jaxmat.materials.viscoplastic_flows module

class VoceHardening

Bases: Module

Voce hardening model for stress-strain behavior.

\[ \sigma_Y(p)=\sigma_0 + (\sigma_\text{u}-\sigma_0)(1-\exp(-bp)) \]

References

• Voce, E. (1955). “A Practical Strain-Hardening Function.” Metallurgia, 51, 219-226.

sig0: float

Initial yield stress \(\sigma_0\).

sigu: float

Saturation stress at large strains \(\sigma_\text{u}\).

b: float

Rate of hardedning \(b\).

class NortonFlow

Bases: Module

A Norton viscoplastic flow with overstress.

\[\dot{\beps}^\text{vp} = \left\langle\dfrac{f(\bsig) - \sigma_y}{K}\right\rangle_+^m\]

where \(f(\bsig)-\sigma_y\) is the overstress, \(\langle \cdot\rangle_+\) is the positive part.

K: float

Characteristic stress \(K\) of the Norton flow.

m: float

Norton power-law exponent

class AbstractKinematicHardening

Bases: Module

An abstract module for Armstrong-Frederic type kinematic hardening.

nvars: AbstractVar[int]

The number of kinematic hardening variables

sig_eff(sig, X)

Effective stress \(\bsig-\sum_i \bX_i\) where \(\bX_i\) is the \(i\)-th backstress.

class LinearKinematicHardening

Bases: Module

Linear kinematic hardening model.

\[\dot{\bX} = \dfrac{2}{3}H\dot{\bepsp}\]

References

Prager, W. (1956). A new method of analyzing stresses and strains in work-hardening plastic solids.

H: float

Linear kinematic hardening modulus

nvars = 1

sig_eff(sig, X)

Effective stress \(\bsig-\sum_i \bX_i\) where \(\bX_i\) is the \(i\)-th backstress.

class ArmstrongFrederickHardening

Bases: AbstractKinematicHardening

Armstrong-Frederick kinematic hardening model.

References

• Armstrong, P. J., & Frederick, C. O. (1966).

  “A Mathematical Representation of the Multiaxial Bauschinger Effect for Hardening Materials.” CEGB Report RD/B/N731.

• Chaboche, J. L. (1991). On some modifications of kinematic hardening to

  improve the description of ratchetting effects. International journal of plasticity, 7(7), 661-678.

C: Array

Kinematic hardening modulus

gamma: Array

Nonlinear recall modulus

nvars: AbstractVar[int] = 2

The number of kinematic hardening variables

sig_eff(sig, X)

Effective stress is here:

\[\bsig-\frac{2}{3}C\sum_{i=1}^\text{nvars}a_i\]