Beam Under Dynamic Loading

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Eur. J. Mech. A/Solids 20 (2001) 615630

2001 ditions scientifiques et mdicales Elsevier SAS. All rights reservedS0997-7538(01)01159-7/FLA

Analysis of the behavior of a Shape Memory Alloy beam under dynamical loading

Manuel Collet , Emmanuel Foltte, Christian Lexcellent

LMARC - UMR6604, universit de Franche-Comt 24, rue de lEpitaphe, 25000 Besanon, France

(Received 2 October 2000; revised and accepted 17 April 2001)

Abstract Shape Memory Alloys (SMAs) are widely studied as new materials with potential for use in various passive or active vibration isolationsystems. Up to now, few papers deal with a precise description of their proper dynamic behaviour. However, it is important to clearly understand the

dissipation mechanisms in order to optimize the design of a structure. We present here a detailed characterization of a CuAlBe beam. The stressinduced phase transformation austenite martensite produces a strongly nonlinear behaviour. The aim of this study is to confront experimental resultsto a rheological model of the beam. The experimental setup consists in a cantilever beam excited by a light electromagnetic actuator. The response ismeasured by an accelerometer fixed at the free end of the beam. Stepped sine measurements have been performed around the frequency of the firstmode of the beam under different excitation levels. The obtained frequency response functions strongly depend on the global vibration amplitude. Thena specific finite element model has been designed, taking into account the geographic repartition of the two phases inside the beam. The simulationsshow a similar behaviour and allow the interpretation of the experimental observations. 2001 ditions scientifiques et mdicales Elsevier SAS

shape-memory alloys / structural dynamics / non-linear mechanics

1. Introduction

The shape memory alloys (SMA) are good candidates for damping vibration systems because they are theseat of a stress induced phase transformation (SIPT) between a mother phase called austenite and a productphase called martensite (Feng and Li, 1996). One objective of our paper is to understand the effect of SIPTon the dynamic of the system. For instance, SMAs and their variable material properties offer alternativeadaptative mechanisms (Keith et al., 1999). Hence, a SMA spring in parallel with traditional spring materialscreates an absorber with a variable stiffness and a corresponding tuning frequency. The pseudoelasticityassociated to SIPT is used for passive or structural vibration control applications (Saadat et al., 1999). Theloss of stiffness of SMAs during phase transformation also allows them to be used as absorbers or vibrationdampers, for example in seismic applications. In a natural way, shock or impact induced phase transformationare examined (Abeyaratne and Knowles, 1999) with a special focus on the propagation of stress waves andphase transformation fronts (Chen and Lagoudas, 2000), SMA elements can be embedded in a resin epoxy

matrix in order to create a hybrid system. A simple heating by Joule effect permits a shift of the naturalfrequencies (Ostachowicz et al., 1998). As a non-exhaustive state of the art, the interest of using SMA elementsfor dynamic response control is evident. In this paper, as it was previously said, the objective is to integratethe effect of a dynamical loading on the thermomechanical behavior of the shape memory alloy. In the presentwork, the SIPT is investigated while the reorientation of martensite platelets will be examined in future works.The present investigation needs:

Correspondence and reprints.E-mail address: [email protected] (M. Collet).


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616 M. Collet et al.

1. An efficient model in the range of pseudoelasticity (forward transformation A M and reversetransformation M A). Classical mechanical tests such as pure tension for different externaltemperatures on CuAlBe beams allow the identification of the material parameters (Raniecki et al.,

1992). Electrical resistance measurements performed at free stress state lead to the identification ofthe characteristic phase transformation temperatures. A special attention is devoted to the writing ofthe heat equation and the incremental energy dissipation expression in order to take into account thethermomechanical coupling.

2. An implementation of the static model in a simplified version for a bending cantilever beam in order toobtain a reliable and efficient finite element formulation. Hence, a dynamical model of this beam is setup with the introduction of an equivalent complex Youngs modulus which describes the stiffness anddamping of the structure whatever the phase state of the sample is.

