The Development of a User Defined Material Model for NiTi SMA Wires

Transcription

1 The Development o a User Deined aterial odel or NiTi S Wires. Iannucci, P. Robinson and W..H. Wan Hamid Imperial College ondon, United Kingdom 1 Introduction Shape memory alloys (Ss) have attracted enormous attention among researchers since their discovery in 196s, because o their unique characteristics namely shape memory eect (SE) and Superelasticity or Pseudoelasticity. These characteristics are caused by the change o the S crystal structure rom a martensite phase to an austenite phase, and vice versa, upon changes in applied temperature or applied mechanical stress. The shape memory eect (SE) has been utilized in various industrial applications including medical, dental, robotics, automotive and aerospace applications. Examples include cylindrical stents to enlarge blood vessels, teeth braces, robotics arms, heat engines, side mirrors o cars, morphing wings and morphing chevrons o jet engines. Finite element modelling o the SE has been perormed by implementing S constitutive models into FE sotware, BQUS. Gao et al. [1] developed a one dimensional S inite element model by implementing a Brinson constitutive model into a user subroutine in BQUS. The model was tested or isobaric, isothermal and constrained recovery cases, and was successively applied to a selhealing hybrid composite to close cracks. However, very little work has been carried out in modelling the shape memory eect (SE) o the Ss using the commercial explicit FE sotware, s-dyna. The most relevant material model in s-dyna is T_3 T_SHPE_EORY [2], which is ideal or the Superelastic behaviour o the Ss at a deined constant temperature, but not the SE behaviour with temperature changes. This paper describes a inite element material model o the shape memory eect (SE) o S wires in s-dyna. The S wire is modelled as a beam element with one integration point, to mimic the behaviour o a 1-D truss element. One o the earliest one dimensional S constitutive model, the Tanaka model [3-4], is implemented within a s-dyna user deined material (UT) subroutine, to simulate the thermomechanical behaviour o the S wires. The new S material model is irst validated by comparison with analytical solutions or a S wire in series with a spring, beore application to a cantilever beam and a more complex corrugated plate structure. The development o the S model in the explicit commercial FE sotware, s-dyna, allows modelling o structures in a ast dynamic situation, such as during aircrat actuation. Simulations o potential morphing structures can be made to determine the number o S wires, the appropriate S cross-sectional area, the structural design or the iber orientation o composites structures to achieve the intended deormed shapes. The actual deormation o the actuated structures can also be predicted. oreover, the s-dyna sotware has a capability o simulating incompressible luid dynamics (ICFD), which could be used to simulate and analyze the movement o the actuated structures in a realistic air low condition (luid-structure interaction - FSI). 2 User Deined aterial (UT) odel o the S user deined material (UT) model is developed or the shape memory eect (SE) characteristic o NiTi shape memory alloy (S) wires. This unique characteristic has requently been utilized or many applications, such as actuation o morphing structures. nother well-known characteristic o S, Superelasticity or Pseudoelasticity, not considered in the current paper. Because the S wire acts in one direction, a one-dimensional constitutive model [3-4] was implemented in a user deined subroutine. The S wire was modeled as a beam element with one integration point. The truss subroutine was not used because output history variables, hsv, cannot be extracted or truss or beam that uses the resultant ormulation. The Hughes-iu beam element ormulation (EFOR = 1) was used or the S. The S constitutive equation is shown in Eq Copyright by DYNmore GmbH

