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

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last term of equation (4.7) ensures that the free energy is null in the undeformedstate, i.e. when C=1.4.2.1.2 Transformation Free EnergyAdopting the basic idea of Souza et al. (1998), in order to account for differentexperimental features of shape-memory alloys the transformation free energyevaluated as the sum of some contributes.In particular, denoting with E ( C 1)computed by means of the following relation:tΨtis= 1 2t− the Green-Lagrange strain, Ψ tisΨ ( E ) =Ψ ( E ) +Ψ ( E ) +Ψ ( E ) +I ( E )(4.8)t t ch t tr t id t ε L twhere:• Ψchis the chemical energy, due to the thermally-induced martensitictransformation. The phase transformation in the SMA depends on the temperatureand in particular, increasing the gap between the test temperature and thecharacteristic temperatureMf, the transition from austenite to martensite starts foran higher value of the stress. Furthermore, experimental evidences show that forT ≤ M fthe stress do not depend yet on the test temperature. For all these reasons,the chemical energy is assumed to be a positive and monotonically increasingfunction of the temperature. Introducing the quantity ω = tr ( E)and the function:h( ω)⎧1 ifω> 0= ⎨⎩0 ifω≤ 0(4.9)it can be expressed as:62
( )( ) E 1 ( )Ψ = h ω β T − M + −h ω β T − M E (4.10)ch t f t c f twhere the parameters βtandβc, linked to the dependence of the critical stress on thetemperature, are introduced as the thermomechanical behavior of a SMA materialresults strongly sensitive to the deformation mode. In particular, experimentalevidences show that the shape-memory materials, expecially NiTi and CuZnAlalloys, exhibit a tension-compression asymmetry in their phase transformation.In the chemical energy expression (4.11), ⋅ is the positive part of the argument,Mfis the finishing temperature of the austenite-martensite phase transformation andT is the room temperature; finally, ⋅ represents the norm of the argument;• Ψtrrepresents the transformation strain energy introduced in order to accountfor the transformation-induced hardening:1 2Ψtr= h Et(4.11)2where h is a material parameter defining the slope of the linear stress-transformationstrain relation in the uniaxial case and related to the hardening of the material duringthe phase transformation;▪Ψidis the free energy due to the change in temperature with respect to thereference state in an incompressible ideal solid:⎡T ⎤Ψid= ( uo − Tηo ) + c⎢( T −To) −Tln⎥⎣To⎦(4.12)63
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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
Page 18: Fig. 2.2: SuperelasticityFig. 2.3:
Page 21 and 22: The Differential Scanning Calorimet
Page 23 and 24: 2.3 Biocompatibility of Shape-Memor
Page 25 and 26: only in the angioplasty procedure,
Page 27 and 28: micropumps used to unblock blood ve
Page 29 and 30: introduction of a multi-well free-e
Page 31 and 32: 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: whereΨerepresents the elastic stra
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 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
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If equation (6.42) is considered as
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7 3D-1D Constitutive Model with 1D
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*∂f6 3σ9 3 2 2 T Ml− hϑ+mz β
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⎡⎤⎢1 0 0 ⎥⎢⎥1ε t= ϑ
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(7.21)σ3 1.5R=− βc( T − Mf)+2
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It is interesting to evaluate how m
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It can be noted that equations (7.3
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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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