3. An experimental set-up and measurement procedures of dynamical loading around the first flexion modeof the cantilever beam.

The results obtained are presented and interpreted in comparison with the theoretical predictions.

2. Thermomechanical model of pseudoelastic shape memory alloy behavior (Raniecki et al., 1992)

At first, a thermomechanical model in the frame of the thermodynamic of irreversible process is necessary. Asin standard plasticity, thermodynamical potential functions are needed. We will consider here the Helmholtzfree energy and two yield functions for the forward transformation A M and the reverse transformationM A. A one-dimensional version of the so called RL model (Raniecki et al., 1992) is written. Let us recallthe main equations of this model. For the two-phase reference volume element RVE containing the volumefractions (1 ) of mother phase (Austenite A) and of product phase (Martensite M), the free energy functioncan be set-up as in equation (1).

(,,T) =12

E

o(T To)

2+ Cv

(T To) T ln

T

To

+ (,T), (1)

where

= u(1)o T s(1)o

fo(T ) + (1 )it(T), (2)

and with the following definitions:

is the axial strain, u()o and s()o are the internal energy and entropy of the phase respectively ( = 1

for austenite, = 2 for martensite), fo(T ) represents the driving force associated with the temperature-induced martensite transformation of

the stress free state:

fo(T ) = u T s,

u = u(1)o u(2)o , s

= s(1)o s(2)o , (3)

the Young modulus E, the heat capacity Cv, the mass density and the thermal expansion o are chosento be the same whatever the phase state of the material,

it represents the coefficient of internal interaction between the martensite platelets and the mother phase:

it = uo Tso. (4)


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Analysis of the behavior of a Shape Memory Alloy beam under dynamical loading 617

In a classical way, the stress can be obtained as:

=

= E o(T To), (5)where the term represents the deformation pt associated with the phase transformation and o(T To) thethermal deformation t. The specific entropy of the system can be evaluated by:

s =

T. (6)

The driving force of phase transformation under mechanical loading is

f( , , T ) =

=

E( o(T To))

+ fo(T ) (1 2 )it. (7)

In order to establish the Clausius Duhem inequality, let us recall the first and second law of thermodynamicsspecified for an infinitesimal homogeneous process:

du = dq +d

, (8)

ds dq

T=

dD

T 0, (9)

where

u = + T s (10)

is the specific internal energy, dq represents the heat exchange and dD is the increment of the energy dissipation

which cannot be negative. In a routine way, the form of the incremental energy dissipation is written as:

dD = fd 0. (11)

Thus, the ClausiusDuhem inequality precludes the parentmartensite transformation (A M) at stateswhere f < 0 and prevents the reverse transformation (M A) when f > 0. Note that f = 0 implies theequilibrium condition. To specify the kinetic equations of phase transformations, we presume that there existtwo functions ( f, ) ( = 1, 2) such that an active process of parent phase decomposition (d > 0 theforward transformation) can only proceed when (1) = const (d(1) = 0) and an active process of martensitedecomposition (d < 0 the reverse transformation) can only proceed if (2) = const (d(2) = 0). Thesefunctions are given by:

(1)

= f

k(1)

( ), (2)

= f

+ k(2)

(). (12)In order to be in agreement with the kinetic phase transformation given by previous authors like (Koistinen andMarburger, 1959), the previous functions k()( ) can be written as:

k()(1) = (A1 + B1 ) ln(1 ), k( )(2) =

A2 B2(1 )

ln(), (13)

with

A1 =s so

a1, A2 =

s + so

a2, B1 =

2soa1

, B2 =2soa2

. (14)


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Coefficients a1, a2 can be obtained with the conditions () = 0 for stress free state. This implies the requiredexponential laws:

A M = 1 ea1(Mos T ),

M A = ea2(TAos ),

(15)

then

a1 = ln(1 Mf )

Mos Mof

, a2 = ln( Af )

Aof Aos

. (16)

Commonly used values for the forward and reverse transformations are respectively Mf = 0.99 and Af = 0.01.

