FINITE ELEMENT SIMULATION FOR DESIGN OPTIMIZATION 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 in his paper the optimization of shape memory alloy (SMA) spring actuators. The purpose of these actuators is to control active endoscopes. The objective for such endoscopes is to minimize one patient pain during surgical interventions. Shape memory alloys have two main characteristics, namely the phase change and non-linear behaviour in each phase. The proposed model is based on a mixed approach combining an Euler-Bernoulli beam model and a bi-dimensional finite element model. It allows the modelling of plastic behaviour and drives the phase evolution in the beam cross-section. The resulting actuator model is then integrated into an optimization process based on genetic algorithms. The overall approach provides a design tool for shape memory alloy spring actuator subjected to design constraints.

Key words: shape memory alloy, finite element, structure calculation, thermo-mechanical behaviour.

Introduction We present in this paper a method for the design optimization of shape memory alloy (SMA) spring actuators whose purpose is the control of active endoscopes. This requires to address two key points. The first point deals with the definition of a fast and accurate model for the calculation of SMA springs actuators. The second is concerned with the use of this model into an optimization process. In (Dumont, 1

2002), a complete simulator for SMA actuated poly-articulated endoscopes was proposed. The objective of such active endoscopes is to minimize the pain suffered by patients during surgical interventions. The simulator can also be used to design, by virtual prototyping, different parts of endoscope prototypes. Figure 1 shows the designed articulations of such a prototype for an active endoscope, for which we want to design the SMA spring actuators. Take in Figure 1 The initial design of the actual prototype was reported in (Szewczyk, 1999) and the machining of the SMA spring actuators, detailed in Figure 2, were presented in (Kühl, 2002). The actuation is obtained by using Joule effect for changing the temperature of the two antagonist springs, leading to a change in their relative rigidity. This change in rigidity involves the rotation of the links. Take in Figure 2 To address the first key point, we propose a finite element model of the Euler Bernoulli beam to predict the behaviour of the SMA spring actuators. After a brief description of the SMA physical properties, we present a symmetric thermo-mechanical material law based on experimental identification of behaviour of SMA wires. Then, we present the mechanical formulation and a finite element model, involving a type of Euler-Bernoulli beam equation with varying stresses in the cross section. Then, we focus on the computational method, which involves an iterative solution process for the evolution

2

problem coupled with a Newton-Raphson algorithm. Some results showing the characteristic behaviour of SMA cantilever beams are shown. They allow us to address the second key point, which is the shape optimization process of the SMA spring actuators. The optimization method is based on genetic algorithms and is applied to a chosen set of parameters. We illustrate the proposed methodology by showing and discussing the shape of the optimized actuator.

SMA physical properties: description and model Short description Shape memory alloys (SMA) exhibit a shape memory effect (Fischer, 1998) as they undergo present a reversible transformation between two solid phases, as described in (Patoor, 1990). The martensite phase correspond to the crystalline state at low temperature and the austenite phase to the crystalline state at high temperature. In case the material has been subjected to a shape memorization process, the austenite phase corresponds to this memorized shape. The state diagram of such an alloy shows four characteristic temperatures of phase transformation. Heating the SMA under zero stress leads to a transformation from martensite to austenite that occurs between temperature As (austenite start) and A f (austenite finish). Cooling the SMA under zero stress leads to the reverse transformation from 



austenite to martensite that is also characterized by two temperatures, Ms

(martensite start) and M f (martensite finish). These temperatures

for the “Flexinol LT” wires used for our experiments are given by 



 As  68 C , A f  78 C , M s  52 C and M f  42 C .







3

Experimental identification Shape memory alloys are governed by two different constitutive laws in the austenitic and martensitic crystalline states. – In the austenitic phase, the behaviour of a SMA wire is elastic and linear, and is represented by the stress-strain relation   DA A where DA is the elastic (Young) modulus in the direction of the 

stress  and  A denotes the strain in that same direction. For our  

“Flexinol LT” wire, experimental results provide DA  39GPa . 

