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Modeling and optimization of Shape memory-Superelastic antagonistic beam assembly

Majid Tabesh, Mohammad Elahinia Dynamic and Smart Systems Laboratory, MIME Department, University of Toledo, Toledo, OH, 43606

ABSTRACT Superelasticity (SE), shape memory effect (SM), high damping capacity, corrosion resistance, and biocompatibility are the properties of NiTi that makes the alloy ideal for biomedical devices. In this work, the 1D model developed by Brinson was modified to capture the shape memory effect, superelasticity and hysteresis behavior, as well as partial transformation in both positive and negative directions. This model was combined with the Euler beam equation which, by approximation, considers 1D compression and tension stress-strain relationships in different layers of a 3D beam assembly cross-section.

A shape memory-superelastic NiTi antagonistic beam assembly was simulated with this model. This wire-tube assembly is designed to enhance the performance of the pedicle screws in osteoporotic bones. For the purpose of this study, an objective design is pursued aiming at optimizing the dimensions and initial configurations of the SMA wire-tube assembly.

1. INTRODUCTION:

NiTi shape memory alloys (SMA) are the most commonly studied and implemented SMAs. They gained researchers’ attention due to their special properties of shape memory (SM) and superelasticity (SE). In addition, the material is biocompatible (with mechanical properties more comparable to bone than Ti-based compounds or stainless steel) and has a good resistance to wear and corrosion. The shape memory effect is the recovery of large strains (up to 8%) created in the material while in the low temperature range by raising the temperature to above a specific value. That specific temperature (called Austenite finish Af) can be manipulated via changing the composition of the material or thermo-mechanical treatment to get a value around body temperature. Superelasticity is the nonlinear elastic behavior of SMAs at temperature levels above Af. Superelasticity (SE), shape memory effect (SM), high damping capacity, corrosion resistance, and biocompatibility are the significant properties of NiTi, unique for biomedical applications.

NiTi is used in orthopedic implants as compression staples/clamps for the treatment of bone fracture and anterior fusion of the spine. It is also employed in intramedullary nails that are used to apply a controlled force to the bone. Applications of NiTi in fixation bone plates or rods for the treatment of scoliosis as well as all the aforementioned cases are nowadays common (1).

The development phase for such applications requires primary modeling and simulation considering NiTi’s complicated thermomechanical behavior. There are many models that can capture the performance of shape memory alloys; among them the Tanaka model (later extended by Brinson) was utilized in this study to simulate the behavior of SMA (2); namely the shape memory effect and superelasticity. In this work, the 1D model developed by Brinson was modified to capture the shape memory effect, superelasticity and hysteresis behavior, as well as partial transformation in both positive and negative directions. This model was combined with the Euler beam equation that takes into account the 1D compression and tension stress-strain relationships in different layers of the cross section of 3D beam assemblies.

The algorithm utilizes an iterative method to decide whether the state of the material at every node is in linear elastic mode, i.e. pre or post transformation, or in transformation mode to use cosine-form transformation equations. The algorithm finally solves for the deflection of an SMA beam.

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A Shape memory-superelastic NiTi antagonistic beam assembly can be used for enhancing the performance of pedicle screws in osteoporotic bones. A pedicle screw is a particular type of bone screw designed for implantation into a vertebral pedicle. The pedicle screw, which is sometimes used as an adjunct to the spinal fusion surgery, provides a means of gripping a spinal segment. The major drawback of spinal surgical treatments with pedicle screws is the lack of strength in degraded osteoporotic bone. The antagonistic assembly consists of a superelastic NiTi (with lower Af temperature) tubular beam and a shape memory NiTi (with higher Af temperature) circular wire that serves as a strength-enhancement attachment for pedicle screws.

Proper functionality of this application necessitates an optimization routine to be performed so that the optimum configuration in terms of several variables forming the assembly such as geometry and dimensions can be defined. For the purpose of this study, an objective design is pursued aiming at optimizing the dimensions and initial configurations of the SMA wire-tubing assembly.

