In
materials science, a general rule of mixtures is a
weighted mean
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
used to predict various properties of a
composite material
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 ...
.
It provides a theoretical upper- and lower-bound on properties such as the
elastic modulus
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 ...
,
mass density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
,
ultimate tensile strength
Ultimate tensile strength (UTS), often shortened to tensile strength (TS), ultimate strength, or F_\text within equations, is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials ...
,
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
, and
electrical conductivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
.
In general there are two models, one for axial loading (Voigt model),
and one for transverse loading (Reuss model).
In general, for some material property
(often the elastic modulus
), the rule of mixtures states that the overall property in the direction parallel to the fibers may be as high as
:
where
*
is the
volume fraction
In chemistry and fluid mechanics, the volume fraction φ''i'' is defined as the volume of a constituent ''V'i'' divided by the volume of all constituents of the mixture ''V'' prior to mixing:
:\phi_i = \frac
Being dimensionless, its unit is ...
of the fibers
*
is the material property of the fibers
*
is the material property of the matrix
It is a common mistake to believe that this is the upper-bound modulus for Young's modulus. The real upper-bound Young's modulus is larger than
given by this formula. Even if both constituents are isotropic, the real upper bound is
plus a term in the order of square of the difference of the Poisson's ratios of the two constituents.
The inverse rule of mixtures states that in the direction perpendicular to the fibers, the elastic modulus of a composite can be as low as
:
If the property under study is the elastic modulus, this quantity is called the lower-bound modulus, and corresponds to a transverse loading.
Derivation for elastic modulus
Upper-bound modulus
Consider a composite material under
uniaxial tension . If the material is to stay intact, the strain of the fibers,
must equal the strain of the matrix,
.
Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of t ...
for uniaxial tension hence gives
where
,
,
,
are the stress and elastic modulus of the fibers and the matrix, respectively. Noting stress to be a force per unit area, a force balance gives that
where
is the volume fraction of the fibers in the composite (and
is the volume fraction of the matrix).
If it is assumed that the composite material behaves as a linear-elastic material, i.e., abiding Hooke's law
for some elastic modulus of the composite
and some strain of the composite
, then equations and can be combined to give
:
Finally, since
, the overall elastic modulus of the composite can be expressed as
:
Lower-bound modulus
Now let the composite material be loaded perpendicular to the fibers, assuming that
. The overall strain in the composite is distributed between the materials such that
:
The overall modulus in the material is then given by
:
since
,
.
Other properties
Similar derivations give the rules of mixtures for
*
mass density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
:
*
ultimate tensile strength
Ultimate tensile strength (UTS), often shortened to tensile strength (TS), ultimate strength, or F_\text within equations, is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials ...
:
*
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
:
*
electrical conductivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
:
See also
When considering the empirical correlation of some physical properties and the chemical composition of compounds, other relationships, rules, or laws, also closely resembles the rule of mixtures:
*
Amagat's law
Amagat's law or the Law of Partial Volumes describes the behaviour and properties of mixtures of ideal (as well as some cases of non-ideal) gases. It is of use in chemistry and thermodynamics. It is named after Emile Amagat.
Overview
Amagat's ...
– Law of partial volumes of gases
*
Gladstone–Dale equation – Optical analysis of liquids, glasses and crystals
*
Kopp's law
Kopp's law can refer to either of two relationships discovered by the German chemist Hermann Franz Moritz Kopp (1817–1892).
#Kopp found "that the molecular heat capacity of a solid compound is the sum of the atomic heat capacities of the elemen ...
– Uses
mass fraction
*
Kopp–Neumann law – Specific heat for alloys
*
Vegard's law – Crystal lattice parameters
References
{{reflist
External links
Rule of mixtures calculator
Materials science
Laws of thermodynamics