % , M phase variants form which are somewhat different than those in tension (Gall et al., 1997) due to crystallographic texture of the bar stock and the reduction of symmetry of mar-tensite variants compared to the parent phase (Patoor et al , 1995, 1996). This induces tension-compression asymmetry. The substantial difference of the area enclosed by the hysteresis loops in the first (tension) and third (compression) quadrants of the stress-strain response are further evidence of interaction energy differences in tension and compression associated with different morphologies of the transformed phases. Latent heat is also released; upon the initial heating of the specimen, heat is transferred out of the specimen gage section so that for the subsequent cycling, the entire temperature profile shifts down until steady state heat transfer conditions are approached for the given cycling condition. It is observed that the transformation stress of the first cycle is higher and the average temperature of the specimen is higher than that of the subsequent cycles.

The stress-strain response of cycles 6 -7 (Step 2) and the corresponding temperature changes are shown in Fig. 3. In Fig. 3, the stress-strain response of cycle 4 is also plotted as dashed line for comparison. For cycles 6-7, that the strain rate is lower than that of cycles 1-5, and the transformation proceeds at close to an isothermal condition since the latent heat released has sufficient time to conduct away. Hence, the transformation stress in both tension and compression are both lower than that of cycle 4, which is closer to adiabatic conditions. It is also

Table 1 Test matrix for Specimen 1

Step Cycle Path Strain rate

1 2 3

4

1-5 6-7

8

9

cycling at - 3 % < e < 3% cycling at - 3 % < e < 3% hold strain for 300s @ - 3 % ,

- 2 % , 0, 2% and 3% hold stress for 300s @ - 3 % ,

- 2 % , 0, 2% and 3%

5 X 10-" S-' 10~' S-'

5 X 10-" S-'

5 X 10-" S-'

800

600

400

.--.200 n a. S 0 b-200

-400

-600

-800

;

" / ^ " y /

- vr^ r ^ f ^

- f J - [^^ 7 , 1 , . , . 1 , , , . 1 . . . . 1 , , , . 1 , , . . 1 , , , , 1 . .

-3 - 2 - 1 0 1 8(%)

Fig. 2(a) Stress-strain

0 1

s(%) Fig. 2(b) Temperature responses of Specimen 1, N = 1-5

observed that the initial flow stress for transformation is essentially the same for both strain rates.

The stress-strain and temperature responses of cycles 8 and 9, involving periodic strain or stress hold periods, are shown in Figs. 4 and 5, respectively. In Figs. 4(a) and 5(fl), the response of cycle 4 is shown as a short dashed line, while that of cycle 7 is shown as a long dashed line for comparison. It can be clearly seen that as the specimen is strained at a rate close to that of cycles 1 - 5 , the flow stress level is close to that of cycles 1-5. During periods of strain or stress hold, the temperature of the specimen, which increases during A to M transformation and decreases during M to A transformation, returns towards ambient temperature. During such hold periods, the stress or strain levels shift towards those representative of the nearly isothermal transformation case of cycles 6-7 .

These data involving hold periods demonstrate the important first order role of the thermomechanical coupling in inducing strain rate dependence of the flow stress during pseudoelastic straining. Since temperature can substantially increase only during the A^M transformation rather than in elastic regimes, it is the apparent work hardening behavior that exhibits a dependence on strain rate. The schematic transformation strip for Ni-Ti shown in Fig. 6 shows that the required stress for A to M transformation increases as the temperature increases due to latent heat release, resulting in a higher flow stress at higher strain rates as adiabatic conditions are approached. Although the mechanical behavior displays rate dependence of flow stress, creep and relaxation, in this case these phenomena are almost exclusively linked to coupling due to chemical energy mismatch between the phases and the latent heat of the transformation.

10 / Vol. 121, JANUARY 1999 Transactions of tlie ASI\ E


N - 4 N = 6 - 7

35

—.30

o o S> 25 3

g.20 E o *- 15

10


-

- ^S: i:SX2==:i'0~,—"<V,,~~5^='"''"'~'**'~'Z!S?'