– In the martensitic phase, due to a reorientation phenomenon of the  martensitic plates, the SMA wire may exhibit a linear elastic behaviour and a pseudo-plastic behaviour. The elastic part is represented by the stress-strain relation   DM M while the pseudo-plastic behaviour is depicted on Figure 3, with the 

characteristic stress-strain pairs  1 , 1  and  2 , 2 . For the same “Flexinol LT” wire, experimental results give 

DM 12GPa ,



 1  38 MPa,  2  70 MPa, 1  3.55% and 2  4.38% .  

Take inFigure 3





Results for the thermo-mechanical behaviour model The

whole

thermo-mechanical

model

is

based

on

a

homogenisation method as presented in (François, 1991). We note that the percentage of martensite created (Trochu, 1997) is the key parameter, as it controls the constitutive relation   f  ,  ; therefore we use the martensitic fraction

R( ,  ) 1 (1 exp(K(   m  C )))



proposed in (Ikuta, 1990) where  denotes the temperature,  m is the 



4



average temperature (  m 

As  A f 2

if heating,  m 

Ms  M f 2

if cooling)

and C is the Clausius-Clapeyron constant of the material. Then the 

thermo-mechanical 



behaviour

law

takes

the

form

 ,    R ,   M   1 R ,   A  . We show in Figure 4 the

responses in case of mechanical and thermal cyclic loading predicted 

by this thermo-mechanical behaviour model. The results are in good agreement with experimental results. Take in Figure 4

Beam bending model The shape of the SMA springs shown in Figure 2 suggests a cantilever beam model. A method was proposed in (Rejzner, 2000) to deal with such beams. The method is based on a partitioning of the cross section into several subdomains. The stresses are calculated on these subdomains and are related to the bending moment, from which the curvature of the beam is obtained. However this model was developed for constant temperatures and supposes a constant bending moment. As the principle used to deform our actuator is to apply temperature variations, and as the shape of the actuator suggests that the bending moment may not be constant, we propose to generalize the above model. The new model, developed from the Euler-Bernoulli hypothesis, will use the thermo-mechanical uniaxial behaviour defined in the previous section. The Euler-Bernoulli hypothesis assumes that plane transverse sections of the beam remain plane during bending and that the stress is uniaxial. As indicated earlier, the behaviour of

5

SMA is non linear. For this reason, we employ an adapted finite element method to take into account the geometry and the loading of our springs. The method uses beam elements for discretization of the beam and quadrangular elements to represent varying stresses in the beam cross-section. Choice of elements Figure 5 displays the classical two-node, four degree-of-freedom element that we use, where v and  denote the orthogonal node displacement and the node rotation, respectively. This element

 accurately represents the behaviour under a continuously varying moment and shows discontinuity for localized moment. Take in Figure 5 With the Euler-Bernoulli hypothesis, we have the relation (x)  dv(x) dx and the displacement at a point x of the (i, j) element is

simply given by:





v(x)  vi  i x  (3



v j  vi l

2

2

i l



j l

)x 2  (2

 vi  v j l

3



i l

2



j l2

)x 3

With the same hypothesis, the strain tensor has only one component  xx , which is determined by:  xx  y

d 2 v(x) . With this dx 2

choice of element, we notice that the quantity  xx is discontinuous 



across elements. 

Due to the non-linear to behaviour of the SMA, we need to represent variable stresses in the cross section of the beam. We thus discretize along the thickness h of the beam using eight-node 6 

quadratic quadrilaterals as shown in Figure 6, where n is the number of used layers and k is the index of the considered layer. Our first

 attempt was based on four nodes linear quadrangles. In this case, the

 of the algorithm was not guaranteed because of the stress convergence discontinuity across layers, thus we use eight-node quadratic elements ensuring stress continuity across layers. Take in figure 6 In order to calculate the stress  (x, y) at points (x, y) inside the domain, we use the following classical finite element approximation: 



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



where the constants A , B , C , D , E , F , G and H are related to the stresses calculated at the nodes. We obtain a linear system of eight       



equations and eight unknowns written at the nodes. Among these equations, one example, referring to Figure 6, is the following: h h h ( ik )   h (x  0, y  k )  A  Ck  E(k )2 n n n



There is no difficulty in obtaining the other seven equations or in solving the system of equations.

