HAL Id: hal-02157456 https://hal.archives-ouvertes.fr/hal-02157456 Submitted on 16 Jun 2019 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Nickel-titanium pseudo-elastic behavior under equi-biaxial dynamic loading conditions Pierre Quillery, Bastien Durand, Olivier Hubert, Han Zhao To cite this version: Pierre Quillery, Bastien Durand, Olivier Hubert, Han Zhao. Nickel-titanium pseudo-elastic behavior under equi-biaxial dynamic loading conditions. Colloque MECAMAT Aussois 2019, Jan 2019, Aussois, France. hal-02157456
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HAL Id: hal-02157456https://hal.archives-ouvertes.fr/hal-02157456
Submitted on 16 Jun 2019
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Nickel-titanium pseudo-elastic behavior under equi-biaxial dynamic loadingconditions
Pierre Quillery1,?, Bastien Durand1,??, Olivier Hubert1,???, and Han Zhao12,????
1LMT, ENS Paris-Saclay / Université Paris-Saclay / CNRS, 61 avenue du Président Wilson 94230 Cachan, France2UFR Ingénierie, Université Pierre et Marie Curie, Sorbonne Universités, Paris
Abstract. Shape memory alloys (SMA) undergo a solid-solid phase transformation called martensitic trans-formation, involving a "high temperature" phase, austenite and a "low temperature" phase, martensite. Thistransformation can be activated by thermal loading (heating or cooling) or mechanical loading (stress) and ex-plains for example the pseudo-elastic phenomenon where high reversible deformation (>6%) can be reachedduring a tensile loading. Although the uniaxial dynamic pseudoelastic behavior of SMA is relatively well docu-mented today, this behavior under multiaxial stress remains unknown. Such knowledge is however essential forthe validation of multiaxial models to democratize the use of these materials. The stress-strain pseudo-elasticbehavior of a Nickel-Titanium under equi-biaxial dynamic compression is addressed in this work. It is measuredthanks to a new homemade equi-biaxial impact testing set-up using split Hopkinson bar. The use of thermaland optical camera allows strain and heating sources fields to be identified. The stress field is estimated bycombining the strain gauges information placed on a coaxial transmitted bar and a transmitted tube and a finiteelement analysis of the specimen. The deformation appears homogeneous in the biaxial loading region of inter-est where a significant rise in temperature due to the phase change latent heat is observed. The dynamic testingallows on the other hand an equivalent dynamic stress/strain curve under biaxial and quasi-adiabatic conditionsto be plotted. Experiments are finally compared to the results of finite difference axisymmetric model wherethe constitutive law is given by a fully coupled stochastic multiscale model.
1 Introduction
The solid-solid phase transformation can be activated inSMA by a thermal loading (heating or cooling) or a me-chanical loading (stress) and explains for example thepseudo-elastic phenomenon where high reversible defor-mations (>6%) can be reached during a tensile loading.Although the uni-axial dynamic pseudo elastic behaviorof SMA is relatively well documented today, behavior un-der multiaxial stress remains unknown. Such knowledgeis however essential for the validation of multi-axial mod-els, and finally for the democratization of these materialsin current applications. SMA are subject to complex stressstates, due to thermal and mechanical loadings and dueto their geometries. For super-elasticity as well as shapememory effect, the nickel-titanium behavior is non-linearand stress states are multi-axial. The description of first or-der austenite to martensite phase transformation is usuallyenough to model uni-axial behaviors [1]. A multiscale de-scription of the material under multi-axial loadings is how-ever necessary to determine reliable 3D models. In addi-tion, the thermal aspects can not be neglected because theygovern the shape memory effect. Indeed, thermal effects
are preponderant during a dynamic adiabatic loading. Inthis paper, thermal and kinematic fields measurements areused to observe the austenite to martensite phase transfor-mation in biaxial condition. Experiments are compared tothe results of a finite difference axisymmetric model wherethe constitutive law is given by a fully coupled stochasticmulti scale model. The first part is dedicated to the presen-tation of the experimental set-up and protocol, then exper-imental results are presented and compared to simulation,highlighting some dynamic effects not considered in thepresent modeling.
2 Experimental set-up
An home-made biaxial set-up coupled with a split Hop-kinson system is used to submit the specimen to an equi-biaxial loading. Hopkinson systems allow high strainrates, about 10/s to 5000/s to be reached [3, 5]. The con-stitutive material of the bars is known and used in the fieldof linear elasticity, so that the system can be consideredas a way of instrumentation. Strain gauges make it pos-sible to determine the strain waves propagating in the barand thus to evaluate forces and velocities conditions at theinterfaces between bars and specimen. In the biaxial set-up (see figure 1), the impact is carried out by a single in-put bar, via a return angle system and two outgoing bars,
so that the sample is loaded along two orthogonal direc-tions. The angle deflection mechanism works thanks totwo planes inclined at 45◦ relatively to the axis of the barand allowing orthogonal compression.
