<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.

This model is known to cause instabilities in finite element computations (localization bands, …). To eliminate them, a stabilized version is available by adding the keyword stab (or any word starting with stab) 1The stabilized version is not supported in versions <= 9.1.5.. This model is based on the so-called ‘oscillating growth’ criterion presented in [M24]. It is obtained by a bifurcation analysis on the transfer matrix relating the perturbation of internal variables (cumulative plastic strain \(p\), and \(t_a\)) to the perturbation of their rates. The oscillating growth criterion is given by

\[S+w\Theta=0\]

where

\[\begin{split}\begin{aligned} S&=S_i+S_a\\ S_i &= \dot{p}\dfrac{d R_v}{d\dot{p}}\\ S_a &= -t_a \pder{R_a}{t_a}\\ \Theta &= \dfrac{dR}{dp}+\pder{R_a}{p} \end{aligned}\end{split}\]

with \(R_v(\dot{p})\) is the strain rate (viscosity) dependent term (e.g. for a norton flow rule, \(R_v=K\dot{p}^{1/n}\)), \(R(p)\) is the plastic strain hardening function (without DSA term), and \(R_a\) is the strain ageing hardening term.

The evolution equation is modified as

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

where

\[\begin{split}w_s = \begin{cases} & -\dfrac{S_i+S_a}{\Theta} \qquad \text{(stabilized)}\\ & w \qquad \text{(non stabilized)} \end{cases}\end{split}\]

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 lagrange_rotate
 **elasticity isotropic
   young 188911.
   poisson 0.3
 **potential gen_evp ev
  *criterion hosford a 8.
  *flow hyperbolic_simple
   K  5.67
   eps0 1e-08
   cutoff 709.
  *isotropic isotropic_sum
   *-nonlinear_sum
    R0 684.0
    Q1 256.0
    b1 215.8
    Q2 409.4
    b2 7.37
   *-isotropic_dsa
    P1 650.
    beta 0.33
    w 0.005
    t0 2.
    stab
***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

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

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

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

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

\[C(p)= {\tt cm1}+ {\tt cm2} p\]

A stabilized version is available by adding the keyword stab 2The stabilized version is not supported in versions <= 9.1.5..

Warning

  • isotropic_dsa_p can only be used as a brick within a gen_evp potential.

  • The stabilized version of isotropic_dsa_p 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_p
    w  0.0004
    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 \(R_a\)).

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

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 as a brick within a gen_evp potential.

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

  • The stabilization is not implemented for this model.

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
    beta  0.33
    alphap  0.0
    P1  1.
    P2  1.15
    cm1  102.
    cm2  1.
***return