Numerical algorithms Formulation of the mechanical problem In order to use the finite element method, we first derive a weak formulation of our original problem as in (Gallimard, 1994) or in (Lemaître, 1996).

7

Let: –



denote the domain occupied by the structure and  be the

boundary of  





–

the given body forces f g denote 

–

Fg denote the given tractions applied on the part 2 of the

boundary  



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



–   (t, x)  A( , x),   t

denote  the

constitutive

depending on the loading history 

The problem to solve reads: – Find U(t, x),  (t, x) defined on 0,T   such that: - U is 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 (kinematically  admissible to 0)  Tr   (U * ) d f g U *d  Fg U *dS  0   2   

 







- the constitutive relation is satisfied pointwise: x  , t  0,T ,  (t, x)  A( , x),   t  - the structure is initially unloaded: x  , U(0, x)  0,  (0, x)  0



8

relation,

Formulation of the discrete problem Our approach is based on a classical iterative method for solving an evolution problem on the domain  during the time interval 0,T . The time interval 0,T  is discretized into sub-intervals t i ,t i1 , on 



which the displacement is assumed to evolve linearly: 



t  t i ,t i1 , U h (t, x) 



U h (ti1 , x) U h (t i , x) (t  t i ) t i1  t i

The discretized displacement Uh (t, x) is written as Uh (t, x)  N(x) uh (t) , where N(x) are the finite element interpolation functions and uh (t) is 



the nodal displacement vector. The strain expression is simply 



h (t, x)  B(x) uh (t) ,

B(x)

where

are the

finite element strain-

displacement transformation functions given in terms of partial  N(x)



derivatives

of

and

the

constitutive

relation

becomes

 hi1  AB uh ,   t i1 . Assuming that the solution {uhj } and the related t

 tj h



{ } are known for each time t j , j  i , at time t i1 the discretized 

problem to solve reads: 

 J(uhti1 )  

 B  T



ti1 h

d 



 N

T

f g (t i1 )d 



 N F (t T

g

i1

)dS  0

2

This nonlinear equation is handled by the Newton-Raphson method with backtracking (Press, 1992).

Algorithm For solving this equation at time t i1 , we have to construct a sequence of vectors {ukti1 }, k  0,1,...,n . The iterative algorithm is 

initialized with uhti and an elastic behaviour step, in order to deal  

9

correctly with possible unloading. Thus

u0ti1

is searched as

u0ti1  uhti  u0 and the first step is to solve: 

- Find u0 such as:



 0ti1   hti  K e Bu0 where K e is the elastic operator 

J(u0ti1 )  

 B  T

d 







ti1 0

 N

T

f g (t i1 )d 

 N F (t T

g

i1

)dS  0

2



ti1 The iterative process is then performed to search uk1  ukti1  uk1 by



solving  u k



uuk

the

 J(ukti1 )    

 B

equation

T

  uk

 duk1  0  uuk 

where

 K t B is the tangent behaviour operator. 

