# The Fundamental Principles of Composite Material Stiffness Predictions. David Richardson

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1 The Fundamental Principles of Composite Material Stiffness Predictions David Richardson

2 Contents Description of example material for analysis Prediction of Stiffness using Rule of Mixtures (ROM) ROM with Efficiency Factor Hart Smith 10% rule Classical Laminate Analysis Simplified approach Overview of misconceptions in material property comparison between isotropic materials and composites

3 Lamina Axis Notation Diagram taken from Harris (1999)

4 Example Material for Analysis M21/35%/UD268/T700 A common Aerospace uni-directional pre-preg material called HexPly M21 from Hexcel E f = 235 GPa ρ f = 1.78 g/cm3 E m = 3.5 GPa ρ m = 1.28 g/cm3 W r = 35% (composite resin weight fraction) Layup = (0/0/0/+45/-45/0/0/0)

5 Stage 1 Convert fibre weight fraction of composite to fibre volume fraction Fibre weight fraction used by material suppliers Fibre volume fraction needed for calculations

6 Fibre Volume Fraction Fibre mass fraction of M21 = 65% (0.65) Data sheet says material is 35% resin by weight, therefore 65% fibre by weight Calculation of fibre volume fraction The resulting volume fraction is 57.2% V f W f f W f f W m m

7 Methods of Stiffness Prediction Rule of Mixtures (with efficiency factor) Hart-Smith 10% Rule Used in aerospace industry as a quick method of estimating stiffness Empirical Formulae Based solely on test data Classical Laminate Analysis LAP software

8 Rule of Mixtures A composite is a mixture or combination of two (or more) materials The Rule of Mixtures formula can be used to calculate / predict Young s Modulus (E) Density Poisson s ratio Strength (UTS) very optimistic prediction 50% usually measured in test Strength very difficult to predict numerous reasons

9 Rule of Mixtures for Stiffness Rule of Mixtures for Young s Modulus Assumes uni-directional fibres Predicts Young s Modulus in fibre direction only E c = E f V f + E m V m Ec = E c = 136 GPa

10 Rule of Mixtures: Efficiency Factor The Efficiency Factor or Krenchel factor can be used to predict the effect of fibre orientation on stiffness This is a term that is used to factor the Rule of Mixtures formula according to the fibre angle See following slide

11 Reinforcing Efficiency a n = proportion of total fibre content θ = angle of fibres η θ = composite efficiency factor (Krenchel)

12 Efficiency (Krenchel) Factor Diagram taken from Harris (1999)

13 Prediction of E for Example Ply E f = 235 GPa E m = 3.5 GPa V f = E(θ) = (Cos 4 θ ) + ( ) Predicted modulus versus angle plotted on following slide

14 Tensile Modulus (GPa) Prediction of Tensile Modulus (Efficiency Factor) Angle (degrees)

15 Efficiency Factor for Laminate Layup = (0/0/0/+45/-45/0/0/0) η = Cos 4 θ 0 = η = 1 45 = η = = η = 0 Laminate in X-direction (6/8 1) + (2/8 0.25) ( ) Laminate in Y-direction (6/8 0) + (2/8 0.25) ( )

16 Prediction of E for Example Ply E f = 235 GPa E m = 3.5 GPa V f = E x = ( ) + ( ) E x = = GPa E y = ( ) + ( ) E y = = 9.9 GPa

17 Ten-Percent Rule Hart-Smith 1993 Each 45 or 90 ply is considered to contribute one tenth of the strength or stiffness of a 0 ply to the overall performance of the laminate Rapid and reasonably accurate estimate Used in Aerospace industry where standard layup [0/±45/90] is usually used E x = E 11. ( % plies at 0 ) σ x = σ 11. ( % plies at 0 ) G xy = E 11. ( % plies at ± 45 )

