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

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If the above solution is not admissible, i.e. Et≥ εL, equation (5.9) is rewritten inresidual form assuming γ > 0 as⎧ C −1 −1 −1 −1 −1R =− Ctn ,+ Ut exp( Δ ζ Ut fUt ) Ut= 0⎪Δζ⎨ R = dev( Y) − R=0⎪γ⎪⎩R = Et− εL= 0(5.12)The non-linear equation system of eight scalar equations is solved with a Newton-Raphson method to compute a new value ofCtand γ .5.1.4 Newton-Raphson MethodThe non-linear evolutionary problem is solved using an iterative method. Inparticular, the solution is herein given only for the case of saturated phase transition,i.e. Et= εL, because the case of evolving phase transition can be simply obtainedeliminating from the governing system the last row.The iterative Newton-Raphson method requires the linearization of equation (5.12)asC C C C⎧ d( R ) = R, C: dC t t+ R, ΔζdΔ ζ + R,γdγ⎪ Δζ Δζ Δζ⎨dR ( ) = R, C: dCt t+ R, ΔζdΔ ζ + R,γdγ⎪ γ γ γ γ⎪⎩dR ( ) = R, C: dCt t+ R, ΔζdΔ ζ + Rd, γγ(5.13)hence the computation of the matrix:82
⎡R R Rt⎢⎢RR Rt⎢⎣R R RtC C C, C , Δζ, γΔζ Δζ Δζ, C , Δζ, γγ γ γ, C , Δζ, γ⎤⎥⎥⎥⎦(5.14)where the subscript comma indicates derivation with respect to the quantityfollowing the comma, i.e.R means the derivation of the first six scalar equationsC, CtCR with respect toDenoting as1 3( 1 1)Ctand so on.devI the fourth-order deviatoric identity tensor defined asI dev = I − ⊗ with I = 1 1 the fourth-order identity tensor with the genericcomponent equal to I ijkl= δ ikδ jl; the derivatives appearing in equation (5.14) assumethe following form:C• ( ) ( ) ( )( G)( C )R = B G U + U⎛⎛∂⎜⎞( U ) + G B⎝⎝⎠−1 −1 lt −1, Ctilhk lt ttjt ijhkilttjlt tjhk⎜⎜∂⎟t hk⎞⎟⎠(5.15)withBrjhk =• ( C−, Δζ) = ( t )1−−1 ⎛ ⎞⎛2∂U⎞t ⎜∂Ct⎟⎜ ⎟ =∂C⎜ ∂C⎟⎝ t ⎠rjhk⎜ t ⎟⎝ ⎠rjhk−1 −1( exp( ζ ))lt t tlt(5.16)G = Δ U fU(5.17)∂ ( G)lt( ζ Q)−( ) ( t )R U Q U1 1ij il gb∂ Δtjgb(5.18)with:83
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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
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 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: where d is a user-defined parameter
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
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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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