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

The inelastic step is solved by the return-mapping algorithm.First of all, the assumption γ = 0 is done, i.e. it is supposed that there is an evolvingphase transformation state in which 0 ≤ η < 2 ε . The equation (7.34) is rewritten3 Lin a residual form as:Δζ 3 dev m devR = X11 + X11− R= 0(7.36)2 3A new value of ϑ is evaluated solving this non-linear equation with a Newton-Raphson method.If the above solution is not admissible, i.e. ϑ > 2 ε 3L, equation (7.34) is rewrittenin a residual form assuming γ > 0 as:⎧Δζ3 dev m devR = X11 + X11− R=0⎪ 2 3⎨⎪γ 3R = ϑ − εL= 0⎪⎩2(7.37)The non-linear equation system of two scalar equations is solved with a Newton-Raphson method to compute a new value of ϑ and γ .As stated, the non-linear evolutionary problem is solved using an iterative method. Inparticular, only the case of the saturated phase transition, i.e. ϑ = 2 ε 3L, is treated,since the case of evolving phase transition can be simply obtained eliminating fromthe governing system of equations the last row.The iterative Newton-Raphson method requires the linearization of equation (7.37)as:Δζ Δζ Δζ⎪⎧ d( R ) = R, ηdη+ R,γdγ⎨ γ γ γ⎪⎩ dR ( ) = Rd, ηη + Rd, γγ(7.38)hence the computation of the matrix:127