The calculation of one time increment stops if J(ukti1 )  tol Fi1 where:



–

tol denotes a user prescribed  tolerance

–

Fi1  

 B  T



ti h

d 

 N

T

f g (t i1 )d 



 N F (t T

g

i1

)dS

is the

2

initial residual value constructed from the stress at time t i 

and from the tractions and body forces at time t i1 

Then uhti1  ukti1 is the solution a time t i1 .  Preliminary



results andcapability of an actuator

Computational results We demonstrate the performance of our method on two problems. The first problem is concerned with the creation of martensite under a mechanical load F  200N , for a beam at 90 C as shown on figure 7. This temperature is higher than the austenite finish temperature for 



10

this SMA, i.e. A f  78 C . In Figure 7, the top picture shows the martensitic fraction in the beam cross-section: the blue colour

 martensite and the red colour represents austenite. The represents bottom picture shows the stresses across the beam section: the green colour represents zero stress, and the red colour and the blue colour represent traction and compression, respectively. Take in Figure 7 The second problem deals with the behaviour of the beam under cyclic loads at various temperatures. The results are shown on figure 8. The temperatures, due to the actual characteristics 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, the stiffer the beam: this is due to the austenitic transformation at high temperatures. Take in figure 8 Since Rejzner (Rejzner, 2000) showed that the dissymmetry in the constitutive relation may be neglected, all calculations were performed under the assumption that the material law is symmetric both in traction and in compression. However, it was reported in (Sehitoglu, 2000) that such an assumption was not valid for nickeltitanium alloys, such as the “Flexinol LT”. This issue should be further investigated in order to verify this assumption.

11

Evaluation of actuator capability The simulation of one of the beams constituting the SMA spring actuators was showed to be consistent. Let us now focus on the actuator capability. We define the capability by answering two questions: – Is it possible, by heating one of the two SMA springs, to reach a given flexion angle? The objective for practical purposes, i.e. for the foreseen active endoscope is to reach an angle  15 . – Which torque can the actuator deliver when this angle is  reached? This value of the torque value, usually called the residual torque, qualifies the capability of the actuator to face both the flexion of the sheath and the additional loads that are due to the contact with the explored environment. In figure 9, we propose a schematic view of an actuator. In the initial configuration, the two wires have length linit corresponding to a desired preloading to ensure rotation. After a rotation by an angle  of the pin joint, the new positions ofthe wire ends are:

 –

     Ar Br   rB sin   x  linit  rB cos   y 2   2 

for the right

wire 

–

       Al Bl  rB sin   x  linit  rB cos   y  2    2 

wire. 

12

for the left

The distance from the (AB) line to the pin joint axis passing by O leads to moment calculation. This distance is:

–

d O, AB 

AO  AB AB

.

Take  in figure 9 Let us consider an actuator made of two antagonistic unique beams. Figure 10 shows the typical behaviour, in a load toward deflexion representation, of one beam in martensitic phase   35 C and of one beam in austenitic phase  130 C . The initial deflexion applied to  the beams, of 0.5mm here, leads to a preload. This preload is

 necessary to ensure the operation of the actuator. By heating the first

 beam, the deflexion goes lower and the beam follows the red trajectory. Conversely, the deflexion of the second beam goes higher and the beam follows the blue trajectory. It is then straightforward, based on geometrical considerations, to determine the torque delivered by the actuator. Only the load cycles both at high temperature and at low temperature need to be known to determine this torque. It should be noticed that the representative point in hot phase is located on the unloading part of the curve. Take in figure 10 Methodology for determining the actuator capability In order to determine the actuator capability, it is necessary to obtain the two load-displacement characteristics, shown in Figure 10, 13

one at low (ambient) temperature and the other at high temperature. These are obtained by employing the finite element beam model, presented above, loaded by a controlled external load. Let  15 be the desired maximal angle for the pin joint. The characteristic at low temperature, corresponding to the right-hand side  beam in figure 9, is calculated in order to reach the angle  under an

increasing load. For high temperature, the characteristic needs to be determined at unloading. Thus, the characteristicat high temperature, corresponding to the left-hand side in figure 9, is calculated in two steps. First, a loading is applied in order to reach the prescribed displacement induced by the preload. Second, a unloading is applied in order to reach the displacement corresponding to the angle  .

Shape optimization of a SMA spring actuator 

Specification of the parameters to optimize We provide below the set of parameters, which characterize the dimensions and shape of SMA springs, that need to be determined by the optimization process: –

l is the length of each blade of the spring. The maximum length is fixed to 2mm due the available space.