. . , I , , , , 1 , , . , I , , .

8(%)

Fig. 3(b) Temperature responses of Specimen 1, M = 6, 7

3.2 Pure Torsional Loading. Specimen 2 was subjected to pure torsional strain control. The Mises equivalent strain is defined by e^ The strain rate in this case is simply the equivalent shear strain rate, eeq • Here, y is the engineering shear strain of the thin walled gage section. The specimen was strained for 5 cycles between ±3% at an equivalent strain rate of eeq = 5 X lO""* s~'. The stress-strain and temperature responses of cycles 1-5 are shown in Fig. 7. The equivalent stress is defined by (Teq = Tv3. The results are very similar to those of Specimen 1. In contrast to tension-compression, the stress-strain response is nearly symmetric since either positive or negative shear are evidently equivalent in producing transformation. A fiber texture imparted by drawing of bar stock would not be expected to impart asymmetry of the shear response in tubular specimens with the torque applied with respect to the fiber axis.

3.3 Combined Proportional Loading. Specimen 3 was subjected to combined proportional loading with A'y/(Ae = 1. The equivalent stress, strain and strain rate are defined by

Cm — V cr + 3r^ e + ^ e ' + ^ (1)

The specimen was strained for 5 cycles at a strain rate of fieq = 5 X 10"" s"' between the limits e = ±2.12% and y/S = ±2.12% to achieve a maximum eeq = 3%. The stress-strain and temperature responses are shown in Fig. 8. Notice that in the plot the sign of Ce, is modified such that it is positive when the axial strain, e, is positive and negative when the axial strain

is negative. The sign of Ceq is also modified to follow the sign of (7.

For comparison, the stress-strain and temperature responses of cycle 4 of all specimens 1, 2, and 3 are shown in Fig. 9. The plot of Specimen 1 is shown as short dash, that of specimen 2 is shown as long dash, and that of specimen 3 is shown as solid line. Since the temperature response is an indication of transformation product, either in its extent or morphology, for a pure torsional case such as Specimen 2, the transformation product at either positive and negative shear strain are approximately the same. This can also be inferred by the symmetry of the stress-strain response and near symmetry of the temperature response in shear. However, there is a distinct tension-compression asymmetry. This tension-compression asymmetry is consistent with previous results (cf. Vacher and Lexcellent, 1991; Patoor et al., 1995, 1996; Orgeas and Favier, 1996; Gall et al., 1997; Gall and Sehitoglu, 1998). Deducing from Fig. 9{b), for pure axial loading, it appears that either there is more transformation product (A to M) in compression than in tension, or there is a different morphology in terms of the variants and their arrangement (number of variants per grain, number of interfaces, etc.). The latter explanation is suggested by the work of Patoor et al. (1995) and Gall et al. (1997). However, for combined loading, the effective flow stress in compression is between that of pure axial and pure torsional loading. The temperature is highest for combined loading in compression. For torsional loading, the effective flow stress is higher than that of tension in a pure axial case while the temperature doesn't differ much. The difference in temperature response may be due to some other mechanisms not yet identified, such as the average

-1 0 1

8{%)

Fig. 4(a) Stress-strain response for Af = 4, 7, and 8

35

—«30

o o ^"^ & 25 3

c S.20 E a> • - i-i

10

'

-

•

i \ \

\

i""

; X ' •

A

, , 1 , , , ,

iL /f/j

'X ^

• • • .

