Composites (Simple Bounds): Derivation of calculations

Derivation of calculations

Any two materials can, in principle, be combined to make a composite, and they can be mixed in many geometries (Figure 1). In this section we restrict the discussion to fully dense, strongly bonded composites such that there is no tendency for the components to separate at their interfaces when the composite is loaded, and to those in which the scale of the reinforcement is large compared to that of the atom or molecule size and the dislocation spacing, allowing the use of continuum methods.

Figure 1. Schematic of hybrids of the composite type: unidirectional fiber, laminated fiber, chopped fiber, and particulate composites. Bounds and limits, described in the text, bracket the properties of all of these.

On a macroscopic scale - one which is large compared to that of the components - a composite behaves like a homogeneous solid with its own set of mechanical, thermal and electrical properties. Calculating these precisely can be done on an individual basis, but is difficult. It is much easier to bracket them by bounds or limits: upper and lower values between which the properties lie. The term "bound" will be used to describe a rigorous boundary, one which the value of the property cannot - subject to certain assumptions - exceed or fall below. It is not always possible to derive bounds; then the best that can be done is to derive "limits" outside which it is unlikely that the value of the property will lie. The extreme values of the bounds or limits are then used as estimates for extremes of configurations.

Physical properties

Density

When a volume fraction f of a reinforcement r (density ρ_r) is mixed with a volume fraction (1-f) of a matrix m (density ρ_m) to form a composite with no residual porosity, the composite density is given exactly by a rule of mixtures (an arithmetic mean, weighted by volume fraction):

The geometry or shape of the reinforcement does not matter except in determining the maximum packing-fraction of reinforcement and thus the upper limit for f. The maximum value that can be practically achieved is 70%; hence this is used as the upper limit that the user can specify.

Mechanical properties

Young's modulus, E and Flexural modulus E_flex

The modulus of a composite is bracketed by the well-known Voigt and Reuss bounds. The upper bound, , is obtained by postulating that, on loading, the two components suffer the same strain; the stress is then the volume-average of the local stresses and the composite modulus follows a rule of mixtures:

Here E_r is the Young's modulus of the reinforcement and E_m that of the matrix. The lower bound, , is found by postulating instead that the two components carry the same stress; the strain is the volume-average of the local strains and the composite modulus is

More precise bounds are possible but the simple ones are adequate to illustrate the method.

UD long fiber laminates

We identify E and E_flex parallel to the fibers with the upper bound and the transverse values with the lower bound .

Quasi-isotropic laminates

We identify E and E_flex with:

Particulate composites

We identify E and E_flex with the lower bound .

Shear and Bulk Modulus.

The upper and lower bounds for the shear and bulk modulus can be found using similar equations to those above, just selecting the respective shear and bulk modulus for the reinforcement and the matrix.

UD long fiber laminates, shear parallel to fibers; and particulate composites

We identify the shear modulus and bulk moduli with the lower bounds:

And

Quasi-isotropic laminates

We identify shear and bulk modulus with the modified bounds:

And

Poisson's ratio

UD long fiber laminates, quasi-isotropic, and particulate composites

We identify Poisson's ratio with:

Tensile strength, yield strength, and flexural strength

Estimating strength is more difficult. The non-linearity of the problem, the multitude of failure mechanisms, and the sensitivity of strength and toughness to impurities and processing defects make accurate modeling difficult. The literature contains many calculations for special cases: reinforcement by unidirectional fibers, for example, or by a dilute dispersion of spheres. We wish to avoid models which require detailed knowledge of how a particular architecture behaves, and seek less restrictive limits.

UD long fiber laminates, in-plane loading parallel to fibers

As the load on a continuous fiber composite is increased, load is redistributed between the components until one suffers general yield or fracture (Figure 2a). Beyond this point the composite has suffered permanent deformation or damage but can still carry load; final failure requires yielding or fracture of both. The composite is strongest if both reach their failure state simultaneously. Thus the upper bound for a continuous fiber ply parallel to the fibers (the axial strength in tension, subscript a) is a rule of mixtures:

where σ_y,m is the strength of the matrix and σ_y,r is that of the reinforcement. If one fails before the other, the load is carried by the survivor. Thus a lower bound for strength in tension is given by:

The current version of the Synthesizer Tool uses the upper bound expresson for the unidirectional fiber model.

Quasi-isotropic laminates

We identify the upper limit for strength with:

(based on the assumption that failure means the fracture of approximately 1/4 of the fibers that lie parallel to the tensile axis, with a contribution from the matrix) and the lower limit with

The current version of the Hybrid Synthesizer uses the upper limit expression for the quasi-isotropic model.

