Density. Density 22/10/2015. Rules of Mixture for Elastic Properties. Materi ke-6

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1 //5 Materi ke-6 Rules o Mixture or lastic Properties Nurun nayiroh, M.Si Rules o Mixture' adalah ekspresi ateatika yang eberikan beberapa properti dari koposit dala hal siat, kuantitas dan penataan bahan penyusunnya. Rule o Mixture bisa didasarkan pada sejulah asusi yang sederhana, dan penggunaannya dala desain harus diaplikasikan dengan sangat hati-hati! MATRIAL KOMPOSIT Density Untuk koposit yang uu, volue total, terdiri dari asa penyusun M a, M b, M c,... Densitas kopositnya adalah: M ρ a Mb M c... M a M b... Dala hal densitas dan volue dari bahan penyusun, diperoleh: vaρa vbρb vcρc ρ... Density Tapi v a / a adalah raksi volue dari bahan penyusun a, sehingga: ρ ρ ρ ρ... a a b b c c Untuk kasus khusus pada atriks yang diperkuat dengan iber, aka: ρ ρ ρ ρ ( ρ ( ρ ρ ρ since

2 //5 Rule o ixtures density or glass/epoxy coposites Macroechanics v.s. Microechanics 3 5 ρ Macroechanics The study o coposite aterial behavior where aterial is assued hoogeneous Obtain average apparent properties o the aterial kg/ ρ Microechanics The study o coposite aterial behavior where interactions o constituent aterial is exained in detail phasize heterogeneity ibre volue raction Deinitions Heterogeneous -- aterial properties vary ro point to point Hoogeneous -- aterial properties are the sae everywhere Isotropic -- aterial properties are the sae in every direction Anisotropic -- aterial properties depend on direction Microechanical odels or stiness

3 //5 Unidirectional Ply (Lapisan searah Unidirectional ibres are the siplest arrangeent o ibres to analyse. They provide axiu properties in the ibre direction, but iniu properties in the transverse direction. Unidirectional ply We expect the unidirectional coposite to have dierent tensile oduli in dierent directions. These properties ay be labelled in several dierent ways: ibre direction, transverse direction, Unidirectional ply By convention, the principal axes o the ply are labelled,, 3. This is used to denote the act that ply ay be aligned dierently ro the cartesian axes x, y, z. 3 Unidirectional ply - longitudinal tensile odulus We ake the ollowing assuptions in developing a rule o ixtures: Fibres are unior, parallel and continuous. Perect bonding between ibre and atrix. Longitudinal load produces equal strain in ibre and atrix. 3

4 //5 Unidirectional ply - longitudinal tensile odulus A load applied in the ibre direction is shared between ibre and atrix: F F F The stresses depend on the cross-sectional areas o ibre and atrix: σ A σ A σ A where A ( A A is the total cross-sectional area o the ply Unidirectional ply - longitudinal tensile odulus Applying Hooke s law: ε A ε A ε A where Poisson contraction has been ignored But the strain in ibre, atrix and coposite are the sae, so ε ε ε, and: A A A Unidirectional ply - longitudinal tensile odulus Dividing through by area A: (A / A (A / A But or the unidirectional ply, (A / A and (A / A are the sae as volue ractions and -. Hence: (- Note the siilarity to the rules o ixture expression or density. In polyer coposites, >>, so tensile odulus (GPa Rule o ixtures tensile odulus (glass ibre/polyester ibre volue raction tensile odulus (GPa 5 5 CSM Rule o ixtures tensile odulus (T3 carbon ibre ibre volue raction quasi-isotropic 4

5 //5 Unidirectional ply - transverse tensile odulus For the transverse stiness, a load is applied at right angles to the ibres. The odel is very uch sipliied, and the ibres are luped together: atrix L ibre σ σ It is assued that the stress is the sae in each coponent (σ σ σ. Poisson contraction eects are ignored. L L σ σ L L The total extension is δ δ δ, so the strain is given by: ε L ε L ε L so that ε ε (L / L ε (L / L σ σ L L But σ σ σ, so that: But L / L and L / L - So ε ε ε (- and σ / σ / σ (- / ( or ( 5

6 //5 (GPa Rule o ixtures - transverse odulus (glass/epoxy ibre volue raction I >>, Note that is not particularly sensitive to. I >>, is alost independent o ibre property: / (- (GPa Rule o ixtures - transverse odulus ibre volue raction The transverse odulus is doinated by the atrix, and is virtually independent o the reinorceent. carbon/epoxy glass/epoxy The transverse rule o ixtures is not particularly accurate, due to the sipliications ade - Poisson eects are not negligible, and the strain distribution is not unior: (source: Hull, Introduction to Coposite Materials, CUP Stiness o short ibre coposites For aligned short ibre coposites (diicult to achieve in polyers!, the rule o ixtures or odulus in the ibre direction is: η ( L The length correction actor (η L can be derived theoretically. Provided L >, η L >.9 For coposites in which ibres are not perectly aligned the ull rule o ixtures expression is used, incorporating both η L and η o. 6

7 //5 In short ibre-reinorced therosetting polyer coposites, it is reasonable to assue that the ibres are always well above their critical length, and that the elastic properties are deterined priarily by orientation eects. The ollowing equations give reasonably accurate estiates or the isotropic in-plane elastic constants: 3 G 5 4 ν G where and are the values calculated earlier tensile odulus (GPa Rule o ixtures tensile odulus (glass ibre/polyester ibre volue raction tensile odulus (GPa 5 5 CSM Rule o ixtures tensile odulus (T3 carbon ibre ibre volue raction quasi-isotropic Rule o ixtures elastic odulus glass ibre / epoxy resin Rule o ixtures elastic odulus HS carbon / epoxy resin GPa ibre volue raction rando GPa ibre volue raction quasi-isotropic plain woven 7

8 //5 Rules o ixture properties or CSM-polyester lainates Larsson & liasson, Principles o Yacht Design Rules o ixture properties or glass woven roving-polyester lainates Larsson & liasson, Principles o Yacht Design olue Fraction in Large Particle Coposites lastic odulus is dependent on the volue raction Rule o ixtures equation - elastic odulus, - volue raction, - atrix, p- particulate upper bound (iso-strain lower bound (iso-stress c p p c p p p Rule o Mixtures Actual alues - atrix Upper bound * * * * * * * - particulate Lower bound conc. o particulates

9 //5 9 Other rules o ixtures Shear odulus: Poisson s ratio: Theral expansion: G G G ( ( ν ν ν [ ] ( ( ( ( ν ν ν