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So far
we have discussed how large deformation
makes a model response non-linear
from a physics standpoint but linear or
non-linear behavior
is a mathematical interpretation of a
physical system
so in this lesson we will learn how
the large deformation is treated
mathematically
while doing so we will also investigate
what mathematical approximations do we
make
when we assume a system to be linear.
This knowledge is important as the
analyst must be aware
of the simplifications that they make so
they are aware of the consequences.
Now let's learn the difference between
the large
and small deformation. Of course
big or small physically there's no
difference but mathematically
we can simplify some class of models.
Before we jump into it let's take a look
at a new term called as the deformation
gradient.
Let's consider a random shaped solid
and let's focus on a point on it.
When you apply some load on this object
it deforms and displaces
and the point now assumes a new position
in space.
We can track the change in positions of
this point
by inspecting its vector in space
in the undeformed state it is in a
vector space capital x
and after deforming it moved to a new
location small x.
So by definition the displacement of
this point
is nothing but the difference of the
initial vector from the final vector.
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If you observe this relation you realize
that it does not tell you
whether the body is simply translated to
the new position
or if it deformed on its way there.
It's important to be able to identify
this as the stresses will be developed
in the object
only when there is a deformation in it
so we define a new term called as the
deformation gradient.
The deformation gradient is the
derivative of the vector
in deformed configuration with respect
to the undeformed configuration.
This definition alone may not convey the
physical significance of this term.
So let's look at the simple example
here the object is simply drifting from
one place to another
this is simple rigid body motion and no
portion of the body
is deforming this means
that the vector x is just a linear
function of
the capital x.
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If you compute the deformation gradient
for this body
it reduces to an identity tensor.
If this was not a rigid body motion and
instead
there are portions of the body that are
stretching
in that case the vector small x is no
longer
a linear function of capital x which
means that the deformation gradient
is not an identity tensor
so clearly we can distinguish between
rigid body motion
and deformation based on the deformation
gradient.
For this reason deformation gradient
will be
the fundamental term that we use in
calculating different measures of
strains.
Now coming back to the discussion on
large strain
let's start by looking at a measure of
strain called as the lagrange strain
there are other measures of strain that
are commonly used
but we'll focus on just the lagrange
strain for now.
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The lagrange strain is defined by this
relation.
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While this advanced math may seem a
little intimidating
it is enough to know that the physical
significance of this term
is that it quantifies the change in
length
of a unit material.
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Since f is written as a function of
displacement field
we'll substitute that relation here and
expand the definition
of large strain once we do that
this is what the individual components
of the lagrange strain tensor look like.
If you observe carefully each component
of the strain tensor
has a linear and a quadratic component
now these relations can be used to
distinguish
between the large and small deformations
when the deformations are very small the
derivatives of the displacements will be
small
and when we square these terms they get
even smaller
to an extent where they are negligible
if we ignore these terms we can reduce
the strain
tensor to just the linear terms so they
can take a linear form.
Comparing these two strains one can
clearly see the fundamental difference
between
the strain tensors using small and the
large deformations formulation.
We ignore the higher order terms in case
of small deformation
which is valid only when the
deformations are small.
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By doing this the equations will remain
linear
and we can use linear solvers to analyze
such systems
which are much less expensive compared
to their non-linear counterparts.
But note that as the deformations
increase
these terms get bigger and may no longer
be neglected
this is the main difference between the
small and large deformations.
If the deformations are small the
quadratic terms are negligible
but once they start increasing the
errors due to simplification
would start growing quadratically