IDMME 2004

Bath, UK, April 5-7, 2004

MIXED BEAM MODEL TO CALCULATE THE BEHAVIOUR OF SHAPE MEMORY ALLOY SPRING ACTUATORS Georges Dumont IRISA/INRIA, SIAMES Project, Campus de Beaulieu, 35042 Rennes cédex, France, +33299842574, [email protected]

Christofer Kühl ENSAM, 151 boulevard de l’hôpital, 75013 Paris, France, +33144246417, [email protected]

Abstract: We address the calculation of shape memory alloy (SMA) spring actuators. The purpose of these actuators is to control active endoscopes. We have already proposed a complete simulator dedicated to SMA actuated polyarticulated endoscopes. The objective for such endoscopes is to minimize one patient pain during surgical interventions. The proposed model is based on a mixed approach between an Euler-Bernoulli beam model and a bi-dimensional finite element model. The quality of the results allows the integration of the actuator model into an optimization process, which will not be described here. We first propose an overview of SMA modelling and exhibit some of the main characteristics of such alloys. The phase change is one of these characteristics and the non-linear behaviour of each phase is the second. The chosen model must be able to deal with plasticity, and to pilot phase fraction in the beam cross-section. We present the algorithms used to calculate the response. The results show the ability to track phase fraction during a cyclic loading and shows the different behaviours of SMA actuators at different temperatures.

Key words: shape memory alloy, finite element, structure calculation, thermomechanical law. 1 Introduction We address the calculation of shape memory alloy (SMA) spring actuators. The purpose of these actuators is to control active endoscopes. In [1], we have proposed a complete Pin joint axis

ring avoiding translation Spring actuators

Figure 1: Detail of pin joint and of SMA actuators 1

IDMME 2004

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simulator dedicated to SMA actuated poly-articulated endoscopes. The objective for such endoscopes is to minimize one patient pain during surgical interventions. We also use the simulator to design, by virtual prototyping, the different parts of an endoscope prototype. Figure 1 shows a part of this prototype of active endoscope, for which we want to design the SMA spring actuators. This prototype has been reported in [2] for the initial design and in [3] for the machining of the actuators, which are detailed in Figure 2. The actuation is obtained by changing the temperature of two antagonist springs, resulting in change of relative rigidity. This change in rigidity involves to the rotation of the links. Joule effect conducts to such a change in temperature.

Figure 2: View of the SMA spring actuators

In this paper, we propose a beam model to calculate the behaviour of the SMA spring actuators. In the following, we propose a small overview of the SMA physical properties and develop a symmetric thermo-mechanical material law. In a second part, we present the calculation model, based on an Euler-Bernoulli beam equation with varying stresses in the cross section. Then, we focus on the calculation methods and algorithms that we have used to solve the problem. The obtained results show characteristic behaviour of SMA cantilever beams, and allow us to address the optimization process. 2 SMA physical properties: description and model 2.1 Short description The shape memory effect [4] is observed for some alloys that present a reversible transformation between two solid phases. The martensite phase is present at low temperature and the austenite phase is present at high temperature. In case the material is educated, this austenite phase corresponds to the memorized shape. The process of phase transformation is presented in [5]. Figure 3 represents the state diagram of such an alloy and shows four noticeable temperatures of transformation of phases under zero stress loading. –

M s is the martensite start temperature: transformation A  M begins. Characteristic for “Flexinol LT” wires gives Ms  52 C . 

 

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A  M stops.

–

M f is the martensite finish transformation: transformation Characteristic for “Flexinol LT” wires gives M f  42 C .

–

As is the austenite start transformation : transformation M  A begins. Characteristic for “Flexinol LT” wires gives As  68 C . 

–

M  A stops. Characteristic A f is the austenite finish transformation : transformation 

for “Flexinol LT” wires gives A f  78 C . 