2. MODELING OF A SMA BEAM

In order to characterize the behavior of shape memory alloy beams, the one dimensional model proposed by Brinson (3) is modified and used in an algorithm. The model considers strain and temperature as the field variables and volume fraction of martensite in the material as the internal variable. The martensirte transformation can be triggered with either changing the temperature or applying stress so the fraction of martensite was divided into stress-induced and temperature induced martensite. The temperature induced or twinned martensite, ζt, consists of a self accommodated combination of martensitic variants. On the other hand, the stress induced martensite, ζs, represents the amount of material transformation into a single martensitic variant corresponding with the direction of the loading. Since in pure bending of beams the upper and lower layers undergo stress in two opposite directions, the stress induced martensite is further divided into two variables to capture the variants of martensite which correspond to positive and negative stress directions. With this distinction the model can predict pseudoelastic and shape memory effects in SMAs. The constitutive equation relating the thermomechanical variables of strain ε, volume fraction of stress induced martensite ζs, and temperature T is:

𝜎 − 𝜎0 = 𝐷 𝜀 − 𝜀0 + 𝛺 𝜁𝑠 − 𝜁𝑠0 + 𝛩 (𝑇 − 𝑇0 ) (1)

The 0 subscript here denotes the initial condition of that variable. Θ is the coefficient of thermal expansion for the SMA material. D is representative of the elastic modulus of SMA material and is taken different for martensite and austenite structures. It is reasonably assumed that the overall modulus of the SMA structure depends on the volumetric fraction of martensite, ζ, as below:

D = Da + ζ (Dm − Da) (2)

Where Da and Dm are respectively the modulus value of the fully austenitic and fully martensitic SMA material.

Ω is the transformation coefficient and can be shown to be:

Ω = −εLD (3)

εL L is the maximum transformation strain of an SMA and is found to be a constant at temperatures below austenite finish temperature, Af. The volume fraction of martensite is:

ζ = ζt + ζs (4)

The stress induced martensite, ζs, is in turn decomposed into positive, ζsp, and negative, ζsn, constituents which respectively correspond to the positive and negative directions of the applied stress:

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ζs = ζsp + ζsn (5)

The transformation equations and transformation stresses have been developed based on an empirically-derived cosine model for defining the relationship between martensite fraction and stress and temperature during transformation

(3). They should be slightly modified to incorporate the decomposition of stress induced martensite into

positive and negative parts.

The transformation to positive or negative stress-induced martensite, ζsi, i = p,n; can be described as below:

if T > Ms and σsicr − CMi T − Ms < 𝜎 < σfi

cr − CMi T − Ms (5)

ζsi =1 − ζsi 0

2cos

π

σsicr − σfi

cr × σ − σficr − CMi T − Ms +

1 + ζsi0

2

ζt = ζt 1 − ζs

1 − ζs0

, ζsj = ζsj 0

1 − ζsi

1 − ζsi0

, ζs = ζsi + ζsj

For transformation to austenite we have:

if T > As and CA T − Af < 𝜎 < CA T − As (6)

ζsp =ζsp 0

2 cos aA T − As −

σ

CAp

+ 1

ζsn =ζsn 0

2 cos aA T − As −

σ

CAn

+ 1

ζt =ζt0

2 cos aA T − As −

σ

CAi

+ 1

ζ = ζt + ζsp + ζsn (7)

Where aM =π

Ms −M f , aA =

π

Af−As

In these equations Ms, Mf, As, Af, are respectively martensite start and finish transformation temperatures and austenite start and finish transformation temperatures. The trend in the critical stress value at the start and end of transformation is adequately expressed as a linear function of temperature with a slope of CM for martensite and CA for austenite transformations. It should be noted that the zero value for different variables should be the value at the start of the transformation process.

σscr and σf

cr denote the critical stresses for start and finish of transformation into stress-induced martensite for temperatures below Ms.

It is worth mentioning that dissymmetry effects have been reported (4)(5) for some shape memory alloys between the critical stress for the onset of martensitic transformation in positive and negative directions; therefore the values corresponding reverse transformation are distinguished for positive and negative directions. Despite this fact, Rejzner et al.(6) showed that neglecting this dissymmetry effect does not have significant influence on the prediction of the SMA NiTi beam behavior.

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The aforementioned model was implemented in an algorithm through a MATLAB® code to characterize the behavior of a beam of NiTi shape memory alloy. The material properties for NiTi are directly taken from (3) and restated in table 1.

