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

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cycle in which the maximum value of temperature is greater thanA f, i.e. toreproduce the shape-memory effect also in a bending state of stress.It can be noted that even in a bending analysis the use of the simplified SMA modelin a beam finite element does not induce significant differences in the solution withrespect to the one recovered adopting the full 3D SMA model.35302D solid elementbeam elementBending moment [Nmm]25201510500 2 4 6 8 10 12 14 16Transversal displacement [mm]Fig. 8.12: Pure bending test:shape-memory behavior. Moment versus transversal displacement8.1.2.3 Tension-Compression TestThe last example is referred to the two-dimensional SMA arch clamped along oneedge and free along the other, whose geometry is given in Fig. 8.5.In order to analyze the mechanical response of the arch, it is discretized both withtwo-dimensional finite elements considering a regular mesh characterized by 10elements in the radial direction and 40 element in the tangent one and beam finiteelements considering a mesh of 10 elements.150
The arch is subjected to tension-compression tests under stress control at threedifferent values of temperature, and in particular at151T> Af, As T Af< < andT = M f. Moreover, for the case T < Asalso the shape memory effect is analyzedunder temperature control.In Fig. 8.13 the displacement of the point A along the free edge is plotted versus theapplied force for the two-dimensional and the beam models.In the first case, i.e.T > A , the arch is subjected to an increasing value of the axialfforce until a maximum of 18N , maintaining constant the temperature at 285K ; thenthe load is removed allowing the element to recover the starting conditions.Therefore, if the material is loaded at a temperature aboveAfmacroscopic nonlinearlarge deformations occur and they are totally recovered during unloading,since the product phase, i.e. the single variant martensite, is not stable at this value oftemperature. An hysteretic loop in terms of stress and strain (pseudolelasticity) isdescribed.In the second case, i.e. As < T < Af, the arch is subjected to an increasing value ofthe axial force until a maximum of 10N , maintaining constant the temperature at240K ; then the load is removed.Finally, in the third case, i.e.the temperature is maintained constant at 223K .For the cases in whichT ≤ M , the maximum value of the axial force is 7N ,sfT < A also the strain recovery induced by a heating cycle isindicated. In fact, after the loading-unloading cycle, a residual strain remains. With athermal cycle in which the temperature reaches the maximum value of 400K andthen it reduces to the starting value, the shape memory alloy is able to recover theoriginal configuration going back to the point defined by zero stress and strain level.The constitutive model is able to catch the asymmetric response of the materialduring tension-compression test in both the case of pseudoelasticity and shapememory effect. In particular, setting all the model parameters equal in tension and in
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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
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Fig. 3.2: Representation of the pol
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ecause, as time progresses, the spe
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It can be introduced the rotated ra
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traction vector. For the Cauchy’s
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Nt per unit area acting on the boun
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transforms according to a= Qa. Velo
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∂∂t=−∂F∂t−1 −1 −1F
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where the direct dependency upon X
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3.4.2 Isotropic Hyperelasticity - M
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such phenomena as the Bauschinger e
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superposed to intermediate configur
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whereΨerepresents the elastic stra
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( )( ) E 1 ( )Ψ = h ω β T − M
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4.2.2 Clausius-Duhem InequalityIn o
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−1∂Ψe−TS− 2FtFt= 0∂Ce(4.
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qφ = T : dt− ⋅gradT≥0(4.30)T
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fact the values of ϕ andenergy usu
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4.2.5 Final Format of the Constitut
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1 ( C )*1t−1α = ( β T − M ( )
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5 FINITE STRAIN CONSTITUTIVE MODEL:
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2 3ζ 2 ζ 3exp( ζ g) = 1+ ζ g+ g
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where d is a user-defined parameter
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⎡R R Rt⎢⎢RR Rt⎢⎣R R RtC C
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γ• R,Ct=12EEtt(5.28)• R γ ,
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The spatial virtual work equation s
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[ E E E 2 E 2 E 2 E ]δE = δ δ δ
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ˆNLB = B + B (5.51)a a ain whichBa
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Kirchhoff stress and converting int
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DF( φ)[ u]d=dεε = 0( φ εu)∂
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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: 908070Bending moment [Nmm]605040302
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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