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

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at the value of T = 285Kand after the maximum value of the bending load 83 Nmmis reached, the load is removed allowing the element to recover the initial conditions.BhmLFig. 8.10: Geometry and boundary conditions of the cantileverFig. 8.11 shows the bending moment-displacement response of the beam discretizedwith two-dimensional finite elements together with the ones of the beam discretizedusing three-node beam finite elements. The results regard the transversaldisplacement of the point B along the free edge of the elements. It is interesting toobserve: (1) the ability to completely recover the deformation during the unloadingcycle being at a temperature aboveAf, i.e. the ability to reproduce the superelasticeffect also in a bending state; (2) the ability of the proposed 3D-1D SMA model toreproduce the bending behavior with good convergence properties and a reducedcomputational effort.148
908070Bending moment [Nmm]60504030201002D solid elementbeam element-100 2 4 6 8 10 12 14 16 18 20Transversal displacement [mm]Fig. 8.11: Pure bending test: superelastic behavior. Moment versus transversal displacement• Shape-memory effectIn this analysis, the beam element of the previous sections is first subjected to anincreasing value of the bending moment at a constant temperature equal to149Mf, untila maximum load value is reached. Then the following loading and temperaturehistory is assumed:t[s] 0 1 2 3 4m[Nmm] 0 30 0 0 0T[K] 223 223 223 400 0In Fig. 8.12 the applied bending load is plotted versus the transversal displacement ofthe point B belonging to the free edge, computed using the two different constitutivemodels. It is interesting to observe: (1) the presence of a residual strain at the end ofthe loading-unloading pattern being at temperature equal toMf; (2) the ability tocompletely recover the initial state (zero stress, zero strain) during the temperature
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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
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Kirchhoff stress and converting int
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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 and 150: 8.1.2 Comparison between 3D and 3D-
Page 151 and 152: PhLFig. 8.7: Geometry and boundary
Page 153: 250200150100Axial force [N]500-50-1
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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