Table 1: Material properties for NiTi

Moduli Transformation Temperatures (℃)

Transformation constants

Maximum residual strain (%)

𝐷𝑎 = 67 × 103 MPa 𝑀𝑓 = 9 𝐶𝑀 = 8 MPa/℃ 𝜀𝐿 = 6.7 𝐷𝑚 = 26.3 × 103 MPa 𝑀𝑠 = 18.4 𝐶𝐴 = 13.8 MPa/℃

𝛩 = 0.55 MPa/℃ 𝐴𝑠 = 34.5 𝜎𝑠𝑐𝑟 = 100 MPa

𝐴𝑓 = 49 𝜎𝑓𝑐𝑟 = 170 MPa

For the purpose of this study the properties of the material were considered the same in positive and negative directions although the model is capable of incorporating non-symmetric effects. The beam is considered an Euler-Bernoulli beam which implies the cross-sections remain plane and orthogonal to the centerline of the deformed beam. The stresses rather than the normal stress are also neglected. If the beam undergoes a deflection of w = w(x), the strain in cross-section layers and equilibrium in terms of the axial force, N, and bending moment, M, are:

εx = −y∂2w

∂x2 , N = σ y b(y) dyh

2

−h

2

, M = yσ y b(y) dyh

2

−h

2

(8)

y and x denote respectively the transverse and longitudinal directions. b(y) is the function representing the shape of the cross section. For the purpose of simplicity the cross section of the beam is considered symmetrical with respect to the centerline so that it could be regarded as the neutral axis.

The algorithm discretizes the beam into nodes in longitudinal and transverse directions. According to the loading, a strain εx = εx (y) linearly distributed through y is initially estimated for each cross section. Then the values of stresses corresponding to this state of strain are calculated at every layer of every cross section. The state of stress is checked against the equilibrium equations for N and M and is revised iteratively to achieve the equilibrium conditions. Revision is made through linear shifting of the stress over the cross section according to a slight change in strain, see Figure 1.

The algorithm utilizes an iteration method to decide whether the state of the material at every node is in linear elastic mode, i.e. pre or post transformation or in transformation to use cosine-form transformation equations.

The final deflection is obtained by integrating the second derivative of w, ∂2w

∂x2 = −εx

y, over the longitudinal direction.

Figure 2 shows the one-dimensional stress-strain graph for NiTi at different temperatures above Af. As depicted the critical stress for onset of transformation raises with increasing temperature. Figure 3 illustrates the stress and strain distribution over the cross section of a rectangular cantilever beam.

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Figure 1 The iteration flowchart for solving the state of stress and phase transformations in an SMA cantilever beam.

Figure 2 Superleastic stress-strain plots at different temperatures.

Figure 3 Stress and stain plots at a cross section of a rectangular cantilever beam 1*10*100 mm under a tip force of 200 N. the beam is loaded and unloaded at a temperature below Ms and recovered at a temperature above Af.

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The beam has a cross section of 1*10 mm and a length of 100 mm with a force of 200 N acting vertically at its tip. The shape memory beam is loaded and unloaded at a temperature below Ms. a mesh of 10*100 is selected for this problem after performing mesh optimization. It is obvious that some middle layers, whose stress state allow, go under transformation during loading so that there is a coexistence of untransformed, transforming and transformed material. The state of stress changes after unloading with some initially tensioned layers now in compression and vice versa; much like residual stresses in an unloaded beam of elasto-plastic material. By raising the temperature of the beam up to above Af the stress-strain state in the beam recovers back to original condition. One point is that since the length of the beam is taken as constant during loading-unloading-recovery, there would be some residual constant stress if the temperature coefficient of expansion for the material is not considered negligible.

The similarity of loading-unloading behavior of SMA beams to elasto-plastic beams suggested the use of dimensionless notions and numbers in the Plastica theory

(7). The following non-dimensional numbers were introduced:

f =FLh

2YI ; the dimensionless force with magnitude F (MPa) being the force applied at the tip of the beam, L (mm), h

(mm), and I (mm4) respectively the length of the beam, half of the height of the beam cross section, and its second

moment of area; the cross section is supposed to be at least symmetrical to the transverse axis. Y is regarded as the stress required for the onset of transformation at the top or bottom layers hence for the change of the linear stress-strain relationship into nonlinear form. i.e:

Y = σscr + CM T − Ms (9)

β = 2L σs

cr + CM T − Ms

EAh

w =δ

L ; where δ (mm) is deflection at the tip.

