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

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( ) 2 2Jdet C= det F = > 0(3.16)Alternatively the symmetric Green-Lagrange strain tensor, E , given as1 1E= ( C− 1 ) or EIJ = ( CIJ −δIJ)(3.17)2 2may be used.The Green-Lagrange strain tensor components may be expressed in terms of thedisplacements gradient asE1 ⎡ ∂ui ∂ui ∂ui ∂u⎤i= ⎢δ+ δ + ⎥2 ⎣ ∂X J∂XI ∂XI ∂XJ⎦1= ⎡δiIuiJ ,δiJuiI ,uiI ,uiJ,2⎣ + + ⎤⎦1= ⎡UI, J+ UJ, I+UK, IUK,J⎤2⎣⎦IJ iI iJ(3.18)3.2.1.3 Spatial Strain TensorsIn the current configuration, a common deformation measure is the left Cauchy-Green deformation tensor or Finger deformation tensor, b , expressed asTb= FF or b = F F(3.19)ij iI jIThe tensor b is symmetric and definite positive at eachx ∈ω,T( ) and 0 for all 0T T Tb= FF = FF = b u⋅ bu> u ≠ (3.20)32
Furthermore,( ) 2 2Jdet b= det F = > 0(3.21)The Euler-Almansi strain tensor, e , is related to the inverse of b as1 −1 1−1e= ( 1− b ) or eij = ( δij −bij)(3.22)2 2or inverting by−1−( 2 ) or b ( δ 2e) 1b= 1− e = −(3.23)ij ij ij3.2.2 Push-Forward, Pull-Back OperationAs already seen, vector and tensor quantities may be resolved along triads of basisvectors belonging to either the reference or the current configurations. Additionally,there are two-point tensors which are associated with both configurations, oneexample being the deformation gradient. The transformations between material andspatial quantities are typically called a push-forward operation and a pull-backoperation and are denoted by χ* (•)and χ − 1( )*• , respectively.In particular, a push-forward is an operation which transforms a vector or tensorquantity based on the reference configuration to the current configuration. Since theEuler-Almansi strain tensor e is defined with respect to spatial coordinates, it can becomputed as a push-forward of the Green-Lagrange strain tensor E , which is givenin terms of material coordinates. From equation (3.22), it can be concluded that:33
Page 1 and 2: University of CassinoFaculty of Eng
Page 3 and 4: Contents1. INTRODUCTION ...........
Page 5 and 6: 5.2.1.2. Finite Element Approximati
Page 7 and 8: 1 INTRODUCTIONScience and technolog
Page 9 and 10: The implementations of the model in
Page 11 and 12: 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 it is a two-point tensor since
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 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 and 134:
The inelastic step is solved by the
Page 135 and 136:
For brevity, only the saturated pha
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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