MATS324 Composites Design and Manufacture

Composites Design and Manufacture (BEng) - MATS 324
Young's moduli, Poisson's ratios, shear and bulk moduli

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Young's modulus (E) .. this page
Poisson's ratio (n) .. this page
Shear modulus (G) .. this page
Bulk modulus (K) .. this page
Anisotropic elasticity theory .. separate page

Hooke's Law [1] originally stated that a change in length was proportional to the load, but now it has been generalised to denote that the stress (σ) is proportional to the strain (ε) with the constant of proportionality known as Young's modulus (E), so that σ = Eε. Traditional structural engineering materials are isotropic and hence the value of E is independent of direction. However, the Young's modulus of a composite material is anisotropic (varies with direction) and can be estimated using the rule-of-mixtures:

Equation 1

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
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

For a unidirectional continuous fibre composite aligned with the stress, the physical assumption underlying the equation above is compatibility of strain between the fibre and the matrix resulting in the constituent phases carrying load in proportion to their respective volume fractions and moduli. Further, the above equation has a number of underlying assumptions [2, 3]:

the composite ply is macroscopically homogeneous and linearly elastic
the fibres are homogeneous and linearly elastic
- if the fibres are orthotropic, then a more general form of the equation may be required.
the matrix is homogeneous and linearly elastic

both the fibre and the matrix are free of voids

the interface is completely bonded

there is no transition region (no third party material) between the matrix and the reinforcement

there is equal strain in the fibre and the matrix
Poisson's strains are neglected (the Poisson's ratio of the fibre and the matrix are the same)
- if the Poisson's ratios differ, then the neglected term has been shown to be of the order of the square of the difference between the two values [3]
the assumptions described below for fibre length distribution factor and fibre orientation distribution factor also apply.

The rule of mixtures can be used for other "elastic" properties (e.g. density, Poisson's ratio, coefficient of thermal expansion and hygrothermal properties).

The fibre length distribution factor (FLDF) assumes that:

the matrix and fibre remain elastic and the interface bond is perfect,
the shear stress at the fibre ends is maximum and falls to zero after half the "critical length", and
the tensile stress at the fibre ends is zero and rises to a maximum after half the "critical length", as shown in the Figure below.

Figure: Diagrammatic representation of the stress distribution in a short fibre,
with shear stress at the fibre matrix interface shown in red, tensile stress in black and E_fε shown in green.

FLDF can be calculated using the Cox equation [4]:

Equation 2

[4]

where:

	Equation 3	[4]
and:
	Equation 4	[5]

and where G_m is the shear modulus of the matrix, L is the fibre length, A_f is the cross sectional area of the fibre, R_f is the radius of the fibre and R is the mean separation of the fibres. If the fibre is shorter than the critical length (l_c), it will never carry a load high enough to cause fibre fracture. The composite will instead fail in shear, either at the fibre/matrix interface or in the matrix itself. Alternative Equations for the prediction of β have been proposed by Nayfeh [6] and McCartney [7]. Nairn [8] has extended the shear-lag analysis from concentric cylinders to a generalized form (with an new β_cor as Equation 44 in his paper) with transverse variations of shear stress described by arbitrary shape functions. The new analysis permits modelling of imperfect interfaces both between concentric cylinders [8a] and within multilayered structures [8b]. Modification of the prior shape functions permits extended shear-lag analysis to work for any fibre volume fraction (previous models were unacceptable at low fibre volume fraction). The full shear-lag analysis can model stress transfer for both isotropic and anisotropic fibres.

The critical length (when there is no debonding) is given by the expression:

Equation 5

[4]

where R is the fibre radius, σ₁₁ is the tensile stress in the fibre and σ'₁₂ is the shear strength (of the interface or of the matrix as appropriate).

and the fibre orientation distribution factor can be calculated using the Krenchel equation [9]:

Equation 6

[9]

For a unidirectional ply, the rule-of-mixtures does assume that:

the fibres are continuous with uniform cross-section and lie parallel to each other.
there is perfect bonding between the fibre and the matrix and hence Cox shear-lag theory applies.
a longitudinal load produces equal strain in both the fibre and the matrix.

and typical values used in the equation would be:

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) varying up to 1 (continuous fibres)
η_o= 0.0 for unidirectional perpendicular to the fibres
η_o= 1/4 for biaxial on the bias angle (at ±45º to the fibres)