2 T E (1) The time-dependant variables,, T and are the stress, strain, temperature and martensite volume raction o the S, respectively. E, and are the modulus o elasticity, thermoelastic coeicient and phase transormation coeicient, repectively. In a total ormulation, Eq. 1 is expressed as: E ) ( T T ) ( ) (2) ( The thermal expansion part is neglected because the contraction rom recovery strain is much higher (1.6% - 4.%) than the thermal expansion caused by temperature changes (2.6E-5 5.5E-5). This is a common assumption in S constitutive model development. The modulus o elasticity o the S has smaller value in the martensite phase, E, and higher value in the austenite phase, E. The modulus o elasticity is a unction o the martensite volume raction. eanwhile, the transormation coeicient is a unction o the modulus o elasticity. These relations are shown in Eq. 3 and Eq. 4. E E E E (3) E (4) For the Hughes-iu beam element ormulation (EFOR = 1) in s-dyna, the time step changes with a change in the element length and the wave speed. This wave speed, which is a unction o modulus o elasticity and density, changes during the phase transormation o the S because the modulus o elasticity changes. Thereore, the time step also changes. The time step and the wave speed are calculated by Eq. 5 and Eq. 6, respectively [5]. t e c (5) E c (6) where, c, E, and ρ are the element length, the wave speed, the modulus o elasticity and the density, repectively. The martensite volume raction or reverse transormation (heating) and martensitic transormation (cooling) are given by Eq. 7 and Eq. 8, respectively. a( s T ) b T e, (7) a ( st ) b T 1 e, (8) where s and s are austenite and martensite starting temperatures respectively, while a, b, a and b are material constants deined by: a ln(.1) s, b ln(.1) C (, s ) a ln(.1) s, b ln(.1) C ( (9) where and are austenite and martensite inishing temperatures respectively. t each time step, Newton Raphson iteration method was applied to solve Eq.2 and Eq. 7/8 using Eq. 1. ( n) ( ) n 1 n (1) n For the reverse transormation (heating), at each time step, Eqs. 2 and 7 were solved iteratively to obtain the stress and martensite volume raction o the S. With initial conditions o zero stress, zero strain and unity martensite volume raction, the stress unction is expressed as: E E 1 (11) s ) 217 Copyright by DYNmore GmbH

3 First derivative o this unction is given by Eq. 12. It is important to note here that the strain, ε is treated as a number because it is solved by s-dyna every time the stress o the S is obtained. It is also crucial to mention here that the strain is the total strain o the S, which is introduced as one o history variables, hsv(), and not the incremental strain, eps(1). E E 1 E 1 (12) Partial dierentiation o the martensite volume action and the elasticity modulus in Eq. 12 are obtained by dierentiating Eq. 7 and Eq. 3 with respect to stress, respectively. These are expressed by Eq. 13 and Eq. 14, respectively. E b e a( s T ) b b E E E Eb aterial Constants S material properties / variables Values - ass density, ρ 7.89E-6 kg/mm 3 cm(1) Elastic modulus at martensite phase, E 4 GPa cm(2) Poisson s ration, ν.3 cm(3) Elastic modulus at austenite phase, E 75 GPa cm(4) artensite starting temperature, s 47 o C cm(5) artensite inishing temperature, 43 o C cm(6) ustenite starting temperature, s 6 o C cm(7) ustenite inishing temperature, 65 o C cm(8) Stress inluenced coeicient at martensite phase, C.82 Gpa/ o C cm(9) Stress inluenced coeicient at austenite phase, C.82 Gpa/ o C cm(1) aximum recoverable strain, ε.16 cm(11) Diameter o S wire, Ø.1/.2/.5 mm cm(12) ength o S wires (early stage o model developement), l - cm(13) Stiness o linear spring (early stage o model developement), k - cm(14) aximum number o iteration (convergence criteria), n max 5 cm(16) Bulk modulus, B GPa cm(17) Shear modulus, G GPa cm(18) FE simulation case: 1. spring-s case (actuated structure) 1 or 2 2. constant load case cm(19) Relaxation time (or constant load case), t relaxation 1, ms cm(2) onitoring:. Switch o print statements 1. Switch on print statements or 1 Table 1: UT material constants and material properties o S wire. For the martensitic trasormation (cooling), Eqs. 2 and 8 were solved iteratively at each time step, to obtain the stress and martensite volume raction o the S. With initial conditions o zero martensite volume raction, non-zero initial stress and strain which were obtained rom the inal values at the end o previous heating, the stress unction is expressed as: E E ( ) (15) This unction dierentiated with respect to stress gives: E ( ) E E (13) (14) 1 (16) 217 Copyright by DYNmore GmbH

