[Music]
we looked at the
motivation for using hyperelastic
materials
and how they differ from metals now
we'll see how one can capitalize on
their key characteristics
and develop solution methods for
capturing their response
to recap two of the key characteristics
they undergo
large elastic deformations with very
little to no permanent deformation
also all the work done in deforming them
is stored as internal energy and is
fully recovered
upon unloading
this makes the deformation of
hyperelastic materials a
thermodynamically reversible process
so their behavior follows the second law
of thermodynamics
now let's shift our focus on how to
represent
this behavior in the form of a
mathematical equation
as noted earlier the total work done in
deforming this material
is stored as internal energy and all of
it is recovered upon unloading
so the area under the stress strain
curve
which is nothing but the strain energy
stored in the material
when it is subjected to a certain strain
remains constant
as long as the strain in the material is
held constant
due to this reason the strain energy
function
is a perfect choice to be a hyperelastic
material model
this energy function is usually defined
as a function of
strain tensor in the material
and the stress developed in the body at
any given strain
is calculated from this strain energy
function
the strain energy function is often
expressed
as an additive split of the deviatoric
and the volumetric strain energies
it's important to note that the stress
developed in the body
depends on the type of strain that is
present in it
for instance the stress developed in the
part
at a tensile strain of 20 is different
from
the stress developed in the same body at
a shear strain of 20
so it's important to note the mode of
deformation that the body is
experiencing
in general there are three different
modes of deformation
that can characterize the deviatoric
behavior of hyperelastic materials
namely uniaxial tension
uniaxial compression and shear
deformation
as you can see from this chart the value
of stress
developed in the part at the same amount
of strain
differs depending on the type of
deformation
so how do we calculate the stress from
the energy function
and how do we differentiate between
different types of strain
to understand this we must know
what the strain tensor looks like for
each of this deformation cases
this can be done in two ways
the first method uses principle
stretches
let's start with this cube with the
sides of size
l is being stretched in one direction
when it is stretched to a size l y the
cube will shrink
in the other two directions to conserve
the volume of the component
due to incompressibility
the ratios of the final length of the
cube
to its initial length are called as the
principal stretches
physically they tell us to what extent
and in what direction did the material
change its dimensions
during deformation
for instance at zero deformation state
both the initial and final lengths of
the cube are the same
so the stretch ratios are equal to one
if the part is elongated in one
direction
its final length is going to be more
than the initial length
so a stretch ratio that is more than one
tells you that it is tension
similarly when a part is compressed in
one direction
its final length is going to be smaller
than its initial length
and hence the stretch ratio is less than
one
and since these stretches are defined in
the principal directions
there are no shear deformations in this
direction
if we measure the changes in the
dimensions in the three directions as
delta x delta y and delta z
then we can write them in this form and
simplifying it
shows us that stretch is nothing but one
plus
engineering strain
this way one can relate any type of mode
of deformation
in the form of principle stretches and
use them as a way of representing the
strain tensor
principle stretches are not the only way
of representing the strain tensor for
all the modes of deformation
another very popular way of doing this
is by using the strain invariants
the strain invariants as the name
suggests
do not vary as the reference coordinate
system changes
so whether a body is undergoing tensile
or shear or combined loading its state
of strain
can be described using strain invariants
using one of the two methods discussed
above one can distinguish
between different modes of deformation
due to this reason they are commonly
used in defining hyperelastic material
models
let's look at some of these commonly
used models
to show you how the hyperelastic models
are formulated
this is neohukian model the simplest
form of hyperelastic material model
this function uses a linear relation
between the strain energy potential
and the first strain invariant
note that although the equation is
linear
invariant itself is a non-linear
function of string
so this is still a non-linear model
it has only two material constants the
initial shear modulus
mu so it has a units of stress
and the incompressibility constant which
is nothing but
2 by initial bulk modulus
note that when two different materials
are both
modeled using neo who can model the
values of mu
and d for both the cases will be
different
and this is how we can differentiate
between different materials
now let's take a look at another form of
equation
which is slightly more complicated
the monitoring model is another
commonly used form of hyperelastic
models
as you can see it involves more
invariants
and more material constants
due to its mathematical nature it's
capable of
modelling material behavior that's
slightly more non-linear
once again the material constants are
related to
the initial shear modulus of the
material by this equation
and yet again the incompressibility
parameter
is related to the initial bulk modulus
as
two by d
the moonee rivlin model that we see over
here
is just one form of it in fact
you can add more terms to it to have
five terms and even a nine
term one eleven model
they come in handy when the material is
highly nonlinear
and may require more polynomial terms to
capture the behavior
the ogden model is another commonly used
form
and this one is defined using principal
stretches
it can be formulated by one or more
terms
usually the non-linearity of the model
increases as we use more terms
the derivative component of the model
has two material constants per term
and the volumetric term uses one
constant per term
the initial shear modulus of the
material is nothing but
the sum of the material constants mu
from each term
and alpha is just a dimensionless
non-linearity term
d once again is nothing but 2 by
initial bulk modulus
earlier i had mentioned that we'll focus
on fully or
nearly incompressible hyper-elastic
materials in this section
but let's look at a hyper elastic model
that is used
specifically for modeling compressible
foam materials
just to get an idea of the form
this is the ogden 4 model which is used
to model
compressible foam materials
its formulation looks very similar to
the ogden model
but you can notice that both the
deviatoric and the volumetric terms
are tightly coupled this
makes the model suitable for capturing
the deformation
but the volumetric deformation is not
negligible compared to the deviatoric
deformation
this is just one example of such a model
with tight coupling between the
deviatoric and volumetric terms
in fact there are several other forms
that are available to model such
materials depending on how non-linear
they are
or even whether they are incompressible
such as rubbers
or compressible such as foams
in most of the forms that we have seen
we notice that some of the models such
as
ogden or moon even can have multiple
terms in it
but how do we decide when and how many
terms
are we supposed to use in modeling a
material
to answer this let's look at what these
terms actually do
let's take this response as example
say the first term has a convex upward
response
where the material starts with a stiff
response and reduces at higher strength
and the second term has a concave
upwards response
where the material stiffness increases
with the string
so the combination of these two mid
terms
will look like this as
one model is more dominating at the
start we can see that the material
starts off with a stiffer response
then the stiffness reduces for a bit
followed by
an increase again in the stiffness
most of the elastomers that have such
behavior have
very complex microstructure with
multiple polymer chains
acting as load bearing components
and each term in the model represents
one such chain structure
so in most cases when you see a response
like this
it indicates that we'll need to pick a
multiple term model
to represent its behavior
before we conclude this lesson let's
have a short discussion
on the incompressibility term
in case of incompressible forms a proper
usage
of the incompressibility term is very
important
this term is related to the initial bulk
modulus of the material
and based on this definition as the
value of d
decreases the material tends to be more
incompressible
this term is usually positive and does
not have
a range but a value of 0 indicates that
the material
is fully incompressible
in some cases if we need to introduce a
small amount
of compressibility to make it nearly
incompressible
we need to pick a very small value of d
one recommendation is to set the initial
bulk modulus
to a few orders of magnitude about 100
to 1000 times
higher than the initial shear modulus
and calculate the value of d
accordingly
you