 –

h is the height of each blade in millimeters. Due to  realization constraints, h  0.1;0.5.

 

14

–

n is the number of blades. This number is between 2 and 8 in order to ensure that the structure fits in the available



space. –

p preload is the preload parameter. This parameter is between 0.1 and 6. Each initial blade deflexion is then provided by



init  ppreloadh .

Optimization algorithm  Classical optimization methods, see e.g. (Minoux, 1986), require the computation of first-order (or of second-order) derivatives of the function to optimize. However, there is no real analytic function here to quantify the capability of a SMA spring actuator. Therefore genetic algorithms (Goldberg, 1989) are considered here to solve this problem. The evaluation method used within this algorithm is as follows: – Once a set of parameters is selected by the genetic algorithm, we evaluate the distance between the blades. This distance should never be smaller than 0.1mm in order to be able to machine the actual spring. The overall size of 

the spring determines the preload. – The low and high temperatures responses are calculated by our finite element model. – The torque for  15 is calculated. If this torque is high enough to bend the sheath, the residual torque C is used 

15



to construct the objective function Fobjective  (1 C) with

 15 .



 – If this torque is not high enough to bend the sheath, we determine the angle  that can be reached. The objective function, in this case, is simply Fobjective   .

 – The objective function is then corrected by an estimate of  the stress concentration due to the radius of curvature at the junction between the blades. Result for shape optimization The evaluation time for the proposed function is approximately one minute for a total optimization time of eight hours on a standard personal computer. The parameters for the genetic algorithm are: – Steady-state algorithm – Population of 150 individuals – Replacement percentage: 85% – Crossing percentage: 90% – Mutation percentage: 5% – Stopping criterion: evolution lower than 95% on last 10 generations.

16

The obtained parameters for the optimized spring, presented in figure 11, are: –

l  2mm. This is not surprising, the longer the blade, the higher the deflexion, and the lower the



stress –

h  0.29mm for each blade. The empty space between blades is 131m long which is higher than



the necessary 100m  – The number of blades is n  6 

– The preload is p preload  0.9  Take in figure 11



Such a designed spring ensures a total deflection (  15 ) of the pin joint. The residual torque is C  33.103 mN , which is

 enough to support approximately 170 rings. The estimated maximal stress is 550MPaand is compatible with the maximal admissible stress ( 895MPa ), provided by the manufacturer. The

 constitutive law of this optimal SMA spring is described in figure 12.



Take in figure 12

Conclusion A model based on a beam with variable stresses across the section is proposed. This model is valid for non-constant temperatures. The 17

change in phases for shape memory alloys and the pseudo-plastic behaviour for martensite requires the development of a non linear time-dependent model. The resulting model is solved by an iterative Newton-Raphson method. Our numerical results are in agreement with the experimental observations of wires, and have been validated for mechanical and thermal cyclic loading. Predictions of the behaviour of spring actuators show to be reasonably accurate as well. However, comparisons with experimental values should be performed to complete this validation. The proposed method is then incorporated into an optimization process, based on genetic algorithms, for optimal design of SMA spring actuators. The overall approach has been successfully applied to the design of such an actuator. We believe that an actuator compatible with the prescribed design constraints could not have been designed without the help of our optimization tool.

References Dumont G., Kühl C. and Andrade G. (2002), « A dynamical simulator for designing active endoscopes », In proceedings of the 5th World Congress on Computational Mechanics, WCCM V, Vienna University of Technology, Vienna, Austria. ISBN 3-9501554-0-6, 712 July, http://wccm.tuwien.ac.at. François D., Pineau A. and Zaoui A. (1991), « Comportement mécanique des matériaux », Editions Hermès, 1991 also published as « Mechanical behaviour of materials », Kluwer Academic Publishers, 1998.