4 Partial dierentiation o the martensite volume action and the elasticity modulus in Eq. 16 are obtained by dierentiating Eq. 8 and Eq. 3 with respect to stress, and expressed by Eq. 17 and Eq. 18, respectively. E b e a ( st ) b b ( 1) E E E E b ( 1) ll equations were introduced as history variables within the UT subroutine, with Eq. 7 and Eqs in a heating loop, while Eq. 8 and Eqs in a cooling loop. Eq. 1 was used in both loops to calculate the stress o the S or each time increment. The material properties o the S wire were obtained rom the manuacturer datasheet, SES [6], and were assigned to the material model through material constants in the UT keyword deck. These are listed in Table 1. The S material model is irst tested or a case o a S wire connected to a linear spring in series, which is validated by analytical solution presented in the next section 3. The modelling approach is applied to a cantilever beam, and inally to a pre-curved corrugated plate. vectorized UT has been developed or the pre-curved corrugated plate actuated by multiple S wires. This increases the simulation speed as all beam elements o the S are processed simultaneously at each time step. 3 nalytical Formulation To validate the S material model implementation, analytical solutions or the S wire connected to a linear spring in series were solved in TB. One complete heating-cooling cycle was applied on the S wire, and the stress was calculated iteratively using Eq. 1. In contrast to the FE model, the analytical strain was treated as a variable which is a unction o the S stress, as expressed in Eq. 19. For the iteration, the stress unctions and their derivatives o the martensite-to-austenite transormation (heating) and the austenite-to-martensite transormation (cooling) are given by Eqs and Eqs , respectively. The respective partial derivatives o the martensite volume raction and the stiness are similarly deined in the previous Eqs and Eqs l l F kl kl (19) kl E E 1 (2) E E 1 E E kl 1 (21) kl E E ( ) (22) E kl E E E E (17) (18) 1 (23) For the cantilever beam, it is well known that the tip delection is given by: tip 3 F 3EI (24) Hence to validate the S stress and strain or the cantilever beam case, the analytical S-spring model was solved in TB by replacing the stiness (k) value with the stiness calculated by Eq Copyright by DYNmore GmbH

5 F tip 3EI 3 (25) 4 FE Simulation Results The FE simulation results o the user deined material (UT) model or the S wire are presented in the ollowing sections. The material model was initially validated prior to detailed actuator simulations o a cantilever beam and a pre-curved corrugated plate. Several cases are presented or the pre-curved corrugated plate, such as the number o S wires per cell and the coniguration o S wires in each cell. 4.1 Validation o S aterial odel Fig.1: S wire connected to a linear spring in series. The UT was initially validated by an analytical solution as described in section 3. For a S wire connected to a linear spring in series (Fig.1), the linear spring was modelled as a discrete element with a spring stiness o 3.5 N/mm, while the S wire was modelled as a beam element with one integration point (QR/IRID = 1). The Hughes-iu beam element ormulation (EFOR = 1) with a circular/tubular cross section (CST = 1) was applied. The inner (TT1 and TT2) and outer (TS1 and TS2) diameter o the cross section were deined as mm and.5 mm, respectively. One end o the S wire and the spring were ixed in all degree o reedoms, and one complete heating-cooling cycle was applied on the S wire. Figure 2 shows the stresses and strains in the S, respectively. The solid lines represent the FE simulation results while the dotted lines represent the analytical solutions. s the temperature increases above austenite starting temperature s, the S wire contracts (the strain decreases) and extends the linear spring, hence the S stress increases. These are shown by the red lines. ter this actuation, the temperature decreases and consequently reduces the stiness o the S wire rom E to E. s a result, the linear spring extends the S wire, hence the S strain increases and the S stress decreases, as shown in blue. The results show excellent agreement between analytical and numerical solutions. Fig.2: Validation o S stress and strain by analytical solutions. This model was run on a computer with 8 CPUs and 32 GB memory capacity, and was solved in 31 minutes (CPU time). In a preliminary check, the kinetic energy o the S wire increased slightly at the beginning o heating, by approximately 4.5 µj, and then decreased back until zero at the end o heating. The kinetic energy slightly increased again at the start o cooling, by approximately 4.9 µj, and drops to zero at the end o cooling. This assured that the S wire reached equilibrium state at the end o heating and cooling cycles. The S internal energy decreased by approximately 4.25 mj 217 Copyright by DYNmore GmbH

