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The Strain Tensor And Its Weird Formula | Dr. Simulate | YouTubeToText
YouTube Transcript: The Strain Tensor And Its Weird Formula
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Summary
Core Theme
The strain tensor is a crucial concept in continuum mechanics that mathematically quantifies material deformation by isolating it from rigid body motion (translation and rotation), which are also captured by the displacement gradient.
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when I saw the formula for the strain
tensor the very first time I found that
it looks unnecessarily complicated why
is it not possible to Simply Define The
Strain tensor as the gradient of the
displacement field well it turns out
that this would result in some
unphysical Behavior take for example
this simulation where I apply some
displacement in X1 direction to the top
of a solid cube if I run the same
simulation but with the wrong definition
for the strain tensor the results look
much less realistic in this video you
[Music]
why continum mechanics studies the
deformation of mattera under external
influences to do so it is necessary to
somehow quantify the deformation of meta
mathematically the displacement field
which is a vector field that describes
the displacement of each point carries
information about the deformation of
mattera however the displacement field
does not only carry information about
deformation it also carries information
about the translation and rotation of
mattera take for example this
displacement field which leads to a
translation of
meta or this displacement field which
leads to a rotation of
meta for both examples The Meta is not
deforming but just moving in space this
is called rigid body motion we observe
that the displacement field is not
really suited to quantify the
deformation of meta to measure the
deformation of meta we desire a physical
quantity that is zero where the mattera
is not deformed and nonzero elsewhere
this quantity is the strain tensor
Epsilon The Strain measures how
displacement is changing in space for
problems in one dimension The Strain is
the derivative of the displacement field
in three dimensions however the
displacement field is a vector field
with three components U1 U2 and u3 which
can each change in space in all three
directions X1 X2 and
X3 this means we can compute a total of
nine derivatives these nine derivatives
Define the gradient of the displacement
field which is denoted by nabla U
recognizing that the strain in one
dimension is the derivative of the
displacement field a seemingly Natural
Choice for defining the strain tensor in
three dimensions would be that the
strain tensor is the gradient of the
displacement field spoiler we shouldn't
do that but let's anyways think this idea
idea
through let's take a look at the
previous displacement field
examples in the case of rigid body
translation the displacement gradient is
z zero great this is what we wanted the
meta is translating in space but it is
not deforming and the strain is
zero but what happens for the other
example for rigid body rotation the
displacement gradient is non zero this
means that the displacement gradient
does not only contain information about
the deformation of meta but also
information about the rotation of
mattera now the following question iises
can we take the displacement gradient
and somehow remove all contributions
that belong to
rotation if this would be possible then
all we would have left would be a good
measure of
deformation guess what it is possible to
do that all we need is a short mathematical
interlude a tensor a can be additively
decomposed into two parts a symmetric
and a school symmetric part let's assume
that a is a 3X 31 or here the formula
for the symmetric part is 12 * a + a
transposed if we expand this formula we
can see why this is called the symmetric
part of a the upper right and lower left
off diagonal elements of the symmetric
part are
equal the formula for the school
symmetric part is 12 * a minus a
transposed again expanding the formula
we can see why this is called the school
symmetric part of a the upper right and
lower left of diagonal elements of the
school symmetric part are equal in
magnitude but change in
sign if you like pause the video and
quickly verify that the symmetric plus
a
let's go back to the displacement
gradient we have seen that the rotation
of meta resulted in a nonzero displacement
displacement
gradient now take a look at the
following no matter how The Meta is
rotated the displacement gradient always
pattern it is always goo
symmetric it turns out that the rotation
of mattera is directly linked to the SK
symmetric part of the displacement
gradient this means by subtracting the
skoo symmetric part of the displacement
gradient we can remove all contributions
that arise from rotations so let's do it
let's write down the displacement
gradient minus the SCH symmetric part of
it after some rearrangements we get this
here we have it this is the formula for
the strain tensor now we know why this
formula looks strange at the first
clance the strain tensor is the
displacement gradient minus all
contributions that arise from
rotations what is left is the symmetric
part of the displacement
gradient I personally find this
derivation super intriguing because it
once more shows that every formula that
we use in physics does not come from
nowhere there's always a reason for a
formula looking kind of weird and in
this case there's no black magic in
defining the strain tensor all we needed
was a measure for the deformation of
mattera the displacement gradient was a
good choice to start with but it also
included rotation which we had to remove
to arrive at the proper measure of
deformation The Strain
tensor let's finally go back to the
simulation that I've have shown in the
very beginning of the video here you can
see the boundary value problem that I
solved numerically for simulating the
deformation of the solid block under
Shear if we changed the definition for
the strain t tensor such that it equals
the displacement gradient that is we
include rotation into the strain tensor
then we get a much less realistic
behavior of the solid the reason for
this is that now the rotation of meta
would also contribute to the stress in
The Meta therefore we get an
unrealistically stiff meta so we should
stick to the correct definition of the strain
strain
tensor at the end there's one important
thing that must be mentioned about the
strain tensor The Strain tensor that has
been discussed in this video is only
suitable for measuring deformations that
are very small this strain tensor is
therefore called the small strain tensor
or infinitesimal strain tensor all
displacements and strains shown in this
video have been Amplified for
illustration purposes but note that the
small strain tensor should only be used
for strains that do not exceed one or
maximum 2% for larger deformations
different tensors are required to
correctly quantify The Strain but this
we will have to postpone to future
videos on nonlinear Continuum
mechanics that's it for this video in an
upcoming video in the series of videos
on continer mechanics I will show you
how the strain tensor and its individual
elements can be visualized stay tuned
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