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

⎧∂f∂ ε⎪ C ⎡⎤⎣ ⎦⎪⎪ΔζJ⎨R = J + m − R=⎪⎪ R = − =⎪⎪⎩XTR*tR = X−( σ − Δ ζ ) + β T − Mf+ h εt+ γ = 0∂X∂εt3220J2γεtεL0(6.42)The non-linear equation system, consisting of eight scalar equations is solved with aNewton-Raphson method to compute a new value of εtand γ .From a computational point of view, the presented time-discrete model shows aproblem due to the fact that the transformation stress X depends on thetransformation strain norm, which can be zero, so that this quantity remainsundefined. To overcome this difficulty, the Euclidean normεtcan be substitutedwith a regularized normεt, defined as( d+1)ddεt = εt − εt+d −1( ) ( d−1d)d(6.43)where d is a user-defined parameter which controls the smoothness of theregularized norm. For large values of the transformation strain tensor, the regularizednorm tends to the Euclidean one; for small values of ε t, the parameter d measuresthe difference betweenε tand ε t. In this way, the quantity ε tis alwaysdifferentiable, even in the case εt= 0 with d > 0 .109