View a brief overview of Multicontinuum Theory.
The fundamental premise underlying continuum mechanics is that any physical quantity of interest can be evaluated at a material point by averaging the quantity over a representative volume element that surrounds the point of interest. The size of this representative volume element (or RVE) must be very small compared to the overall physical dimensions of the material body, yet large enough to provide an accurate statistical representation of the quantity within the body's material microstructure. For typical unidirectional, fiber reinforced composites, the fibers tend to have a somewhat random spacing within the matrix material. Therefore, the RVE used to characterize a material point must be large enough to contain numerous fibers to provide an accurate statistical representation of any quantity averaged over the RVE. However, for the sake of computational expediency, it is common practice to assume a uniform fiber distribution that represents the actual random fiber distribution in some statistically meaningful sense. This assumption of uniform fiber spacing permits the RVE to economically be represented by a single unit cell with periodic boundary conditions.
The concept of a multicontinuum simply extends the notion of a continuum to reflect distinctly different materials that coexist within the RVE used to characterize a material point. Such an extension is natural in any case where there are two or more clearly identifiable constituents with drastically different material properties. Hence, a unidirectional fiber-reinforced composite material may be viewed as two interacting continua (a fiber continuum and a matrix continuum) that coexist within a suitably chosen representative volume element of the composite material. In such a multicontinuum representation of a unidirectional fiber-reinforced composite material, there are three different volume averages relevant to the mechanics of the composite material. These three volume averages are described below.
a) Physical quantities of interest are averaged over the entire RVE that represents the composite material. These quantities are traditionally referred to as 'homogenized' composite quantities and represent the overall averages of the physical quantities as they vary over the fiber and matrix constituents of the microstructure within the RVE. Variables used to represent these composite average quantities are given a distinguishing superscript 'c'. In this document, these quantities will be referred to as composite average quantities.
b) Physical quantities of interest are averaged specifically over the fiber continuum within the RVE of the composite material. These quantities are referred to as fiber average quantities. Variables used to represent these fiber average quantities are given a distinguishing superscript 'f'.
c) Physical quantities of interest are averaged specifically over the matrix continuum within the RVE of the composite material. These quantities are referred to as matrix average quantities. Variables used to represent these matrix average quantities are given a distinguishing superscript 'm'.
For woven composite materials, the fiber and matrix volume average quantities are still the fundamental quantities of interest, but there is additional complication because the woven RVE contains a mesostructure (i.e., warp tows, fill tows, and pure matrix pockets) in addition to a microstructure (fiber and matrix). In this case we must distinguish between matrix average quantities that occur in the warp tows, fill tows, and pure matrix pockets. Furthermore, we must distinguish between fiber average quantities that occur in the warp and fill tows. This necessitates that we identify certain intermediate level constituents relevant to the mesostructure of the woven composite, for example the warp tow constituent, the fill tow constituent, and the pure matrix pocket constituent. Volume average quantities must also be computed for these mesoscale constituents.
In traditional continuum mechanics (as applied to fiber-reinforced composite structures), attention is focused on the development of relationships between various composite average quantities (e.g. stress, strain). Multicontinuum Theory (or MCT) augments traditional continuum mechanics by expanding the focus to include two additional issues: 1) the development of relationships between various constituent average quantities of interest, and 2) the development of relationships that link composite average quantities to constituent average quantities [1-12].