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: Δξ
s = ξ
s ΔF / ( F - R
f2)
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.
References
- 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.
- 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.
- 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.
- 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.