η_o= 3/8 for random in-plane

η_o= 1/2 for biaxial parallel to the fibres

η_o= 1.0 for unidirectional parallel to the fibres

For a continuous fibre carbon/resin composite (E_f = 210 GPa, E_m = 3 GPa with V_f = 70% for unidirectional or V_f = 50% for bidirectional), the following Figure illustrates how the Young's modulus calculated from the rule-of-mixtures varies with change in the angle between the applied stress and the fibre direction:

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

The Reuss model suggests that up to 0.5% strain, there is equal stress in both the fibres and the matrix. The Reuss model for equal stress in the context of transverse stiffness is described in Hull and Clyne [10 - page 63].
The Voigt model suggests that above 0.5% strain, there are equal increases in strain in both the fibres and the matrix. [NB: I cannot currently trace where I found the transition point at 0.5% strain :-(]

Poisson's ratio

Poisson's ratio is denoted by the Greek letter nu: n. It has a value determined by:

n = -(strain normal to the applied stress)/(strain parallel to the applied stress)

Tensile deformation is taken as positive and compressive deformation is taken as negative. The minus sign in the definition of Poisson's ratio normally results in positive values of Poisson's ratio and a thermodynamic constraint which restricts the values to -1 < ν < 1/2.

For an orthotropic composite, there may be a different Poisson's ratio associated with each plane. Maxwell’s reciprocal theorem [11] states that two strains must be equal if the two stresses are of equal magnitude and sense. It is implicit in the symmetry of the stiffness/compliance matrices for a square symmetric material (E₂= E₃) that ν₁₂E₂ = ν₂₁E₁, and hence ν₁₂ = ν₁₃, ν₂₁ = ν₃₁ and ν₂₃ = ν₃₂. Lemprière [12] generalised the above thermodynamic constraint for the case of orthotropic materials where both the stiffness and compliance matrices are positive-definite (i.e. E₁, E₂, E₃, G₂₃, G₁₃, G₁₂ > 0) to yield the following results:

(1-ν₂₃ν₃₂), (1-ν₁₃ν₃₁), (1-ν₁₂ν₂₁), (1-ν₁₂ν₂₁-ν₁₃ν₃₁-ν₂₃ν₃₂-2ν₂₁ν₃₂ν₁₃) > 0

and thus

ν_ij ≤ (E_i/E_j)^1/2 and ν₂₁ν₂₃ν₁₃ < 1/2.

Craig and Summerscales [13] measured the Poisson's ratios in all three planes for two glass-fibre laminates

C1: 13 layers of Fothergill and Harvey Y119 unidirectional rovings
A2: 12 layers of TBA ECK25 woven rovings in Crystic 625TV resin

and confirmed that the Lemprière criteria were valid for both materials:

	Unidirectional	Unidirectional	Woven	Woven
	Panel C1: ν, E or G	Panel C1 √E_i/E_j	Panel A2: ν, E or G	Panel A2 √E_i/E_j
ν₁₂	0.308	1.606	0.140	0.942
ν₂₁	0.123	0.623	0.109	1.061
ν₁₃	0.354	1.687	0.408	1.285
ν₃₁	0.124	0.593	0.247	0.778
ν₂₃	0.417	1.051	0.380	1.364
ν₃₂	0.414	0.952	0.297	0.733
E₁ (GPa)	20.3		15.5
E₂ (GPa)	7.9		17.5
E₃ (GPa)	7.1		9.4
G₁₂ (GPa)	3.45		3.0

If one Poisson's ratio in an orthotropic material is negative, then no restriction is placed on the other two values. Dickerson and Di Martino [14] published data for cross-plied boron/epoxy composites in which the Poisson's ratios range from 0.024 to 0.878 in the orthotropic case and from -0.414 to 1.97 for a ±25º laminate.

Materials and structures with negative Poisson's ratios do exist and are termed auxetic. A variety of often re-entrant or chiral structures achieve this effect, as in the animation below from Rod Lakes (University of Wisconsin) webpage on Negative Poisson's ratio materials:

............

Re-entrant (left) and chiral (right) auxetic (negative Poisson's ratio) structures (from Rod Lakes webpage)

Shear modulus

The inter-relationship of the elastic constants for isotropic materials is given by:

	G = E/2(1+n)	Equation 7	[1]
	K= E/3(1-2n)	Equation 8	[1]

where E is the Young’s modulus, G is the shear modulus, K is the bulk modulus and n is Poisson’s ratio.

