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

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4.2 Constitutive ModelThe micromechanical behavior of crystal metals is assumed as the basis for thederivation of a new constitutive model for shape memory alloys. Nevertheless, itmust be pointed out that the plastic mechanism of deformation simply based oncrystallographic slips of the atomic planes results really different from thediffusionless thermoelastic martensitic transformation governing the phase transitionin the shape memory alloys. However, as for the modeling of the crystal metalplasticity, the SMA constitutive model considers the assumption of the localmultiplicative decomposition of the deformation gradient in the form:FX ( , t) = F( X, t) F( X , t)(4.3)etfor each material pointX ∈Ω, the reference configuration of the body. Feis relatedto the deformation caused by stretching and rotation of the material, Ftaccounts forthe inelastic flow associated to the phase transition in the SMA and F is the totaldeformation gradient. The multiplicative split of the deformation gradient in the formpresented in expression (4.3) is consistent with a large part of the literature on finitedeformationinelastic models (Lee,1969; Mandel, 1974).This decomposition introduces the idea of an intermediate configuration which canbe considered as the one associated to the body when the loads are removed and thetemperature is reduced to the initial one, thus releasing the thermoelastic strains.As a consequence, the intermediate configuration is obtained from the currentconfiguration by elastic destressing to zero stress. It differs from the initialconfiguration by a residual or inelastic deformation and from the currentconfiguration by a reversible or elastic deformation. Accordingly, it is implicitlyassumed that the local intermediate configuration defined bydeformation gradients F eandF is stress-free. Theconfiguration is not unique; in fact, arbitrary local material rotations can be58−1eFpare not uniquely defined because the intermediate
superposed to intermediate configuration preserving it unstressed. However, thedecomposition can be made unique by additional specifications dictated by the natureof the considered material model.Fig. 4.2: Multiplicative decomposition of the deformation gradientThe model has been conceived within the framework of Generalized StandardMaterials (Halphen and Nguyen, 1975): internal variables are introduced in order todescribe the phase transition processes and two convex potentials are defined, fromwhich the constitutive relations and the evolution law for the transformation strainare derived.In general, the state of a continuum material depends on the whole history of itsmechanical variables, and the modeling of its behavior may be based on differentkinds of approaches. In order to obtain a formalism which is directly accessible to themethods of functional analysis, the approach of thermodynamics of irreversibleprocesses is adopted by introducing state variables (Coleman and Noll, 1963;Coleman and Gurtin, 1967). This approach was initiated first by chemists, and wasapplied to continuum mechanics by Eckart and Biot around 1950. Thethermodynamic potential allows to define associated variables chosen for the studyof the phenomenon. This naturally leads to the state laws. Furthermore, the59
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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
Page 13 and 14: 2 SHAPE MEMORY ALLOYS2.1 Introducti
Page 16 and 17: 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: such phenomena as the Bauschinger e
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 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
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⎧∂f∂ ε⎪ C ⎡⎤⎣ ⎦⎪
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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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