FLOW

Description

This object class defines the model for inelastic flow in plastic and viscoplastic models and potentials.

Syntax

The syntax to specify a flow object will consist of giving the keyword for a particular flow desired, followed by a list of appropriate coefficients which are dependent on the model chosen. The flow type may be chosen among the following laws:

CODE	DESCRIPTION
norton	Norton power law
hyperbolic	hyperbolic sin function applied to power law of overstress
strain_hardening	Norton type viscoplastic rate with hardening
gsell	viscoplastic model for some polymers
modified_visco	viscoplastic rate using strain hardening or softening
sellars_tegart	Coupled with hardening to study variable strain rate
function	A function in terms of overstress and cumulated plastic strain
abq_strain_hardening	flow class available through ABAQUS
norton_exp	exponential (saturating) Norton law
double_norton	two term Norton law
flow_sum	summation of Norton terms
flow_sum_inv	summation of inverse Norton rates
inv_exp	viscoplastic according to \(\dot\epsilon=A\exp\left({(p-p_0)/\alpha\sigma^n}\right)\)
interface_control	interface reaction flow
plasticity	time independent plasticity
use global_function	uses a function defined elsewhere
exponential_crystal	physically based exponential flow law esp. useful for crystal deformation

The default type of flow is plasticity.

norton

This law corresponds to the classical Norton creep power law. The coefficients are chosen to normalize the stress term, and then apply the exponent:

\[\dot{\lambda} = <f/K>^n\]

with \(f\) positive. The coefficients K and n must be non-zero.

norton_exp

This flow provides a limit stress approximating creep at low stress levels and plasticity at high stresses:

\[\dot{\lambda} = <f/K>^{n}\exp(\alpha <f/K>^{n+1})\]

which uses the coefficient names: K, n, and alpha.

double_norton

This model is a two term variation of the Norton law in order to model changes in flow mechanisms over a wide stress range:

\[\dot{\lambda} = <f/K>^{n_1} + <f/K_2>^{n_2}\]

with the required coefficients K, n1, K2, n2.

plasticity

This flow type indicates time-independent plasticity (\(f=0\)). The use of plasticity may limit integration to implicit only in the case of some complex models. There are no coefficients.

inv_exp

This law provides an exponential dependence on the inverse of the effective overstress: \(\dot\lambda=A\exp\left(-\frac{p+p_0}{\alpha\sigma^n}\right)\) with the coefficients n, A alpha, and p0, where p0 is optional (default \(0\)).

interface_control

This flow rule is expressed:

\[\dot\lambda= \frac{1}{d^2} \frac{k_1\sigma}{1+ \frac{k_2}{d \sigma^m} }\]

with the coefficients k1, k2, m, d. d represents a grain size.

flow_sum_inv

This model is similar to the flow_sum rule, but sums the inverse of the different flow rules:

\[\dot{\lambda} = \left[ \sum_i \frac{1}{\dot\lambda_i} \right]^{1/2}\]

where the \(\dot{\lambda}_i\) terms are given by the various sub-laws (different than sum_flow and flow_sum_inv).

exponential_crystal

This law is a physically based flow law for crystalline slip which can be used with the crystal POTENTIAL models (actually it can be used anywhere viscoplasticity is allowed).

\[\dot{\gamma} = {\tt gamma0} \exp\left[ - {\tt F0\_RT} \left[ 1 - \left< \frac{f}{K} \right>^{\tt n1} \right]^{\tt n2} \right]\]

Note that the terms for this equation may be composed of function coefficient types in the event that a functional form is desired. For example: F0_RT function  6.8e4/(1.9868*temperature);

strain_hardening

This flow law is a Norton type viscoplastic rate with a hardening effect.

\[\dot{\lambda} = \left<\frac{f}{K}\right>^n (v+v_0)^m\]

The flow law accepts coefficients K, n, m, and v_0. Because when \(v=0\) there will be no flow rate in the absence of \(v_0\) (and therefore \(v\) will never become anything other than null), the coefficient should have a non-zero value (but possibly very small).