Composites (Simple Bounds): Assumptions and calculations

Assumptions used by model

This model uses continuum methods to predict the upper and lower bounds in performance expected from a composite. The unidirectional (UD) model represents the upper bound; the particulate model, the lower bound; and the quasi-isotropic (QI) model, the performance mid-way between the upper and lower bound. As a result, this model should be considered as a scoping tool to identify potential composite compositions, which could exhibit unique combinations of properties. It is not intended to be used as a detailed laminate analysis tool.

The anisotropic structure of continuous fiber reinforced composites can lead to significant differences in properties in the in-plane and through-thickness directions. In all cases the values quoted on the datasheet for the synthesized record represent the performance parallel to the principal reinforcement direction. For example, the properties quoted for a unidirectional fiber reinforced composite reflect the performance that is expected when the applied load is parallel to the fiber direction.

In determining the calculations for the composites (simple bounds) model, the following assumptions have been made:

All models

• The composite material is fully dense (i.e. contains no porosity).

• The orientation and distribution of reinforcement within the matrix is perfect (i.e. no misalignment of fibers and uniform distribution of fibers and particulates).

• Good fiber-matrix interfacial bonding is achieved (i.e. no tendency for delamination).

• The scale of the reinforcement is large compared to that of the atom, molecule size, and the dislocation spacing (this allows the use of continuum methods).

Unidirectional composite model

• The fibers have a high degree of alignment for the unidirectional model

Short fiber composite model

• There is perfect orientation of fibers in the Aligned model and completely random orientation of fibers in the Random model.

• The random model represents a 2-D short fiber mat.

• The shear strength of the interface can be approximated by that of the matrix, which in turn can be approximated as such:

• The calculations that are used for the properties in the other composites models that do not depend on reinforcement geometry are also applicable to short fiber composites.

• Properties are quoted in the fiber direction for the aligned fiber composite.

• All the fibers present have the same aspect ratio.

• Price and eco properties are simple rule of mixtures, they only take into account the constituent materials (no processing).

Calculations used by model

• Unidirectional composite model
• Quasi-isotropic composite model
• Particulate composite model
• Short fiber composite model
  • Aligned
  • Random
• Summary of calculated properties and data required

Unidirectional composite model calculations

The equations used by the unidirectional fiber reinforced composite model are summarized below. For more information on the derivation of these equations see derivation of calculations.

For a definition of the symbols used, see symbols.

• Density

  

• Price

  

• Young's modulus (parallel to fibers)

  

• Flexural modulus (bending with axis normal to fibers)

  

• Shear modulus (shear parallel to fibers)

  

• Bulk modulus

  

• Poisson's ratio

  

• Yield strength (parallel to fibers)

  

• Compression strength (parallel to fibers)

  

• Flexural strength

  

• Tensile strength

  

• Thermal conductivity (parallel to fibers)

  

• Thermal expansion (transverse to fibers)

  

• Specific heat capacity

  

• Electrical resistivity (parallel to fibers)

  

• Dielectric constant

  

• Dielectric loss tangent

  

• Embodied energy (primary production)

  

• CO2 footprint (primary production)

  

Quasi-isotropic composite model calculations

The equations used by the quasi-isotropic fiber reinforced composite model are summarized below. For more information on the derivation of these equations see derivation of calculations.

For a definition of the symbols used, see symbols.

• Density

  

• Price

  

• Young's modulus

  

• Flexural modulus

  

• Shear modulus

  

• Bulk modulus

  

• Poisson's ratio

  

• Yield strength

  

• Compression strength

  

• Flexural strength

  

• Tensile strength

  

• Thermal conductivity

  

• Thermal expansion

  

• Specific heat capacity

  

• Electrical resistivity

  

• Dielectric constant

  

• Dielectric loss tangent

  

• Embodied energy (primary production)

  

• CO2 footprint (primary production)

  

Particulate composite model calculations

The equations used by the particulate reinforced composite model are summarized below. For more information on the derivation of these equations see derivation of calculations.

For a definition of the symbols used, see symbols.

• Density

  

• Price

  

• Young's modulus

  

• Flexural modulus

  

• Shear modulus

  

• Bulk modulus

  

• Poisson's ratio

  

• Yield strength

  

• Compressive strength

  

• Flexural strength

  

• Tensile strength

  

• Thermal conductivity

  

• Thermal expansion

  

• Specific heat capacity

  

• Electrical resistivity

  

• Dielectric constant

  

• Dielectric loss tangent

  

• Embodied energy (primary production)

  

• CO2 footprint (primary production)

  

Short fiber composite model calculations

The equations used by the short fiber reinforced composite model are summarized below.

For a definition of the symbols used in Composites (simple bounds) models, see symbols.

Aligned

• Density

  

• Price

  

• Young's modulus

  

  where

  

• Flexural modulus

  

• Yield strength

  

  

  

  

  

• Thermal conductivity

  

  where

  

• Thermal expansion

  

• Specific heat capacity

  

• Electrical resistivity

  

  where

  

• Dielectric constant

  

• Dissipation factor

  

• Embodied energy (primary production)

  

• CO2 footprint (primary production)

  

Random

• Density

  

• Price

  

• Young's modulus

  

  where

  

  and

  

• Flexural modulus

  

• Yield strength

  

  

  

  

  

  

• Thermal conductivity

  

  where

  

  and

  

• Thermal expansion

  

• Specific heat capacity

  

• Electrical resistivity

  

  where

  

  

• Dielectric constant

  

• Dissipation factor

  

• Embodied energy (primary production)

  

• CO2 footprint (primary production)

  

Summary of calculated properties and data required

(* = Particulate model only, ** = Unidirectional and quasi-isotropic models only)

Calculated Property	Reinforcement properties required by calculation	Matrix properties required by calculation
Density	Density	Density
Young's modulus	Young's modulus	Young's modulus
Flexural modulus	Young's modulus	Young's modulus
Shear modulus	Shear modulus	Shear modulus
Bulk modulus	Bulk modulus	Bulk modulus
Poisson's ratio	Poisson's ratio	Poisson's ratio
Yield strength	Yield strength Young's modulus*	Yield strength Young's modulus*
Tensile strength	Tensile strength Young's modulus*	Tensile strength Young's modulus*
Compressive strength	Compressive strength Young's modulus*	Compressive strength Yield strength Young's modulus*
Flexural strength	Yield strength** Compressive strength** Flexural strength* Young's modulus*	Yield strength** Compressive strength** Flexural strength* Young's modulus*
Specific heat capacity	Specific heat capacity Density	Specific heat capacity Density
Thermal expansion coefficient	Thermal expansion coefficient Young's modulus	Thermal expansion coefficient Young's modulus
Thermal conductivity	Thermal conductivity	Thermal conductivity
Electrical resistivity	Electrical resistivity	Electrical resistivity
Dielectric constant	Dielectric constant	Dielectric constant
Dielectric loss tangent	Dielectric loss tangent	Dielectric loss tangent
Embodied energy	Embodied energy Density	Embodied energy Density
CO2 footprint	CO2 footprint Density	CO2 footprint Density

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