In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, homogenization is a method of studying
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
with rapidly oscillating coefficients,
such as
:
where
is a very small parameter and
is a 1-periodic coefficient:
,
.
It turns out that the study of these equations is 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 concept which is used in
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
. Under this assumption, materials such as
fluids
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
,
solids
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural ri ...
, etc. can be treated as homogeneous materials and associated with these materials are material properties such as
shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackrel ...
,
elastic moduli
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
, etc.
Frequently, inhomogeneous materials (such as
composite materials
A composite material (also called a composition material or shortened to composite, which is the common name) is a material which is produced from two or more constituent materials. These constituent materials have notably dissimilar chemical or ...
) possess
microstructure
Microstructure is the very small scale structure of a material, defined as the structure of a prepared surface of material as revealed by an optical microscope above 25× magnification. The microstructure of a material (such as metals, polymers ...
and therefore they are subjected to loads or forcings which vary on a length scale which is far bigger than the characteristic length scale of the microstructure. In this situation, one can often replace the equation above with an equation of the form
:
where
is a constant tensor coefficient and is known as the effective property associated with the material in question. It can be explicitly computed as
:
from 1-periodic functions
satisfying:
:
This process of replacing an equation with a highly oscillatory coefficient with one with a homogeneous (uniform) coefficient is known as ''homogenization''. This subject is inextricably linked with the subject of
micromechanics Micromechanics (or, more precisely, micromechanics of materials) is the analysis of composite or heterogeneous materials on the level of the individual constituents that constitute these materials.
Aims of micromechanics of materials
Heterogeneo ...
for this very reason.
In homogenization one equation is replaced by another if
for small enough
, provided
in some appropriate norm as
.
As a result of the above, homogenization can therefore be viewed as an extension of the continuum concept to materials which possess microstructure. The analogue of the differential element in the continuum concept (which contains enough atom, or molecular structure to be representative of that material), is known as the "
Representative Volume Element
In the theory of composite materials, the representative elementary volume (REV) (also called the representative volume element (RVE) or the unit cell) is the smallest volume over which a measurement can be made that will yield a value representati ...
" in homogenization and micromechanics. This element contains enough statistical information about the inhomogeneous medium in order to be representative of the material. Therefore averaging over this element gives an effective property such as
above.
Classical results of homogenization theory
were obtained for media with periodic microstructure modeled by partial differential equations with periodic coefficients. These results were later generalized to spatially homogeneous random media modeled by differential equations with random coefficients which statistical properties are the same at every point in space. In practice, many applications require a more general way of modeling that is neither periodic nor statistically homogeneous. For this end the methods of the homogenization theory have been extended to partial differential equations, which coefficients are neither periodic nor statistically homogeneous (so-called arbitrarily rough coefficients).
The method of asymptotic homogenization
Mathematical homogenization theory dates back to the French, Russian and Italian schools.
The method of asymptotic homogenization proceeds by introducing the fast variable
and posing a formal expansion in
:
:
which generates a hierarchy of problems. The homogenized equation is obtained and the effective coefficients are determined by solving the so-called "cell problems" for the function
.
See also
*
Asymptotic analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as beco ...
*
Γ-convergence
*
Mosco convergence
In mathematical analysis, Mosco convergence is a notion of convergence for functional (mathematics), functionals that is used in nonlinear, nonlinear analysis and set-valued analysis. It is a particular case of Γ-convergence. Mosco convergence is ...
*
Effective medium approximations
In materials science, effective medium approximations (EMA) or effective medium theory (EMT) pertain to analytical or theoretical modeling that describes the macroscopic properties of composite materials. EMAs or EMTs are developed from averagin ...
Notes
References
*
*
*
*
*{{Citation
, last1 = Braides
, first1 = A.
, last2 = Defranceschi
, first2 = A.
, title = Homogenization of Multiple Integrals
, series = Oxford Lecture Series in Mathematics and Its Applications
, place =
Oxford
Oxford () is a city in England. It is the county town and only city of Oxfordshire. In 2020, its population was estimated at 151,584. It is north-west of London, south-east of Birmingham and north-east of Bristol. The city is home to the ...
, publisher =
Clarendon Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
, year = 1998
, isbn = 978-0-198-50246-3
Asymptotic analysis
Partial differential equations