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

5.1.2 Time Integration of the Constitutive ModelThe time-discrete counterpart of the constitutive model is given by⎧ ⎡λ1 ⎤ −1 −1 −1⎪ S= 2⎢( I1−3)− μ4 2 ⎥Ct + μCt CCt⎪ ⎣⎦⎪1⎪ ( C )*1t−1( ( ) ) 2⎪α = β T − Mf + h Ct− 1 + γ⎪2 1( Ct−1)⎪2⎪⎪Y= CS−Cαt⎨γ ≥ 0⎪−1 −1 −1 −1 −1⎪ Ctn ,= Ut exp( ΔζUt fUt ) Ut⎪⎪Ut= Ct⎪ f = 2 dev( Y) Ct/ dev( Y)⎪⎪Et≤ εL⎪ F( Y) = dev( Y) −R≤0⎪⎩ Δζ≥0 Δ ζF( Y) = 0(5.9)where Δ ζ = ( ζ − ζ n ) is the consistency parameter, time integrated over the interval[ t , nt ] .From a computational point of view, the time-discrete model as presented in equation(5.9) shows a problem due to the fact that the transformation stress α is proportionalto the inverse of the Green-Lagrange strain norm and to its derivative so that, whenthe strain is zero, this quantity remains undefined. To overcome this difficulty, theEuclidean normEtcan be substituted with a regularized norm Et, defined as( d+1)dd( ) ( d−1) dEt = Et − Et+ d(5.10)d −180