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Flow Rule (Nitinol Material Model)

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 = εul ξs( 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 εul ξs)

t = 2 G (e(bar) - εul ξsn(bar))

In the above formulations:

εul scalar parameter representing the maximum material plastic strain deformation [EUL]
ξs parameter between 0 and 1, as a measure of the plastic straining
θ volumetric strain = ε11 + ε22 + ε33
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(1-2ν)], G = E / [2(1+ν)]

The linear flow rule in the incremental form can be expressed, accordingly:

Loading: Δξs = ( 1.0 - ξs) ΔF / ( F - Rf1)

Unloading: Δξs = ξs ΔF / ( F - Rf2)

And the exponential flow rule, used when a nonzero β is defined:

Loading: Δξs = β1( 1.0 - ξs) ΔF / ( F - Rf1)2

Unloading: Δξs = β2ξs ΔF / ( F - Rf2)2

  • In general, shape-memory-alloys are found to be insensitive to rate-effects. 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 loading-unloading, since similar expressions (with compressive property parameters) can be used for the compressive loading-unloading conditions.
  • The incremental solution algorithm here uses a return-map 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.


  1. Auricchio, F., “A Robust Integration-Algorithm for a Finite-Strain Shape-Memory-Alloy Superelastic Model,” International Journal of Plasticity, vol. 17, pp. 971-990, 2001.
  2. Auricchio, F., Taylor, R.L., and Lubliner, J., “Shape-Memory-Alloys: Macromodeling and Numerical Simulations of the Superelastic Behavior,” Computer Methods in Applied Mechanics and Engineering, vol. 146, pp. 281-312, 1997.
  3. Bergan, P.G., Bathe, K.J., and Wunderlich, eds. “On Large Strain Elasto-Plastic and Creep Analysis,” Finite Elements Methods for Nonlinear Problems, Springer-Verlag 1985.
  4. Hughes, T., eds. “Numerical Implementation of Constitutive Models: Rate-Independent Deviatoric Plasticity,” Theoretical Foundation for Large-Scale Computations for Nonlinear Material Behavior, Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1984.

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