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

More documents

Recommendations

Info

DE[ u]= ε(6.5)Similarly, the right and left Cauchy-Green deformation tensors defined respectivelyin equations (3.14) and (3.19) can be linearized to give[ ] D [ ] 2DCu = bu = ε + 1(6.6)6.1.3 Linearized Volume ChangeThe volume change is given by the JacobianJ = det F . Physically, it expresses theratio between the elementar volume element defined in the current configuration andthe one defined in the reference configuration, i.e.:dvJ = (6.7)dVThe directional derivative of J with respect to an increment u in the spatialconfiguration is:( )DJ[ u] = D det F DF[ u ](6.8)Recalling the expression of the directional derivative of the determinant of a tensorand applying equation (6.3) for the linearization of F , it results:DJ[ u]⎛Jtr⎜⎜⎝−1= F ⎟= Jtr∇u∂u⎞∂X⎟⎠(6.9)96
In terms of the linear strain tensor ε , the above equation can be expressed as:DJ[ u]= Jtrε (6.10)In the initial configuration, it follows that:DJ[ u]= trε (6.11)6.1.4 Linearized Elastic Free EnergyThe expression of the elastic free energy, adopted to describe the elastic SMAbehavior in a finite strain regime is the following:1⎡ ⎛λ + 2μ ⎞ 2 ⎛3λ+ 2μ ⎞ ⎛9λ+ 6μ⎞⎤Ψe= I1 I1 μI22⎢⎜ ⎟ − ⎜ ⎟ + + ⎜ ⎟4 2 4⎥⎣⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎦(6.12)where I1and I 2are the principal invariant of the elastic Cauchy-Green tensor. Inparticular, the linearized expression of these quantities are derived as:[ ]*I = trC = tr 2E + 1 = 2trE + 3= 2trε+ 3= 2I+ 3 (6.13)1 e e e e11 2 12( ( ) )⎡ ( )2 2 ⎣1 2( 42tr ( Ee)3 4 tr EeI )21( ( εe) ( εe)εe)( 4 4 ⎤)I = tr C − I = tr E + 1+ E −I⎦2 22 e 1 e e 1= + + −2 2 * *2 1= 2tr −3−2 tr − 4tr = 4I −4I−3(6.14)97
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 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: DF( φ)[ u]d=dεε = 0( φ εu)∂
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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?