\

L u -1 0 1

S(%)

Fig. 4(b) Temperature response for A/ = 8 of Specimen 1

Journal of Engineering Materials and Technology JANUARY 1999, Vol. 121 / 11


-1 0 1

8(%)

Fig. 5(a) Stress-strain response for N = 4,7, and 9

35

. -^30

o 0 S. 25 3

c g.20 E <u • - 15

10

: /

- \

7 . 1 , , . , J

\

' ^

1 1 1 1 1 1 1 1

f^ t^y^

- 3 - 2 - 1 0 1 2 3 8(%)

Fig. 5(b) Temperature response for A/ = 9 of Specimen 1

- 1 0 1 2 Y/V3 (%)


-1 0 1 Y/V3 (%)

Fig. 7(b) Temperature responses of Specimen 2, A/ = 1-5

number of interfaces in each grain or some other feature of comparable complexity. Crystallographic texture should be examined as a possible contributor to these stress state differences as well. It is difficult to see how redefining effective stress and strain quantities to match the multiaxial mechanical behavior (cf. Sittner et a l , 1995, 1996) can describe the coupled thermo-mechanical behavior observed here.

Ideal isothermal loading A A

Cooling during ~-stress/sti^in Y hold A

Specimen heating due to latent heat generation

M, M, A, A,

Fig. 6 Transformation strip; as specimen is floated upon ioading, stress level for SIM increases

4 Nonproportional Loading Histories

Nonproportional loading tests were performed in order to investigate variant coalescence (cf. Marketz and Fischer, 1996) of transformed M phase, variant re-orientation and complex combinations of simultaneous forward and reverse transformation. Specimen 4 was strained according to the test matrix outlined in Table 2.

For cycles 1-7, circle 1 is the circular strain path with radius, r = 2.12% as shown in Fig. 10(a). For cycles 8-14, circle 2 is the circular path with r = 3% as shown in Fig. 10(b). At cycle 15, the strain was imposed from (e, y/vS) = (0, 0) to (2.12%, 0%), then circle 1, from (2.12%, 0%) to (3%, 0%), then circle 2 and back to (0, 0) . For cycle 15, the strain was held for 300 s at 90°, 180°, 270° and 360° with respect to the positive e-axis. For cycle 16, the straining followed the same path as cycle 15; in addition, strain was held for 300 s at 45°, 135°, 225°, and 315° with respect to the positive e-axis.

The stress path and the temperature profiles of cycles 4 and 7 are shown in Fig. 11. Again, for the lower strain rate, the transformation is close to an isothermal condition, while for a higher strain rate, latent heat is generated dominately in compression, which is consistent with the results of proportional loading. As a result, the stress path of the lower strain rate is enclosed within that of the higher strain rate. The tension-compression asymmetry of the stress space response is still quite evident in Fig. 11.

For such a circular path, the direction of transformation strain rate is not in the direction of either the deviatoric stress or stress rate. To assess the direction of the transformation strain rate.

12 / Vol. 121, JANUARY 1999 Transactions of the ASME


- 1 0 1 2 &eq(%)

35

^ ^ 3 0

u 0

e 25 3 >4-»

& ^ 2 0

E a> • - 15

10

- \

" -

. . 1

Fig. 8(a)

1

Stress-strain

/ / / /& / / / /

/ / / j ^

\ ^ j ^ j /

l_I_l.^l-l.l-i_l.l..i^.^..l.L4-l.i 1 1 1 1 - 1 0 1 2

Seq{%)

Fig. 8{b) Temperature responses of Specimen 3, N = 1-5

the arc length, s, along the stress path is defined in the axial-torsional subspace as (McDowell, 1985)

j ' = X (As„- As„)"^ s„ = trni + Vi THa (2)

The derivative of transformation strain is evaluated with respect to s as

ds

de

ds

\_da^

E ds n, + ds G ds) -^ ns (3)

based on a moving five point parabola fit for dependent variables as a function of arc length, where n, and n^ are unit base vectors. Elastic moduli E and G relate to axial and shear cases, respectively. The unit normal vector in the direction of the transformation strain rate was obtained by

n = de'/ds

\\de'/ds\\ (4)

