FE Update: Simulating Plastics in Drop and Crash Tests
12/11/2008
by
Hubert Lobo If you want a crash simulation involving
plastics to yield useful results, it is important to model the material
behavior appropriately. The high strain rates have a significant effect
on the properties and failure can be ductile or brittle in nature,
depending on a number of factors And although the LS-Dyna solver is
mentioned and used frequently, the ideas here are applicable to other
FEA software as well. A few fundamentals
Polymers are complex materials with mechanical properties that vary with
stress level, time (rate), temperature, and other parameters. This means
plastics perform in a nonlinear way that is not easily captured by
conventional material models with roots in metals theory. Consider just
two effects. Dependency of the stress-strain relation on
stress level. It is unique for plastics. Hyperelastic materials
(elastomers) have highly nonlinear elastic behavior but show no
plasticity. Metals, on the other hand, show a highly linear elastic
behavior, with plasticity becoming relevant only after yielding. But
the stress-strain behavior of plastics is neither hyperelastic nor
linear. Contrary to metals, plastic strain begins prior to yield. In
addition, the elastic behavior is nonlinear. Trying to model this
behavior using metals theory poorly approximates the actual behavior and
leads to several compromises. For instance, trying to accurately predict
the onset of true plastic behavior under predicts material stiffness at
low stresses. Attempting to be true to the material’s elastic modulus
predicts too much plastic strain as one is forced to assume the onset of
plastic strain much before it actually occurs. And in the second effect.
The rate-dependent behavior of a polymer brings
additional complications. Up to the vicinity of yield, some plastics
exhibit significant rate-dependency of modulus while others do not. This
contrasts with metal behavior in which the expected behavioral trend is
toward no dependency of modulus with strain rate, as exemplified by the
frequently used MAT24 material model in LS-Dyna. (Other FEA programs
feature a similar material model) As a consequence, polymers with a
modulus rate dependency cannot be described by a MAT24 model. Applying
this model to polymers ends up in a significant error in stiffness
predictions. Nonetheless, it is possible to conduct meaningful
simulations by selecting models that closely match the behavior shown by
material data. Many plastics show a remarkable consistency with
respect to rate dependency. An idea gaining wider acceptance is
described in the Eyring equation, which describes a linearly increasing
relationship between yield stress versus log strain rate. In contrast,
the Cowper-Symonds equation, used extensively for metals and implemented
in MAT24, does not capture the behavior of plastics, which leads to
inaccuracy in modeling-rate dependencies.
Another problem arises with fiber-reinforced plastics. In addition to
increasing stiffness, fibers also change how the plastic fails. With
such materials, failure often changes from ductile to brittle. Finally,
with some plastics, an increase in strain rate causes a gradual change
from ductile to brittle failure. This variation in post yield behavior
with strain rate is not easily captured in available material models.
The problem
LS-Dyna’s MAT24 is the one of the most widespread material models. It is
used to simulate tests such as, crash, drop, and other rate-dependent
phenomena. It’s simplest and most commonly used capability couples a
Cowper-Symonds equation (it describes the change of yield stress with
strain rate) with an elasticplastic curve this way: The elastic
rate-independent region occurs up to an arbitrarily or otherwise
determined yield point, beyond which the stress-strain curve at the
lowest strain rate of interest is described by an elastic-plastic model.
This produces a curve of stress versus plastic strain — the plasticity
curve. The left of the accompanying image, How strain rate affects
modulus, shows the classic yield formation often seen in metals but not
easily identified in plastics. The accuracy of the classic (metal) model
depends on three conditions: The stress-strain relationship is linear up
to the chosen yield point, the initial linearity is not rate dependent,
and the shape of the plasticity curve is uniform and independent of
strain rate. But for most plastics, these are simply not true.
Plastics should be modeled with a nonlinear elastic region followed
by an elastic-plastic period, so the location for the transition is
usually identified somewhere along the increasing part of the
stress-strain curve to indicate the onset of plastic strain. However,
this not possible with current crash material models. Compromises are
needed.
Selecting a modulus based on the initial region for MAT24 shows
fidelity to the linear-elastic region and results in predicting too much
plastic strain because the material is still elastic at stresses far
exceeding the “linear-elastic” region. On the other hand, using a secant
modulus (the slope of a line drawn from the graph origin to the plastic
point) to describe behavior up to the plastic point results in a
material model that predicts too little stiffness in the elastic region.
There is no recourse with MAT24 other than to choose a value for the
elastic modulus that locates in a plastic point somewhere between these
extremes, often leaning toward the initial linear region so as to be as
close as possible to the stress-strain data.
After assigning a modulus, it is a simple matter to discrete the
static stress strain and convert the data into plastic strains following
elastic-plastic-model rules. For instance, pick a series of points on
the stress strain curve and use the modulus value in an equation to
convert total strain into plastic strain, generating a plasticity curve.
