Approach: use the rule of mixtures equation to give the theoretical density

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Melissa Virginia Phillips
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1 Exercises Instructions: there is no time limit for these exercises. Work through as many of the questions that you can in the time available. 1. A polymer matrix composite was found to have a volume fraction of Assuming that the density of the polymer was 900 kg m -3 and that of the glass fibre reinforcement was 2600 kg m -3 calculate the theoretical density of the composite. Approach: use the rule of mixtures equation to give the theoretical density Remember that: V f = V fibre /V composite Answer: 2600*0,25+(1-0,25)*900 = 1325 kg m Using the equations you have learned about during the lectures, calculate the theoretical axial Young s modulus of a unidirectional carbon fibre reinforced-epoxy composite containing 45% by volume fibre. (Assume an axial Young s modulus for the fibre of 230 GPa and 3,5 GPa for the epoxy matrix.) Again using the rule of mixtures approach, you can calculate the Young s modulus of the composite from: Answer: 230*0,45+ (1-0,45)*3,5 = 105,43 GPa 3. Analysis of a flax fibre revealed that it had the following chemical composition and properties: Constituent V f (%) Axial modulus, E (GPa) Cellulose

2 Other polysaccharides 10 8 Lignin 5 4 The microfibril angle was measured to be 5 o relative to the fibre axis. Using a rule of mixtures approach, calculate the effective axial tensile modulus of the fibre assuming that the void content (lumen) of the fibre is 10% Here again you can use the rule of mixtures, but rather than having only 2 components, you have 3. You can calculate the Young s modulus of the fibre with a slightly modified form of the equation: First calculate the Young s modulus of the cell wall using the equation then allow for the fact that there is a void space in the middle of the fibre. This has the effect of reducing the real cross sectional area by 10%, so the final answer needs to account for this by multiplying by 0,9. Answer: Then Efibre = 0,85* ,1*8 + 0,05*4 = 114,9 GPa But, void content = 10% therefore effective modulus is 114,9*0,9 = 103 GPa Part A: 4. a) Derive an expression using the equal stress approach (i.e. Reuss model) for the dependence of the transverse stiffness of a unidirectional composite lamina on the transverse elastic properties of phases (reinforcement and matrix) as a function of volume fraction. b) Using the Halpin-Tsai equation for the transverse elastic properties of a composite laminate, calculate the transverse stiffness of a unidirectional glass fibre-unsaturated polyester laminate of 50% fibre volume fraction. Assume that the transverse stiffness of glass fibre is the same as the axial stiffness and equal to 76 GPa and that of the unsaturated polyester matrix to be 3 GPa. Also assume = 1. Comment on the difference between the values for the transverse moduli calculated by the different methods First assume that the composite is composed of a block of reinforcement and a block of matrix (c in the diagram)

(Source: DoITPoMS - TLP Library Mechanics of Fibre-Reinforced Composites - www.doitpoms.ac.uk) Then, the overall strain in the composite will be given by the rule of mixtures (once again): i.e. Composite strain = Vf x reinforcement strain + (1-Vf) x matrix strain You can then use Hooke s law (E= stress/strain) to replace strain (i.

3 (Source: DoITPoMS - TLP Library Mechanics of Fibre-Reinforced Composites - Then, the overall strain in the composite will be given by the rule of mixtures (once again): i.e. Composite strain = Vf x reinforcement strain + (1-Vf) x matrix strain You can then use Hooke s law (E= stress/strain) to replace strain (i.e. strain = stress/e) and since the stress is the same in the reinforcement, matrix and composite you can obtain an expression for the composite s Young s modulus in terms of the Young s moduli and volume fractions (f)of the constituents: Part B: Use the Halpin-Tsai equation given in the lecture slides. The difference between the two models is due to the fact that the matrix and reinforcement work solely in series in the equal stress model (and so gives an underestimate of composite stiffness, whereas in the Halpin-Tsai model, an attempt is made to account for the geometric effects of the fibre on the matrix. Answers: Reuss model: 5,77 GPa Halpin-Tsai model: 8,15 GPa 5. A chopped strand mat (CSM) glass fibre reinforced vinyl ester laminate was found to have a fibre volume fraction of 0,2. Assuming that the axial and transverse Young s modulus of the glass fibre is the same and equal to 76 GPa, and that of the vinyl ester matrix is 3 GPa, estimate the in-plane stiffness of the laminate. (Hint: you will need to use the Halpin-Tsai equation to calculate E2, the transverse stiffness of the laminate. When doing this assume that = 1)

4 Simply use the Rule of mixtures and Halpin-Tsai equations to calculate the axial and transverse Young s moduli and then apply to the following equation to get an estimate of the in-plane stiffness of the random (CSM) laminate: Answer: approx. 9,3 GPa 6. A unidirectional composite is composed of the materials listed in the table below. Assuming the fibre volume fraction id 0.6, what is the theoretical tensile strength of the composite? What assumptions have you made? Young s modulus (GPa) Tensile strength (MPa) Strain to failure (%) E glass fibre Unsaturated polyester In this case as the failure strain of the matrix is lower than that of the reinforcement, the axial tensile strength is given by the volume weighted strength of the fibres: Answer: 1200 MPa 7. You are designing a tensile member and your calculations have shown that the maximum permissible tensile deformation over the 1 metre length of the member is 1 mm when it is subjected to a load of 5 kn. You have decided to use a unidirectional high strength (HS) carbon fibre-reinforced epoxy composite for the member and after consultation with the manufacturer you have learned that the maximum fibre volume fraction can be 0.6. Calculate the cross sectional area of the member. (Assume E1 for the carbon fibre to be 230 GPa, tensile strength 3400 MPa. Stiffness of the epoxy is 5 GPa) Calculate the allowable strain - i.e. 1/1000 From the rule of mixtures you can calculate the composite Young s modulus

5 As you know the strain, you can calculate the stress in the composite at that strain As you know the stress and the load (5 kn) you can simply calculate the cross-sectional area from the relationship stress = F/A Answer: 35,72 mm 2 8. Using one of the many laminate programs available on the web (the program provided by Prof. John Nairn at Oregon State University, is a good one: determine the axial and transverse (E11 and E22) stiffness s of: 1. A unidirectional carbon fibre reinforced epoxy composite of 0,6 fibre volume fraction 2. A three layer cross-ply laminate based on a carbon fibre reinforced epoxy composite of 0,6 fibre volume fraction (i.e. construction 0/90/0) 3. A nine layer cross-ply laminate based on a carbon fibre reinforced epoxy composite of 0,6 fibre volume fraction (i.e. construction 0/90/0/90/0/90/0/90/0) What do you notice about the stiffness as the number of plies increases? Have a go with some other material combinations!! Mark Hughes, 23 rd March 2016

6 Appendix Halpin-Tsai model

Assignment #1 Due Date 22 September 2009; 5PM Group I: Odd questions, Group II : Even questions

Assignment #1 Due Date 22 September 2009; 5PM Group I: Odd questions, Group II : Even questions 1. A burnout test was performed to determine the volume fraction of constituents in a glass-fibre- reinforced