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

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5.1.2 Time Integration of the Constitutive ModelThe time-discrete counterpart of the constitutive model is given by⎧ ⎡λ1 ⎤ −1 −1 −1⎪ S= 2⎢( I1−3)− μ4 2 ⎥Ct + μCt CCt⎪ ⎣⎦⎪1⎪ ( C )*1t−1( ( ) ) 2⎪α = β T − Mf + h Ct− 1 + γ⎪2 1( Ct−1)⎪2⎪⎪Y= CS−Cαt⎨γ ≥ 0⎪−1 −1 −1 −1 −1⎪ Ctn ,= Ut exp( ΔζUt fUt ) Ut⎪⎪Ut= Ct⎪ f = 2 dev( Y) Ct/ dev( Y)⎪⎪Et≤ εL⎪ F( Y) = dev( Y) −R≤0⎪⎩ Δζ≥0 Δ ζF( Y) = 0(5.9)where Δ ζ = ( ζ − ζ n ) is the consistency parameter, time integrated over the interval[ t , nt ] .From a computational point of view, the time-discrete model as presented in equation(5.9) shows a problem due to the fact that the transformation stress α is proportionalto the inverse of the Green-Lagrange strain norm and to its derivative so that, whenthe strain is zero, this quantity remains undefined. To overcome this difficulty, theEuclidean normEtcan be substituted with a regularized norm Et, defined as( d+1)dd( ) ( d−1) dEt = Et − Et+ d(5.10)d −180
where d is a user-defined parameter which controls the smoothness of theregularized norm. For small values of the parameter d , the regularized norm of theGreen-Lagrange strain tensor tends to the Euclidean one so that d indirectlymeasures the difference betweenEtand Et. In this way, the quantity Etisalways differentiable, even in the case Et = 0 with d > 0 .5.1.3 Solution AlgorithmThe solution of the modified time-discrete counterpart of the model is approachedwith a predictor corrector procedure, typical used for the classical theory ofplasticity. The algorithm consists in evaluating an elastic trial state, in which theinternal variables remain constant and in verifying the admissibility of the trial limitTRfunction. Then, if the trial state is admissible ( f ≤ 0 ), the step is effectively elastic,i.e. there is no transformation strain development and it represents the solution;TRotherwise, if the trial state is not admissible ( f > 0), the step is inelastic and a newsolution has to be evaluated. The transformation strain tensorthrough integration of the evolutionary equations.The inelastic step is solved using the following procedure.Ethas to be updatedFirst of all, the assumption γ = 0 is done, i.e. it is assumed that there is an evolvingphase transformation state in which 0 ≤ Et< εL. The equation (5.9) is rewritten inresidual form asC −1 −1 −1 −1 −1⎧ ⎪R =− Ctn ,+ Ut exp( Δ ζ Ut fUt ) Ut= 0⎨Δζ⎪⎩ R = dev( Y) − R=0(5.11)A new value ofa Newton-Raphson method.Ctis evaluated solving these seven scalar non-linear equations with81
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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
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 and 82: 1 ( C )*1t−1α = ( β T − M ( )
Page 83 and 84: 5 FINITE STRAIN CONSTITUTIVE MODEL:
Page 85: 2 3ζ 2 ζ 3exp( ζ g) = 1+ ζ g+ g
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 and 134: The inelastic step is solved by the
Page 135 and 136: 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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