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

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8 Numerical Results8.1 IntroductionIn Chapters 4, 6 and 7 several constitutive models in large and small deformationregime able to reproduce the SMA macro-behavior are presented, paying attention onthe time-discrete models and their algorithmic implementation within a finiteelementscheme in the code FEAP. Some numerical applications are developed inorder to assess the ability of the model to reproduce the shape memory alloysmechanical response and to verify the efficiency of the proposed procedures.8.1.1 Comparison between Large and Small DeformationModelsFirst of all, the following material properties for shape-memory alloys are assumed:E = 53000MPa ν = 0. 36 T = 245K ε = 0.04 h=1000MPa(8.1)β β σ σ0L-1 -1t= 2.1MPaKc= 2.1MPaKc= 72MPat= 56MPa Mf= 223K( ± )where E and ν are the SMA Young modulus and the Poisson coefficient,respectively. A 2D four node quadrilateral element is implemented.8.1.1.1 Uniaxial Response-Superelastic EffectA two-dimensional clamped element with dimensions L= 100mmand h=20mm(see Fig. 8.1) is subjected to a tensile loading-unloading history, inducing a fulltransformation of austenite into martensite. The temperature is kept constant at thevalue of T = 285Kand after the maximum value of the axial load 7 KN is reached,the load is removed allowing the element to recover the starting conditions. In order134
to analyze the two-dimensional element response a regular mesh characterized by 10elements in the axial direction and 1 element in the transversal one is considered.Fig. 8.2 shows the uniaxial force-displacement response of a point along the freeedge of the two-dimensional element for the case of small and finite strains. It isinteresting to observe: (1) the ability to completely recover the deformation duringthe loading-unloading cycle being at a temperature aboveAf, i.e. to reproduce thesuperelastic effect in a uniaxial state of stress; (2) the similar prediction of the smalland finite strains models except in the part of the curve corresponding to the end ofthe phase transition and the saturated phase transformation.hPLFig. 8.1: Geometry and boundary conditions of the cantilever135
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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
Page 89 and 90: ⎡R R Rt⎢⎢RR Rt⎢⎣R R RtC C
Page 91 and 92: γ• R,Ct=12EEtt(5.28)• R γ ,
Page 93 and 94: The spatial virtual work equation s
Page 95 and 96: [ E E E 2 E 2 E 2 E ]δE = δ δ δ
Page 97 and 98: 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: NNNv1v2v3N θ1= ξξ ( − 1)21= ξ
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 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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