 M+A

M



A



Mf

Ms

As

Af

T

Figure 3: State diagram of a Shape Memory Alloy      2.2 Experimental identifications The constitutive law of SMA is different in the austenitic and in the martensitic phases. – In austenitic phase, the behaviour of a SMA wire is elastic and linear, and is represented by the equation:

A   D A

(1)

where DA is the elastic modulus in the direction of the stress  and where A is the strain in this direction. For “Flexinol LT” wires, experimental results give DA  39GPa.



– In martensitic phase, the behaviour of a SMA wire is much more like an elastic-plastic  behaviour, linked to reorientation phenomenon of the martensitic  plate. It is  represented by the equation:

M   D M

(2)

for the elastic part, and shows a pseudo-plastic behaviour, presented on Figure 4. For the same “Flexinol LT” wire, experimental results give DM 12GPa, 1  38MPa,  2  70MPa, 1  3.55% and 2  4.38% .



 



 3

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IDMME 2004

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Mechanical cycle loading in martensitic phase 150

Stress (Mpa)

100

2 50

1 2

1

0



-50

 

-100

-150 -5

 -4

-3

-2

-1

0

1

2

3

4

5

Strain (%)

Figure 4: Cyclic loading in martensitic phase 2.3 Results for the thermo-mechanical behaviour model We propose a behaviour model. This model is based on a homogenisation method as presented in [6], and uses the martensitic fraction evolution R(T, ) 1 (1 exp(K(T  Tm  C ))) proposed in [7]. Figure 5 shows the results obtained for our thermo-mechanical behaviour model. It allows calculating mechanical and thermal cyclic loading and shows good accordance with experimental results.



Thermal cyclic loading

Mechanical cyclic loading 4.5

400 T=20° T=75° C T=80° C T=120° C C

350

3.5

Strain (%)

Stress (Mpa)

300 250 200 150

3 2.5 2 1.5

100

1

50

0.5

0 0

sigma=40M sigma=60M pa sigma=80M Pa Pa

4

1

2

3

4

5

6

0 20

7

Strain(%)

40

60

80

100

120

140

Temperature (°C)

Figure 5: Thermo-mechanical constitutive law 3 Beam bending model The shape of the SMA springs presented in Figure 2, suggests a cantilever beam model. In [8], a method is proposed to deal with such beam. This method is based on a partitioning of the cross section into several domains. The stresses are calculated on these domains and are related to the bending momentum. Then the curvature of the beam is obtained. This model is suitable for constant temperatures and supposes a constant bending momentum. The principle used to deform our actuator is to apply temperature variations, and the shape of the actuator suggests that the bending momentum is not constant. For these reasons, we propose a 4

IDMME 2004

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generalized model under the hypothesis of Euler-Bernoulli. As the behaviour of SMA is non linear, we use an adapted finite element method to take into account the geometry and the loading of our springs. This method uses beam elements to support discretization and quadrangular elements to represent varying stresses in the beam cross section. 3.1 Choice of elements The chosen element is the classical two nodes, four degrees of freedom element as presented in Figure 6, with the notation v for node displacement and  for node rotation.

vi



y

x

vj



lij

i

j node j

node i

Figure 6: Beam finite element With the Euler-Bernoulli hypothesis, we have the relation (x)  dv(x) dx and the displacement of a current point of the (i, j) element is simply:

v j  vi v v     (3)  2 i  j )x 2  (2 i 3j  2i  2j )x 3 2 l l l l l l  accurately the behaviour under a continuously varying This element represents momentum, thus it shows discontinuity for localized momentum. The only necessary component xx of the associated discontinuous strain tensor is then determined by: v(x)  v i   i x  (3



d 2v(x)

(4) xx  y dx 2 We need, due to behaviour of the SMA, to represent variable stresses in the cross section



of the beam in order to use local constitutive law. We introduce a complementary discretization of the thickness h of the beam. This discretization is based on eight nodes quadrangles and is presented in Figure 7. Tests have been made with four nodes linear quadrangles, but the convergence of the algorithm was not guaranteed, thus we use eight nodes quadratic elements.  ( ik 1 ) 

h (k  1) n  

k

( i

k



h n

1 2 

k

)

