Is using periodic boundary condition in homogenization theory at microscopic scale for identical micro-structure unit cell a good approximation or not?
What are the other techniques that we can apply?
If the mechanical properties of your heterogeneous material vary in a periodic manner in your structure, then periodic boundary conditions are the consistent conditions to use on the unit cell.
If your properties do not vary periodically, then you have to identify a RVE and you have three different ways to impose boundary conditions :
- uniform stresses
- uniform strains
- periodic conditions
The first two will yield respectively a lower bound and an upper bound on your homogenized properties. The periodic conditions do not furnsih any bounds but will yield properties which are between the bounds obtained by uniform stresses or strains.
When the size of your RVE gets bigger all methods should yield the same value. It is generally observed that periodic BCs converge more rapidly to the homogenized value with the size of the RVE than the other two types of BCs (depending if you have a strong contrast between your different materials).
So it depends if you are interested in computing lower/upper bounds or just a good estimate.
I hope this will help.
If the mechanical properties of your heterogeneous material vary in a periodic manner in your structure, then periodic boundary conditions are the consistent conditions to use on the unit cell.
If your properties do not vary periodically, then you have to identify a RVE and you have three different ways to impose boundary conditions :
- uniform stresses
- uniform strains
- periodic conditions
The first two will yield respectively a lower bound and an upper bound on your homogenized properties. The periodic conditions do not furnsih any bounds but will yield properties which are between the bounds obtained by uniform stresses or strains.
When the size of your RVE gets bigger all methods should yield the same value. It is generally observed that periodic BCs converge more rapidly to the homogenized value with the size of the RVE than the other two types of BCs (depending if you have a strong contrast between your different materials).
So it depends if you are interested in computing lower/upper bounds or just a good estimate.
I hope this will help.
Yes, if media has periodic structure. But even for periodic structure you can use alternative techniques
The periodic boundary condition in homogenization theory was demonstrated and I dont see any reason to the contrary.
Concerning the other methods I remember to see many years ago a comparison with mechanical of materials methods with homogenization which address that problem. I think it has a publication from Noboru Kikuchi and some of their students and/or colleague (but after so many years may be this information is not correct). Any way with time I am sure you will find that studies.
Jeremy's answer is perfect.
Look at papers from Eshelby, Hashin-Strickman, Mori-Tanaka;
also, Jianmin Qu and Mohammed Cherkaui have written a book which includes the information on the bounds titled Fundamentals of micromechanics of solids.
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