β is a parameter which reflects the geometric and material properties of the beam. It is equal to the curvature of the section bearing the ultimate elastic moment – prior to the start of transformation at the outermost layers- and is an indication of the flexibility of the beam.

In order to capture the trend of deflection in the shape memory and super-elastic beams under bending, several simulations where performed for loading of shape memory cantilever beams at different temperature levels. Figure 4 shows the plot of dimensionless force versus dimensionless tip displacement for various β values. The plots are obtained by only loading SMA beams at different temperatures. It is possible to evaluate the performance of a superelastic beam under tip loading with a custom cross section in terms of tip displacement using this figure. These curves can be concluded into the following dimensionless equation:

f = aw + H x − b c sin dw + e (10)

Where

a =297.4

β, b = 0.005 × β − 0.012 or 0.002 × β1.33 , c = −3.552 × β + 87 or

493

β (11)

d = −0.005 × β + 0.61 or 0.72

β0.11 , e = −0.002 × β + 3.14 or {

3.18

β0.01},

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Figure 4 Dimensionless force f vs. dimensionless tip displacement δ/L of an SMA

beam

In order to estimate the amount of tip deflection of a SMA with arbitrary symmetrical cross section under a specific tip force and at a specific temperature, one should evaluate the dimensionless values β and f, and then calculate the amount of δ using the above equations.

It should be noted that small displacement Euler-Bernoulli beam theory

is accurate only for the first few percent of w; furthermore the dimensionless force-tip displacement equation was supposed to cover the range of β mentioned and works for both superelastic and shape memory beams.

These equations are utilized to predict the behavior of a shape memory beam with circular cross section which is inserted into an outer superelastic beam with tubular cross section.

3. ANTAGONISTIC SHAPE MEMORY-SUPERELASTIC BEAM ASSEMBLY

Bone screws have been used in spinal instrumentation since the 1960s. A pedicle screw is a particular type of bone screw designed for implantation into a vertebral pedicle. The pedicle screw, which is sometimes used as an adjunct to the spinal fusion surgery, provides a means of gripping a spinal segment. The screws themselves do not fixate the spinal segment, but act as firm anchor points that can then be connected with other spinal instrumentations. One major drawback in spinal surgery using pedicle screws is the adverse effect of osteoporosis. Osteoporotic bone, due to the degradation of supporting bone structures, cannot provide enough of an anchoring foundation for the screw to be implanted into; therefore pedicle screw placement in osteoporotic bone entails the risk of intra-operation or post-operation screw pull-out or loosening.

One proposed solution to enhance the performance of pedicle screw in osteoporotic bone utilizes shape memory-superelastic NiTi antagonistic beam assembly.

The assembly consists of a superelastic NiTi (with lower Af temperature) tubular beam and a shape memory NiTi (with

higher Af temperature) circular beam; as illustrated in Figure 5. Moreover as in Figure 5, the initial memorized shape of

both the tubing and the wire are bent off the centerline in opposite directions. The basis of operation is the difference

in transformation temperatures between the wire and the tubing. At a named low temperature- body temperature

37℃ for the present biomedical purpose- the tubing is superelastic and the wire is shape memory; at a higher

temperature level- suppose 70℃ - the wire also becomes superelastic above which both wire and the tubing exist in

austenite state trying to achieve their initial memorized shape. This process is depicted in Figure 6. At low temperature

the tubing is in stiffer austenite form while the wire is in flexible martensite form so the tubing tends to raise the

assembly upwards towards its memorized situation. At high temperature the wire also becomes austenitic and acts

against the current situation, pushing the assembly down.

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Figure 5 Initial memorized shapes of the assembly components.

Figure 6 Assembly at (a) low temperature, (b) high temperature

Figure 7 (a) the assembly in straight condition, (b) desired high temperature form, (c) desired low temperature form

Functioning of the smart pedicle screw requires the amount of raise at low temperature to be as much higher as possible and the amount of recessing at low temperature to be such that the assembly reaches the original straight condition or even bends downward; Figure 7 explains the desired functionality illustratively.