It is noted that E and G were assumed constant (corresponding to the A phase) in these calculations; in spite of this approximation, little error is involved since the transformation strain rate dominates the elastic strain rate since the effective flow stress remains nearly constant during the transformation. The direction of the transformation strain rate along the stress path of cycles 4, which is typical of all cycles 1 to 14, is shown as arrows in Fig. 12, with tails attached to the stress point where the derivative is evaluated at the midpoint of the moving five point parabola. Clearly, an associated flow rule based on a macroscale von

800

600

400

"5" 200 Q. S 0

<T

J3-200

-400

-600

-800

~ _ _

- //C '• T ^ : 1^ r . 1 , . . . r 1

-Axial - Torsional - Combined

y ^ V ^

^ /] ___ w ^^ ^

I I J Ll„l 1 1

„_—• - ^

.,^-'*=^^^^^''~''^^

"

^ ^ . , l . J , . . U . I i , l l i i •

-1 0 1 2 8eq(%)

Fig. 9(a)

35 -

1 0 -

Axial Torsional Combined

- 3 - 2 - 1 0 1 2 3 8eq(%)

Fig. 9(d)

Fig. 9 Comparison of (a) stress-strain and (b) temperature response of Specimens 1, 2, and 3, A/ = 4

Mises 2 potential (a circle in this stress subspace) would be highly inaccurate without additional internal variables introduced to either a shift or distort the flow potential away from the Mises form.

The stress path and temperature profiles of cycle 16 are shown in Fig. 13. During the strain hold periods, the temperature shifted towards the ambient temperature and the stress relaxed approximately in opposition to the direction of the transformation strain rate. But upon straining immediately after strain holds, the transformation strain rate was almost identical to that shown in Fig. 12.

The stress response of the circular strain path clearly shows that the SMA phase transformation does not follow the von Mises criteria. Recent self-consistent micromechanics simulations of Patoor et al. (1995) suggest that the transformation surface in stress space follows a Drucker-Prager form for proportional loading. We investigate here if such a transformation

Table 2 Test matrix of Specimen 4

Step

1 2 3 4 5 6

Cycle

1-5 6-7 8-12

13-14 15 16

Strain

circle 1 circle 1 circle 2 circle 2

path

circles with stops 1 circles with stops 2

Eq. strain rate

5 X lO""* S-' 1 0 - ' S-'

5 X 10"^ s^' 10"' s"'

5 X 10"" S-' 5 X 10"" S-'

Journal of Engineering IVIaterials and Technology JANUARY 1999, Vol. 121 / 13


- 2 - 1 0 1

s{%) Fig. 10(tp)

Fig. 10 Circuiar strain paths (a) 1 and (b)2

is descriptive of more general nonproportional loading paths which may involve martensite reorientation and coalescence. Patoor et al. (1995, 1996) proposed the representation

1 -I- b-F(J2, J,) = 72

where b and K are respectively given by

K'

b = ^ al

2 al + at -i 1 + lb

and J2 and 73, in this axial-torsional case, are given by

1 ^. = 3 . ^ + r^ J, = — a^ + - ar^

27 3

(5)

(6)

(7)

Here, a^ and a, are the critical stresses necessary to start SIM transformation in compression and tension, respectively, recognizing the possible existence of tension-compression asymmetry. Notice that if a^ - a,, then b = 0 and F reduces to the von Mises criterion. Assuming associativity of the transformation strain rate, its direction can then be expressed in the (a, TV3) subspace in terms of the unit vector

V\ 1 dF

da da

_ 2 IdF

T-1/2 9J^

n = r?ini + rjjnj

1 ,-3/2 , SJ2

2 da

(8)

(9)

-800 -600 -400 -200 0 200 400 600 800

a(MPa) Fig. 11 (a) Axial-torsionai stress subsapce

Fig. 11 (b) Temperature response of Specimen 4, /V = 4, 7

— = ^ + b dr dr • dr

^ J2'" h dr

--"fy^Kf)"

(10)

(11)

The stress path data of cycle 11 of the circular strain path test were introduced into Eq. (8) . Constants a^ and a, were

