Cauchy strain tensor
Strain quantifies the deformation of a solid object.
If the body has been deformed, e.g. by pulling or bending,
its constituent particles have moved a bit.
Let be the original location of a particle,
and its new location after the deformation.
We can thus define the displacement field :
We restrict ourselves to infinitesimal strain,
where is so tiny that the material’s properties are unchanged,
and a slowly-varying strain,
where the particle’s neighbourhood has been distorted,
but not completely changed.
A key challenge when quantifying deformation
is that we need to somehow exclude movements of the entire body:
for example, you can bend a twig in your hands while walking or dancing,
but we are only interested in the twig’s shape change,
not in your movements.
The above definition of includes both,
so we should be careful how we extract the strain from it.
We use the Eulerian description of deformation,
where the new position is the reference,
and the old position is expressed as a function of :
Let us choose two nearby points in the deformed solid,
and call them and ,
where is a tiny vector pointing from one to the other.
Before the displacement, those points respectively had these positions,
where we define as the “old” version of :
Because the new positions are our reference,
we would like to write without .
To do so, we use the definition of , yielding:
Using the fact that is tiny by definition,
we expand the middle term to first order in :
With this, we can now define the “shift”
as the difference between and like so:
In index notation, we write this expression as follows,
with simply being the partial derivative
with respect to the th coordinate:
Where are called the displacement gradients,
and are just one step away from the desired definition of strain.
Note that these gradients are dimensionless,
so we can more formally define a slowly-varying displacement
as one where .
Now, to solve the problem of macroscopic movements,
we take another tiny vector starting in the same point as .
Here is the trick: if the whole body is uniformly translated or rotated,
the scalar product is unchanged,
but if there is a non-uniform distortion, it changes.
We thus define the scalar product’s difference like so:
Where is the old version of .
Since these vectors are all tiny, we apply the product rule:
It is more informative to switch to index notation here.
Inserting and yields:
At last, we define the Cauchy infinitesimal strain tensor
such that it has as components:
Which allows us to rewrite the shift of the scalar product in the following compact way:
The Cauchy strain tensor is a second-rank tensor,
and can alternatively be expressed like so:
Where is the transpose. Being defined from the scalar product,
all macroscopic movements of the body are removed from the tensor,
which turns out to make it symmetric, i.e. .
So far we have used Cartesian coordinates,
but we can choose any three vectors , and ,
and project onto this basis.
For example, the component then becomes:
And so forth, for the other eight components.
The basis in which is diagonal is the one formed by its eigenvectors,
and their directions are the principal axes of strain
at that point in the solid.
Because is symmetric, such a basis always exists.
Given a vector , its relative length change
due to the deformation is simply given by:
To find the angle change
between two vectors and ,
we start with the product rule:
We isolate this for , using the fact that
thanks to the projection :
By recognizing the length change ,
we arrive at the following expression:
Now, everything so far has been about tiny vectors,
so the change of the line element
is easy to express using the displacement field :
Next, we calculate the change of the differential volume element
by treating it as the volume of a tiny parallelepiped
spanned by , and :
We can reorder the factors like so
(write it out in index notation if you are not convinced):
By applying a couple of vector identities,
we can rewrite this more compactly as follows:
Here, we recognize the definition of ,
leading to the following infinitesimal volume change:
Finally, for the surface element ,
we use that the volume element :
By comparing this to the previous result for ,
we arrive at the following equation:
Since is dot-multiplied at the front of each term,
we remove it, and isolate the rest for :
- B. Lautrup,
Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition,