Elastohysteresis model implemented in the finite element ...
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Elastohysteresis model implemented in the finite element sofware HEREZH++ G´ erard Rio 1, a , Denis Favier 2 , and Yinong Liu 3 1 Universit´ e Europ´ eenne de Bretagne, LIMATB, Rue de Saint Maud´ e - BP 92116, 56321 Lorient cedex, France 2 Universit´ e de Grenoble/CNRS, 3SR, BP 53, 38041 Grenoble Cedex 09, France 3 The University of Western Australia. School of Mechanical Engineering, Crawley, WA 6009, Australia Abstract. This short paper describes the simulation results obtained with the elasto-hysteresis model implemented in the finite element software Herezh++ and compares them with the experimental data of the Roundrobin SMA Modelling benchmark. 1 The elastohysteresis model As proposed in [1,2], the permanence of the simultaneous existence of reversible processes and hysteresis in the thermomechanical behaviour of shape memory alloys suggests to express the total stress σ as the addition of two partial stresses, the first σ r being hyperelastic while the second one is related to hysteresis of elastoplastic type [3–5]. A 1-D illustration of the ”elastohysteresis model” is shown in figure 1a. In the present work, the partial hyperelastic stress tensor is calculated from the hyperelastic potential proposed by Org´ eas [6,7], which leads in the 1-D case (shear test) to the stress-strain curve shown in figure 1b. In the case of shape memory alloys [2], the potential ω = ω(V,Q ε ,ϕ ε ) is expressed as function of 3 invariants of the Almansi strain tensor: the relative variation of volume V, the intensity of the deviatoric deformation Q ε , and ϕ ε a measure of the angle giving the direction of the deformation tensor in the deviatoric plane. The hysteresis is described by an incremental model of hypoelastic type :˙ σ =2μ ¯ D + βφΔ t R ¯ σ and an algorithm to manage discrete memory points σ R introduced by P.Gu´ elin [3] and used in the model. ¯ D is the deviatoric strain rate tensor and Δσ = σ(t) - σ R . More details can be found in [2,4–6,9]. Figures 1b-e show the meaning of the parameters used in this work. 2 Process to parameter identification Firstly, eight parameters independent of the temperature are used in this model, including 6 parameters involved in the hyperelastic potential and 2 parameters of the hysteresis scheme. The 6 parameters independent of the temperature of the hyperelastic potential (cf. fig. 1 b ) are: the 3 slopes: μ 1 μ 2 μ 3 , the 2 curvatures for the transitions: α 1 α 2 and the bulk modulus K. The 2 hysteresis parameters indepedent of the temperature (cf. fig. 1 c )are: the initial slope μ and the Prager parameter np which controls the transition to the saturation of the hypoelastic scheme). The bulk modulus K is chosen from the literature. The 7 other parameters independent of the temperature are determined from one isothermal tensile test performed at a temperature above Af. a e-mail: [email protected] ESOMAT 2009, 08005 (2009) DOI:10.1051/esomat/200908005 © Owned by the authors, published by EDP Sciences, 2009 This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License (http://creativecommons.org/licenses/by-nc/3.0/), which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited. Article available at http://www.esomat.org ou http://dx.doi.org/10.1051/esomat/200908005
Transcript of Elastohysteresis model implemented in the finite element ...
Elastohysteresis model implemented in the finiteelement sofware HEREZH++
Gerard Rio1,a, Denis Favier2, and Yinong Liu3
1 Universite Europeenne de Bretagne, LIMATB, Rue de Saint Maude - BP 92116, 56321 Lorient cedex,France
2 Universite de Grenoble/CNRS, 3SR, BP 53, 38041 Grenoble Cedex 09, France3 The University of Western Australia. School of Mechanical Engineering, Crawley, WA 6009, Australia
Abstract. This short paper describes the simulation results obtained with theelasto-hysteresis model implemented in the finite element software Herezh++ andcompares them with the experimental data of the Roundrobin SMA Modellingbenchmark.
