Theory - Composite Materials

The Create Composite options in Patran synthesize material properties for four classes of composite construction techniques.

Two of these construction methods can be implemented in more than one way. There are five Halpin-Tsai options and two Short Fiber composite options.

Table 1‑1
Construction Method	Algorithm	Intended Application
Laminate	Classical Lamination Theory.	Laminated shells and solids.
Rule of Mixtures	Volume-Weighted Averaging.	3D composites with multiple phases, arbitrary orientations, and arbitrary volume fractions.
Halpin-Tsai	Halpin-Tsai Equations.	2-Phase Composites.
Short Fiber	Monte-Carlo integration combined with volume-weighted averaging.	Short fiber composites whose orientation distribution can be described by a Gaussian curve or surface.

The Laminate model is used to describe laminated solids and shells. In this construction, adjacent layers (or laminae or plies) are arranged as shown in Figure 1‑1, and the orientation of each layer is defined by a single constant angle

. Each layer may be a unique material and have a unique constant thickness. The Laminate model uses Classical Lamination Theory (CLT) to calculate the membrane, bending, and membrane-bending coupling stiffness matrices for a laminated shell.

The two fundamental assumptions of CLT are: (1) that surface normals remain normal when the laminate deforms:

where

is the strain,

is the midsurface strain,

is the curvature, and z is the distance from the neutral surface; and (2) that each layer is in a state of plane stress, implying that the transverse stresses are all zero:

The constitutive equation for an orthotropic ply in a state of plane stress is given by:

where:

The constitutive matrix

for a layer in the laminate frame is given by:

with:

where

is the matrix transforming the laminate frame strain into the ply frame and

is the angle from the laminate frame to the ply frame shown in Figure 1‑1. Combining the expression for the kinematic assumption, (1‑1), with the constitutive equation for the kth ply

yields:

Substituting (1‑11) into the integral expressions for force per unit length

and moment per unit length

:

leads to:

The midsurface strains

and curvatures

are not a function of z, and

is constant wi thin each ply, so the above expressions may be simplified to:

where

is the coordinate of the top of the kth ply (or higher z coordinate of the kth ply, see Figure 1‑1. The shell constitutive equations relating the midsurface strains

and curvatures

to the in-plane forces

and moments

are documented in (1‑15) and (1‑16). From these two expressions the stiffness matrices for membrane behavior

, bending behavior

, and membrane-bending coupling behavior

can be extracted:

If no laminate offset is specified, then Patran assumes that the middle surface is the neutral surface, and the above expressions for shell stiffness are used. The Patran offset is not the distance from the middle surface to the neutral surface, but rather the coordinate of the bottom of the stack (or lowest z coordinate, see Figure 1‑1) relative to the neutral surface, which, by default is the negative of half the laminate thickness. If a non-default offset is specified, implying that the neutral surface does not coincide with the middle surface, then the following corrections must be made to the bending matrix

and the membrane-bending coupling matrix

:

where d is the coordinate of the neutral surface relative to the middle surface and is related to the user-input offset as:

Thus, if a laminate having three layers of thickness .02 is specified without an offset, the default offset is taken to be -.03, since d = 0. If, however, the neutral surface is taken to be, for example, the interface between the first and second ply, corresponding to d = -.01, then the user-input offset should be -.04, yielding the corrected bending and coupling matrices:

Patran also calculates the resultant in-plane forces

and moments

corresponding to a uniform temperature increase of one degree:

where

is the vector of thermal expansion coefficients in the laminate frame for the kth layer.

All other laminated composite material properties are calculated using the same algorithms as those implemented by the Rule-of-Mixtures option, whose description starts on the next page.