( )

node i



( kj 1 )

k 1 ij

k  i

 ijk





1

( j 2 ) ( )

k  j

 

node j







 Figure 7: Thickness discretization for stresses 5

1 h (k  ) 2 n

IDMME 2004

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The used interpolation to calculate  (x, y) inside the domains, is the following:

 (x, y)  A  Bx  Cy  Dx 2  Ey 2  Gx 2 y  Hxy 2

(5)

where the constants are related to the stresses calculated at the nodes by a linear system of  eight equations and eight unknowns written at the nodes. Among these equations, one example, referring to Figure 7, is the following:



h h h ( ik )    (x  0, y  k )  A  Ck  E(k ) 2 n n n There is no difficulty in obtaining the other seven equations.

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4 Calculation and algorithms



4.1 Formulation of the mechanical problem The formulation of the problem refers to the finite element method with non-linear behaviour. Our model finds his basis in [9] and in [10] Let:











f g denote the given applied forces in the volume of 

–

  Fg denote the given applied forces on a surface part 2  

– Find U(t, M), (t, M) defined on 0,T    such as : -



–

–  (t, M)  A(, M),  t  denote the material law, depending on the history of the   loading  The problem to solve is:





 denote the structure,  is the boundary surface of 

 – U g denote the boundary conditions on 1   , where 1 is the complementary part of 2 





–

-

U is regular and kinematically admissible: t  0,T , U(t, M)  Ug (t, M) on 1  



is statically admissible: t  0,T , U * such as

  

U * (t, M)  0 on 1  Tr   (U * )d   f g  U * d   Fg  U * dS  0 

 2



- the material law isverified:

 M  , t  0,T,  (t, M)  A(, M),  t 

- the structure is initially completely discharged:



M  , U(0, M)  0,  (0, M)  0

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IDMME 2004

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4.2 Formulation of the discrete problem The used method is a classical iterative method for solving the non-linear evolution problem on the domain  during the time interval 0,T . The time interval 0,T  is discretized in sub-intervals t i1,t i , on which the displacement is supposed to have a linear evolution. By using a Newton-Raphson method with backtracking [11], the problem is to solve:   

J(uhi1)    BT  hi1d  

 N

T

f g (t i1 )d 

 N

T

Fg (t i1 )dS  0

(9)

 2



In this equation, we have used the classical notations: –



–

  

N(M) are the finite element interpolation functions: Uh (t, M)  N(M)uh (t) B(M)

are the finite element strain-displacement transformation functions:

h (t, M)  B(M)uh (t)



4.3 Algorithm We have to construct a sequence of vectors u0,u1,...,un . The iterative algorithm is initialized with an elastic behaviour, in order to deal correctly with possible unloading. u0 is searched as u0  uhi  u0 . The first step is to solve:



- Find u0 such as:



˜ 0   Ke Bu0 with K e the elastic operator i h

   BT˜ 0d  





 N

T

f g (t i1)d 

T

Fg (t i1)DS

 2





 N

The calculation for this increment is finished if J(un )  tol Fi1 , where: – tol denotes the tolerance –



Fi1    BT  hi d  

 N

T

f g (t i1 )d 

 N

T

Fg (t i1)dS  0

 2



If not, another iteration is performed to search un 1  un  un 1 by solving the equation:





  T  un 1  0 J(un )   B d    u  n u un   This solving induces the calculation of the tangent elastic operator.

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5 Results and limitations We propose two kinds of complementary results. The first one, Table 1, shows the evolution of the martensitic fraction for a beam at 90 C for a cyclic loading from F  200N to F  200N . This temperature is higher than the austenite finish temperature for this SMA,

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A f  78 C . In this table, the top image shows the martensitic fraction in the beam cross sections: the blue colour represents martensite and the red colour represents austenite. In this table, the bottom image shows the stresses across the beam sections: the green colour represents zero stress, the red colour represents traction and the blue colour represents compression.