1 The elastohysteresis model
As proposed in [1,2], the permanence of the simultaneous existence of reversible processesand hysteresis in the thermomechanical behaviour of shape memory alloys suggests to expressthe total stress σ as the addition of two partial stresses, the first σr being hyperelastic whilethe second one is related to hysteresis of elastoplastic type [3–5]. A 1-D illustration of the”elastohysteresis model” is shown in figure 1a. In the present work, the partial hyperelasticstress tensor is calculated from the hyperelastic potential proposed by Orgeas [6,7], which leadsin the 1-D case (shear test) to the stress-strain curve shown in figure 1b. In the case of shapememory alloys [2], the potential ω = ω(V,Qε, ϕε) is expressed as function of 3 invariants ofthe Almansi strain tensor: the relative variation of volume V, the intensity of the deviatoricdeformation Qε, and ϕε a measure of the angle giving the direction of the deformation tensorin the deviatoric plane. The hysteresis is described by an incremental model of hypoelastic type: σ = 2µD + βφ∆t
Rσ and an algorithm to manage discrete memory points σR introduced byP.Guelin [3] and used in the model. D is the deviatoric strain rate tensor and ∆σ = σ(t)−σR.More details can be found in [2,4–6,9]. Figures 1b-e show the meaning of the parameters usedin this work.
2 Process to parameter identification
Firstly, eight parameters independent of the temperature are used in this model, including 6parameters involved in the hyperelastic potential and 2 parameters of the hysteresis scheme. The6 parameters independent of the temperature of the hyperelastic potential (cf. fig. 1 b ) are: the3 slopes: µ1 µ2 µ3, the 2 curvatures for the transitions: α1 α2 and the bulk modulus K. The 2hysteresis parameters indepedent of the temperature (cf. fig. 1 c )are: the initial slope µ and thePrager parameter np which controls the transition to the saturation of the hypoelastic scheme).The bulk modulus K is chosen from the literature. The 7 other parameters independent of thetemperature are determined from one isothermal tensile test performed at a temperature aboveAf.
a e-mail: [email protected]
ESOMAT 2009, 08005 (2009)DOI:10.1051/esomat/200908005© Owned by the authors, published by EDP Sciences, 2009
This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License (http://creativecommons.org/licenses/by-nc/3.0/), which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.
Article available at http://www.esomat.org ou http://dx.doi.org/10.1051/esomat/200908005
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d)Fig. 1. Significations of the parameters of the behavior.
Table 1. Values of the behavior parameters and function.
hyper-elasticity K µ1 µ2 µ3 α1 α2 Qe0(T = 273.5K)(MPa for modulus) 270000 14500 300 11000 0.001 0.005 0.0574
f(ϕ) nQs γQs nµ1 γµ1
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243 K 5.71 MPa/K 620 MPa 9
hysteresis np µ(MPa for modulus) 2. 10500.
Q0(T ) 150 + 145 tanh(T−24510
)
Secondly, three parameters of the elastohysteresis model are dependent on the tempera-ture, i.e. two parameters of the hyperelastic potential (Qs(T ) and Qe(T ) and one parame-ter of the hysteresis scheme Q0(T ). The evolutions of these parameters with the temperatureare determined from isothermal tensile tests at different temperatures and from physical con-siderations. The evolution of Qs(T ) described in fig. 1 d involves 4 parameters, includingT0rev dQs/dT Qsmaxi and α. Parameter α obeys the Clausius Clapeyron relation. The evolu-tion of Qe(T ) is such that Qe(T ) = Qe0 + (Qs −Qs(273))/(3. ∗ h1 ∗ (µ3 + µ2)) with h1 = 1.2 .The evolution of the hysteresis width Q0(T ) is described here by the function presented in table1.