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Fischer F.D., Oberaigner E.R., Reisner G., Sun Q.P, Tanaka K. (1998), «Shape Memory Alloys (SMAs): their properties and their modelling», Revue Européenne des Eléments Finis, Vol. 7, n°8, pp.934. Gallimard L. (1994), « Contrôle adaptatif des calculs en élastoplasticité et en viscoplasticité », Thèse de Doctorat, École Normale Supérieure de Cachan. Goldberg D.E. (1989),

« Genetic algorithms in search,

optimization and machine learning », Addison Wesley Longman, Inc.

Ikuta K. (1990),

« Micro/miniature

shape memory alloy

actuator », In Proceedings of IEEE International Conference on Robotics and Automation (ICRA’90). Los Alamitos, CA, USA. IEEE Comput. Soc. Press, vol.3, p. 2156-2161, 13-18 May. Kühl C., Dumont G., Mognol P., Gouleau S. and Furet B. (2002), « Active Catheter Prototyping: from virtual to real », In proceedings of IDMME’02: 4th International Conference on Integrated Design and Manufacturing in Mechanical Engineering, Clermont-Ferrand, 14-16 May. Lemaître J. and Chaboche J.L. (1996), « Mécanique des matériaux solides », 2ème édition, Editions Dunod, 1996 also published as « Mechanics of Solid Materials », first edition, Cambridge University Press, 1994.

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Minoux, M. (1986), "Mathematical Programming: Theory and Algorithms", John Wiley. Patoor E. and Berveiller M. (1990), « Les alliages à mémoire de forme », Editions Hermès. Press W. H., Teukolsky S. A., Vetterling W. T. and Flannery B. P. (1992), « Numerical rescipes in C, The art of scientific computing, Second edition », Cambridge University Press. Rejzner J. (2000), « 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. Szewczyk J., Troisfontaine N. and Bidaud P. (1999), « An active tubular polyarticulated micro-system for flexible endoscopes », In proceedings of International Advanced Robotics Program (IARP’99): International Workshop on Micro Robots, Micro Machines and Systems, Moscow, 24-25 November. Sehitoglu H., Canadinc D., Zhang X., Kotil T., Karaman I., Gall K., Maier H.J. and Chumlyakov Y. (2002), « Overview of shape memory single crystals », Invited talk presented at the NATO ARW conference, ENSAM, Metz, France, 23-26 April. Trochu F. and Qian Y.Y. (1997), « Nonlinear Finite Element Simulation of Superelastic Shape Memory Alloy Parts», Computers & Structures, Vol. 62, n°5, pp.799-810, Elsevier Science Ltd.

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Pin joint axis

Ring avoiding translation

Spring

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

Figure 2: View of the SMA spring actuators

Figure 3: Cyclic loading in martensitic phase

21

Figure 4: Thermo-mechanical constitutive law

vi

y

x

vj

lij

i

j node j

node i

Figure 5: Beam finite element

( ik 1 ) h (k  1) n  



(

k



1 k 2  i

( kj 1 )

k 1 ij

k

)

h ( ) n node i

k  i





k ij



1 2 

( j ) ( )

 





 Figure 6: Thickness discretization for stresses

22

k  j

node j 

1 h (k  ) 2 n

Figure 7 : Creation of martensite under mechanical load

Figure 8: Response to cyclic loading at   65 C ,   70 C ,   73 C and   80 C . 

y



Bl 

Br rB

 O

Al



Br’



Bl’ linit



hd

Ar Co dfix ntr  ôle Figure 9: Schematic view of an actuator

23

x

Figure 10: Limit cycles for each wire

Figure11: Shape of the optimized SMA spring

Limit cycles 18 16 14 12

results at 35¡C Interpolation at 35¡C

10

lo ad (N)

Results at 130¡C Interpolation at 130¡C

8

Lower bound 6

Preload Higher bound

4 2 0 -0,1

0

0,1

0,2

0,3

0,4

0,5

-2 deflexion (m m)

Figure12: Characteristics for the optimized SMA spring

24