Through adoption of the logarithmic strain definition, the deviatoric and volumetric components of the strain and stress tensors and their relations can be correctly expressed in a decoupled form.
First, we consider the total plastic & elastic strain vectors to be presented by:
ε(bar)^{p} = ε_{ul} ξ_{s}( n(bar) + α*m(bar))
ε(bar)^{e(bar)} = ε(bar)  ε(bar)^{p}
The Kirchhoff stress vector can then be evaluated from:
τ(bar) = p m(bar) + t(bar)
p = K (θ  3 α ε_{ul} ξ_{s})
t = 2 G (e(bar)  ε_{ul} ξ_{s}n(bar))
In the above formulations:
ε_{ul} 
scalar parameter representing the maximum material plastic strain deformation [EUL] 
ξ_{s} 
parameter between 0 and 1, as a measure of the plastic straining 
θ 
volumetric strain = ε_{11} + ε_{22} + ε_{33} 
e(bar) 
deviatoric strain vector 
t(bar) 
deviatoric stress vector

n(bar) 
norm of the deviatoric stress = t(bar) / (sqrt(2) σ(bar))

m(bar) 
the identity matrix in vector form: {1,1,1,0,0,0}^{T}

K and G 
bulk and shear elastic moduli: K = E / [3(12ν)], G = E / [2(1+ν)]

The linear flow rule in the incremental form can be expressed, accordingly:
Loading: Δξ_{s} = ( 1.0  ξ_{s}) ΔF / ( F  R_{f}^{1})
Unloading: Δξ
_{s} = ξ
_{s} ΔF / ( F  R
_{f}^{2})
And the exponential flow rule, used when a nonzero β is defined:
Loading: Δξ_{s} = β^{1}( 1.0  ξ_{s}) ΔF / ( F  R_{f}^{1})^{2}
Unloading: Δξ_{s} = β^{2}ξ_{s} ΔF / ( F  R_{f}^{2})^{2}
 In general, shapememoryalloys are found to be insensitive to rateeffects. Thus, in the above formulation “time” represents a pseudo variable, and its length does not affect the solution.
 All the equations are presented here for tensile loadingunloading, since similar expressions (with compressive property parameters) can be used for the compressive loadingunloading conditions.
 The incremental solution algorithm here uses a returnmap procedure in the evaluation of stresses and constitutive equations for a solution step. Accordingly, the solution consists of two parts. Initially, a trial state is computed; then if the trial state violates the flow criterion, an adjustment is made to return the stresses to the flow surface.
References
 Auricchio, F., “A Robust IntegrationAlgorithm for a FiniteStrain ShapeMemoryAlloy Superelastic Model,” International Journal of Plasticity, vol. 17, pp. 971990, 2001.
 Auricchio, F., Taylor, R.L., and Lubliner, J., “ShapeMemoryAlloys: Macromodeling and Numerical Simulations of the Superelastic Behavior,” Computer Methods in Applied Mechanics and Engineering, vol. 146, pp. 281312, 1997.
 Bergan, P.G., Bathe, K.J., and Wunderlich, eds. “On Large Strain ElastoPlastic and Creep Analysis,” Finite Elements Methods for Nonlinear Problems, SpringerVerlag 1985.
 Hughes, T., eds. “Numerical Implementation of Constitutive Models: RateIndependent Deviatoric Plasticity,” Theoretical Foundation for LargeScale Computations for Nonlinear Material Behavior, Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1984.