800

600

400

5*200 Q. S 0 CO "5-200

-400

-600

-800

r

-

- ^ = s i ^ 4 ^

: (} A J/

- i)v ^ ^ r ^ V \ \ v j ^ ^ *

"1 1 — 1 1 -800 -600 -400 -200 0 200 400 600 800

a(MPa) Fig. 12 Transformation strain rate directon along the stress path of Specimen 4, A/ - 4



800

600

400

« 200 DL

S 0 eo ^ - 2 0 0

-400

-600

-800

r

- / ^ ? ^ = = * ^

\ ifjX 1 j JJ

" \ V, / • x^^^^^V^ - ^ —

T • • • • 1 • • • •< 1

-800 -600 -400 -200 0 200 400 600 800

a(MPa) Fig. 13(a) Axial-torsional stress subspace

Fig. 13(t)) Temperature responses of Specimen 4, W = 16

based on a 0.2% offset definition of yield from the uniaxial tension-compression test as outlined in Section 3. The experimentally based evaluation of the function F is shown in Fig. 14(a). The corresponding transformation strain rate directions are shown in Fig. 14(^), superimposed on the stress path at each point where derivatives are taken.

The function F varies substantially over the cycle. Moreover, the transformation strain rate direction of the experiment (Fig. 12) and that obtained from Eqs. (5) and (8) (Fig. 14(fo)) are in poor agreement. Thus, it can be concluded that the phase transformation process does not follow the Drucker-Prager criterion in the nonproportional case. Recent works of Sittner et al. (1995, 1996) have also shown that the stress response is dependent on the nonproportional strain path history. These results appear to invalidate any generalized J2 or simple combined 72-/3 associative form of SMA transformation at the macroscale for more general loading paths.

5 Micromechanical Simulations We employ a simple computational micromechanics scheme

which uses 5000 grains within a representative volume element (RVE) to study the distribution of the transformation process and the origin of the discrepancies for nonproportional loading outlined in the previous section. The lattice parameters of the 24 different martensite (M) phase variants are used (Patoor et al., 1995). Each of the 5000 grains is assigned a different random orientation, so the transformation strain of each M phase variant of a single grain is given by

different variants (n = 1, 2, . . . , 24) in a unrotated state, R,- is the orientation (rotation) matrix assigned randomly to different grains (; = 1, 2 , . . . , 5000). The local transformation criterion is defined in the forward and reverse case (Patoor et al., 1996) as

-B(T - To) + a-.€'-" + YH""'Z'" \, + X-o — (13)

where B(To - T) is the driving force associated with the difference of chemical energy of the phases, H'"" is the interaction matrix which governs interaction energy, z" is the martensite volume fraction for the nth variant, X.„ and Xo are Lagrange multipliers that are introduced in imposing constraints on the sum of volume fraction of the n variants in the system and Fc is the critical driving force for SIM. In our modeling, we simplify by assuming that each grain can have only one variant and neglect transformation-induced internal stresses, so 1,H""'z" = 0. We further assume that each grain is subjected to the same stress and assume isothermal conditions, leading to a criterion based on mechanical driving force as proposed previously by Patel and Cohen (1953). Hence, Eq. (13) becomes

o-.e'f" = a ±/3 (14)

pertaining to forward and reverse transformations, respectively. In our study, we assume that a follows a gaussian distribution, such that the average of the 5000 grains is 17.5 MPa with a standard deviation of 7.5 MPa, and assign fi = 7.5 MPa. This effectively enhances rounding of the transformation stress-strain curve in the early stages of transformation and introduces macroscale work hardening without explicit inclusion of the interac-

Fig. 14(a) Function F evaiuated along the stress path

800

600

400

5*200 Q.

