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

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The strain-energy function vanishes in the reference configuration, i.e. where F=1.This assumption is expressed by the normalization condition:W = W() 1 = 0(3.93)From the physical observation that the strain-energy function must increase withdeformation, it therefore follows that:W = W( F ) ≥0(3.94)which restricts the ranges of admissible functions occurring in expressions for thestrain energy. The strain-energy function attains its global minimum for F=1 asthermodynamic equilibrium. Relations (3.93) and (3.94) ensure that the stress in thereference configuration, called residual stress, is zero. The reference configuration isstress-free. For the behavior at finite strains the scalar-valued function W mustadditionally satisfy so-called growth conditions. This implies that W tends to +∞ ifeitherJ = det F approaches +∞ or 0 + , i.e.⎧⎪W( F) →+∞ as detF→+∞⎨+⎪⎩W ( F) →+∞ as detF→0(3.95)Physically, that means that is required an infinite amount of strain energy in order toexpand a continuum body to the infinite range or to compress it to a point withvanishing volume.54
3.4.2 Isotropic Hyperelasticity – Material DescriptionThe hyperelastic constitutive equations discussed so far are unrestricted in theirapplication. In particular they are applied to the isotropic case. Isotropy is defined byrequiring the constitutive behavior to be identical in any material direction. Thisimplies that, for the representation theorem for invariants (Gurtin, 1981, Trusdell andNoll, 1992) the relationship between W and C must be independent of the materialaxes chosen and, consequently, W must only be a function of the invariants of C asW ( CX ( ), X) = W ( I , II , III , X)(3.96)C C Cwhere the invariants of C are defined here asIIICCIII= trC=C:1= trCC= C : CC= det C = J2(3.97)As a result of the isotropic restriction, the second Piola-Kirchhoff stress tensor can berewritten from equation (3.92) as∂W ∂W ∂I ∂W ∂II ∂W∂IIIS = 2 = 2 + 2 + 2∂C ∂I ∂C ∂II ∂C ∂III∂CC C CC C C(3.98)(3.99)Introducing the derivatives of the invariants with respect to C , enables the secondPiola-Kirchhoff stress to be evaluated as∂W ∂W ∂WS= 2 1+ 4 C+2JC∂I ∂II ∂III552 −1C C C(3.100)
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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
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: where the direct dependency upon X
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
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♦The limit function is assumed fu
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6.3.1.1 Time Integration of the 3D
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⎧∂f∂ ε⎪ C ⎡⎤⎣ ⎦⎪
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RRRRRRRR*( f )2 2 2Xd, X= I+ hΔ ζ
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If equation (6.42) is considered as
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7 3D-1D Constitutive Model with 1D
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*∂f6 3σ9 3 2 2 T Ml− hϑ+mz β
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⎡⎤⎢1 0 0 ⎥⎢⎥1ε t= ϑ
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(7.21)σ3 1.5R=− βc( T − Mf)+2
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It is interesting to evaluate how m
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It can be noted that equations (7.3
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The inelastic step is solved by the
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For brevity, only the saturated pha
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Fig. 7.2. Timoshenko’s beam theor
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NNNv1v2v3N θ1= ξξ ( − 1)21= ξ
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to analyze the two-dimensional elem
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300 N . A regular mesh characterize
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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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