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

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⎛⎞⎜⎟1 1.5RAs = Mf− ⎜ + hε⎟Lβ ⎜t⎛ m ⎞ ⎟⎜⎜−1.5⎟⎟⎝⎝6 ⎠ ⎠(7.25)Af= M −f1.5Rmβt( −1.5)6(7.26)and, in an equivalent way, starting from compressive tests it results:⎛⎞⎜⎟1 1.5RAs = Mf+ ⎜ −hε⎟Lβ ⎜c⎛ m ⎞ ⎟⎜⎜+ 1.5⎟⎟⎝⎝6 ⎠ ⎠(7.27)Af= M +f1.5Rmβc( + 1.5)6(7.28)From equations (7.25), (7.26), (7.27) and (7.28) it can be concluded that in order toguarantee the uniqueness of the starting and finishing temperature of the transitionfrom single variant martensite to austenite at zero stress level, i.e.A sandAf, intension and compression it is necessary that the parameters β tand β csatisfy aparticular ratio depending on m .It follows that, on the basis of experimental data, it is possible to identify the valuesto assign to the proposed constitutive model parameters in order to reproduce thebehavior of the material under consideration.122
It is interesting to evaluate how many experimental tests are necessary to completelyidentify the constitutive model parameters with a sufficient degree of accuracy:Two loading and unloading tensile and compressive uniaxial tests should beperformed at two different high enough temperature values in order to assess theposition of the phase transformation straight lines which define the starting and thefinishing of the austenite to single variant martensite and of the reverse transitions.Considering the results of these tests in terms of stress-strain, the pointscorresponding to the starting and the finishing of the phase transition at the twodifferent temperatures can be defined in the diagram σ − T . In this way the straightlines defining the forward and the reverse phase transformation can be obtained.These lines have a slope which is linked, in tension, to the model parameter βt. Inparticular, if p represents the slope of these lines it can be pointed out that:2βt= p(7.29)3The same result can be obtained in compression.The material constant ε Lcan be easily derived from a uniaxial loading-unloadingtest estimating the maximum inelastic strain attained in the case of elastic unloadingwhen the material is fully martensite.Furthermore, the lines corresponding to the start and the end of the A→ S and of theS→ A transitions are far from each other of a quantity equal to 3 hε 2L, so thatalso the parameter h can be evaluated.The lines corresponding to the single variant martensite to austenite transition crossthe temperature axis in two points indicating the two characteristic temperaturesAsandAf. For temperature aboveAfthe pseudo-elasticity range occurs: at the end ofeach loading-unloading cycle the material recovers all the deformation, and the123
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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
Page 77 and 78: fact the values of ϕ andenergy usu
Page 79 and 80: 4.2.5 Final Format of the Constitut
Page 81 and 82: 1 ( C )*1t−1α = ( β T − M ( )
Page 83 and 84: 5 FINITE STRAIN CONSTITUTIVE MODEL:
Page 85 and 86: 2 3ζ 2 ζ 3exp( ζ g) = 1+ ζ g+ g
Page 87 and 88: 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: (7.21)σ3 1.5R=− βc( T − Mf)+2
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 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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