6 during heating, then maintained at a constant value once the martensite-to-austenite transormation completed, and then increased back during cooling. n opposite trend was observed or the internal energy o the linear spring. This showed that the loss o the S internal energy due to work done on the linear spring during heating (actuation), was converted to potential energy, stored in the linear spring. This stored potential energy was used to retract the S wire during cooling. The hourglass energy be zero at all time, which ensured that there were no hourglass problems in the S material model. 4.2 Cantilever Beam ctuated by a S Wire The previous section showed the interaction between the S wire and the discrete element, a linear spring. Beore applying the S wire on a more complex structure, a cantilever beam actuated by the S wire was modelled in s-dyna to show the interaction between the S material model and shell elements. The S wire has a diameter o.5 mm, a length o 1 mm, and a maximum recoverable strain o 1.6%. The aluminium cantilever beam was modelled with shell elements with a density o 2.81E-6 kg/mm 3, a Young s modulus o 71.7 GPa and a lexural stiness (EI) o 5975 knmm 2. The length, width and height o the cantilever beam were 225 mm, 1 mm and 1 mm, respectively. One end o the cantilever beam was ixed in all degree o reedoms while the other end was ree and was actuated by the S wire in a vertical direction, as shown in Fig. 3a. One complete heating-cooling cycle was applied to the S wire. (a) (b) Fig.3: cantilever beam (a) beore and (b) ater actuation, (c) tip delection o the cantilever beam. The actuation o this cantilever beam model was run on a computer with 4 CPUs and 32 GB memory capacity, and was completed in 1 hour and 5 minutes (CPU time). The deormation o the cantilever beam ater S actuation is shown in Fig. 3b. The resulted vertical tip delection o the cantilever beam is 1.57 mm, as shown by the ringe levels. The tip delection as a unction o temperature is shown in Fig. 3c. The stress developed in the S wire and the S strain or one complete heatingcooling cycle are depicted in Fig. 4a and 4b, respectively. They show excellent agreement with analytical solutions, which were solved in TB with spring stiness ( F / ) o N/mm. The recovery stress ater the completion o the heating cycle is smaller (around 12.6 Pa) compared to the previous example, because the stiness in the actuation direction is smaller. In contrast, the recovery strain achieved at the end o the heating cycle is slightly higher in magnitude (1.58%), which is close to the maximum recoverable strain o the S wire (1.6%). (c) 217 Copyright by DYNmore GmbH

7 (a) Fig.4: (a) Stress and strain o the S wire or the actuation o the cantilever beam. (b) 4.3 Pre-curved Corrugated Plates (a) (c) (b) Fig.5: Pre-curved corrugated aluminium plates actuated by one S wire per cell: (a) isometric view and (b) side view, and by two S wires per cell: (c) isometric view and (d) side view. To demonstrate the applicability o the developed S material model on a more complex structure, the model was urther tested or actuation o pre-curved corrugated plates. The plates were modelled with shell elements, with a plate width o 1 mm, a length o 225 mm, a thickness o 1 mm, and 15 cells. The corrugated plates were modelled using aluminium material with material properties similar to the previous cantilever beam. The S wires have a diameter o.5 mm, with material properties listed in Table 1. Fixed boundary condition in all degree o reedoms was applied on one end o the plates while the other end was ree. symmetrical boundary condition was applied on the side edges o the plates. One complete heating-cooling cycle was applied on the S wires, and the vertical tip (d) 217 Copyright by DYNmore GmbH

8 delection o the ree end was evaluated. Two cases are presented or the actuation o the pre-curved corrugated plates. Firstly, each cell was actuated by one S wire and two S wires, to investigate the eect o increasing the number o S wires per cell. Secondly, the conigurations or arrangements o the S wires in the cells were varied to study their eect on the tip delection. Fig. 5 shows simulation results o the pre-curved corrugated aluminium plate actuated by one S wire per cell (5a) and two S wires per cell (5c). The ormer was run on a computer with 4 CPU and 32 GB memory capacity, while the later was run on the High Perormance Computer (HPC) o Imperial College ondon with 24 CPUs and 2 GB memory capacity. They were completed in 63 hours and 39 minutes (elapsed time), and 46 hours and 28 minutes (elapsed time), respectively. The simulation on HPC was completed about 13 times aster than the simulation on the normal computer, because o higher number o CPUs. The resulted vertical tip delections due to the S wires actuation are 56.3 mm and 63.8 mm, respectively. It shows that increasing the number o the S wires by two (hence increase in power requirement) increases the tip delection by only 13.3%. (a) (c) (b) Fig.6: Pre-curved corrugated aluminium plates actuated by 2 S wires per cell in parallel: (a) isometric view and (b) side view, and by 2 S wires per cell in V coniguration: (c) isometric view and (d) side view. In addition to the analysis o the number o S wires per cell, their conigurations or arrangements were also varied. Two S wires in each cell were arranged in parallel and V conigurations, as depicted in Fig. 6a and 6c, respectively. In this analysis, S wires with a diameter o.1 mm and a maximum recoverable strain o 3.9% were used. This was intentionally perormed to allow the parallel coniguration to be compared with the previous pre-curved corrugated plate actuated by S wires having a larger diameter, but a smaller maximum recoverable strain (Fig. 5c and 5d). The simulation results show maximum vertical tip delections o 38.1 mm and 35.7 mm or the parallel and V conigurations, respectively. Thereore, the S wires with the parallel coniguration are preerable because a slightly larger tip delection can be achieved. However, the maximum tip delection (38.1 mm) or the parallel coniguration is much smaller than the maximum tip delection achieved by the (d) 217 Copyright by DYNmore GmbH