eo "^-200

-400

-600

-800

-

MiHUi;. . / ^

^ V J ^ Vnrrf^ 1 1 , , , , 1 , , . , 1 1 . . . , 1

e'r" = Rje'-" R,- (12)

where e ' " denotes the transformation strain tensor of the 24

-800 -600 -400 -200 0 200 400 600 800

a(MPa) Fig. 14(b) Transformation strain rate prediction based on F for Specimen 4, N = 11



tion energy term. As shown by finite element analyses of Mar-ketz and Fischer (1995), the internal stresses that develop in the microstructure during the transformation assist transformation in such a way that it starts simultaneously in a larger number of grains; the average mechanical driving force is not substantially changed by the internal stresses due to grain mis-orientation. We introduce the distribution for a to reflect effects of multi-variant intragranular martensite transformation and associated interactions as well. The elastic moduli were approximated to correspond to the A phase. The following algorithm was employed for each stress increment, starting with zero transformation product at zero applied stress:

(i) Apply a stress increment, Atr, then update the stress from the previous step, i.e., o- = o-oid + Acr.

(ii) Impose the same updated stress on all grains and apply the transformation criterion (14).

(iii) Transform grains that meet the criterion (14) from A to M (or M to A) by adding (or removing) a transformation strain ej'" for a single variant over the entire grain during this step. Only one M variant is allowed per grain, so variant reorientation or coalescence is not explicitly treated.

(iv) Sum up the transformation strain of all grains and calculate average transformation strain over the entire RVE, then return to step (i) .

5.1 Modeling Proportional Loading. Figure 15 shows the result of three proportional loading cases: axial tension-compression, cyclic torsion and combined axial-torsional cycling. In each case, the equivalent stress is applied such that the equivalent strain cycles between —3% and 3%. In the case of combined loading, A7/Aev3 = 1. The size of the stress increment is constant, Affeq = 2 MPa. Although in all three cases the shape of the stress-strain curves resembles experimental data in Fig. 9, this model predicts a higher flow stress in tension with no significant tension-compression asymmetry. Pa-toor et al. (1995) argue that the low symmetry of M phase contributes to the asymmetry of the mechanical response. Recent work of Gall and Sehitoglu (1997), however, has shown for Ni-Ti SMAs that initial texture of cold drawn bar stock significantly manifests the tension-compression asymmetry in conjunction with the low symmetry of the M phase. Since we assigned a random orientation distribution of grains in this simulation, of course the model cannot predict any texture-induced tension-compression asymmetry. Aside from this asymmetry, however, the model represents nonproportional loading behavior qualitatively well, enabling a micromechanical study of the physical origin of the noncollinearity of the direction of inelastic strain rate and the deviatoric stress.

800

600

400

I02OO Q. S 0

cr J3-200

-400

-600

-800

r Axial Torsional Combined

- f y'

•_ fe.-==^-=''*^

- 1 ,\,,,.\,.,,\..

800

600

400

W 200 Q.

S 0 Cl H-200

-400

-600

-800

r

-

-

1 M 1 l l 1 1 1

^-^S^i^^^W^

1 -800 -600 -400 -200 0 200 400 600 800

CT(MPa)

Fig. 16(a) Imposed circular stress path with predicted polycrystal transformation strain rate direction superimposed

-3 -1 0 1 8eq(%)

Fig. 15 Modeling of pure axial, pure torsional and combined proportional axlal-torsional cyclic loading

-2 0 2

8(%) Fig. 16(6) Predicted strain path in axial torsional strain subspace

5.2 Modeling Nonproportional Loading. Figure 16(a) shows an applied circular stress path with radius, r = 450 MPa. It moves from (a, TV3) = (0, 0) to (450, 0) , runs counterclockwise back to (cr, rVs) = (450, 0), then back to (0, 0) . The calculated transformation strain rate direction is also shown in Fig. 16(a), superimposed on the stress path. The resulting strain path is shown in Fig. 16(b). The transformation strain rate direction is very similar to that of the experimental data shown in Fig. 12. The transformation strain rate direction shown in Fig. 16 is the average of contributions from the individual grains.