These requirements necessitate an optimization routine to be performed so that the optimum configuration in terms of several variables forming the assembly could be defined. These variables include: the diameters of the wire and the tubing, the transformation temperatures of both the wire and the tubing, the austenite/martensite modului, the initial memorized deflection of the wire and the tubing, etc.

The physical and shape memory properties of SMA are virtually opposed by the supplier of the material and literally cannot be optimized. So for the purpose of this study, two NiTi alloys are considered with fixed properties tabulated in table 2. Tubing is supposed to be out of alloy B (superelastic) and the wire out of alloy A (shape memory). The optimization is then focused on the geometrical aspects’ of the assembly; namely diameters and initial deflections.

Table 2: Material properties for NiTi alloy A (SE) and B (SM)

Moduli Transformation Temperatures (℃)

Transformation constants Maximum residual strain

(%)

A NiTi: Ni 55.0%wt

𝐷𝑎 = 67 × 103 MPa 𝑀𝑓 = 12 𝐶𝑀 = 8 MPa/℃ 𝜀𝐿 = 6.7

𝐷𝑚 = 26.3 × 103 MPa 𝑀𝑠 = 31.4 𝐶𝐴 = 13.8 MPa/℃ 𝛩 = 0.55 MPa/℃ 𝐴𝑠 = 47.5 𝜎𝑠

𝑐𝑟 = 69 MPa 𝐴𝑓 = 62 𝜎𝑓

𝑐𝑟 = 193.05 MPa

B NiTi: Ni 55.8%wt

𝐷𝑎 = 67 × 103 MPa 𝑀𝑓 = −35 𝐶𝑀 = 8 MPa/℃ 𝜀𝐿 = 5

𝐷𝑚 = 26.3 × 103 MPa 𝑀𝑠 = −25.6 𝐶𝐴 = 13.8 MPa/℃ 𝛩 = 0.55 MPa/℃ 𝐴𝑠 = −9.5 𝜎𝑠

𝑐𝑟 = 55 MPa 𝐴𝑓 = 5 𝜎𝑓

𝑐𝑟 = 130MPa

4. OBJECTIVE DESIGN: The objective process is schematically drawn in Figure 8. δB1and δA1are the initial tip deflection of tubing B and wire A respectively. For the purpose of simplicity the two mating beams are modeled as two serially connected springs with

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nonlinear stiffness properties. Some approximations are included herein: firstly the shear force between the wire and the tubing is neglected therefore the problem reduces to two entangled beams having the same curvature at every cross section. Secondly the hysteresis effect during unloading of the superelastic beam and the linear elastic unloading of the shape memory beam are also disregarded and the behavior of the beams are modeled as nonlinear elastic materials with stiffness properties shown and formulated in Figure 4. In fact, it could be proved that the hysteresis or linear loading of the material tends to help to further achieve the objective functionality of the assembly and neglecting them is a conservative assumption.

Figure 8 Schematic procedure representing functioning of the antagonistic beam assembly

Hence the overall deflection of the system at low temperature, ΔL , can be solved via this system of nonlinear equations:

FA2 − FB2 = 0

δA2 + δB2 − δB1 − δA1 = 0 (12) ΔL = δB1− δB2 = δA2 − δA1 (13)

By the same token for the overall deflection of the assembly at high temperature, ΔH

FA3 − FB3 = 0

δA3 + δB3 − δB1 − δA1 = 0 (14) ΔH = δB1− δB3 = δA3 − δA1 (15)

For the solution of the above equations, the length of the tubular and circular beams are taken to be L=10 mm, the yielding stresses Y are defined as the critical stress for start of the transformation to Austenite and the moduli E are taken as the stiffness of the material while in austenite phase. It should be mentioned that the higher ΔL the more engagement of the smart pedicle screw with its surrounding bone and hence the higher the pull-out resistance of the

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screw. ΔH should be as low as possible so that the screw becomes revisable. Therefore, the goal is to find a configuration of initial deflections, δB1and δA1, and the corresponding diameters - outer diameter of the tubing dB and the outer diameter of the wire dA - by which ΔL maximizes and ΔH minimizes.

