The expressions given below are reliable as long as the state of stress in the
test specimen is homogeneous. When localization (necking) begins, this condition
of homogenity no longer holds.
First of all, make sure your experimental data from a uniaxial tension test
is expressed in terms of true stress vs. true strain, not
engineering stress or strain.
True strain = ln(1 + engineering strain) where ln designates the natural log
True stress = (engineering stress) * exp(true strain)
= (engineering stress) * (1 + engineering strain)
where exp(true strain) is 2.71 raised to the power of (true strain).
In the equations above, tensile values are positive and compressive values are negative.
Be aware that experimental data always includes some degree of error and thus tends
to be somewhat noisy or erratic. When using mat_24, one should input a
smoothed stress-strain curve utilizing a minimal number of points.
Input of noisy experimental data may cause spurious behavior, particularly in the
case of the default, 3-iteration plane stress plasticity algorithm for shells.
Full iterative plasticity can be invoked for shells, at greater expense, for
material models 3, 18, 19, and 24 by setting MITER=2 in *control_shell.
The effective stress values input in defining a stress vs. effective
plastic strain curve in a LS-DYNA plasticity model should be
absolute values of true stress, starting with the value at initial yield.
The effective plastic strain values input in defining a stress vs. effective
plastic strain curve in a LS-DYNA plasticity model should be absolute values
of residual true strains after unloading elastically.
Some adjustment to the effective plastic strains may be necessary to ensure that
effective plastic strain is zero at the initial yield stress. Such an adjustment
is necessary if the elastic portion of the true stress vs. true strain curve is nonlinear.
Calculating effective plastic strains epspl from the uniaxial true stress vs. true strain curve:
1. Make a determination of the point on the true stress vs. true strain curve at which
first yield occurs. Call this point (epsy0, sigy0).
2. Calculate trial epspl = |total true strain| - |true stress|/E
Note that as the stress value increases, the recoverable elastic strain = |true stress|/E
increases as well.
3. Calculate final epspl = trial epspl - (|epsy0| - |sigy0|/E). This adjustment ensures that
epspl is zero when the effective stress = |sigy0| and that your first point in the
effective stress vs. epspl curve is (0,|sigy0|).
In the case where the user elects to input only an initial yield stress
SIGY and the tangent modulus Etan in lieu of a von Mises stress vs. effective
plastic strain curve (in the case of *mat_piecewise_linear_plasticity/mat_024),
Etan = (E * Eh)/(E + Eh)
Eh is the slope of the von Mises stress vs. effective plastic strain curve.
Eh = (E * Etan)/(E - Etan)
Or,described in terms of uniaxial, total true stress vs. total true strain,
Eh = (|sig| - SIGY)/(|eps| - |sig|/E)
where (eps,sig) is a data point taken from the bilinear, total true stress vs. total true strain curve
and sig is > SIGY.
Etan is the slope of the postyield portion of the bilinear, total true stress vs. total true strain curve.
E should not be less than (Etan)max where (Etan)max is the maximum, post-yield slope of the
true stress vs. TOTAL true strain curve. Young's modulus is usually taken as the
pre-yield slope of the true stress vs. total true strain curve but adjustments may need
to be made to ensure than E is not less than (Etan)max.
Actually, this requisite condition of E > Etan is met if a stress vs. epspl curve is given
with the first point corresponding to the initial yield stress and zero plastic strain.
For example, if Eh (slope of stress vs. epspl curve) = 3253 and E were set to a relatively low value,
say 10, Etan is then equal to Eh*E/(Eh + E) = 9.97.
In mat_024, if cards 3 and 4 are used to define the curve and E < max slope,
the job will stop with a fatal error (conservative).
If curve LCSS is defined and E < max slope:
In 971 executables predating 9/8/2006, no warning or error is given.
A fatal error message was put in place in ver. 971 on 9/8/2006.
Later, on 6/25/08, Tobias Erhart replaced the fatal error with a warning. Tobias later remarked
(12/12/12) that he feels this warning message can be ignored. The reason cited is that it is
questionable to compare the Youngs modulus (slope in stress vs. TOTAL strain) with the slope
in yield curve (stress vs. PLASTIC strain). A better comparison would be to compare Youngs modulus
with the tangent modulus, because both "live" in stress vs. total strain, i.e.
E should always be greater than Etan = (E * Epl) / (E + Epl)
but this is always the case, because the limit value for big plastic slopes is:
lim(Epl->inf) {(E*Epl)/(E+Epl)} = E
The warning message is often seen for polymer materials and "we never experienced any problems".
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jpd