Composites Design and Manufacture (Plymouth University teaching support materials) Some basic equations and definitions

Lecture PowerPoint

Review papers

Subject Index

Anisotropy

Degree of anisotropy	Principal axes	Properties	Example
Isotropic	Orthogonal	Constant regardless of direction	Metals
Square symmetric	Orthogonal	Two different principal axes	Unidirectional fibres or woven cloth
Orthotropic	Orthogonal	Three different principal axes	Unidirectional weave with light weft
Anisotropic	Any angle	Constant relative to axes	Filament wound tube : Many crystals
Aeolotropic	Any angle	May change with position	Timber

Further reading:

B Hutchinson, Critical Assessment 16: anisotropy in metals, Materials Science and Technology, September 2015, 31(12), 1393-1401.

Fibre volume fraction (V_f)

	Thickness measurement method	Resin burn-off method
Equation:

where:

• A_F = the areal weight of the fabric,
• M = mass of the component
• n = the number of layers,
• ρ_f = density of the fibre, and
• t = the thickness of the laminate
• subscripts are c = composite, f = fibre, F = fabric and m = matrix.

The above formulae (albeit with different symbols) appear in CRAG method 1000 Methods of assessment of fibre volume fraction of fibre reinforced plastics [1].

Reference 1: 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.  MooDLE

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 [1]

where:

	Parameter	Typical values/comments
Ec	Young's modulus of the composite
Ef	Young's modulus of the fibre	70 GPa (glass), 140 GPa (aramid) or 210 GPa (carbon) for standard grade fibres
Em	Young's modulus of the matrix	1-3 GPa (polymers) or 70 GPa (aluminium)
Vf	fibre volume fraction	indicative fibre volume fractions are given in the Table below, and also see the discussion of compressibility of reinforcements
Vm	matrix volume fraction	(1-Vf-Vv)
Vv	void volume fraction	see the discussion at void formation and transport and division of void content into sub-components, especially for plant fibre composites.
κ	fibre area correction factor	set at unity (1) for circular cross-section fibres
ηd	fibre diameter distribution factor for natural fibres	set at unity (1) for most man-made fibres
ηl	fibre length distribution factor	0 (if significantly less than the critical length) or 1 (continuous fibres)
ηo	fibre orientation distribution factor	Fibres 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) Platelets [2] 8/15 (3-D random platelets) 1 (prefectly oriented material in-plane)

and typical V_f values would be:

Reinforcement form	Vf without consolidation	Vf with high pressure consolidation
Random in-plane	10%	30%
Woven	30%	60%
Unidirectional	50%	80%

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

BS EN 14272:2011 [3] defines a modification factor, k_a, which is a knock-down factor for modulus of elasticity and strength dependent on the surface appearance classes of plywood veneer [4,5].  For appearance classes E, I and II, k_a is unity, and these basic mechanical property values are used as the basis of calculations.  For class III, k_a is 0.85.  For class IV, and where inner layers are not appearance graded, k_a is 0.75.

References

1. AS Virk, W Hall and J Summerscales, Modulus and strength prediction for natural fibre composites, Materials Science and Technology, July 2012, 28(7), 864-871.
2. Z Li, RJ Young, NR Wilson, IA Kinloch, C Vallés, Z Li, Effect of the orientation of graphene-based nanoplatelets upon the Young's modulus of nanocomposites, Composites Science and Technology, 8 February 2016, 123, 125–133.
3. Plywood - calculation method for some mechanical properties, BS EN 14272:2011, British Standards Institution, 2012.  ISBN 978-0-580-69385-4.
4. Plywood - classification by surface appearance - hardwood, BS EN 635-2:1995, British Standards Institution, 1995.  ISBN 0-580-24769-4.
5. Plywood - classification by surface appearance - softwood, BS EN 635-3, 1995, British Standards Institution, 1995.  ISBN 0-580-24772-4.

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_g determines the upper use temperature, especially for stressed components which will creep above this 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.

Recycling numbers

1. polyethylene terephthalate (PET or PETE)
2. high density polyethylene (HDPE)
3. polyvinyl chloride (PVC, V or vinyl)
4. low density polyethylene (LDPE)
5. polypropylene (PP)
6. polystyrene (PS)
7. other

Types of composite

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.

At a higher level these layers may be organised in four distinct ways
The terminology is not always consistent, e.g. laminate may be used for monolithic composite materials.

monolithic	material	all layers aligned parallel
laminate	structure	orientation changes between layers
hybrid	structure	more than one type of fibre (e.g. carbon/glass)
sandwich	structure	composite skins and lightweight core

Laminate stacking sequence

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 ("the magnitude of the ply orientation angle below the laminate mid-plane is a mirror image of the ply orientations above the mid-plane with signs reversed" [1]).
• 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

Rana and Fangueiro [2] suggest that when specifying lay-ups (laminate ply stacks) and design details, use the following basic guidelines:

• Use balanced laminates to avoid warping.
• Use manufacturing techniques that produce a minimum fiber content of 55% by volume.
• Use a minimum of 10% of plies in each of the principal directions (0°, 90°, ±45°) to provide a minimum acceptable strength in all directions.
• Use a maximum of four adjacent plies in any one direction to avoid splitting on contraction from cure temperature or under load.

For aircraft applications and other design issues:

• Place 45° plies on the outside surfaces of shear panels to increase resistance to buckling
• Avoid highly directional laminates in regions around holes or notches because stress concentration factors are significantly higher in this ply lay-up
• Add ply of woven fiberglass barrier between carbon and aluminum alloy for galvanic protectionDrop plies where required progressively in steps with at least 6 mm (0.25 in) landing to improve load redistribution
• Where possible, cover ply drops with a continuous ply to prevent end-of-ply delamination
• Maintain three-dimensional edge distance and four-dimensional pitch for mechanical fasteners to maximize bearing strength
• Where feasible, avoid honeycomb in favor of stiffened construction, because honeycomb is prone to moisture intrusion and is easily damaged
• Avoid manufacturing techniques that result in poor fiber alignment, because wavy fibers results in reduced stiffness and compression strength
• Minimize the number of joints by designing large components or sections because joints reduce strength and increase weight and cost

1. Z Gurdal, RT Hafka and P Hajela, Design and Optimization of Laminated Composite Materials, John Wiley & Sons, 1999.  ISBN 0-471-25276-x.  PU CSH Library
2. S Rana and R Fangueiro (editors), Advanced Composite Materials for Aerospace Engineering: processing, properties and applications, Woodhead Publishing, Cambridge, 2016.

Ashby material indices to minimise mass

Function	Form	Stiffness	Strength
Tension	tie bar	ρ/E	ρ/σy
Bending	beam	ρ/E1/2	ρ/σy2/3
Bending	panel	ρ/E1/3	ρ/σy1/2