The Edit Mixing Rule Dialog Box
The Edit mixing rule dialog box is used to control the mixing rules in the material contents table of a Multiphase Material or Effective Material.
To open the Edit mixing rule dialog:
Right-click the desired row in the Materials contents table and select Edit mixing rule ().
Select a row and click the Edit mixing rule button () under the table. The button is not active if more than one row are selected, or if no row is selected.
If there are several material properties in the table, you can step through the table without closing the dialog using the Previous Row (Discard Changes) (), Previous Row (Store Changes) (), Next Row (Discard Changes) (), and Next Row (Store Changes) () toolbar buttons. The current row of the is identified by the Property Group and Parameter fields.
The multiphase and effective material support several different mixing rules. Some of the rules are only available for specific material parameters. The mixing rules for composites are exclusive of the Effective Material, and the Krieger-type viscosity is only available for the Multiphase Material.
Basic Mixing Rules
The basic mixing rules are available to all material properties.
Volume Average
The volume average mixing rule uses an arithmetic mean weighted by the volume fractions,
(9-1)
where n is the number of phases, Vf,i is the volume fraction of phase i, X is the effective material property, and Xi is the material property of phase i.
Mass Average
The mass average mixing rule uses an arithmetic mean weighted by the mass fractions,
(9-2)
where wi is the mass fraction of phase i,
(9-3)
Here, ρ is the effective density and ρi is the density of phase i.
The mass average mixing rule is not available for density.
Harmonic Volume Average
The harmonic volume average mixing rule uses a harmonic mean weighted by the volume fractions,
(9-4)
Harmonic Mass Average
The harmonic mass average mixing rule uses a harmonic mean weighted by the mass fractions,
(9-5)
The harmonic mass average mixing rule is not available for density.
Power Law
The power law mixing rule uses a power law distribution,
(9-6)
Heaviside Function
The Heaviside Function mixing rule uses an arithmetic mean weighted by a smooth step function of the volume fraction:
(9-7)
where is the fraction used for phase i, is the fraction corresponding to the constrained phase, and lmix is a user-defined mixing parameter (default: 0.8) defining the size of the transition zone between phases for the smooth Heaviside function H.
Mixing Rules for Composite Materials
The effective structural material properties of composites depend on the material properties of the constituents and their geometric arrangements. These can be approximated either analytically or numerically. The numerical approach is applicable to a broader range of geometrical configurations, but is more complicated to use. The Effective Material includes a list of analytical approaches that are simple and handy, aimed at cases where an orthotropic fiber acts as reinforcement of an isotropic matrix. These methods are taken from Ref. 1, Ref. 2, and Ref. 3.
In this section, the properties of the fiber reinforcements are denoted by subscript f, while properties of the matrix are denoted by m. The mixing rules for composites are only available to a specific set of material properties. These rules should be used together with a Volume Averaged mixing rule for density.
Voigt–Reuss Model
The Voigt–Reuss model, also known as Rule of Mixtures, is an analytical method to compute homogenized material properties of composites. This method works well for cases where continuous orthotropic fibers are embedded in the isotropic matrix.
E11
E22
E33
G12
G23
G13
ν23
ν13
α11
α22
α33
β11
β22
β33
k11
k22
k33
σ11
σ22
σ33
Modified Voigt–Reuss Model
The Voigt–Reuss model provides results in the longitudinal directions that are in good agreement with experiments, but not in the transverse directions. The modified Voigt–Reuss model, or modified rule of mixture, provides corrections in the transverse direction. The following correction factors are used:
(9-8)
(9-9)
(9-10)
 
E11
E22
E33
G12
G23
G13
ν23
ν13
Chamis Model
The Chamis model works well for continuous orthotropic fibers embedded in the isotropic matrix. Unlike the Voigt–Reuss model, it can be used for randomly scattered unidirectional fibers.
E11
E22
E33
G12
G23
G13
ν23
ν13
α11
α22
α33
β11
β22
β33
k11
k22
k33
Halpin–Tsai Model
The Halpin–Tsai model is a semi-empirical method to computed homogenized properties of composites based on self-consistent micromechanics as well as experimental curve fitting. This method works well for continuous orthotropic fibers or discontinuous orthotropic fibers embedded in the isotropic matrix, including short fibers.
The Halpin–Tsai model includes empirical factors ζ for the components of the elastic modulus and are a measure of the reinforcement that depends on fiber geometry. Some typical values for the orthotropic Young’s modulus are depicted in the table value. The value of ζ for the components of G is typically 1.
Table 9-5: Typical values of ζ for Young’s Modulus E.
ζ for discontinuous fibers
ζ for continuous fibers
E11
E22
In the table above, l11 is the fiber length in the 11 direction, w22 is the fiber width in the 22 direction, and t33 is the fiber thickness in the 33 direction.
The effective material properties are given by:
E11
E22
E33
G12
G23
G13
ν23
ν13
Halpin–Tsai–Nielsen Model
The Halpin–Tsai–Nielsen model extends the Halpin–Tsai model by accounting for the maximum packing fraction of the reinforcement, (default 0.82). Typical values of the maximum packing factor are listed in Table 9-7.
The Halpin–Tsai–Nielsen model introduces an auxiliary variable,
(9-11)
and the effective properties are given by Table 9-8.
E11
E22
E33
G12
G23
G13
ν23
ν13
Hashin–Rosen Model
The Hashin–Rosen model is based on a composite cylinder assemblage (CCA). In order to compute homogenized material properties, additional properties are computed first.
The bulk modulus of fiber, Kf, and matrix, Km, under longitudinal strain are
(9-12)
(9-13)
The bulk modulus of the composite K under longitudinal strain is
(9-14)
In this model, the effective shear modulus of the composite G23 is approximated from the quadratic equation
(9-15)
where
(9-16)
(9-17)
(9-18)
and
The effective material properties are then given by Table 9-9.
E11
E22
E33
G12
G23
G13
ν23
ν13
Other Mixing Rules in the Multiphase Material
Krieger-Type Viscosity
The effective dynamic viscosity μ in suspensions may be modeled using the Krieger-type model as
(9-19)
When Krieger type viscosity is selected, enter a value or expression for the Maximum packing concentration (dimensionless). The default is 0.62.
The Krieger type mixing rule is available only for the dynamic viscosity parameter in the basic property group.
References for Mixing Rules
1. K.K. Chawala, “Composite Materials: Science and Engineering,” Springer, 2012.
2. J. Aboudi, S.M. Arnold, and B.A. Bednarcyk, “Micromechanics of Composite Materials: A Generalized Multiscale Analysis Approach,” Elsevier, 2013.
3. J.C. Halpin and J.L. Kardos, “The Halpin-Tsai Equations: A Review,” Polymer Eng. Science., vol. 16, no. 5, pp. 344–352, 1976.