18 Tensile Modulus (GPa) Prediction of Tensile Modulus ROM 80 Hart-Smith Angle (degrees)

19 Calculation of E 11 for Ply Using Rule of Mixtures E 11 = E f V f + E m V m E 11 = E 11 = 136 GPa Layup = (0/0/0/+45/-45/0/0/0) 6/8 = 75% of plies in zero degree direction

20 Ten-Percent Rule E x = E 11 ( % plies at 0 ) E x = 136 (0.1 + ( )) E x = 136 (0.775) E x = GPa E y = E 11 ( % plies at 0 ) E y = 136 (0.1 + (0.9 0)) E y = 136 (0.1) E y = 13.6 GPa

21 Classical Laminate Analysis 4 elastic constants are needed to characterise the in-plane macroscopic elastic properties of a ply E 11 = Longitudinal Stiffness E 22 = Transverse Stiffness ν 12 = Major Poisson s Ratio G 12 = In-Plane Shear Modulus

22 Elastic Constant Equations E 11 = Longitudinal Stiffness (Rule of Mixtures Formulae) E 22 = Transverse Stiffness (Inverse Rule of Mixtures Formulae (Reuss Model)) ν 12 = Major Poisson s Ratio (Rule of Mixtures for Poisson s Ratio) G 12 = In-Plane Shear Modulus (Inverse Rule of Mixtures for Shear)

23 Calculation of E 11 for Ply Using Rule of Mixtures E 11 = E f V f + E m V m E 11 = E 11 = 136 GPa

24 Calculation of E 22 for Ply Using Inverse Rule of Mixtures Formulae (Reuss Model) E 22 = 8 GPa

25 Calculate Poisson s Ratio (ν 12 ) for Ply Using Rule of Mixtures formula However, we do not know Poisson s ratio for carbon fibre Poisson s ratio for epoxy matrix We would need to find these for accurate prediction We will assume a Poisson s Ratio (ν) of 0.3

26 Calculate Shear Modulus (G 12 ) of Ply Using Inverse Rule of Mixtures formula G for carbon fibre = 52 GPa (from test) G for epoxy = 2.26 GPa (from test) Both calculated using standard shear modulus formula G = E/(2(1+ν)) G 12 for composite = 5 GPa

27 Resulting Properties of Ply E 11 = 136 GPa E 22 = 8 GPa ν 12 = 0.3 G 12 = 5 GPa

28 Matrix Representation 4 material elastic properties are needed to charaterise the in-plane behaviour of the linear elastic orthotropic ply We conveniently define these in terms of measured engineering constants (as above) These are usually expressed in matrix form due to large equations produced and subsequent manipulations required The stiffness matrix [Q] The compliance matrix [S] (inverse of stiffness)

29 Off-axis Orientation & Analysis The stiffness matrix is defined in terms of principal material directions, E 11, E 22 However, we need to analyse or predict the material properties in other directions As it is unlikely to be loaded only in principal direction We use stress transformation equations for this Related to Mohr s stress circle The transformation equations are written in matrix form They have nothing to do with the material properties, they are merely a rotation of stresses.

30 Single Ply [6 x 6] stiffness matrix [C] or [6 x 6] compliance matrix [S] Often reduced stiffness matrix [Q] for orthotropic laminates [3 x 3] Orthotropic = 3 mutually orthogonal planes of symetry 4 elastic constants characterise the behaviour of the laminate E1, E2, υ12, G12

31 Stiffness & Compliance Martricies Stiffness Matrix [Q] Calculates laminate stresses from laminate strains Compliance Matrix [S] Calculates laminate strains from laminate stresses (inverse of compliance)

32 Transformation Matrix The stress transformation equation that relates known stresses in the z, y coordinate system to stresses in the L, T coordinate system. These are related to the transformation performed using Mohr s stress circle.

33 Transformed Stiffness Components

34 Transformed Compliance Components

35 Individual Compliance & Stiffness terms

36 CLA Derived Formula Formula from Engineering Design with Polymers and Composites, J.C.Gerdeen et al. Suitable for calculating Young s Modulus of a uni-directional ply at different angles.