Figure 1: Set-up principle for dynamic equibiaxial com-pression test.
The Hopkinson formulae [3] firstly make it possibleto determine the two orthogonal forces. We seek to de-termine the transition matrix from forces to stresses usingequation (1), where mi j parameters have to be identifiedusing a finite element elastic modeling of the specimenstructure. (
σxx
σyy
)=
(mxx mxy
myx myy
) (Fx
Fy
)(1)
Gauges are glued on incident and on the two trans-mitter bars to measure and calculate the strains thanks tousual Wheatstone bridge facilities (recording frequency is500 KHz). The equibiaxial behavior is investigated in thestudy (angles of return angle system is fixed at 45◦). Thecruciform specimen is designed to concentrate stress inthe central area (region of interest - ROI) and avoid buck-ling. The ROI is a 3 mm diameter circle with a thickness of0.5 mm especially machined in the center of a 2 mm thickspecimen. The maximum overall dimension is fixed bythe Hopkinson bar system and the biaxial set-up. Externaldimension of specimen are 8 mm × 8 mm, the cruciformgeometry is shown in figure (2a).
During the test, the biaxial specimen is tracked by ahigh speed numerical camera to accurately calculate itsstrain field by digital image correlation (DIC). The record-ing is realized by a SA5 fast-cam at 50000Hz and ona 512px × 271px area. The whole specimen surface isrecorded to evaluate the free-body motions due to the Hop-kinson bar system (kinematic boundary conditions). DICprinciple consists to calculate the difference between twoimages, a reference one and a distorted one. An imageis seen numerically as a function characterizing the graylevel of each pixel. If we call f (x) the function of the ref-erence image and g(x) the function of the deformed image,the determination of the displacement field u(x) is obtainedby minimizing equation (2).
g(x + u(x)) = f (x) (2)
Correlation code (Correli-RT3) has been used consid-ering the displacement field as continuous. This code al-lows the identification of displacement on each node of afinite element triangular mesh. The obtained displacementis regularized by an elastic solution [4]. A thermal cameraplaced in front of the other side of sample which has beenpainted in black allows the temperature field to be mea-sured. A calibration of the infrared sensor with a blackbody allows the link between the digital level (functionof electromagnetic radiation and emissivity) measured bythe camera and the temperature to be obtained. Figure (2b)shows the full range calibration and the restricted area cho-sen to increase the recording frequency to 15000 Hz. Thisobservation area, of about 64 × 8 pixels, enables to trackthe specimen during the testing and to compensate the freebody displacement.
(a) Reference image and area for DIC
(b) Observation zone for a 15000 Hz infrared recording(64 px × 8 px)
Figure 2: Optical and thermal image settings
3 Experimental results
Figure 3a shows the strain measured by the gauges as func-tion of time. After time shifting to virtually transport thesignals from the strain gauges to the multi-axial set-upinterfaces, and after Hopkinson formulae application, wecan plot the forces and the velocities at these interfaces(figures 3b and 3c). During the steady state, all veloci-ties are equal, and the forces applied by the incident andthe transmitted bars are equal too. In our case, we haveto compare the incident bar force to the sum of the twotransmission bars forces to check at the equilibrium [6].
(ms)
(ms)
Forc
e (k
N)
Vel
oci
ty (
m/s
)
(ms)
Total transmitter force
(a)
(b)
(c)
Figure 3: Measurements from bars: (a) strain from gauges,(b) velocities, and (c) forces in bars
These requirements are confirmed in figure 3c. How-ever we observe that force in the internal transmitter baris a few lower than the force in the external transmitterbar denoting that biaxial loading is not perfectly equipro-portional. This difference can be explained by a frictionin the biaxial set up, higher along x direction than alongy direction. Average stress components σxx and σyy arecalculated form forces thanks to equation 1 (parametersused are: mxx = myy = 0.2941 mm−2 and mxy = myx =
(b)
(a)
Figure 4: Average strain components and temperature overthe ROI from DIC and infrared observations
−0.0794 mm−2). Strain components εxx, εyy and εxy arecalculated in the ROI from DIC measurements. εxx andεyy are plotted as function of time in figure4a. Strains donot seem perfectly synchronized. The lower magnitude ofεyy comparing to εxx is in accordance with the lower stresslevel along y axis. At the beginning, the strain along ycan be related to a Poisson effect. At the end, the strain isreaching about 3.5%.
Due to friction in the return angle system and microclearance in the setup, the loading are not perfectly equi-biaxial and perfectly reproducible. Figure 5 shows thestresses and strains multiaxial loading for two tests. Foreach case, the global loading is biaxial even if at the be-ginning, in the time to fill clearance it’s look like unixial.
