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

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C = ζf( C, C ) = ζg( CC , ) C(5.1)t t t twhere the tensor f = f( C, C ) = 2 dev( Y) C / dev( Y)represents a function of twottarguments C andCt. Usingf= fC C it can be obtained a representation written− 1ttin terms of the function g=g( CC , ) . It can be emphasized that the tensor f istsymmetric, while the same do not hold for g .If the function ζ g was a constant g the evolution equation (5.1) could beanalytically integrated by means of the exponential map:C = exp( g( t−t )) C (5.2)t n t,nThe direct transfer of this integration rule to an implicit numerical integration of theevolution equation C t= ζ gCtwhere the term ζ g is held constant within the timeincrement Δ t = tn+1− tndoes not lead to a suitable integration rule because thesymmetry requirements are not fulfilledexp( ζ g Δt) C ≠C exp ( ζ g Δt)(5.3)Tn+ 1 n+ 1 tn t, n n+ 1 n+1To guarantee symmetry it can be used an updating formula ζ = ζΔtCC, C = exp( ζ g)C (5.4)−1t t n t tThe latter integration formula has the disadvantage that the exponent of a nonsymmetrictensor has to be computed. This requires to work with a seriesrepresentation78
2 3ζ 2 ζ 3exp( ζ g) = 1+ ζ g+ g + g + …2 62 3−1 ζ −1 2 ζ −13= 1+ ζ fCt + ( fCt ) + ( fCt) + …2 6(5.5)2 3⎛ −1 −1 ζ −1 −1 2 ζ −1 −1 3⎞−1= Ut⎜1+ ζ Ut fUt + ( Ut fUt ) + ( Ut fUt ) ⎟Ut⎝2 6 ⎠It follows that:( ζ )exp( ζ g) = U exp U fU U(5.6)−1 −1 −1t t t tThe latter derivation shows that exp( ζ g ) can be alternatively represented by( ζ )− −U exp U fU U− , where in contrast to the former format, the exponent of a1 1 1t t t tsymmetric tensor is included. This fact has the important advantage that theexponential function can be computed in closed form by means of the spectraldecomposition:3 3∑A T= A → A = A N ⊗ N ⇒ exp( A ) = exp( A ) N ⊗ N (5.7)∑b b b b b bb= 1 b=1In the latter relation Ab, with b= 1,2,3 represent the eigenvalues of A andN , with b = 1,2,3 its eigenvectors.bUsing (5.5) the integration rule (5.4) is finally given by( ζ )CC C = U exp U fU U C ⇒ C = U exp( ζ U fU ) U (5.8)−1 −1 −1 −1 −1 −1 −1 −1 −1t t, n t t t t t t t,n t t t t79
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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
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 and 82: 1 ( C )*1t−1α = ( β T − M ( )
Page 83: 5 FINITE STRAIN CONSTITUTIVE MODEL:
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 and 134: 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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