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 and 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.