<FLOW> multiplicative

Description

This model allows to introduce various types of (isotropic) hardening mechanisms via the K coefficient of the norton or hyperbolic viscoplastic flow rules:

\[\dot{\lambda} = <f/K>^n ~\text{or}~ \dot{\lambda} = {\tt eps0}\left[ \sinh<f/{\tt K}>^{\tt n} \right]^{\tt m}\]

In this case the K value in the previous equations will be changing according to equations similar to the ones usually defined for the yield radius in <ISOTROPIC> models.

Syntax

The syntax is the following:

*flow multiplicative \(~\,\) *-flow <FLOW> \(~\,\) *-isotropic <ISOTROPIC>

where:

<FLOW>

is a norton or hyperbolic type

<ISOTROPIC>

is an isotropic type of model. All types are allowed, including isotropic_sum, thus allowing an almost unlimited number of K evolution equations.

Example

The example below uses a norton law as the internal flow rule and a nonlinear isotropic to define the K coefficient evolution.

***behavior gen_evp
**elasticity isotropic
     young 260000.
     poisson 0.3
**potential gen_evp ev
 *criterion mises
 *flow multiplicative
  *-flow norton
     K 140.0
     n  5.0
  *-isotropic nonlinear
     R0 0.0
     Q 100.0
     b 0.1
 *isotropic constant
     R0 130.0
***return

In the above model K will depend on the cumulative plastic strain \(\lambda\) according to the following equation:

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

where K is the value given in the *flow norton definition, and R0, Q, b the *isotropic nonlinear coefficients. The actual equation used for \(K(p)\) is therefore:

\[K(p) = 140 + 100\left(1 - e^{-0.1 p}\right)\]