In order to be self consistent, the heat equation corresponding to the energy conservation of the two phasessystem must be given. Remembering equations (9) and (11):

dD = T ds dq = fd, (17)

with s = T ( , , T ), the explicit form finally becomes:

Cv dT dq

+ T oE +

u (1 2 )uo

d + T oE d = 0, (18)

Cv dT dq

f( , , T ) + T (oE + s)

d + T oE d = 0. (19)

This last equation will now be simplified in the particular case of a vibrating beam.

3. Dynamical model of the cantilever beam

We consider a bending cantilever beam as shown in figure 1. In order to obtain a reliable and efficient finiteelement formulation, the following steps are needed:

the thermomechanical model previously described is simplified by defining the specific assumptionsrelated to that particular case;

an equivalent complex Young modulus taking into account the phase tranformation is defined; the vibrating equations of the bending beam are written by using the complex Young modulus; a finite element formulation based on an iterative computation procedure is built up.

3.1. Simplified thermomechanical model

The dynamical analysis of a system described by the previous thermomechanical model is very intricate ifwe try to solve it without any physical considerations. A simpler model usable in a numerical application canbe obtained by considering the following assumptions:

Figure 1. SMA cantilever beam.


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The term so in equation (4) is neglected. That means that the coefficient of internal interaction betweenthe martensite platelets and the mother phase is not temperature dependent, implying:

it = uo, B1 = B2 = 0.

The coefficients a1 and a2 are supposed to be equal, which implies that A1 = A2. No asymmetrical behaviour of the phase transition in tensioncompression is taken into account. This

hypothesis is classical in dynamical analysis where we know that this kind of behaviour which is importantfrom a static point of view is smoothed in the frequency domain.

The coefficient o is neglected. The phase transformation which occurs during a period of vibration is supposed to be adiabatic: dq = 0.

Putting all these assumptions in (19) and writing that d() = 0 (12) for both forward ( = 1) and reverse( = 2) transformations leads to the systems given in (20) and (21).

forward transformation (A M): f > 0, d 0

T oE

E

d =

Cv f(,,T) + T (oE + s

)

s +o E

2E

2it +

A

1

dT

d

; (20)

reverse transformation (M A): f < 0, d 0

T oE

E

d =

Cv f( , , T ) + T (oE + s

)

s +o E

2E

2it +

A

dT

d

. (21)

In the both cases, if [0, 1] then f( , , T ) becomes independent of , so f( , T ) , T and are constantalong all the specified phase transformation. The figures 2, 3 and 4 show respectively the adiabatic evolution of, T and versus . We immediately underline that this complex behaviour can be simplified by consideringsome pieces of affine functions as shown in figure 5. All characteristical points explicitely depend on the meantemperature To, the maximal strain m and the mechanical constants. They are calculated by using equations(20) and (21). This description can be completed for compression by using symmetrical properties of thetransformation.

3.2. Equivalent complex Young modulus

Looking at the simplified model offigure 5, it appears clearly that a phase transformation will induce a local

stiffness reduction. Moreover, the hysteresis phenomenon will increase the damping effect. If we consider agiven vibrating state, the equivalent stiffness and damping can be modelised by a complex Young modulus:

E = E(1 + i). (22)

Defining the strain energy densities W1, W1/4 over one period of vibration and one quarter of periodrespectively:

W1 =

T0

: d, W1/4 =T /4

0 : d,


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Figure 2. vs diagrams for different values of m: m = 0.018 (-.-), m = 0.035 ( ), m = 0.009 (-).

Figure 3. T vs diagrams for different values of m: m = 0.018 (-.-), m = 0.035 ( ), m = 0.009 (-).


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Figure 4. vs diagrams for different values of m: m = 0.018 (-.-), m = 0.035 ( ), m = 0.009 (-).

Figure 5. Simplified model of the non linear adiabatical phase transition.

it can be shown that these quantities are related to E and by the following relations:

W1 = E 2m, W1/4 =

E

2

1 +

2

2m, (23)

with m the maximal strain reached during one cycle as shown in figure 5. Two different cases need to bedistinguished: a partial transformation (m 2) ( 1) and a complete transformation (m > 2) ( = 1).


