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

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⎡R⎢⎣ RRΔζΔζ, η , γγ γ, ηR,γ⎤⎥⎦(7.39)where the subscript comma indicates derivation with respect to the quantityΔfollowing the comma, i.e. R ζ, γmeans the derivation of the first scalar equation R Δζwith respect to γ and so on.The derivatives appearing in equation (7.39) assume the following form:RRRRΔζ, ϑΔζ, γγ, ϑγ, γ⎛1 d⎛3⎛3⎞⎞⎞⎜⎛⎞⎜ ϑ ⎜1− ⎜dϑ + d⎟⎟⎟⎟*⎛ 3 m ⎞⎜⎛βT − M2 2f+ γ ⎞⎜ ⎜ ⎝ ⎠ ⎟⎟⎟= h1⎝⎠⎜⎟⎜ ⎟⎟⎜+ 2 3 ⎟− − −⎝ ⎠⎜⎜ ε ⎟tεt⎟⎜⎝ ⎠ ⎜ ⎜⎟⎟⎟⎜⎜⎟⎟⎝⎝⎠⎠(7.40)= 0== 03 sgn2( ϑ )7.3.2 Consistent Tangent ModulusThe time-discrete model is completed by the calculus of the consistent tangentmodulus which preserves the quadratic convergence of the Newton-Raphson method.The consistent tangent modulus can be evaluated as a linearization of the stress σl:Edσdεlt= (7.41)128
For brevity, only the saturated phase transformation problem is treated; in fact, theevolving phase transition can be obtained simply removing all terms related to theparameter γ .Recalling equation (7.34) the linearization of the elastic constitutive equation gives:⎛dε dϑ⎞ ⎛ dϑ⎞dσ l= E⎜ − ⎟dε = E⎜1−⎟dε⎝dε dε ⎠ ⎝ dε⎠(7.42)If equation (7.37) is considered as a function of the quantities ϑ , γ and ε , thecorresponding linearization gives:Δζ Δζ Δζ Δζ⎧ ⎪d( R ) = R, ϑdΔ ζ + R, γdγ + R,εdε= 0⎨ γ γ γ γ⎪⎩ dR ( ) = Rd, ϑΔ ζ + Rd, γγ + Rd, εε = 0(7.43)where:2 ⎛ 3 m ⎞ΔζR,ε= E3 ⎜+2 3 ⎟⎝ ⎠Rγ, ε= 0(7.44)Accordingly, it can be written that:ζ ζ−1⎡2 ⎛ 3 m ⎞⎤Δ Δ⎡dϑ⎤ ⎡R, R ⎤ϑ , γ ⎢ E3 ⎜+2 3 ⎟⎥⎢ dεγ γdγ⎥ =−⎢R, ϑR⎢ ⎝ ⎠⎥⎣ ⎦ ⎢⎣ , γ ⎥⎦ ⎢0⎥⎣⎦(7.45)The linearization of the quantity ϑ gives:d( ϑ)= ϑ dϑ+ ϑ dγ = dϑ+ ϑ dγ(7.46), ϑ , γ , γ129
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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
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fact the values of ϕ andenergy usu
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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:
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
Page 133: The inelastic step is solved by the
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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