Caution:	The elasticity matrix [C] calculated for laminated composites using the Rule-of-Mixtures equations is based on a volume weighted averaging scheme and is insensitive to the order of plies in a lay-up. In other words, for plies of the same material and thickness, a 90-0-90 degree stack will yield the same elasticity matrix as a 90-90-0 degree stack. Thus the elasticity matrix should be used for laminate problems with membrane behavior only (no bending and no coupling behavior). Additional simplifications, which may or may not be warranted for the user’s application, are made when the nine engineering constants (elastic moduli, Poisson’s ratios, and shear moduli) are evaluated from the elasticity matrix. In order to calculate nine unique engineering constants, Patran assumes that the 12 terms of the elasticity matrix that correspond to normal-shear coupling and shear-shear coupling behavior are zero. This reduces an elasticity matrix that is, in general, anisotropic, to orthotropy, on the premise that the engineering constants can only be meaningful if the laminated composite is effectively orthotropic. The resulting engineering constants can only be used, therefore, for laminate problems in which the response is characterized by membrane behavior only, when the laminate is effectively orthotropic.

The Rule-of-Mixtures model is used to describe three-dimensional solids having an arbitrary number of material phases with arbitrary orientations and volume fractions. Orientations are defined for each phase using a triad of space-fixed rotation angles

in a 3-2-1 sequence. These angles rotate the composite material frame to the phase frame. The orientation of each phase is defined by starting with the phase frame aligned with the composite frame and rotating the phase material frame

degrees about the 3-axis of the composite material frame, then rotating the phase frame

degrees about the 2-axis of the composite frame, and finally rotating the phase frame

degrees about the 1-axis of the composite frame. Rule-of-Mixtures composites are, in general, fully anisotropic.

Scalar quantities, such as density, are calculated using a simple volume-weighted averaging method, as in

where

is the density of the kth phase,

is the volume fraction of the kth phase, and n is the number of phases. The composite structural damping coefficient is also calculated in this way. For vector and matrix quantities, however, it is necessary to transform the phase properties into the composite material coordinate frame before performing volume-weighted averaging. Thus, the expression for the composite elasticity matrix is given by

with

where

is the elasticity matrix for the kth phase in the phase frame,

is the matrix that transforms the strains in the kth phase from the laminate frame to the phase frame, and the dij are the terms of the matrix of direction cosines

from the composite frame to the phase frame in terms of the rotation angles,

,

, and

, for the kth phase. Similarly, the composite thermal conductivity matrix is calculated using the expression

where

is the thermal conductivity matrix of the kth phase in the phase frame. The composite thermal and moisture expansion coefficient vectors are given by

where

is the composite flexibility matrix,

is the thermal expansion coefficient vector for the kth phase, and

is the moisture expansion coefficient vector for the kth phase. The nine engineering constants (elastic moduli, Poisson ratios, and shear moduli) are calculated from the composite flexibility matrix as follows:

Note that only nine of the 21 Sij’s are used to calculate the nine engineering constants. The potential anisotropy of the composite material is partially ignored: it is assumed to be at most orthotropic. Thus, these nine constants should be used in subsequent analyses only if the composite is known to be orthotropic. Patran also calculates the 2D plane stress constitutive matrix

from the 3D composite elasticity matrix

:

The user must be sure that the composite material is appropriate for a 2D plane stress analysis before using the

matrix. The composite specific heat, or heat capacity per unit mass, is calculated using a mass weighted averaging scheme:

where

is the specific heat for the kth phase,

is the mass fraction for the kth phase, and

is the density of the composite material. Finally, the reference temperature for the composite material is taken to be the reference temperature for the first phase material specified by the user.

The Halpin-Tsai models are used to describe 2-phase composites in which the matrix phase is isotropic. Halpin-Tsai materials may be transversely isotropic, orthotropic, or isotropic, depending on the geometry of the material reinforcing the matrix. The composite material frame corresponds with the fiber (or non-matrix) phase frame. Five different Halpin-Tsai material models exist in Patran: continuous fiber, discontinuous fiber, continuous ribbon, discontinuous ribbon, and particulate. These provide empirical relations for the engineering constants using, generally, Rule-of-Mixtures equations having the form

and Halpin-Tsai equations of the form:

where

is the composite elastic property (which may be an elastic modulus, a Poisson ratio, or a shear modulus),

and

are the corresponding properties for the fiber and matrix material, respectively,

and

are the volume fractions for the fiber and matrix phase, respectively, and

is a user-specified empirical constant. Each Halpin-Tsai model specifies a set of equations for the engineering constants and each equation in the set has a default value for

which may be overridden by the user. These models are summarized below from J.C. Halpin’s text, Revised Primer on Composite Materials: Analysis, Technomic Publishing Co., Lancaster, PA, 1984, pp. 123-142.