F  200N

F  0N

F  200N

Important stress at fixed boundary There exists a residual There is martensite creation under implies apparition of martensite. displacement. This displacement is stress. The beam becomes softer. explained by the pseudo-plastic    behaviour in the martensitic phase

Table 1: Response to cyclic loading at 90 C





The second, Table 2, shows the behaviour of thebeam under cyclic loads for different temperatures. The chosen temperatures, due to the real characteristic of the material, are   65 C ,   70 C ,   73 C and   80 C . For temperatures  between As  68 C and A f  78 C , the hysteresis shifts with respect to the cyclic loading. The stabilization of this hysteresis occurs after ten cycles. The results show also that the higher the temperature is, the harder the beam   appears: this  is the result of austenitic transformation  for high temperatures. As Rejzner [8] has shown that the material law dissymmetry may be neglected, all the calculations were performed under the hypothesis that the material law is symmetric in traction and in compression. Nevertheless, it is reported [11] that this hypothesis is not valid for nickel-titanium alloy, as the “Flexinol LT”. An investigation may be now considered in order to deal with this dissymmetry during the performed simulations.

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Table 2: Response to cyclic loading at   65 C ,   70 C ,   73 C and   80 C . 6 Conclusion   the section A model based on a beam withvariable stresses across is proposed. This model is valid for non-constant temperatures. The change in phases for SMAs and the pseudo-plastic behaviour for martensite implies to develop a non-linear model evolving in time. The method used to solve the obtained equations is iterative and the non-linearity is handled by a NewtonRaphson method. The results show good accordance with the experiment for wires, and have been validated for mechanical and thermal cyclic loading. Application to behaviour prediction of spring actuators leads to good results to. Nevertheless, comparisons with experiments on these spring actuators should be done to complete this validation. These simulations results are the basis of an optimization process for designing SMA spring actuators that is currently in progress.

References [1] Dumont G., Kühl C., Andrade G., « A dynamical simulator for designing active endoscopes », Proceedings of the 5th World Congress on Computational Mechanics, WCCM V, Autriche, 2002. [2] Szewczyk J., Troisfontaine N., Bidaud P., « An active tubular polyarticulated micro-system for flexible endoscopes », Proceedings of IARP: International Workshop on Micro Robots, Micro Machines and Systems, 1999. [3] Kühl C., Dumont G., Mognol P., Gouleau S., Furet B., « Active Catheter Prototyping: from virtual to real », Proceedings of IDMME’02: 4th International Conference on Integrated Design and Manufacturing in Mechanical Engineering, Clermont-Ferrand, 14-16 May 2002.

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[4] Fischer F.D., Oberaigner E.R., Reisner G., Sun Q.P, Tanaka K., «Shape Memory Alloys (SMAs): their properties and their modelling», Revue Européenne des Eléménts Finis, Vol. 7, n°8, 1998, pp.9-34. [5] Patoor E., Berveiller M., « Les alliages à mémoire de forme », Editions Hermès, 1990. [6] Fançois D., Pineau A., Zaoui A., « Comportement mécanique des matériaux », Editions Hermès, 1991 or « Mechanical behaviour of materials », Kluwer Academic Publishers, 1998. [7] Ikuta K., « Shape memory alloy micro/miniature actuator », Technical meeting micromachining and micromechatronics, pp. 77-86, 1989. [8] Rejzner J. « Modélisation des alliages à mémoire de forme soumis à des sollicitations multiaxiales ou à des gradients de contraintes », Thèse de doctorat, Université de Franche-Comté, 2000. [9] Gallimard L., « Contrôle adaptatif des calculs en élastoplasticité et en viscoplasticité », Thèse de Doctorat, Ecole Normale Supérieure de Cachan, 1994. [10] Lemaître J., Chaboche J.L. « Mécanique des matériaux solides, 2ème édition », Editions Dunod, 1996 or « Mechanics of Solid Materials », Cambridge University Press, 1994. [11] Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P., « Numerical rescipes in C, The art of scientific computing, Second edition », Cambridge university press, 1992 [12] Sehitoglu H., Canadinc D., Zhang X., Kotil T., Karaman I., Gall K., Maier H.J., Chumlyakov Y. « Overview of shape memory single crystals », Presentation at the NATO conference, Metz, 2000.

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