The solution methodology for these, there have been many solution methodologies,

it's not an easy problem.

But most solution methodologies use some sort of perturbation theory.

In this perturbation theories what one does is expresses the field of interest,

for example, strain field is written as an average perturbation.

And of course, all the average of the perturbation has to go to 0.

That's for the bottom the pop represents a volume average field.

Now it turns out once you write this perturbation

one can express the perturbation in terms of a fourth rank localization tensor

applied on the average strain tensor.

This localization tensor is very important to us in homogenization.

It essentially has all the information we need for homogenization.

Because one can show that the effective tensor can be written in

this form where essentially there's a reference value of the stiffness tensor.

And then there's an average of the perturbation and the stiffness tensor.

And then there's a volume average of the perturbation projected

by the localizations tensor, this is an exact result.

In essence, if you somehow know the localized tensor a, you actually

know the rest of the quantities there and you should be able to compute the Ceff.

Again, this result has not been derived here,

the interrogations are mathematically very dense or detailed.

And then need to be followed through some standard textbooks than this field.

So the challenge of the homogenization theory essentially comes down to finding

this localization tensor or sometimes it's also called qualitization tensor.

Again, the solution to the polarization tensor without going into the derivation,

turns out that one way to get the solution is to use this Green's function method.

And if you use the Green's function method, you get a implicit equation or

expression or a recursive expression.

Because the quantity of interest a is on both sides of the equation,

so it's not easy to explicitly solve for it.

In some way, you are to guess the value of a.

Feed it into this equation, find a nu get, a new value, and

then use it again in an way until you find the solution to this implicit equation.

So that's the theory, once you have the expression for

a, we can put it back in the series expression that we had on the previous

slide and recover this expression.

One way to write this, because it's not a implicit equation,

the equation here is not implicity equation.

One way to write this expression is to keep subfuting it in a recursive manner,

and essentially recover the series.

This is an infinite series, has an infinite set number of terms.

But if the perturbation, the reference medium for

c is selected properly this could be a convergence series

at least in particular problems it could be a convergent series.

If it is a convergent series, we had the advantage that you can truncate it at some

point and have an explicit equation, and then you have an explicit equation.

Now let's look at one of these terms in

because we are interested in using this expression.

So if you look at this term,

the definition of the term looks like that's a volume integral.

One can recast this term in this manner.

The difference between the two expressions is one would call this

a Riemann integral and one would call the other one a Lebesgue integral.