<POTENTIAL> gen_evp#

Description#

The potential object of type gen_evp serves as the basic type for classical plasticity and viscoplasticity models with both isotropic variable and an arbitrary number of kinematic hardening variables (c.f. [M26]). The potential will accept a wide variety of criterion types, associated and non-associated, as well as a variety of flow rules (plastic and viscoplastic). The dissipation potential for this model is written generally:

(463)#\[\Omega_i = f_{cr}({{\bf p}},~\ten \sigma-\Sigma{{\ten X}}_i) + \Omega_r(R) + \Sigma\Omega_{\alpha_i}({{\ten X}}_i)\]

If there are hardening variables, they will be stored in the following order:

(464)#\[{\bf h} = [\ten \alpha_1~~\ten \alpha_2~~\ldots~~\ten \alpha_n~~r ]\]

where the tensorial variables \(\ten \alpha_i\) are the kinematic internal variables (analogue to an offset strain), and \(r\) is an internal variable modeling the isotropic expansion or contraction of the yield domain (analogue to an equivalent strain).

Syntax#

The syntax understood by this potential is summarized below:

**potential gen_evp [ name ] [ *flow <FLOW> ] [ *criterion <CRITERION> ] [ *kinematic <KINEMATIC> [ name ] ] [ *isotropic <ISOTROPIC> ] [ *var_coefs ] [ *store_all ]

The option *store_all is used to make all associated force variables to be stored as well as the internal hardening variables. This allows one to observe directly the effective back stresses or isotropic radius even in the case of coupling.

Other statements may be made about this model:

  • The final form of the hardening will be determined by the options *kinematic and *isotropic which are given by the user. If there is incompatibility with one of the hardening mechanisms with this potential, an error message will be output as the invalid calculation is attempted. Static recovery is allowed in both the isotropic and kinematic variables.

  • The type of flow law, \(\dot{\lambda}\), and criterion, \(f\), will be determined by the options *flow and *criterion. Default values for these options are plasticity (time independent flow) and mises respectively.

  • The flow direction is determined by the criterion chosen, which is not necessarily associated.

  • The names of the internal variables will be dependent on the choices given by the user. If no name is given for the potential, the potential name (henceforth referred to as pn will be “e#” with # being the sequential number of the potential in question (i.e. 1 for the first potential, etc). In the absence of names for the kinematic variables (kn), default names will be of the form: “pna_#v” with # being the sequential number of the kinematic variable in the potential’s kinematic list.

With these comments, the internal variables added by a gen_evp potential instance are the following 1This of course means in addition to those created by other potentials and the gen_evp behavior itself.:

prefix

size

description

default

pnvi

T-2

inelastic strain tensor

yes

pncum

S

cumulated value of the multiplier \(\dot{\lambda}\)

yes

kn

T-2

kinematic strain variable

no

Note that the cumulated value of the \(\dot{\lambda}\) multiplier is only sometimes equal to the equivalent cumulated inelastic strain (for Von Mises criterion for example, but not for Hill). The current version of the potential does not allow for calculation of the true equivalent for cases where \(\int\dot{\lambda}\ne p\).

Example#

An example viscoplastic model with Norton flow, Hill type criterion and kinematic hardening is given here as an example using the gen_evp potential. The dissipation potential (flow term only) and criterion may be written as:

(465)#\[\begin{split}\begin{aligned} \Omega &= \left[(\ten \sigma-\ten X):\tenf M:(\ten \sigma-\ten X)\right]^{1/2} -R + \frac{3D}{2C}\ten X:\ten X + (R-R_o)^2/2Q \\ f &= \left[(\ten \sigma-\ten X):\tenf M:(\ten \sigma-\ten X)\right]^{1/2} -R \end{aligned}\end{split}\]

which leads to the evolution rules:

(466)#\[\begin{aligned} &\dot{\ten \varepsilon}_{vp} = \dot{\lambda}\pder{\Omega}{\ten \sigma} = \dot{\lambda}\pder{f}{\ten \sigma} = \dot{\lambda}\frac{2}{3}\frac{\tenf M:(\ten \sigma-\ten X)}{\sigma_{eq}} \end{aligned}\]

with \(\sigma_{eq}=\left[(\ten \sigma-\ten X):\tenf M:(\ten \sigma-\ten X)\right]^{1/2}\).

***behavior gen_evp
 **elasticity isotropic
    young 260000.0
    poisson 0.3
 **potential gen_evp ev
  *flow norton
    n  7.0
    K  400.
  *criterion hill
    hilla 1.     hilld 1.
    hillb 2.     hille 1.
    hillc 3.     hillf 1.
  *kinematic nonlinear  x1
    C  30000.0   m  1.0
    D  500.0     M  20000.0
  *isotropic constant
    R0 130.0
***return