For orthotropic materials within a single plane, Huber proposed that the shear modulus would be predicted by:

Equation 9

[15,16]

Craig and Summerscales [13] used Huber’s equation [15, 16] to validate the elastic constants measured in the plane of the reinforcement for both unidirectional and woven fibreglass panels and presented data for the Poisson's ratios in all three orthogonal planes.

Bulk modulus

By extension of Equation 7, Summerscales [17] proposed that the bulk modulus for a square symmetric material could be predicted from the following equation:

Equation 10

[17]

This equation reduces to the isotropic form when E₁= E₂= E₃ and n₁₂ = n₂₁ = n₂₃.

Table 1 [based on reference 18] gives values for Young’s moduli E, Poisson’s ratios n and bulk moduli K_ref(all moduli in GPa) for various materials. The abbreviation (iso) is used to indicate isotropic properties. The unidirectional carbon fibre composites in the Table are square symmetric. Equations 9 and 10 have been used to predict the bulk moduli K_calcfor the isotropic and square symmetric materials respectively.

Material	E₁	E₂	E₃	n₁₂	n₂₁	n₂₃	K_ref	K_calc	Ref.
Epoxy resin	2.4	(iso)	(iso)	0.34	(iso)	(iso)	-	2.5	[19]
Crown glass	71.3	(iso)	(iso)	0.22	(iso)	(iso)	41.2	42.4	[20]
Heavy flint glass	80.1	(iso)	(iso)	0.27	(iso)	(iso)	57.6	58.0	[20]
Graphite	-	-	-	-	-	-	33.0	-
*Carbon fibre/epoxy composites*
High-modulus (HM) parallel	287	7.75	7.75	0.3	0.01	0.55	-	11.3	[21]
High-strength (HS) perpendicular	177	11.7	11.7	0.33	0.02	0.47	-	13.6	[21]
High-strength (HS) parallel	172	11.6	11.6	0.36	0.024	0.48	-	14.0	[21]
*Carbon fibres*
Tenax HTA5131 carbon fibre	238	28	28	-	-	-	-	-	[22]

Using the mercury porosimetry technique, Matthews and Ridgway [23] have estimated the bulk modulus of woven Tenax HTA carbon fibre Injectex and twill fabrics (Brochier SA, Dagneux - France) at 20 GPa and 7 GPa respectively. Both the Injectex and twill fabrics have the same fabric areal weight (290 gm^-2) and are woven with fibres from the same batch.

Using the rule-of-mixtures, the unidirectional HS fibre composites might be assumed from the axial modulus to be ~70-75% fibre by volume of 238 GPa fibre. This would suggest a contribution of ~0.75 GPa from the resin to the bulk modulus, leaving a fibre bulk modulus of ~13 GPa (average of parallel and perpendicular composite moduli minus resin contribution). Note that this value lies between those estimated by Matthews and Ridgway.

The discrepancy between the predicted and measured bulk moduli for the carbon fibres may be due to any or all of the following assumptions:

the woven fabric has a unit cell size of a similar magnitude to the dimensions of the porosimeter cell. The limited sample of the cloth tested may not be representative of the material.
the Injectex fabric does have a small proportion of polymer fibre to promote flow enhancement by restricting the collapse of the tow to the normal lenticular cross-section.
the estimation of bulk modulus from the porosimeter data involves an element of interpretation.
the difficulty of making accurate measurements of fibre Poisson’s ratios, especially in the transverse direction, and the implications of this for the prediction of the bulk moduli.
the absence of a coherent data set of engineering properties for the specific fibres used.

The proposed equation for the prediction of the bulk modulus of anisotropic fibres gives values which are of the same order of magnitude as estimates from experimentally determined values.

Anisotropic elasticity theory

For information on implementing the above in tensor/matrix form, see the anisotropic elasticity theory page.