Since the 5000 grains possess different orientation and critical SIM driving force, different grains transform from the A phase to different variants of the M phase and vice versa at different locations along the stress path. Only a relatively small fraction of grains exhibit forward or reverse transformation during each stress increment. For example, the response of grains No. 80 and 2964 are shown in Fig. 17 with regard to their specific points of transformation from one of the M phase variants (numbers in parentheses indicate variant numbers) to another M phase variant (or to the A phase). Also shown in this figure is the direction of the transformation strain rate corresponding to the point along the stress path where the transformation occurs. Consider Fig. 17(b) with grain 2964 as an example; it transforms from the A phase to M variant 10 during the initial axial loading, then transforms from the M variant 10 to the A phase, then to the M variant 4 and so on.

For this simple model, each grain can transform only into one type of M phase variant. In order to transform into another M phase variant, it has to transform back to the A phase first.



In reality, different M phase variants and the A phase can coexist in a single grain in any proportion. However, since there is interaction energy between different variants, (Marketz and Fischer, 1995,1996 and Patoor et al., 1994) it takes more energy to transform the A phase into the M phase if there already exists different M phase variants within the same grain. Our simplified model does capture that once a type of variant of M phase already exists in the structure due to prior loading, there is less A phase available to transform to another type of variant once the applied stress changes direction. These phenomena contribute to the path history dependence of SMAs during nonpropor-tional loading, thereby invalidating any simple Ji-J^ type theory with an associated flow rule. It would appear that mac-roscale models must retain enough information regarding the orientation distribution of grains and the history dependence and heterogeneity of transformation events to reflect the degree of distortion of the macroscopic transformation strain rate potential that must exist in the vicinity of the stress point to rectify these experimental measurements.

5.3 Thermomechanical Coupling. To model thermome-chanical coupling, the chemical energy driving force is included, i.e.,

-B{T - To) + a:er P (15)

The boundary condition for transient heat conduction from the specimen (conduction to or away from the gage section) can be approximated by

800-

600 r

400 -

m 200 CL

= 0 CO

"J-200

-400

-600

-800

A->M(16)

VM{1«i->A

; M(19)->A A->M(19)

•M(16)V>M(3) M(1)->A-^M(19)

M(3)->A->M(1)

-800 -600 -400 -200 0 200 400 600 800

a{MPa)

800

600

400

,_ « 200

i . 0 «*> "^-200

-400

-600

-800

Fig. 17(a)

1

M(10)->A->M(4) r *7^^^

/ ^ ; M(5)->A A->lA(10)

• \ ; f M(7)->A->M(5)

r \ -^^- - ' ^ M(4)->A->M(7)

-1 , , . . 1 , , . , 1 1 1

-800 -600 -400 -200 0 200 400 600 800 a(MPa)

Fig. 17(b)

Fig. 17 Transformation history of (a) grain No. 80 and (b) Grain No. 2964 among an ensembie of 5000 grains in polycrystal

800

600

400

- - . 2 0 0

Q.

S 0 t5.200

-400

-600

-800

-

"

_

-

-

r 1

1 / I^U

1

3

L I . J , J

- Isothermal

. U _ l J ^ 1 .1 iJ , i ^J I..I.L.

1

/ / 1

' 1

-2 0 1 8(%)

30

0 25

i> 3 ^ 2 0

%. E

. » 15 1-

10

Fig. 18(a)

-

3

•

Predicted stress-strain

\

, , 1 , . . . 1 . . . .

1 \

I

2

-3 0 1

8(%)

Fig. 18(b) Temperature response of tlie specimen gage section with fully coupled thermomechanical theory and energy equation

dt = -h{T - To) (16)

where h is the heat transfer coefficient, and T^ is assumed to be the ambient temperature. Assuming uniform temperature, T, within the gage section, the energy equation can be approximated as

dt dt (17)

where C is heat generated per volume fraction transformed, denoted here by z. The temperature is then updated using Eqs. (16) and (17) at the end of each stress increment as an extension of the algorithm outlined previously. The same stress path of pure axial loading as outlined in Section 5.1 was enforced, the stress was held for 100 time steps at both extrema and at 100 MPa during unloading from tension and at —100 MPa during unloading from tension and at -100 MPa during unloading from compression. For these calculations, the following constants were used: T„ = 20°C, B = 0.25 MPa°C-', h = 0.025 s~' and C = 50°C. The stress-strain and temperature responses are shown in Fig. 18. In Fig. 18(fl) the response of the previous isothermal axial tension-compression case is also shown.