This model assumes the 2-phase geometry shown in Figure 1‑2. The fibers are uniform, continuous, cylindrical, and transversely isotropic. The resulting composite is therefore transversely isotropic. This is the only Halpin-Tsai model supported by some earlier versions of Patran.

Rule-of-Mixtures equations are used to determine

for the composite, with the default value for the empirical constant

being 1.0 in both cases. (The default value of

for any Rule-of-Mixtures equation in the five Halpin-Tsai models is always 1.0.) Halpin-Tsai equations are used to determine

, so that the expression for

, for example, is given by:

in which ETm is the transverse matrix modulus and ETf is the transverse fiber modulus. The default empirical constants for

are given by:

where

is the Poisson ratio for the isotropic matrix material and the ugly expression

is a correction term for composites with high fiber volume fractions. (Remember, these are empirical relations. They were not derived for the sole purpose of looking elegant and sophisticated.) The composite transverse Poisson ratio

is then determined from the known transverse isotropy:

Some earlier versions of Patran used a Halpin-Tsai equation to calculate

instead of GTT, and then used relations for transverse isotropy to calculate GTT. Although Halpin’s text does not specify a default value for

, Patran provides the value:

which should be selected with some caution. GTT is then calculated using the expression:

This model assumes the fibers are uniform, discontinuous, cylindrical, and transversely isotropic. The resulting composite is therefore transversely isotropic.

A Rule-of-Mixtures equation is used to determine

for the composite, with the default value for the empirical constant

being 1.0. Halpin-Tsai equations are used to determine EL, ET, GLT, and GTT, so that the expression for EL, for example, is given by:

in which ELm is the longitudinal matrix modulus and ELf is the longitudinal fiber modulus. The default empirical constants for EL, ET, GLT, and GTT are given by:

where

is the fiber length-to-diameter ratio. As with the Uniform Continuous Fiber model, the transverse Poisson ratio

is determined from (1‑47).

This model assumes the fibers are uniform, continuous, and orthotropic, with a rectangular cross section. The resulting composite is orthotropic.

Rule-of-Mixtures equations are used to determine

for the composite, with the default value for the empirical constant

being 1.0 in both cases. Halpin-Tsai equations are used to determine

, so that the expression for E2, for example, is given by:

in which E2m is the transverse matrix modulus, and E2f is the transverse fiber modulus. The default empirical constants for E2, E3, G12, and G23 are given by:

where

is the ribbon width-to-thickness ratio.

The transverse Poisson ratio

is calculated from the expression:

where

is the transverse Poisson ratio for the fiber material and

is the matrix Poisson ratio. The remaining two engineering constants,

, are not provided for in the theory, but are calculated in Patran by making the approximation that

, from which:

This model assumes the fibers are uniform, discontinuous, and orthotropic, with a rectangular cross section. The resulting composite is orthotropic.

A Rule-of-Mixtures equation is used to determine

for the composite with the default value for the empirical constant

being 1.0. Halpin-Tsai equations are used to determine E1, E2, E3, G12, and G23, so that the expression for E3, for example, is given by:

in which E3m is the cross-ply matrix modulus and E3f is the cross-ply fiber modulus. The default empirical constants for E1, E2, E3, G12, and G23 are given by:

where

is the ribbon length-to-thickness ratio and

is the ribbon width-to-thickness ratio.

As with the Uniform Continuous Ribbon model, the transverse Poisson ratio

is calculated from (1‑60), and the remaining two engineering constants, G13 and

, are calculated by making the approximation that

, yielding

by (1‑61).

