ISOTROPIC#

Description#

This object class defines models of isotropic hardening for use in a variety of material behaviors and potentials. Isotropic hardening causes an isotropic expansion or contraction of the yield surface in tensorial stress space. The isotropic hardening may be a function of the simulated multiplier which is normally equal to the cumulated plastic strain equivalent \(p\), or in terms of its own internal variable. The latter must be used for cases of static recovery.

The initial value \(R_0\) for the yield radius is noted here to be the onset of plasticity, and not the engineering yield stress. It may thus be found to be significantly lower than expected.

Syntax#

The syntax to specify a isotropic hardening object will consist of giving the keyword for a particular law desired, followed by a list of appropriate coefficients which are dependent on the model chosen. The possible isotropic laws are the following:

CODE

DESCRIPTION

constant

initial value only (perfect plasticity).

linear

(or iso_linear) linear function of \(p\).

power_law

power law of \(p\).

by_point

hardening defined by a list of experimental points in an external datafile.

iso_table

hardening defined by a table of points.

linear_pp

linear hardening followed by perfect plasticity.

nonlinear

nonlinear, exponentially saturating function of \(p\).

nonlinear_v1

nonlinear with an internal variable and coefs 1.

nonlinear_v2

nonlinear with an internal variable and coefs 2 (equivalent to nonlinear).

nonlinear_bsi

nonlinear with kinematic interaction (for use with the mises_2m1c potential only).

linear_nonlinear

combined linear and nonlinear saturating hardening.

nonlinear_sum

multiple term nonlinear hardening.

nonlinear_double

multiple term nonlinear hardening. This lawa is rigorously equal to nonlinear_sum, but is retained for backwards compatibility. Although the name suggests that only two terms are allowed, more can be added.

nonlinear_recovery

function

hardening according to a user-defined function of \(p\).

Only the laws with an internal variable may have static recovery and be used in state interactions. Isotropic hardening is available with positive values of the Q or H parameters (depending on the model), while softening occurs with negative values of these coefficients. The saturation rate coefficients b are necessarily positive. All parameters may have the normal dependencies through the COEFFICIENT mechanism.

constant

This gives the radius as a single coefficient R0.

linear

(or iso_linear) Law with initial radius and a linear evolution depending on the cumulated inelastic strain \(p\) (i.e. \(R={\tt R0} + {\tt H}p\)) and using the coefficients R0 and H.

power_law

Gives a power-law evolution according to

(428)#\[R = R_0 + K \left( e_0 + p \right) ^{n}\]

with the coefficients R0, K, e0, n, and \(p\) the cumulated inelastic strain. Negative values for e0 are set to zero. The case \(n<1\) with \(e_0=0\) gives an infinite derivative at the onset of inelastic deformation and is therefore not allowed. Use a very small value for e0 instead.

by_point

Gives the isotropic curve as input by a list of points (see the example). The first column should be sigeq and the second the cumulated plastic strain equivalent.

iso_table

Gives the isotropic curve as input by a list of points directly in a table (see example). When the current value of \(p\) depasses the last value specified in the table, the isotropic hardening no longer changes, as if the isotropic hardening saturates.

linear_pp

linear hardening with an initial radius R0 and slope H up to Ru (i.e. \(R={\tt R0} + {\tt H}p\) for \(R<{\tt Ru}\)), followed by perfect plasticy at \(R={\tt Ru}\).

nonlinear

This law gives a nonlinear saturating evolution:

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

where the saturation radius will be \({\tt R0}+{\tt Q}\) for large \(p\). The law will reach \(95\%\) of the saturation value at \(p\approx 3.0/{\tt b}\) so that higher values of b give a more rapid saturation.

linear_nonlinear

linear hardening with an initial radius R0 followed by a combined linear and nonlinear evolution with coefficients H, Q and b, according to

(430)#\[R = {\tt R0} + {\tt H}p + {\tt Q}\left(1-e^{-{\tt b}p}\right).\]
nonlinear_sum or nonlinear_double

This law allows fine tuning of the isotropic hardening by use of multiple-term evolution. For instance, two terms might be combined analogous to short and long term mechanisms. There is no upper limit for the number of terms N. The radius is calculated as:

(431)#\[R = R_0 + \sum_{i}^{N} Q_i \left(1 - e^{-b_i p}\right)\]

with coefficients R0, Q1, Q2, ..., QN, and b1, b2, ..., bN.

nonlinear_1
(432)#\[m_{iso} = 1 - \frac{R-{\tt R0}}{\tt Q}\]

Example#

An example of a three-term nonlinear hardening with a very rapid initial hardening, a very slow hardening (third term) and an intermediate softening (negative Q2). Note: these are not necessarily reasonable values.

*isotropic nonlinear_sum
 R0    50.
 Q1    30.  b1  2000.
 Q2   -10.  b2    20.
 Q3    40.  b3     2.

The next example shows the use of *isotropic by_point. Between two data points, interpolation is linear.

**potential gen_evp ep
 *criterion mises
 *flow plasticity
 *isotropic by_point
  sigeq   epcum
  130.0   0.0000
  140.0   0.0001
  145.0   0.0002
  150.0   0.0004
  160.0   0.0009
  170.0   0.0017
  180.0   0.0028
The following gives a point-by-point specification of an isotropic hardening for an anisothermal case. Between two data points, interpolation is linear (as with all Z-set tables). For \(p>0.1\) it uses the values of 0.1 because that was the last specified. Two alternative data inputs are possible, using *isotropic by_point or *isotropic table2 syntax.
The examples are provided below :
*isotropic by_point
 sigeq   epcum   temperature
 220.    0.0     1000.
 200.    0.0     1200.
 230.    0.1     1000.
 210.    0.1     1200.

*isotropic table2
 temperature 1000.
  0.0 220.
  0.1 230.
 temperature 1200.
  0.0 180.
  0.1 190.