Heat is generated as the specimen is loaded in tension. As the stress reaches its maximum, the temperature also reaches a maximum, (marked by 1 in both Figs. 18(a) and 18(&)). During the stress hold period, the specimen gage section cools down and the strain shifts to the right, approaching the curve associated with the isothermal case. Upon unloading, heat is



absorbed. The start of the stress hold at 100 MPa is marked by 2; as the gage section heats back to ambient temperature, the strain also shifts back towards the isothermal case. The compressive behavior is similar. The results here also match the experimental results qualitatively (Fig. 5), indicating that a simple linear chemical energy term can be added to the transformation driving force to model coupled thermomechanical cycling, along with the energy equation, and that apparent viscous effects for this particular material arise solely from the coupling of the chemical and mechanical driving forces for the transformation.

6 Conclusions This study has investigated differences in mechanical behav

ior among various proportional and nonproportional cyclic loading paths for a Ni-Ti SMA, along with the history of temperature in each case. Several significant findings are evident. First, the asymmetry of forward and reverse cyclic flow stress depends on stress state. Cyclic shear did not exhibit asymmetry of flow stress, while proportional histories involving tension and compression with respect to the specimen axis did. Initial crystallo-graphic texture along with the low symmetry of the M phase is the likely explanation (Gall et al , 1997; Gall and Sehitoglu, 1997; Patoor et al., 1995, 1996). It is shown that during phase transformation, latent heat will be released/absorbed, and the temperature history and coupled transformation driving force will depend on the rate of the applied loading relative to heat transfer to or away from the specimen gage section. Constitutive modeling should include this coupled thermomechanical effect as the differences in temperature and in the transformation driving force can be substantial over modest changes of mechanical rate of loading.

In the case of multiaxial nonproportional loading, the phase transformation does not follow a J2-J:^ transformation criterion with an associated flow rule selected to fit results of microme-chanical simulations for proportional loading (Patoor et al., 1995, 1996). From simple numerical simulations on 5000 grains, it can be concluded that the heterogeneity of transformation among grains manifests as a sequence of transformation of different M phase variants plays a major role in the overall macroscopic behavior of the SMA (i.e., the stress-strain response is path history dependent). This heterogeneity engenders a strong curvature of the macroscopic flow potential surface in the vicinity of the loading point, accounting for large rotation of the transformation strain rate relative to the deviatoric stress or normal vector to an isotropic flow potential in the case of nonproportional loading. Addition of texture-induced anisot-ropy in the macroscopic potential would not by itself have modeled the proper transformation strain rate either, although tension-compression asymmetry may have been captured. It would seem that the heterogeneity of the transformation among grains, in general, imparts an apparent nonassociativity to the macroscopic transformation strain rate with respect to a smooth flow potential inferred either through self-consistent micro-mechanics or experiments under proportional loading. Recent work involving variant coalescence by Marketz and Fischer (1996), which considers all martensite variants and their interaction energies, may lead to more indepth understanding and more accurate modeling. Along these lines, finite element stud

ies may be used for detailed study of interactions among variants and grains and their influence on the interaction energy and system internal stress distribution. This may lead to more appropriate treatment of the interaction energy and transformation strain rate in both self-consistent micromechanical approaches and macroscopic internal state variable approaches.

Acknowledgments

The authors wish to acknowledge the support of NSF through grant No. CMS-9414634 in conducting this research.

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mechanical behavior of an ni-ti shape memory alloy under axial-torsional proportional and...

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