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

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∂φ( X, t)VX ( , t)=∂t(3.42)Observe that the velocity is a spatial vector despite the fact that the equation has beenexpressed in terms of the material coordinates of the particle X . In fact, by inverting(3.3) the velocity can be more consistently expressed as a function of the spatialposition x and time as1VX ( , t) = v( φ − ( x, t), t) = vx ( , t)(3.43)Relationship (3.43) can be written in compact form as:V= v v=V1φor φ −(3.44)where denotes composition.3.3.2 Material Time DerivativeGiven a general scalar or tensor quantity g , expressed in terms of the materialcoordinates X , the time derivative of g( X ,t)denoted by g( X,t) is defined as( ,t)∂g Xg =(3.45)∂tThis expression measures the change in g associated with a specific particle initiallylocated at X , and it is known as the material time derivative of g . The spatialquantities are expressed as functions of the spatial position x , in this case thematerial derivative is more complicated to establish. The complication arises40
ecause, as time progresses, the specific particle being considered changes its spatialposition. Consequently, the material time derivative in this case is obtained from acareful consideration of the motion of the particle as( φ( , t+Δ t), t+Δt) − ( φ( , t), t)g X g Xg ( x, t)= lim(3.46)Δ→ t 0ΔtThis equation clearly illustrates that g changes in time as a result of a change in timebut with the particle remaining in the same spatial position and because of the changein spatial position of the specific particle. Using the chain rule, (3.46) gives thematerial derivative of g ( x ,t)as( , t) ( , t) φ ( , t) ( , t)∂g x ∂g x ∂ X ∂g xg ( x,t)= + = + ( ∇g)v (3.47)∂t ∂x∂t ∂tThe second term, involving the particle velocity in equation (3.47) is often referred toas the connective derivative.3.3.3 Rate of Deformation TensorsVelocity has been expressed as a function of the spatial coordinates as v( x ,t). Thederivative of this expression with respect to the spatial coordinate defines thevelocity gradient tensor l as( )∂v x,tl = =∇v∂x(3.48)To obtain an alternative expression for l , it must be observed that41
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 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: Fig. 3.2: Representation of the pol
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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