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Finally, for a given thermomechanical behaviour defined by (1, 2, 3, 4, 1, 2, 3, 4) and a given vibratingstate characterised by m, the complex Youngs modulus is obtained as follows:

partial transformation (m 2)

E =E21 + (2 + 1)(m 1)

2m

(m 1)2 + (2 1)2

(1 4)2 + (1 4)2 sin( )

2m, (24)

=2

(m1)

2+(21)2

E21 + (2+1)(m1)

(14)

2 + (14)2 sin ( )

(m1)

2+(21)2

(14)2 + (14)

2 sin ( ); (25)

complete transformation (m > 2)

E =E21 + (2 + 1)(2 1) + (2 + m)(m 2)

2m

(2 1)2

+ (2 1)2

(1 4)2

+ (1 4)2

sin( )2m

, (26)

=2

(m1)

2+(21)2

E21 + (2+1)(m2 1)+(2+m)(m

m2 )

(14)2+(14)2 sin ( )

(m1)2+(21)2

(14)2+(14)2 sin ( ), (27)

where and are characteristic angles of the stressstrain diagram given by

tan() =2 1

m

1

, tan() = E. (28)

These expressions can be even more simplified by considering that the stiffness of the transition period is verysmall compared to E. This implies 2 3 and 1 4, leading to:

partial transformation (m 2)

E =E21 + (m 1)(E(1 m + 3) + 2)

2m, (29)

=2

E(m 1)(m 3)

E21 + (m 1)(E(1 m + 3) + 2); (30)

complete transformation (m > 2)

E =E21 + (m 1)(E(1 m + 3) + 2) + (m

m2 )(22 + E(m 2))

2m, (31)

=2

E(m2 1)(m2 1))

E21 + (m 1)(E(1 m + 3) + 2) + (m m2 )(22 + E(m 2))

, (32)

where 3 is linearly dependent on m and To. Figures 6and 7show an example of the evolution ofE and versus the maximal strain m.


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Figure 6. E versus m.

Figure 7. versus m.


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3.3. Non-linear equations of the vibrating beam

By considering a beam as described in figure 1 with the Saint-Venant assumptions, the governing equations

can be written as follows:

2w(x,t)

t2

2M(x,t)

x2= f(x,t) x ]0, L[, (33)

w(0, t ) = 0,w(0, t)

x= 0, (34)

M(L,t) = 0, T (L, t) = 0, (35)

where M =

S xx y dy is the bending moment, T = Mx

the shear force, f the external force density appliedonto the beam. The internal strain is assumed to be linear inside the beam thickness and can be written accordingto Euler Bernoulli approach:

= y

2w

x2 . (36)Let us now consider a harmonic vibration at frequency and introduce the complex Youngs moduluspreviously defined:

= E(1 + i), (37)

the beam equation becomes:

2w(x,) + l

e/2e/2

E

y,2w

x2

1 + i

y,

2w

x2

y2 dy

4w(x,)

x4= F, (38)

where l is the beam span and e its thickness. We have defined an explicit non-linear problem where the complex

stiffness operator depends on 2

wx 2 in each section of the beam. The integral operator can finally be transformed

as an integration on variable . The maximal strain being m =e2 |

2w

x 2|, it becomes:

2w(x,) + 2lm

0E()

1 + i()

2|

2w

x 2|3

d 4w(x,)

x4= F, (39)

with the following boundary conditions:

w(0, ) =w(0, )

x= 0, (40)

2w(L,)

x2 =

3w(L,)

x3 = 0. (41)

3.4. Finite element computation procedure

To discretise the problem given by equations (39), (40) and (41), we introduce a finite element model withclassical P2 elements. In this case, the integral term in equation (39) is constant on each element. The discretisedweak solution is thus given by:

WTK(W)W 2WTMW WTF = 0, W H20 , (42)


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where H20 is the functional space generated by the finite element functions verifying W (0, ) =W (0,)

x= 0.