To take into account the tension-compression asymmetry [8,10], and more generally thedependence of the transformation stress on the 3D loading type (shear, tension, compression ),3 parameters Qs, µ1 and Qe are set as functions of the angle ϕ in the deviatoric plane of thestrain (cf. fig. 1 e ). Qs(ϕ) = Qs f(ϕ), µ1(ϕ) = µ1 f(ϕ) with f(ϕ) = (1 + γ cos(3ϕ))−n [2].In order to express the independence of the transformation energy on loading type [6,10], theproduct Qs.Qe is assumed independent of ϕε.
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Fig. 2. Isothermal tension tests.
3 Simulation, strength and weakness of the SMA model
The elastohysteresis model is implemented in an academic software Herezh++ [12] which canbe used as a standalone software or can be connected with Abaqus [11] by an Umat.
Simulations of tensile tests are done with 1 hexaedral element. Simulation of torsion testsare performed on a wire of diameter D=0.1mm and length L=0.6mm. The mesh contains48 full quadratic elements (399 nodes): 8 hexaedral elements + 8 pentahedral elements in asection, 3 elements in the length which lead to 672 gauss points (5 gauss points along a radius).Boundary conditions are: clamped at upper circular plane surface and imposed rotation at thelower circular plane surface. The average time for one torsion simulation (initial pre-stress + 1global loop of rotation) is 1200s on one processor 2.5 GHz (∼800 equilibrium iterations).
The isothermal tensile tests presented in figure 2 are used for identification. All othertests are simulated. Model and simulations are performed in finite deformations and large dis-placements. The parameter identification process is simple, and leads to a model which takesinto account physical characteristics of the martensitic transformation, including the tension-compression assymetry, the Clausius-Clapeyron relation, and the independency of transforma-tion work on the loading type.
We observe that isothermal loops and subloobs (cf. fig. 2), are well described based on theconcept of discrete memory introduced by Guelin in 1980 [3]. Torsions (cf. fig. 3, 4) are alsowell described for temperature higher than Af and lower than Ms.
Some improvements have to be done on one hand for isothermal tests in the temperaturerange Ms − Af as shown in fig. 3 in the cases (S=70 MPa, T=-10 0C and T=0 0C) and(S=194MPa T=0 0C and T=20 0C), and on the other hand for recovery stress strain tests(cf. fig. 5, 6). The model gives the trends but must be also improved to represent correctly thehysteresis loop observed during a loop of temperature, particularly for the recovery stress tests,when the imposed deformation is positioned on lower plateau.
References
1. D. Favier, P. Guelin, and P. Pegon, Materials Science Forum, 56-58, (1990) 559-5642. D. Favier, Habilitation Thesis (Universite de Grenoble, France, 1988)3. P. Guelin, Journal de Mecanique, 19(2), (1980) 217-2474. B. Wack, J.M. Terriez and P.Guelin, Acta mechanica, 50, (1983) 1-2, 9-375. P. Pegon, Habilitation Thesis, Universite de Grenoble, France, 19886. L. Orgeas, PhD thesis (Universite de Grenoble, France, 1997)
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Fig. 3. Isothermal torsion tests with two constant axial tensions: 70 MPa and 194 MPa, influence ofthe thermal level (range from -20 0C to 50 0C )
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Fig. 4. torsion tests with a constant temperature 50 C, influence of the constant tension level (rangefrom 70MPa to 379MPa)
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Fig. 6. Thermomechanical recovery stress tests: prestrain (2%, 3.5%, 5%) at the upper and at thelower plateau, then cycling with temperature and, finally unloading
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7. L. Orgeas, D. Favier, and G. Rio, Revue Europeenne des Elements Finis, 7, 8, (1998) 111-1368. P.Y. Manach, PhD thesis (Universite de Grenoble, France, 1993)9. G. Rio, P. Y. Manach, and D. Favier, Archives of Mechanics, 47(3), (1995) 537-55610. L. Orgeas and D. Favier, Acta Materialia, 46, (1998) 5579?559111. G. Rio, H. Laurent, and G. Bles. Advances in Engineering Software, 39(12) (2008) 1010?102212. G. Rio. Herezh++, University Bretagne Sud, (2007) iddn.fr.010.0106078.000.r.p.2006.035.20600
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