Anisotropy

Fibre volume fraction (V_f)

where:

• n = the number of layers,
• A_F = the areal weight of the fabric,
• ρ_f = density of the fibre, and
• t = the thickness of the laminate

The above formula (albeit with different symbols) appears in CRAG method 1000 Methods of assessment of fibre volume fraction of fibre reinforced plastics - 'thickness measurement' method, which is included in PT Curtis, CRAG Test Methods for the Measurement of the Engineering Properties of Fibre Reinforced Plastics, Royal Aerospace Establishment Technical Report 88 012, February 1988.

Areal Weight of a Fabric (A_F)

• N_f = number of filaments per tow
• N_T = number of tows in unit width of fabric
• r_f = radius of the fibre cross-section
• ρ_f = density of the fibre

Crimp will increase the areal weight by ~1% at 10˚, 3% at 20˚ or 6.5% at 30˚ maximum crimp angle.

Rule of Mixtures

where:

• E_c = Young's modulus of the composite
• E_f = Young's modulus of the fibre
• E_m = Young's modulus of the matrix
• V_f = fibre volume fraction
• V_m = matrix volume fraction (1-V_f-V_v)
• V_v = void volume fraction
  • See the page on voids for a discussion of the division of void content into sub-components, especially for plant fibre composites.
• κ = fibre area correction factor (set at unity for circular cross-section fibres)
• η_d = fibre diameter distribution factor (set at unity for man-made fibres)
• η_l = fibre length distribution factor
• η_o = fibre orientation distribution factor

• E_f = 70 GPa (glass), 140 GPa (aramid) or 210 GPa (carbon)
• E_m = 1-3 GPa (polymers) or 70 GPa (aluminium)
• V_f = 0.1-0.3 (random), 0.3-0.6 (woven) or 0.5-0.8 (unidirectional)
• η_l = 0 (if significantly less than the critical length) or 1 (continuous fibres)
• η_o = 1/5 (random 3D all-planes), 1/4 (biaxial on the bias angle), 3/8 (random 2D in-plane),
          1/2 (biaxial parallel to the fibres) or 1 (unidirectional parallel to the fibres)

The materials data above is representative and should not be used for 'design' purposes.

Transition temperatures (including glass transition temperature (T_g) ..and.. crystalline melting point (T_m)

In ascending order, the major transition temperatures are normally:

• T_g = glass transition temperature
• T_c = peak crystallisation temperature
• T_m = crystalline melting point
  
  • typically T_m = T_g + 200±50°C
  • nb: no melting point in amorphous materials
• T_p = processing temperature
  
  • typically T_p = T_m + ~30°C for “semi”-crystalline polymers
  • T_g is normally similar to the cure temperature for thermosetting resins
• T_d = degradation/decomposition
temperature
• may limit T_p (especially for PVC)

although these key temperatures do not necessarily occur in all cases (e.g. T_c and T_m are only applicable to partially crystalline polymers).

As the temperature rises through the glass transition temperature, short segments of the polymer backbone which had insufficient energy for movement other than atomic vibration, start to move as a group of atoms.  On cooling through this temperature, it is normal to refer to segmental motion being frozen out.  The mechanical properties of the polymer are then:

• below T_g:  normally elastic and brittle (with good resistance to creep deformation)
• above T_g:  normally viscoelastic and tough (however creep deformation can be a problem)

The crystalline melting point is not applicable to amorphous polymers and is usually only important in thermoplastics.  The crystalline melting point value is normally ~200 (±50) ºC above the glass transition temperature. T_m may be a narrow range of temperatures rather than a single point.

Stacking sequence

There are a number of ways in which fibres can be arranged.  In order of increasing stiffness and strength, these are:

• 3-D random: e.g. injection moulding grades.
• planar random: e.g. moulding compounds, chopped strand mat and continuous random swirl.
• quasi-isotropic: e.g. continuous fibres oriented at 0/-45/90/+45 or 0/60/120.
• bidirectional: e.g. woven fabrics or cross-plied unidirectional at 0/90.
• unidirectional: e.g. pultrusions and aligned fibre laminates.

The normal way to concisely record a laminate stacking sequence is, for example:

• [0º/+45º/-45º/90º]_ns

where the subscripts are:

• n is the number of times the sequence is repeated
• Q indicates an antisymmetric laminate
• s means the laminate is symmetric
• T is the total number of plies, and
• overbar denotes that the laminate is symmetric about the mid-plane of the ply

Thus for n = 2 in the above example, when * denotes the line of symmetry, the sequence will be:

• 0º/+45º/-45º/90º/0º/+45º/-45º/90º *90º/-45º/+45º/0º/90º/-45º/+45º/0º