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

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where*I1and*I2are the principal invariant of the elastic strain tensor.Substituting equations (6.13) and (6.14) into the expression of the elastic free energy(6.12) it follows that:1Ψ ( )( * ) 2 *e= ⎡ λ+ 2μ I1 + 4μI⎤22 ⎢⎣⎥⎦(6.15)that is the elastic energy in a small deformation regime.6.1.5 Linearized Transformation Free EnergyThe transformation strain energy is assumed for a finite strain regime as:* 1 2⎡T ⎤Ψt= β T − MfEt + h Et + ( uo − Tηo) + c⎢( T −To) − Tln⎥+Iε ( E )L t(6.16)2 ⎣To⎦as a function of the Green-Lagrange strainLinearizing the expression (6.16) it follows the expression of the transformationenergy in the small strain regime as:Et.* 1 2⎡T ⎤Ψt= β εt + h εt + ( uo − Tηo) + c⎢( T −To) − Tln⎥+Iε ( ε )L t(6.17)2 ⎣To⎦with εtthe second-order transformation strain tensor.98
6.2 3D Constitutive Model for Stress-Temperature InducedSolid Phase Transformation in a Small Strain RegimeAssuming a small strain regime, the free energy for a polycrystalline SMA materialis, therefore, defined through the following convex potential:( ) ( T)Ψ (, εε, T) =Ψ ε +Ψ ε,(6.18)t e t tThe strain ε and the absolute temperature T are assumed as model control variableswhile the second-order transformation strain tensor ε trepresents the internalvariable. In accordance with experimental evidences, no volume variations duringthe phase transition are observed. The inelastic strain tensor ε trepresents a measureof the strain associated to the phase transition in the SMA. The norm of this quantityshould be bounded between zero for the case of a material without orientedmartensite and a maximum value ε L, for the case in which the material is fullytransformed in single-variant oriented martensite. The scalar quantity εLis related tothe maximum transformation strain reached at the end of the transition during auniaxial test.The Clausius-Duhem form of the entropy inequality, i.e. the second law ofthermodynamics, takes the following form:gradTσ: ε − ( Ψ+ ηT)−q⋅ ≥0(6.19)Twhere η is the entropy, q is the heat flux vector and σ is the stress tensor.Introducing in equation (6.19) the dependence of the energy function from the modelvariables ε , T and εtthe following inequality is obtained:99
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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
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 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: In terms of the linear strain tenso
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 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
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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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