<ISOTROPIC> nonlinear_with_memory

Description

This model can be used to add a memory effect in the nonlinear isotropic equation that depends on the maximum plastic strain amplitude detected during the loading [M8]. To keep track of the plastic strain amplitude we adds 2 new material integrated variables: the center \(\ten{\xi}\) and the radius \(q\) of of the plastic strain embedding surface \(F\):

\[F~=~\frac{2}{3}~J\left(\ten{\varepsilon}_p-\ten{\xi}\right)~-q~=~0\]

Denoting by \(\ten{n}^*\) the normal to F and \(\ten{n}\) the normal to the the yield surface:

\[\ten{n}^* = \sqrt{3/2} \frac{\ten{\varepsilon}_{p} - \ten{\xi}}{J\left(\ten{\varepsilon}_{p} - \ten{\xi}\right)} ~~~~,~~~~\ten{n} = \frac{3}{2~J\left(\ten{s}\right)}~\ten{s}\]

with \(\ten{s}\) the effective stress, we get the following evolution equation for \(\ten{\xi}\) and \(q\):

\[\dot{\ten \xi} = \dfrac{1}{2} (\ten{n}^*:\ten{\dot{\varepsilon}_{p}})~\ten{n}^*~~~~,~~~~\dot{q} = \dot{p}\dfrac{\ten n:\ten n^*}{2 |\ten n:\ten n^*|}\]

Then in the classical nonlinear isotropic hardening equation:

\[R = {\tt R0} + Q \left( 1 - e^{- {\tt b} p} \right)\]

\(Q\) is no longer a coefficient, but now depends on the size of the plastic strain enveloppe \(F\) according to the following equation

\[Q = {\tt Qsat} - ({\tt Qsat}-{\tt Q0}) ~e^{-2 {\tt \mu} q}\]

with 3 new material coefficients \({\tt Q0}\), \({\tt Qsat}\) and \({\tt \mu}\).

Example

***behavior gen_evp
 **elasticity isotropic
      young 260000.
      poisson 0.3
 **potential gen_evp ev
  *criterion mises
  *isotropic nonlinear_with_memory
   R0    50.0
   b     25.
   Q0    100.
   Qsat  400.
   mu    50.
***return

See also the example given for the behavior memory.