37 Example Material (θ = 0 ) E 11 = 136 GPa, E 22 = 8 GPa ν 12 = 0.3, G 12 = 5 GPa 1/E x = 1/ E x for 0 fibres = 136 GPa

38 Example Material (θ = 45 ) E 11 = 136 GPa, E 22 = 8 GPa ν 12 = 0.3, G 12 = 5 GPa 1/E x = 0.25/ /8 + ( 1/5-0.6/136 ) /E x = 0.25/ /8 + ( 1/5-0.6/136 ) 0.25 E x for 45 fibres = 12.2 GPa

39 Example Material (θ = 90 ) E 11 = 136 GPa, E 22 = 8 GPa ν 12 = 0.3, G 12 = 5 GPa 1/E x = 0 + 1/8 + 0 E x for 90 fibres = 8 GPa

40 Properties of Laminate CFRE (0/0/0/+45/-45/0/0/0) 0 = 136 GPa 45 = 12.2GPa 90 = 8 GPa Laminate in X-direction (6/8 136) + (2/8 12.2) = 105 GPa Laminate in Y-direction (6/8 8) + (2/8 12.2) = 9 GPa Note: This simplified calculation ignores the effect of coupling between extension and shear

41 Classical Laminate Analysis The simplified laminate analysis approach taken ignores the effect of coupling between extension and shear Classical Laminate Analysis takes full account of this effect However, this is too long and complex for hand calculations Therefore build a spreadsheet or use software such as LAP (Laminate Analysis Programme)

42 Summary of Results For HexPly M21 Pre-preg with (0/0/0/+45/-45/0/0/0) layup ROM & Efficiency factor Hart-Smith 10% Rule ROM of CLA rotation formula Classical Laminate Analysis LAP E x E y

43 Tensile Modulus (GPa) Prediction of Tensile Modulus CLA ROM 80 Hart-Smith Angle (degrees)

44 Procedure for Structural Analysis Classical Laminate Analysis to provide Relationship between in-plane load and strain Relationship between out of plane bending moments and curvatures Plate properties from the A, B and D matrices Then analysis is equivalent to classical analysis of anisotropic materials FE analysis Standard formulas to check stiffness

45 Material Property Comparison Young s Modulus of materials Isotropic Materials Aluminium = 70 GPa Steel = 210 GPa Polymers = 3 GPa Fibres Carbon = 240 GPa common high strength T700 fibre Glass = 70 GPa

46 Care with Property Comparison E for Aluminium = 70 GPa E for Glass fibre = 70 GPa However. 50% fibre volume fraction E now 35 GPa in x-direction E in y-direction = matrix = 3 GPa 0/90 woven fabric 50% of material in each direction E now 17.5 GPa in x and y direction E at 45 = 9 GPa Glass reinforced polymer composite now has a low stiffness compared to Aluminium!

47 Stress (MPa) 2000 UD CFRE 50% Fv (E = 120 GPa) CFRE is elastic until ultimate failure (no plasticity) /90 CFRE 50% Fv (E = 60 GPa) Elasticity until failure (no plasticity) 1000 Quasi-Isotropic CFRE 50% Fv (E = 45 GPa) Elasticity until failure (no plasticity) 500 Yield Point - Elastic to here Aluminium (E = 70 GPa) Elastic until yield point, then followed by large range of plasticity. Design to yield - therefore beneficial plasticity safety zone. Strain

48 References & Bibliography Engineering Design with Polymers and Composites J.C.Gerdeen, H.W.Lord & R.A.L.Rorrer Taylor and Francis, 2006 Engineering Composite Materials Bryan Harris, Bath University Composite Materials - UWE E-learning resource David Richardson, John Burns & Aerocomp Ltd.

49 Contact Details Dr David Richardson Room 1N22 Faculty of Engineering and Technology University of the West of England Frenchay Campus Coldharbour Lane Bristol BS16 1QY Tel:

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