5. OPTIMIZATION OF THE BEAM ASSEMBLY:

The aforementioned problem is a multi-objective problem which is solved through a compromise method (8). Mathematically the problem can be stated as below:

find X =

δA1

δB1

dA

dB

, such that the objective function f X1 maximizes and f X2 minimizes where f X = ΔLΔH

subjected to the constrains

g1 X = dA −2.6dB −12.7

≤ ∅ , and g2 X = 0.1−dA0.15−dB

≤ ∅ , and X 3 − X(4) ≤ 0 (16)

The upper and lower boundaries on the diameters are due to the limitation of production. The fact that the decision variable X is discrete by itself suggests breaking the continuous search space in to a set of combinatorial feasible points. The result is depicted in Figure 9. The outcome is further revised in Figure 10 by applying the constraints (16) stated above. The compromising constraints that distinguish the optimum region are

ΔH

ΔL ≤ 0.1 and ΔL ≥ 0 (17)

Figure 9 Discrete solution space for the optimization problem

Figure 10 Optimum region

The optimum region shown in Figure 10 is additionally discretized and the points are solved for to obtain the trade-off surface.

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Figure 11 Trade-off surface

The points on the trade-off surface which do not dominate each other make the set of Pareto optimal solutions:

𝑿𝒂 𝑿𝒃 𝑿𝒄 𝑿𝒅

𝜹𝑨𝟏 4.9 5 5 5

𝜹𝑩𝟏 1.7 1.4 1.5 1.6

𝒅𝑨 0.10 0.10 0.10 0.10

𝒅𝑩 0.15 0.15 0.15 0.15

Notably the values for diameters are the lower boundary ones.

The effect of each parameter on the objectives ΔL and ΔH , can be investigate through Figure 12 to Figure 15.

Figure 12 Effect of 𝜹𝑨𝟏 on the objective functions

Figure 13 Effect of 𝜹𝑩𝟏on the objective functions

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Figure 14 Effect of 𝒅𝑨on the objective functions

Figure 15 Effect of 𝒅𝑩on the objective functions

6. CONCLUSION: The unique properties of NiTi shape memory alloys present remarkable opportunities in the development of smart biomedical devices. To this end, the capability to simulate and evaluate the behavior of these materials is essential. An assembly of shape memory wire and superelastic tubing out of NiTi was modeled through a MATLAB®-based code. An optimization process was also conducted to determine the configuration that yields the desired performance of the assembly. This antagonistic assembly is intended to enhance the performance of pedicle screws in osteoporotic bone. The results of this optimization procedure were used to prepare the pilot prototype of the smart pedicle screw.

7. REFERENCES:

[1] Medical applications of shape. L.G. Machado, M.A. Savi. s.l. : Brazilian Journal of Medical and Biological Research,

2003, Vol. 36.

[2] Lagoudas, Dimitris C. Shape Memory Alloys: Modeling and Engineering Applications. s.l. : Springer, 2008 .

[3] One-dimensional constitutive behaviour of shape memory alloys: thermomechanical derivation with non-constant

material functions and redefined martensite internal variable. Brinson, L. C. s.l. : journal of intelligent material systems

and structures, April 1993, Vol. 4.

[4] Study of pseudoelastic behaviour of polycrystalline shape memory alloys by resistivity measurements and acoustic

emission. Vacher, P. and Lexcellent, C. Kyoto : proceedings of ICM VI, 1991. 3: 231-236.

[5] Non-symmetric tension-compression behaviour of NiTi alloy. Orgeas, L. and Favier, D. s.l. : JOURNAL DE PHYSIQUE

4, 1995, Vols. 5, (8): C8-605-C8-610.

[6] Pseudoelastic behaviour of shape memory alloy beams under pure bending: experiments and modelling. Rejzner, J.,

Lexcellent, C. and Raniecki, B. 4, s.l. : International Journal of Mechanical Sciences, April 2002, Vol. 44.

[7] Yu, Tongxi X. and Zhang, Liangchi. Plastic Bending: Theory and Applications. s.l. : World Scientific, 1996.

[8] Collette, Yann and Siarry, Patrick. Multiobjective Optimization: Principles and Case Studies. s.l. : Springer, 2003.