<ISOTROPIC> isotropic_dsa

Description

The plastic strain hardening function is given by

\[R_a = P_1\left(1-\exp{\left[-\left(\dfrac{t_a}{t_0}\right)^\beta\right]} \right)\]

where

• \(P_1\) is the saturation aging strength, when dislocations are fully pinned.

• \(t_0\) is characteristic diffusion time.

• \(t_a\) is the aging time required for the concentration of solute atmospheres in the neighborhood of dislocations to reach a critical level and pin the dislocations, and is governed by the following evolution equation:

  \[\dot{t}_a=1-\dfrac{t_a}{w}\dot{p}\]

• \(\beta\) is a parameter related to the type of diffusion of solute atoms (\(\beta=0.33\) for core diffusion, and \(\beta=0.66\) for lattice diffusion).

In dynamic strain aging (DSA), the pinning of dislocations by diffusing solute atoms can lead to negative strain rate sensitivity, which is a key factor in triggering the Portevin–Le Chatelier (PLC) effect.

• At low strain rates, dislocations move slowly, giving solute atoms more time to diffuse and re-pin them, which increases flow stress.

• At higher strain rates, dislocations move too fast for solute atoms to keep up, reducing the pinning effect, which leads to a lower increase in flow stress.

Example 1

***behavior gen_evp
 **elasticity isotropic
   young 260000.
   poisson 0.3
 **potential gen_evp ev
  *flow norton
   n     7.0
   K     100.
  *criterion mises
  *isotropic isotropic_dsa
    P1    700.
    t0    2.0
    w     0.0002
    beta  0.66
***return

Example 2

***behavior gen_evp
 **elasticity isotropic
   young 260000.
   poisson 0.3
 **potential gen_evp ev
  *flow norton
   n     7.0
   K     100.
  *criterion mises
  *isotropic isotropic_sum
   *-nonlinear
    R0   115.0 %150.
    b    150.
    Q    400.
   *-isotropic_dsa
    P1    700.
    t0    2.0
    w     0.0002
    beta  0.66
***return

<ISOTROPIC> isotropic_dsa_p

This model is an extended version of isotropic_dsa. The saturation aging strength is given as an explicit function of the over-concentration of solute atoms around dislocations. The contribution \(R_a\) is given by

\[R_a = P_1 C(p) \left(1-\exp{\left(-P_2 p^{\alpha}t_a^{\beta}\right)}\right)\]

where \(P_1 C(p)\) is the saturation aging strength, \(P_2\) gives the saturation rate of diffusion around dislocations, and

\[\dot{t}_a=1-\dfrac{t_a}{w}\dot{p}.\]

To reproduce the increase of the stress drop amplitudes with increasing strain, a linear decomposition of the solute atom concentration C(p) is considered:

\[C(p)= c_1+ c_2 p\]

Example

***behavior gen_evp
 **elasticity isotropic
   young 260000.
   poisson 0.3
 **potential gen_evp ev
  *flow norton
   n     7.0
   K     100.
  *criterion mises
  *isotropic isotropic_sum
   *-nonlinear
    R0   684.
    b    150.
    Q    400.
   *-isotropic_dsa_p
    w  0.0004
    t0  1.0
    beta  0.33
    alphap  0.0
    P1  1.
    P2  1.15
    cm1  102.
    cm2  878.
***return

<ISOTROPIC> isotropic_dsa_r

This model is a variant of isotropic_dsa_p. The saturation aging strength is given as an explicit function of the over-concentration of solute atoms which depends on the current isotropic hardening \(R\) (without the contribution \(R_a\)).

\[R_a = P_1 C(R) \left(1-\exp{\left(-P_2 p^{\alpha}t_a^{\beta}\right)}\right)\]

where \(P_1 C(R)\) is the saturation aging strength. A linear decomposition of the solute atom concentration \(C(R)\) is considered:

\[C(R)= c_1+ c_2 R\]

Warning

isotropic_dsa_r can only be used inside an *isotropic isotropic_sum assembly.

Example

***behavior gen_evp
 **elasticity isotropic
   young 260000.
   poisson 0.3
 **potential gen_evp ev
  *flow norton
   n     7.0
   K     100.
  *criterion mises
  *isotropic isotropic_sum
   *-nonlinear
    R0   684.
    b    150.
    Q    400.
   *-isotropic_dsa_r
    w  0.0004
    t0  1.0
    beta  0.33
    alphap  0.0
    P1  1.
    P2  1.15
    cm1  102.
    cm2  1.
***return