Additional teaching support material:

Stephen Grove: Rules of mixture for elastic properties (University of Plymouth)
Polymer Composites: layer property prediction (AEA Technology Materials Solutions)

References

JC Anderson, KD Leaver, JM Alexander and RD Rawlings, Materials Science, second edition,
Thomas Nelson & Sons, Sunbury-on-Thames, 1974. ISBN 0-17-771015-2.
A Brent Strong, Fundamentals of Composites Manufacturing: materials, methods and applications, Society for Manufacturing Engineers, Dearbon MI, 1989. ISBN 0-87263-358-6.
RF Gibson, Principles of Composite Material Mechanics, McGraw-Hill, 1994. ISBN 0-07-023451-5.
HL Cox, The elasticity and strength of paper and other fibrous materials, British Journal of Applied Physics, 1952, 3, 72-79.
MJ Folkes, Short Fibre Reinforced Thermoplastics, Research Studies Press, Letchworth, 1982. ISBN 0-471-10209-1.
AH Nayfeh, Thermomechanically induced interfacial stresses in fibrous composites, Fibre Science and Technology, July 1977, 10(3), 195-209.
LN McCartney, Analytical models of stress transfer in unidirectional composites and cross-ply laminates, and their application to the prediction of matrix/transverse cracking,
In JN Reddy and KL Reifsnider (editors), Proceedings of the IUTAM SYmposium, Blacksburg VA, 1992, 251-282.
JA Nairn, Generalized shear-lag analysis including imperfect interfaces, Advanced Composite Letters, 2004, 13(6), 263-274. Author's proof.
(a) JA Nairn, On the use of shear-lag methods for analysis of stress transfer in unidirectional composites, Mechanics of Materials, 1997, 26(2), 63–80.
(b) JA Nairn and DA Mendels, On the use of planar shear-lag methods for stress-transfer analysis of multilayered composites, Mechanics of Materials, 2001, 33(6), 335-362.
H Krenchel, Fibre Reinforcement, Akademisk Forlag, Copenhagen, 1964.
D Hull and TW Clyne, An Introduction to Composite Materials: second edition, Cambridge University Press, August 1996. ISBN 0-521-38855-4.
NJ Hoff, The Analysis of Structures, John Wiley, New York, 1956, p.373.
BM Lemprière, Poisson's ratio in orthotropic materials, AIAA Journal, 1968, 6(11), 2226-2227.
PD Craig and J Summerscales, Poisson’s ratios in glass fibre reinforced plastics, Composite Structures, 1988, 9(3), 173-188.
EO Dickerson and B di Martino, Off-axis strength and testing of filamentary materials for aircraft application, 10th National Symposium, SAMPE, San Diego, 9-11 November 1966, H23-H50.
MT Huber, The theory of crosswise reinforced ferroconcrete slabs and its application to various important constructional problems involving rectangular slabs, Der Bauingenieur, 1923, 4(12), 354-360 and 4(13), 392-395.
V Panc, Theories of elastic plates, Noordhoff International Publishing, Netherlands, 1975.
J Summerscales, The bulk modulus of carbon fibres, Journal of Materials Science Letters, 2000, 19(1), 15-16.
GWC Kaye and TH Laby, Tables of Physical and Chemical Constants, 16th edition, Longman, Harlow, 1995. ISBN 0-582-22629-5.
NG McCrum, CP Buckley and CB Bucknall, Principles of Polymer Engineering - second edition, Oxford Science Publications, Oxford, 1997. ISBN 0-19-856526-7.
G Bradfield, Use in industry of elasticity with the help of mechanical vibrations, Note on Applied Science 30, HMSO, London, 1964.
BE Read and GD Dean, The Determination of Dynamic Properties of Polymers and Composites, Adam Hilger Limited, Bristol, 1978. ISBN 0-85274-363-7.
JR Lowe (Tenax Fibers GmbH, Wuppertal), private communication, 2 June 1999.
GP Matthews and CJ Ridgway, Simulation of resin intrusion during injection moulding, Proceedings of the 5th International Conference on Flow Processes in Composite Materials, University of Plymouth, 12-14 July 1999,
ACMC Plymouth, pp 91-97. ISBN 1-870918-01-0.

Further reading

MR Wisnom, Size effects in the testing of fibre-composite materials, Composites Science and Technology, 1999, 59(13), 1937-1957.
AS Chen and FL Matthews, A review of multiaxial/biaxial loading tests for composite materials, Composites, 1993, 24(5), 395-406.

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Updated by John Summerscales on 10 August 2011 14:21. Terms and conditions. Errors and omissions. Corrections.