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

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3.3.2. Material Time Derivatives ....................................................................403.3.3. Rate of Deformation Tensors................................................................413.3.4. Stress Measures for Reference and Deformed State.............................433.3.5. Equation of Motion ...............................................................................463.3.6. Objectivity.............................................................................................483.3.6.1. Objective Stress Rate ..................................................................493.4. Constitutive equations-hyperelastic materials ................................................523.4.1. Hyperelasticity ......................................................................................523.4.2. Isotropic Hyperelasticity-Material Description ....................................554. A FINITE STRAIN CONSTITUTIVE MODEL...............................................564.1. Introduction.....................................................................................................564.2. Constitutive Model..........................................................................................584.2.1. Thermodynamic Framework.................................................................604.2.1.1. Elastic Free Energy.....................................................................614.2.1.2. Transformation Free Energy.......................................................624.2.2. Clausius-Duhem Inequality...................................................................654.2.3. Constitutive Equations ..........................................................................664.2.4. Pseudo-Potential of Dissipation and Evolution Law ............................684.2.5. Final Format of the Constitutive Model................................................735. FINITE STRAIN CONSTITUTIVE MODEL: NumericalProcedure...................................................................................................................775.1. Introduction.....................................................................................................775.1.1.Exponential Map....................................................................................775.1.2. Time Integration of the Constitutive Model .........................................805.1.3. Solution Algorithm ...............................................................................815.1.4. Newton-Raphson Method .....................................................................825.2. Principle of Virtual Work ...............................................................................855.2.1.Finite Element Implementation..............................................................885.2.1.1. Reference Configuration Formulation ........................................88
5.2.1.2. Finite Element Approximation...........................................................906. LINEARIZATION OF THE FINITE DEFORMATIONSMA MODEL ...........................................................................................................946.1. Introduction.....................................................................................................946.1.1. Linearized Deformation Gradient .........................................................946.1.2. Linearized Strain Measures...................................................................956.1.3. Linearized Volume Change ..................................................................966.1.4. Linearized Elastic Free Energy.............................................................976.1.5. Linearized Transformation Free Energy...............................................986.2. 3D Constitutive Model for Stress-Temperature Inducedsolid Phase Transformation in a Small strain Regime...............................996.3. Numerical Procedure.....................................................................................1066.3.1. Introduction.........................................................................................1066.3.1.1. Time Integration of the 3D SMA ConstitutiveModel ............................................................................................................1076.3.1.2. Newton-Raphson Method .........................................................1106.3.1.3. Consistent Tangent Matrix........................................................1127. 3D-1D Constitutive Model with 1D Evolution .................................................1157.1. Introduction...................................................................................................1157.1.1. Constitutive Model..............................................................................1157.2. 1D Model and Analysis of Model Parameters ..............................................1197.3. Numerical Procedure.....................................................................................1257.3.1. Time Integration of the 3D-1D SMA ConstitutiveModel ............................................................................................................1257.3.2. Consistent Tangent Modulus ..............................................................1287.3.3. Beam Finite Element...........................................................................1308. Numerical Results...............................................................................................1348.1. Introduction...................................................................................................134
Page 1 and 2: University of CassinoFaculty of Eng
Page 3: Contents1. INTRODUCTION ...........
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 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 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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