The infrared observations allow us to calculate thetemperature increase during the test. Figure 6 illustratesthis brutal increase during the first 1ms. The evolution ofaverage temperature with time in the ROI is reported infigure 4b showing that temperature is increasing from thevery beginning of the test. However, it must be noticedthat the temperature increases along a time range (about6ms) much longer than the duration of mechanical load-ing (<1ms).
The numerical model used to describe the evolution ofthermochemical fields is based on four scales: the vari-ant scale α variant of a phase φ (martensite, R phase oraustenite variants), the crystal g, the representative volumeelement (RVE) which is an assembly of crystals, and thatof the structure. The single crystal model [7] predicts thedistribution of the volume fractions fφα of each variant inthe grain from the calculation of their Gibbs free energyWφα (3). The Gibbs free energy is the sum of chemicaland elastic energies (4,5), calculated from the knowledgeof the enthalpies and entropies of each phase (hφ, sφ), ofthe transformation strain (Green-Lagrange tensor) of eachof variants ε tr
φα (12 variants of monoclinic martensite, 4variants of rhombohedral R phase and 1 variant of cubicaustenite), of the stiffness tensor of the medium Cφ, thetemperature Tφα and the stressσφα (considered as homoge-neous in the RVE to fasten the calculations). The fractionof each phase is estimated via a Boltzmann probability dis-tribution (6), often used in magneto-mechanical problems
Figure 6: Sequence of frames from infrared camera (6ms).
[8], where As is a parameter identified from a calorimetrymeasurement.
Wφα = WTφα + Wσ
φα (3)
WTφα = hφ − Tφα sφ (4)
Wσφα = −
12σφα : C−1
φ : σφα − σφα : ε trφα (5)
fφα =exp(−AsWφα )∑n
φ=1∑mα=1 exp(−AsWφα )
(6)
The modeling of the polycrystal behavior is based onthe scaling up by simple averaging of the quantities bycrystal (using localization procedures if need be). For this,the RVE is described as an aggregate of single crystals viaan Orientation Distribution Function (ODF). The RVE iscomposed of about 100 orientations for the calculation.The physical parameters used for the modeling are gath-ered in 1. A Backward Euler Finite Difference decomposi-tion of the sample has been done in axisymmetic conditionusing Neumann Boundary condition (adiabatic condition),where phase fraction ratio acts as heat sources in the heatequation (7) [1].
ρCpdTdt
= kd2Tdr2 +
kr
dTdr−(hM−hR)
d fM
dt+(hR−hA)
d fA
dt(7)
Using the numerical discretization :
dTdt
=T n+1
j − T nj
∆t(8)
d fM
dt=
fMn+1j − fM
nj
∆tand
d fA
dt=
fAn+1j − fA
nj
∆t(9)
d2Tdr2 =
T n+1j+1 − 2T n+1
j + T n+1j−1
∆r2 (10)
dTdr
=T n+1
j+1 − T n+1j
∆r2 (11)
A scalar representation of experimental and modeledstress-strain behaviors has been plotted in figure 7a (us-ing a von Mises and Levy-Mises standardization for stressand strain. eq.(12) and (13)). We observe first in figure6a that the threshold from elastic to inelastic transition(phase transition characterized by a plateau) seems under-estimated by the modeling. This difference may be ex-plained by the existence of some micro-inertia phenomenaassociated with the phase transition (rotation of crystal-lographic lattice) and ignored by the modeling. This un-derestimation happens although the temperature increaseis overestimated at the very beginning of the loading (fig-ure 6b). Indeed, the experimental temperature emissionis spread over a much longer duration than the emissionduration provided by the model, in direct relation with theduration of the shock. This difference may be explained bythe existence of incubation delay associated with the ger-mination of martensite in austenite. This phenomenon isignored by the modeling. Despite this discrepancy, exper-imental and modeled final temperatures are in accordance,validating the adiabatic thermal boundary conditions.
A α=β=γ=90R α=γ=90, β=120 C11=238, C12=142, C44=232M α=γ=90, β=97.8
ρ Cp As
(kg/m3) (J/kg/K) (m3/J)6450 900 2.54e-06
Table 1: Physical constants used for the multiscale model-ing
5 Conclusion
In this study, the multi-axial dynamic behavior of a NiTialloy has been prospected and compared to the results ofaxisymmetric finite difference modeling including a mul-tiscale and multiphysic constitutive law. Despite verystrong assumption considering the mechanical and thermalboundary conditions, interesting qualitative results havebeen obtained. Improvements may be achieved by more
representative boundary conditions, the introduction ofmicro- inertial phenomena in the modeling, and a time de-lay in martensite production able to model the heat emis-sion more accurately.
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