Note that the K matrix is complex. This nonlinear problem can be solved for each excitation frequency by aniterative procedure which tracks a stationary solution. The existence and uniqueness questions are not discussed

here. The initial value W0 is the elastic solution of the homogeneous problem. Let us consider a given iterationk and the corresponding value Wk .

The function Z(Wk) is computed as shown in (43), 2W

x 2 ibeing the value of

2w

x 2in element i. We can

underline that the computation ofK(W) directly depends on this vector:

Z(Wk) =

2W

x 2 1...

2W

x 2 N

= D Wk. (43)

Defining the testing function Y by:

Y(W) = Z

K(W) 2M

1F

Z(W), (44)

the stationary solution must verify:

Y(W) = 0. (45)

Wk+1 is thus given by Newton iterative steps:

Wk+1 = Wk

Y(Wk)

W

1Y (Wk), (46)

where the Jacobian matrix ( Y(Wk )W

)1 is estimated by a simple approximation around Wk .

3.5. Numerical results

The composition of the SMA used in the numerical applications is Cu 11.7 Al, 0.6 Be (Wt%). Itscharacteristic phase transformation temperatures measured by electrical resistance evolution are MoF = 191 K,MoS = 213 K, A

oS = 205 K, A

oF = 221 K. The material parameters are: E = 7.5e + 10 Pa m

2, =8129 kg m3, u = 2871.6 J m3, s = 11 J m3 K1, uo = 100.3 J m3, = 0.0295, a = 0.055,Cv = 490 J kg

1, o = 17.e 6 K1. These parameters have been identified by performing classical tensiletests at different temperatures in the range of pseudoelasticity, as described in (Raniecki and Lexcellent, 1998)and (Lexcellent et al., 1992). The dimensions of the cantilever beam are: L = 170 mm, l = 40 mm ande = 2.4 mm. The external temperature is To = 293 K. The force is applied at x = 25 mm.

Different tests have been carried out. Figure 8 shows the magnitude of the displacement at x = L forincreasing-decreasing frequency sweeping from 40 to 41 Hz and for different force levels. It must be notedthat the jump phenomenon occurs at different frequencies depending on the frequency sweeping order: for theincreasing tests, the jump frequency is greater than for the decreasing ones. Furthermore, we observe an increaseof the maximum amplitude and a decrease of the corresponding frequency as the force level is increased.

Figure 9 shows the position of the point yo corresponding to f = 0 in each element during each frequencystep for an applied force of 1.7 N. One can observe that the phase transformation occurs only near theembedding (elements 1 to 5) and that the transformation thickness is not so deep (about 1 /4 of the totalthickness).


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Figure 8. Amplitude Magnitude at x = L for ascending-descending sweep frequency sinusoidal force.

Figure 9. Coordinates of the point yo during the computation with F = 1.7 N.


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Figure 10. First natural frequency of the beam.

Figure 10 describes how the first natural frequency of the corresponding linear system changes when thetransition occurs. This frequency decreases when martensite appears in the beam skin. This phenomenoncorresponds to the evolution of Re(E) (figure 6).

4. Experiment

Classical dynamical measurements have been performed around the first flexion mode of the cantilever beam.We first describe the experimental set-up and the measurement procedures. The results are then presented andcommented on.

4.1. Experimental set-up

As it is shown in figure 11, the beam size is 170 40 2.4 mm. The composition of the SMA and

its characteristic temperatures are those given in 3.5. An electromagnetic exciter and an accelerometer arerespectively located at 13.5 and 160 mm of the embedding. The exciter consists of a light coil attached to thebeam and placed inside the induction field of a magnet fixed to the ground. A piezoelectric transducer betweenthe coil and the structure allows the applied force to be measured.

4.2. Measurements

As the dynamical behaviour of the beam is significantly nonlinear, stepped sine technique has been used. 19different force amplitudes between 0.01 and 1 Newton have been applied. After having determined the adapted


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Figure 11. Experimental set-up.