Continuous fiber composites can fail in compression by fiber kinking (Figure 2a, extreme right). The kinking is resisted by the shear strength of the matrix, approximately , requiring an axial stress:

Here θ is the initial misalignment of the fibers from the axis of compression, in radians. Experiments show that a typical value in carefully aligned composites is =0.035, giving the final value shown on the right of the equation. We identify the compressive strength with the lesser of the two failure modes, hence in compression the strength of a unidirectional fiber composite is:

If the misalignment is severe (i.e. θ=1 ) then the compressive strength will be closer to the yield strength of the matrix, σ_y,m.

Figure 2. Failure modes in composites

Particulate composites and UD laminates loaded transversely

The transverse strength of UD composites and of particulate composites (Figure 2b) is equally difficult to model. It depends on interface bond-strength, fiber distribution, stress concentrations and the presence of voids. In general the transverse strength is less than that of the unreinforced matrix, and the strain to failure, too, is less. In a continuous, ductile matrix containing strongly bonded, non-deforming particles or fibers, the flow in the matrix is constrained. The constraint increases the stress required for flow in the matrix. We base the model on the assumption that the matrix has a fixed failure strain. The reinforcement makes it stiffer so it takes a higher stress to reach the failure strain of the matrix.

This consideration leads to an expression that requires a ratio of the Young's moduli of the matrix and reinforcement:

As the particulate model is used to illustrate the lower limit of a composite, the other value that should be taken into account is the harmonic mean of the matrix and reinforcement. The overall strength is then taken as the least of these two expressions:

Thermal properties

Specific heat (all composite configurations)

The specific heats of solids at constant pressure, C_p, are almost the same as those at constant volume, C_v. If they were identical, the heat capacity per unit volume of a composite would, like the density, be given exactly by a rule-of-mixtures:

where C_p,r is the specific heat of the reinforcement and C_p,m is that of the matrix (the densities are involved because the units of C_p are J/kg.K). A slight difference appears because thermal expansion generates a misfit between the components when the composite is heated; the misfit creates local pressures on the components and thus changes the specific heat. The effect is very small and need not concern us further.

Thermal expansion coefficient (all composite configurations)

The thermal expansion of a composite can, in some directions, be greater than that of either component; in others, less. This is because an elastic constant - Poisson's ratio - couples the principal elastic strains. If the matrix is prevented from expanding in one direction (by embedded fibers, for instance) then it expands more in the transverse directions. For simplicity we shall use the approximate lower bound for particulate and quasi-isotropic laminates

(it reduces to the rule of mixtures when the moduli are the same). The upper bound is given by:

where α_r and α_m are the two expansion coefficients and ν_r and ν_m the Poisson's ratios. The Synthesizer Tool at present uses the lower bound for all three model types.

Thermal conductivity.

The thermal conductivity determines heat flow at steady rate. A composite of two materials, bonded to give good thermal contact, has a thermal conductivity that lies between those of the individual components, λ_r and λ_m.

UD long fiber laminates, thermal conduction parallel to fibers.

Not surprisingly, a composite containing parallel continuous fibers has a conductivity, parallel to the fibers, given by a rule-of-mixtures:

This is an upper bound: in any other direction the conductivity is lower.

UD long fiber laminates, conduction transverse to fibers; particulate composites; and quasi-isotropic laminates

The transverse conductivity of a parallel-fiber composite and of particulate composites, assuming good bonding and thermal contact, lies near the lower bound first derived by Maxwell

Poor interface conductivity can make drop below it. Debonding or an interfacial layer between reinforcement and matrix can cause this; so, too, can a large difference of modulus between reinforcement and matrix because this reflects phonons, creating an interface impedance, or a structural scale which is shorter than the phonon wavelengths.

Electrical properties

Resistivity

UD long fiber laminates, electrical conduction parallel to fibers

The conductivities of fiber and matrix add, leading to a resistivity:

UD long fiber laminates, electrical conduction transverse to fibers; particulate composites; and quasi-isotropic laminates

The resistivity's add, giving the upper bound:

Dielectric constant (all composite configurations)

The dielectric constant is given by a rule of mixtures:

provided both ε_r,r and ε_r,m exist. Here ε_r,r is the dielectric constant of the reinforcement and ε_r,m that of the matrix.

Dielectric loss tangent (all composite configurations)

The dielectric loss tangent is also given by a rule of mixtures:

provided both D_ε,r and D_ε,m exist.

Primary material production: energy, CO2 footprint and water

The embodied energy is calculated as a simple rule of mixtures:

And as such only calculates the embodied energy of the constituent materials. It does not include the energy involved in processing.

The same is true of the CO2 footprint. Only the relative proportions of the constituent materials are taken into account so it too is described by a rule of mixtures:

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