9 pre-curved corrugated plate actuated by S wires having.5 mm diameter and 1.6% maximum recoverable strain shown in the previous Fig. 5c and 5d (63.8 mm). This shows that the inluence o the S cross-sectional area on the tip delection o the pre-curved corrugated plate is greater than the inluence o the maximum recoverable strain. 5 Conclusions and Future Work S material model has been successully implemented in s-dyna and veriied with analytical solutions. The model was then tested or actuation o a cantilever beam, which was also validated, beore application on a pre-curved corrugated plate. From actuation o the pre-curved corrugated plates, it can be concluded that increasing the number o S wires per cell or a large diameter S wire (.5 mm) resulted in a small increase in the tip delection. parallel coniguration o S wires in each cell gives slightly higher tip delection compared to a V coniguration. The S model development, veriication, and trials in the explicit commercial FE sotware, s-dyna, has allowed the possibility o modelling a wing design in a ast dynamic situation, such as light control actuation. It could also be employed in other ields that involves dynamic situation, such as structural impact. The next step in this work is to apply the S material model to actuate a morphing wing with light aerodynamic pressure acting on the wing. In addition to that, the S wires under cyclic thermal loading is another important aspect to be investigated. 6 Summary user deined material (UT) model or shape memory alloy (S) wires has been developed and implemented into the explicit inite element analysis sotware, s-dyna. The validation o the thermomechanical behaviour o the S material model by analytical solutions shows excellent agreement. The S material model has been validated against the actuation o a cantilever beam, and inally applied to the actuation o a pre-curved corrugated plate. For the cantilever beam, a very small tip delection was achieved, approximately 1.57 mm, when actuated perpendicularly by a S wire with a length o 1 mm. The actuation o the pre-curved corrugated plates shows the ampliication o the tip delection when actuated by S wires in each cell, compared to the actuation o the cantilever beam. Increasing the number o S wires per cell is not preerable or a large diameter S wires (e.g..5 mm), due to the small increment in the tip delection. I two or more S wires are used in each cell, a parallel coniguration gives slightly higher tip delection compared to a V coniguration. The simulation speed o the pre-curved corrugated plates has been greatly increased by developing a vectorized UT or the S material model. The completion o this research provides a irm oundation or uture exploration o structures actuated by S wires. The developed UT or the S wire can be used as a design tool to virtually design realistic composite morphing wings. It permits a design o structures under ast dynamic condition because the model has been developed within the explicit FE sotware, s-dyna. The model allows the prediction o the number o the S wires and also the appropriate S diameter or cross-sectional area to achieve the targeted coniguration. realistic structural design, an optimum iber orientation o composites, and actual deormation o the actuated structures can also be predicted. The capability o the commercial s- Dyna sotware to perorm optimization tasks using S-OPT and to simulate an incompressible luid dynamics (ICFD) simulation, can be utilized in a uture research to optimize the structural design and to analyze the luid-structure interaction (FSI) between the movement o actuated suraces and a luid low. 7 iterature [1] Gao, X, Qiao, R and Brinson, C: Smart aterials and Structures, 16, 27, [2] ivermore Sotware Technology Corporation (STC): S-DYN Version R8. Keyword User s anual, Volume II, 215. [3] Tanaka, K, Hayashi, T and Itoh, Y: echanics o aterials, 13, 1992, [4] Tanaka, K, Kobayashi, S and Sato, Y: International Journal o Plasticity, 2, 1986, [5] ivermore Sotware Technology Corporation (STC): S-DYN Theory anual, 216. [6] S..E.S. Group (Societa pparecchi Elettricie Scientiici Electrical and scientiic components company): SmartFlex wire and springs datasheet, Copyright by DYNmore GmbH