Figure 12. Frequency response functions for 19 different force amplitudes.

bandwidth by a quick random excitation, stepped sine acquisitions have been performed with 41 excitationfrequencies.

4.3. Results

The Bode plot of the obtained Frequency Response Functions (FRF) is given in figure 12. It clearly appearsthat the behaviour of the beam is nonlinear. The lower peak looks like a classical linear mode, it correspondsto the lower force amplitude (0.01 Newton). As this amplitude increases, it can be seen that the maximumamplitude first increases and that the corresponding frequency decreases. Figure 13 shows the superposition ofthe Nyquist diagrams for three force amplitudes: 0.25, 0.5 and 1 Newton. The fact that the shape of thesediagrams remains circular indicates that the damping nonlinearity effect is weak. On the other hand, the


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Figure 13. Nyquist diagrams for three force amplitudes: 0.25 (o), 0.5 () and 1 Newton ().

Figure 14. FRF for ascending (-) and descending (- -) frequency.


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dramatic phase shift (which is related to the jump phenomenon on the Bode plot) is typical of a stiffnessnonlinearity.

Finally, figure 14 shows the FRF obtained for a force amplitude of 0.8 Newton with ascending (continuous

line) and decreasing frequency (broken line). The observed phenomenon is similar to the prediction which wasmade using the finite element model of the beam.

5. Conclusion

The dynamical behaviour of a CuAlBe beam has been studied. A simplified thermomechanical model hasbeen set-up, leading to a nonlinear finite element model of the bending beam. Experimental measurementshave been performed around the first bending mode. The numerical results are in very good agreement with theexperimental ones: the FE model is able to simulate the jump phenomenon, depending on the force amplitudeand the frequency sweeping order. It appears that the stress induced phase transformation mainly affects the

beam stiffness causing an increase of resonance peak magnitude, whereas the hysteresis phenomenon does notchange the damping significantly. One can conclude that this SMA may not be a good candidate for a controlledbending damper. Nevertheless, this study gives us a nice understanding of the dynamic phase transformation,and will allow us to study various types of SMAs in order to optimise their use for structural active control.

References

Abeyaratne, R., Knowles, J.K., 1999. On a shock-induced martensitic phase transition. J. Appl. Phys. 87 (3), 11231134.Chen, Y.C., Lagoudas, D.C., 2000. Impact induced transformation in shape memory alloys. J. Mech. Phys. Solids 48, 275300.Feng, Z.C., Li, D.Z., 1996. Dynamics of mechanical system with a shape memory alloy bar. J. Intell. Mater., Systems and Structures 7, 399409.Keith, W., Chiu, G., Bernhard, R., 1999. Passive-adaptative vibration absorbers using shape memory alloys, in: SPIE Conference on Smart Structures

and Integrated Systems, Newport Beach, California, pp. 630641.Koistinen, D.P., Marburger, R.E., 1959. A general equation prescribing the extent of the austenite-martensite transformation in pure non-carbon

alloys and plain carbon steels. Acta Met. 7, 5569.Lexcellent, C., Raniecki, B., Berriet, C., 1992. Pseudoelastic behavior modelling of shape memory alloys CuZnSn and CuZnAl monocrystals, in:

13th RFSO International Symposium on Material Science, Roskilde, Denmark, pp. 343348.Ostachowicz, W., Krawczuk, M., Zak, A., 1998. Buckling of a multilayer composite plate with embedded shape memory alloy fibers, in: ICCE/5,

Las Vegas, Nevada, July 511.Raniecki, B., Lexcellent, C., 1998. Thermodynamics of isotropic pseudoelasticity in shape memory alloys. Eur. J. Mech. A/Solids 17 (2), 185205.Raniecki, B., Lexcellent, C., Tanaka, K., 1992. Thermomechanical models of pseudoelastic behaviour of shape memory alloys. Archives

Mech. 44 (3), 261284.Saadat, S., Saliche, J., Noori, M., Hou, Z., Davoodi, H., 1999. Promises for structural vibration control (SVC) using shape memory alloys, in: Trochu,

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Beam Under Dynamic Loading

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