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

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Ni-Ti 49/51 at % Ni -50 to 110 30Fe-Pt approx. 25 at % approx. -130 4PtMn-Cu 5/35 at % Cu -250 to 180 25Fe-Mn-Si 32 wt % Mn, 6 -200 to 150 100wt % SiTab. 2.1: Shape-memory alloys propertiesFusion temperature, °C 1300Densità, g/cm 3 6.45Austenite approx. 100Resistivity,micro-ohms•cmMartensite approx. 70Austenite18Thermical conductivity,W•cm•°CMartensite8.5Fusion latent calor, KJ/Kg•atoms 167Austenite approx. 83Young modulus, GPaMartensiteapprox. 28Austenite 195 to 690Yield Strength, MPaMartensite70 to 140Ultimate Tensile Strength, MPa 895Corrosion resistenceSimilar to Ti based alloys orto inox steel series 300Temperature of transformation, °C -200 to 110Shape Memory Strain8.5% maxPoisson ratio 0.33Tab. 2.2: Properties of Ni-Ti based shape-memory alloys2.2 SMA CharacterizationIn general, in order to describe the SMA behavior in its transformation range, somemethods accounting for the variation of temperature can be used.14
The Differential Scanning Calorimetry (DSC) measures the energy quantity adsorbedor regained by a sample, not stressed, when it is heated or cooled in itstransformation range. The exothermic and endothermic higher and lower values,corresponding to the energy variations, can be easily evaluated and show a clearphysical meaning. A result of DSC test is shown in Fig. 2.4, with the indication ofthe transformation temperatures.Fig. 2.4: DSC curve of a Ni-Ti alloyAnother technique is based on resistivity measures on the sample heated and cooled;using this procedure, significant variations of resistivity up to 20% are noted near thetransformation range.In order to define the mechanical behavior of SMA, the most effective method isthrough the application of load and temperature cycles; in particular, the sample,subjected to a constant value of load and to a temperature cycle, describes atransformation cycle in which the strains in both the directions are measured. Thecurve illustrated in Fig. 2.5 is a direct result of this test.15
Page 1 and 2: University of CassinoFaculty of Eng
Page 3 and 4: Contents1. INTRODUCTION ...........
Page 5 and 6: 5.2.1.2. Finite Element Approximati
Page 7 and 8: 1 INTRODUCTIONScience and technolog
Page 9 and 10: The implementations of the model in
Page 11 and 12: applications are presented in order
Page 13 and 14: 2 SHAPE MEMORY ALLOYS2.1 Introducti
Page 16 and 17: transformation (M→A), indicated r
Page 18: Fig. 2.2: SuperelasticityFig. 2.3:
Page 23 and 24: 2.3 Biocompatibility of Shape-Memor
Page 25 and 26: only in the angioplasty procedure,
Page 27 and 28: micropumps used to unblock blood ve
Page 29 and 30: introduction of a multi-well free-e
Page 31 and 32: spectral decomposition allowing the
Page 33 and 34: Fig. 3.1: Reference and deformed co
Page 35 and 36: δ δ = δ and δ δ = δ(3.6)iI iJ
Page 37 and 38: and it is a two-point tensor since
Page 39 and 40: Furthermore,( ) 2 2Jdet b= det F =
Page 41 and 42: 3.2.3 Polar DecompositionThe deform
Page 43 and 44: which can now be interpreted as fir
Page 45 and 46: Fig. 3.2: Representation of the pol
Page 47 and 48: ecause, as time progresses, the spe
Page 49 and 50: It can be introduced the rotated ra
Page 51 and 52: traction vector. For the Cauchy’s
Page 53 and 54: Nt per unit area acting on the boun
Page 55 and 56: transforms according to a= Qa. Velo
Page 57 and 58: ∂∂t=−∂F∂t−1 −1 −1F
Page 59 and 60: where the direct dependency upon X
Page 61 and 62: 3.4.2 Isotropic Hyperelasticity - M
Page 63 and 64: such phenomena as the Bauschinger e
Page 65 and 66: superposed to intermediate configur
Page 67 and 68: whereΨerepresents the elastic stra
Page 69 and 70:
( )( ) E 1 ( )Ψ = h ω β T − M
Page 71 and 72:
4.2.2 Clausius-Duhem InequalityIn o
Page 73 and 74:
−1∂Ψe−TS− 2FtFt= 0∂Ce(4.
Page 75 and 76:
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
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1 ( C )*1t−1α = ( β T − M ( )
Page 83 and 84:
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
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
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T − Mf Tη = η0− ⎡⎣h( ω)
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assumed to be ruled by the limit fu
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♦The limit function is assumed fu
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6.3.1.1 Time Integration of the 3D
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⎧∂f∂ ε⎪ C ⎡⎤⎣ ⎦⎪
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RRRRRRRR*( f )2 2 2Xd, X= I+ hΔ ζ
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If equation (6.42) is considered as
Page 121 and 122:
7 3D-1D Constitutive Model with 1D
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*∂f6 3σ9 3 2 2 T Ml− hϑ+mz β
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⎡⎤⎢1 0 0 ⎥⎢⎥1ε t= ϑ
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(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
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NNNv1v2v3N θ1= ξξ ( − 1)21= ξ
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to analyze the two-dimensional elem
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300 N . A regular mesh characterize
Page 145 and 146:
8.1.1.3 Superelastic and Shape-Memo
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200180T=285 K160140120Force [N]1008
Page 149 and 150:
8.1.2 Comparison between 3D and 3D-
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PhLFig. 8.7: Geometry and boundary
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250200150100Axial force [N]500-50-1
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908070Bending moment [Nmm]605040302
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The arch is subjected to tension-co
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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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