Finite Strain Shape Memory Alloys Modeling - Scuola di Dottorato in ...

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• Pseudo-Elastic EffectA clamped beam with dimensions L = 10mmand b= h= 1mm(see Fig. 8.7), with band h the dimensions of the rectangular cross section, is subjected to a simpletensile-compressive loading-unloading history, inducing a full transformation ofaustenite into martensite. In order to verify the robustness and the effectiveness ofthe proposed simplified 3D-1D constitutive SMA model, the analyzed element isdiscretized with both two-dimensional finite elements and three-node finite beamelements. In the first case, the SMA phase transformation is represented by the fullthree-dimensional constitutive laws and the mesh is characterized by 10 elements inthe axial direction and 1 element in the transversal one; in the second application thephase transition in the SMA is assumed just ruled by the stress along the applied loaddirection and the structure is discretized in 1 element.In particular, the temperature is kept constant at the value of T = 285Kand after themaximum value of the axial load 400 N is reached, the load is removed allowing theelement to recover the starting conditions.Fig. 8.8 shows the uniaxial force-displacement response for the two kind ofdiscretizations. In particular the results are in terms of the axial displacement of apoint along the free edge of the elements. It is interesting to observe: (1) the ability tocompletely recover the deformation during the loading-unloading cycle being at atemperature aboveAf, i.e. to reproduce the pseudo-elastic effect in a uniaxial stateof stress; (2) the ability of the proposed 3D-1D SMA model implemented in a threenodebeam element to reproduce the exact behavior of the material with goodconvergence properties and a reduced computational effort.144
PhLFig. 8.7: Geometry and boundary conditions of the cantilever400300200Axial force [N]1000-100-200-3002D solid elementbeam element-400-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5Axial displacement [mm]Fig. 8.8: Uniaxial test: superelastic behavior in tension. Axial load versus axial displacement• Shape-Memory EffectAim of the application is to demonstrate the possibility to significantly govern theaxial displacement of the considered elements performing temperature cycles in theSMA. Some tests using the two different constitutive SMA models, the 3D and the145
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University of CassinoFaculty of Eng
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Contents1. INTRODUCTION ...........
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5.2.1.2. Finite Element Approximati
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1 INTRODUCTIONScience and technolog
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The implementations of the model in
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applications are presented in order
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2 SHAPE MEMORY ALLOYS2.1 Introducti
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transformation (M→A), indicated r
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Fig. 2.2: SuperelasticityFig. 2.3:
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The Differential Scanning Calorimet
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2.3 Biocompatibility of Shape-Memor
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only in the angioplasty procedure,
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micropumps used to unblock blood ve
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introduction of a multi-well free-e
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spectral decomposition allowing the
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Fig. 3.1: Reference and deformed co
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δ δ = δ and δ δ = δ(3.6)iI iJ
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and it is a two-point tensor since
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Furthermore,( ) 2 2Jdet b= det F =
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3.2.3 Polar DecompositionThe deform
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which can now be interpreted as fir
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Fig. 3.2: Representation of the pol
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ecause, as time progresses, the spe
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It can be introduced the rotated ra
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traction vector. For the Cauchy’s
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Nt per unit area acting on the boun
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transforms according to a= Qa. Velo
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∂∂t=−∂F∂t−1 −1 −1F
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where the direct dependency upon X
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3.4.2 Isotropic Hyperelasticity - M
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such phenomena as the Bauschinger e
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superposed to intermediate configur
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whereΨerepresents the elastic stra
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( )( ) E 1 ( )Ψ = h ω β T − M
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4.2.2 Clausius-Duhem InequalityIn o
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−1∂Ψe−TS− 2FtFt= 0∂Ce(4.
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qφ = T : dt− ⋅gradT≥0(4.30)T
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fact the values of ϕ andenergy usu
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4.2.5 Final Format of the Constitut
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1 ( C )*1t−1α = ( β T − M ( )
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5 FINITE STRAIN CONSTITUTIVE MODEL:
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2 3ζ 2 ζ 3exp( ζ g) = 1+ ζ g+ g
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where d is a user-defined parameter
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⎡R R Rt⎢⎢RR Rt⎢⎣R R RtC C
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γ• R,Ct=12EEtt(5.28)• R γ ,
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The spatial virtual work equation s
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[ E E E 2 E 2 E 2 E ]δE = δ δ δ
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ˆNLB = B + B (5.51)a a ain whichBa
Page 99 and 100: Kirchhoff stress and converting int
Page 101 and 102: DF( φ)[ u]d=dεε = 0( φ εu)∂
Page 103 and 104: In terms of the linear strain tenso
Page 105 and 106: 6.2 3D Constitutive Model for Stres
Page 107 and 108: T − Mf Tη = η0− ⎡⎣h( ω)
Page 109 and 110: assumed to be ruled by the limit fu
Page 111 and 112: ♦The limit function is assumed fu
Page 113 and 114: 6.3.1.1 Time Integration of the 3D
Page 115 and 116: ⎧∂f∂ ε⎪ C ⎡⎤⎣ ⎦⎪
Page 117 and 118: RRRRRRRR*( f )2 2 2Xd, X= I+ hΔ ζ
Page 119 and 120: If equation (6.42) is considered as
Page 121 and 122: 7 3D-1D Constitutive Model with 1D
Page 123 and 124: *∂f6 3σ9 3 2 2 T Ml− hϑ+mz β
Page 125 and 126: ⎡⎤⎢1 0 0 ⎥⎢⎥1ε t= ϑ
Page 127 and 128: (7.21)σ3 1.5R=− βc( T − Mf)+2
Page 129 and 130: It is interesting to evaluate how m
Page 131 and 132: It can be noted that equations (7.3
Page 133 and 134: The inelastic step is solved by the
Page 135 and 136: For brevity, only the saturated pha
Page 137 and 138: Fig. 7.2. Timoshenko’s beam theor
Page 139 and 140: NNNv1v2v3N θ1= ξξ ( − 1)21= ξ
Page 141 and 142: to analyze the two-dimensional elem
Page 143 and 144: 300 N . A regular mesh characterize
Page 145 and 146: 8.1.1.3 Superelastic and Shape-Memo
Page 147 and 148: 200180T=285 K160140120Force [N]1008
Page 149: 8.1.2 Comparison between 3D and 3D-
Page 153 and 154: 250200150100Axial force [N]500-50-1
Page 155 and 156: 908070Bending moment [Nmm]605040302
Page 157 and 158: The arch is subjected to tension-co
Page 159 and 160: 10864Axial force [N]20-2-4-6-8beam
Page 161 and 162: Some parameters characterizing the
Page 163 and 164: 9876Applied load [N]54321Numerical
Page 165 and 166: The proposed constitutive models se
Page 167 and 168: REFERENCESAdler P.H., Yu W., Pelton
Page 169 and 170: Auricchio F., Taylor R., A return-m
Page 171 and 172: Ciarlet P.G., The Finite Element Me
Page 173 and 174: Huang M.S., Brinson L.C., A multiva
Page 175 and 176: Raniecki B., Lexcellent Ch., Thermo
Page 177: Taylor G.I., Plastic strain in meta
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