Applying the Cowper-Symonds equation allows scaling this plasticity
curve to other strain rates. The equation allows smooth extrapolation
without limits. However, because the equation cannot describe the rate
dependency of the yield phenomenon, it does not accurately scale a
plastic’s rate dependency. One solution (called the LCSR option) allows
applying a table containing a scale factor for each strain rate. This
option allows fidelity to test data because it does not depend on an
equation, but uses actual test data.
However, a serious drawback of MAT24 arises from the fact that with
plastics, failure strains often drop with increasing strain rate. The
model does not accommodate this variation. Instead, the model assumes
that failure strain is constant and independent of strain rate. Failure
in MAT24 is defined as the accumulated plastic strain in an element
reaching a specified failure value. At each time step, if the computed
trial stress lies outside the yield surface (Von Mises), LSDyna scales
the stress back to the yield surface and derives accumulated plastic
strain by using the material model to calculate a corresponding
effective plastic strain (EPS) at the strain rate of the element. If
this accumulated plastic strain exceeds a specified failure value, the
element is removed from the model. The failure value is usually chosen
by the analyst as largest failure strain in the material data. This is a
conservative approach. If the data shows a variation in failure strains
with strain rates, analysts must review the strain-rate experienced by
the part, to assign a value at that corresponding strain rate.
Another option in MAT24, LCSS, is useful when the shape of the
plasticity curve changes with strain rate, a phenomenon seen in some
plastics. In this case, submitting a plasticity curve for each strain
rate lets users describe stress-strain behavior as a function of strain
rate. It may still be a useful exercise to smooth the rate dependency
using the approach outlined earlier. However, LCSS offers no relief in
modeling ductile-brittle transitions because of the limitation of the
failure criteria, which allows specifying only one failure strain,
rather than varying failure strain with strain rate.
LCSS requires extrapolating all plasticity curves to the largest
failure strain for the model. Consequently, simulation loses information
regarding the change in failure strain with strain rate.
Polymers such as polycarbonate, polyethylene, and polypropylene
exhibit long tails of post yield strain and can absorb significant
energy in this phase of deformation. Stress-strain curves for nonbrittle
plastics go through an inflection or local maximum commonly referred to
as the yield point. Do not confuse this with the Von Mises yield which
corresponds to the onset of plastic deformation.
Complications arise when handling post yield behavior. For example,
most post yield behavior is accompanied by necking, a localized
nonuniform deformation in which the cross section of the deformation
zone is unknown. Consequently, stress is also unknown and only crudely
estimated by making assumptions about the cross section. The most common
assumes that the true stress calculation applies in this region as well,
which means the slope of the stress-strain curve gradually increases
with increasing strain.
In the case of olefin-based materials, such as polypropylene and
polyethylene, necking is more closely equated with unraveling of the
dendrite structure, so it is more likely that stress remains constant
during necking. In any case, to model these regions using MAT24, it is
essential to eliminate negative slopes in the model.
A number of fiber-filled plastics have rate dependent modulus
followed by small strains to failure. A small plastic strain accumulates
in the material prior to failure. This behavior is difficult to model
using MAT24 for several reasons. First, the stress-strain curves diverge
almost immediately as seen on the right of How strain rate affects
modulus. Consequently, MAT24 either under predicts the stiffness
at low strain rates or over predicts stiffness at high strain rates,
depending on the choice of the elastic modulus. Although this may
significantly affect simulating most plastics, it is more dramatic for
filled plastics because failure strains are small, typically 2%. Even
though MAT 19, another material model, suffers from being bilinear, it
is better suited than previous models and comes closer to replicating
experimental data. And it can precisely indicate the failure envelope of
a material by using failure strain versus strain-rate dependency. In
addition, the model allows for failure based on tensile plastic strain
only.
MAT89 is an elastic-plastic material model that does not need data
broken into elastic and elastic-plastic regions. The developer of LS-Dyna
recommends it to handle the complex behavior of ductile-brittle
transitions where failure strains can vary anywhere between 100 and 10%
for some plastics. With MAT89, the initial stress-strain curve is
entered as true stress-strain data. LS-Dyna internally checks the slope
of the curve. When the slope falls below the modulus E specified in the
material card, the material is assumed to have yielded. The treatment of
plasticity then follows MAT24, as described earlier. The LCSR scaling of
the stress-strain curve allows scaling this model to high strain rates
in a manner similar to MAT24.
The table of yield stress versus strain rate in the LCSR option is a
better choice for modeling rate-dependency than the Cowper-Symonds
equation for the same reasons described earlier. The key benefit of
MAT89 is a table (called LCFAIL in the software) which lets users enter
failure strains versus strain rate. The feature overcomes the limitation
of MAT24, which restricts its ability to model plastics in which failure
strains change significantly with strain rate.
About the Author
Hebert Lobo
President
Datapoint Labs
Help with this article came from Brian Croop of DatapointLabs and
Suri Bala of Livermore Software
Technology Corp.
Article edited by Leslie Gordon,
Sr Editor, Machine Design
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Article reprinted by permission of Penton Media,
publisher of Machine Design |
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