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

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( ) 1 1T T T TFt dev T Ft = Ft ( M−α − tr( M − α) 1F ) ( )3t= CSCt −FαF t t− F3ttr M− α Ft=T 1 T( ) 1 T( ) 1 T= CSCt −Ft αFt − F ( )3ttr M Ft − F3ttr α Ft = CSCt −CαC t t− F3ttr CS F−(4.54)1 T+ F ( ) 1 ( ) 1 ( ) ( )3ttr CαtFt = CSCt −CαC t t− tr CS C3t− tr Cα3tCt = dev Y CtwithY= CS− Cαt(4.55)The tensor Y represent the new thermodynamic force introduced with the aim towrite all the constitutive model equations in terms of the variables chosen to descriedthe phase transition phenomena.To conclude this theoretical part the material laws can be summarized in whatfollows:second Piola-Kirchhoff stress tensor−1∂ΨeS=2FtFt∂C∂Ψtback-stress α = 2 ∂Crelative stress tensorY= CS−Cαtdev( YC )tevolution equationC t= 2 ζdev( Y)Kuhn-Tucker conditions ζ ≥0 f ≤ 0 ζ f = 0limit function f ( Y) = dev( Y)− Ret−T76
5 FINITE STRAIN CONSTITUTIVE MODEL:Numerical Procedure5.1 IntroductionThe derived material model in the framework of finite strain is implemented into thefinite element program FEAP (Zienkiewicz and Taylor, 1991). A numerical-discretetime integration scheme is adopted because of the strongly non-linear form of theevolution equations. In particular, the time integration is performed adopting abackward-Euler implicit procedure (Simo, 1992; 1998a; 1998b) and the inelasticstrain tensorC trelated to the phase transformation occurring in the SMA isevaluated. The time interval of interest is subdivided in subincrements and theevolutive problem is solved over a generic interval [ , ]tntn + 1, being t > n+ 1t . The nquantities at time n are denoting by the pedex n while the index n + 1 of allquantities evaluated at the time tn + 1is omitted.Once the solution at the time t nis known as well as the strain tensor C at time tn + 1,the stress history is computed from the strain history by means of a procedure knownas return-map.5.1.1 Exponential MapIn order to obtain the time integration of the evolutive equation given by formulas(4.40), some considerations have to be done. An exponential map is adopted inwhich the spectral decomposition is used to compute an exponential function (Reeseand Christ, 2008).In particular, equation (4.40) can be written in the following form:77
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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
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
Page 81: 1 ( C )*1t−1α = ( β T − M ( )
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 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
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The inelastic step is solved by the
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For brevity, only the saturated pha
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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
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8.1.1.3 Superelastic and Shape-Memo
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200180T=285 K160140120Force [N]1008
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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
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Some parameters characterizing the
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9876Applied load [N]54321Numerical
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The proposed constitutive models se
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REFERENCESAdler P.H., Yu W., Pelton
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Auricchio F., Taylor R., A return-m
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Ciarlet P.G., The Finite Element Me
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Huang M.S., Brinson L.C., A multiva
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Raniecki B., Lexcellent Ch., Thermo
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Taylor G.I., Plastic strain in meta
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