This model assumes an isotropic particulate reinforcement of the matrix. The resulting composite is therefore isotropic.

Halpin-Tsai equations are used to determine both E and G, so that the expression for E, for example, is given by

in which Em is the matrix elastic modulus, and Ef is the fiber elastic modulus. The default empirical constants for E and G are given by

The isotropy of the particulate composite uniquely defines the Poisson ratio

.

The elasticity matrix can be expressed in terms of the orthotropic engineering constants as

body

If the composite material symmetry is more general than that of an orthotropic material, i.e., if the material is isotropic or transversely isotropic, then the above equations can be simplified. The flexibility matrix is calculated by inverting the elasticity matrix.

The exact Levin solution for 2-phase composites (V.M. Levin, “Thermal Expansion Coefficients of Heterogeneous Materials,” Mekhanika Tverdogo Tela, Vol. 2, No. 1, pp. 88-94, 1967) is used to determine both thermal and moisture expansion coefficients for all Halpin-Tsai models. The composite thermal expansion coefficient vector is calculated using the expression

and the composite moisture expansion vector is given by the analogous expression

and

is the composite flexibility matrix.

All other material properties are calculated using the methods described for Rule-of-Mixtures materials (which are described immediately preceding this Halpin-Tsai discussion), but the calculations are generally simpler for Halpin-Tsai materials because both phase frames coincide with the composite frame. Thus, it is not necessary to transform phase properties to the composite frame before summing their contribution to the composite properties.

The Short Fiber Composite model is used to compute the material properties of short fiber composites whose fiber orientations can be described by a normal (Gaussian) distribution. The orientations may vary in a single plane, in which case a Gaussian curve

describes the fiber orientations. Here

is the mean orientation and

is the standard deviation of the distribution. The fiber orientations may also vary in two dimensions, however, in which case the fiber distribution is described by a Gaussian surface

where

and

are the mean orientations,

and

are the corresponding standard deviations, and

is the correlation coefficient. Figure 1‑3 illustrates the spherical coordinates used to define a 2D Gaussian distribution in Patran. Here the e1-e2 plane defines the “equator” and

is the azimuthal angle defining, effectively, a “longitude,” while a positive angle

defines a “latitude” in the southern hemisphere.

A Monte Carlo integration scheme is used to sum the contributions of normally distributed “fibers” of a unidirectional material which should usually be a Halpin-Tsai Discontinuous Fiber material or a Halpin-Tsai Discontinuous Ribbon material. In other words, the geometrically appropriate Halpin-Tsai model is used to synthesize the properties of a unidirectional material having the same fiber material, matrix material, and fiber and matrix volume fractions as those of the short fiber composite to be created. The Short Fiber Composite model is then used to “distribute” the properties of the unidirectional Halpin-Tsai material within the specified Gaussian function. The integration is simplified by the approximation that all fibers lie within a

range of the mean orientation, where

is a standard deviation. The default number of iterations is 1000, but it may be overridden to any positive integer. The material properties for each iterate are summed using the Rule-of-Mixtures methods described earlier in this section. Scalar quantities, such as density, are simply assigned the same values as those of the constituent unidirectional material.

Short Fiber Composites are usually, to a first approximation, transversely isotropic or orthotropic, but because of the randomness of the Monte Carlo integration scheme, small shear coupling terms are introduced which tend to make these materials fully anisotropic. Larger iteration counts reduce this effect somewhat, but it cannot be eliminated. Nonetheless, it should not be cause for undue concern: the purpose of this model is to provide material properties with good first-order accuracy. The more complex Eshelby equivalent inclusion (and related) methods, which provide for fiber-matrix and fiber-boundary interaction effects, have been eschewed in favor of this simpler method. This Monte Carlo/Rule-of-Mixtures approach yields good first-order results accounting for the most significant factor in composite stiffness (the fiber orientations) and allows the materials designer to gain an understanding of the relative effects of varying fiber orientation parameters.