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

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5.2.1.2 Finite Element ApproximationThe reference configuration coordinates can be represented asaX = N () ξ X (5.47)I a Iwhere ξ are the three dimensional natural coordinates ξ 1, ξ2and ξ 3,Naarestandard shape functions. Similarly, the displacement field in each element can beapproximated byu = N () ξ u a(5.48)i a iThe reference system derivatives are constructed as followsu = N ξ u (5.49), ,() aiI aI iwhere explicit writing of the sum is omitted and summation convention for a isagain invoked. The derivatives of the shape function can be established by usingstandard routines.The deformation gradient and Green strain may now be computed using equations(3.12) and (3.18), respectively. Finally, the variation of the Green strain is given inmatrix form asˆaδ E= B δu (5.50)awhere B ˆ acan be split into two parts as90
ˆNLB = B + B (5.51)a a ain whichBais identical to the small deformation strain-displacement matrix and theremaining non-linear part is give byBNLa⎡ u1,1 Na,1 u2,1 Na,1 u3,1 Na,1⎢⎢u1,2 Na,2 u2,2 Na,2 u3,2 Na,2⎢ u1,3 Na,3 u2,3 Na,3 u3,3 Na,3= ⎢⎢u N + u N u N + u N u N + u N⎢u N + u N u N + u N u N + u N⎢⎢⎣u1,3 Na,1 + u1,1 Na,3 u2,3N + u N u N + u N1,1 a,2 1,2 a,1 2,1 a,2 2,2 a,1 3,1 a,2 3,2 a,11,2 a,3 1,3 a,2 2,2 a,3 2,3 a,2 3,2 a,3 3,3 a,2a,1 2,1 a,3 3,3 a,1 3,1 a,3⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(5.52)It is immediately evident thatB NLais zero in the reference configuration andtherefore that B ˆ a= Ba.The variational equation may now be written for the finite element problem as:⎛⎞a ⎜∫a a⎟(5.53)⎝V⎠T( ) ˆ Tδu B SdV− f = 0where the external forces are determined asf∫ ∫ (5.54)= N ρ BdV + N tdSa a 0aV∂Vwith B and t the matrix form of the body and traction force vectors, respectively.The tangent term is given by91
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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
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2 SHAPE MEMORY ALLOYS2.1 Introducti
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transformation (M→A), indicated r
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Fig. 2.2: SuperelasticityFig. 2.3:
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The Differential Scanning Calorimet
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2.3 Biocompatibility of Shape-Memor
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only in the angioplasty procedure,
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micropumps used to unblock blood ve
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introduction of a multi-well free-e
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spectral decomposition allowing the
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Fig. 3.1: Reference and deformed co
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δ δ = δ and δ δ = δ(3.6)iI iJ
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and it is a two-point tensor since
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Furthermore,( ) 2 2Jdet b= det F =
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3.2.3 Polar DecompositionThe deform
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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: [ E E E 2 E 2 E 2 E ]δE = δ δ δ
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
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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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