User:Wparnell/Homogenization (physics and engineering)

Homogenization can be defined in many ways.

In the mathematics community it would be defined as the study of partial differential equations with rapidly oscillating coefficients, such as

$$\frac{\partial}{\partial x}\left(A\left(\frac{x}{\epsilon}\right)\frac{\partial u}{\partial x}\right) = 0$$

where $$\epsilon$$ is a very small parameter.

In fact it turns out that the study of these equations are also of great importance in physics and engineering since equations of this type govern the physics of inhomogeneous or heterogeneous materials. Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as homogeneous. A good example is the continuum assumption which is used in continuum mechanics. Under this assumption materials such as fluids, elastic solids, etc. can be treated as homogeneous materials and associated with these materials are material properties such as shear modulus, elastic moduli, etc.

Frequently, inhomogeneous materials (such as composite materials) possess microstructure and therefore they are subjected to loads or forcings which vary on a lengthscale which is far bigger than the characteristic lengthscale of the microstructure. In this situation, one can often replace the equation above with an equation of the form

$$\frac{\partial}{\partial x}\left(A^*\frac{\partial u}{\partial x}\right) = 0$$

where $$A^*$$ is known as the effective property associated with the material in question.