Rule of mixtures - Prof. Supriya PDF

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Rule of mixtures Consider a unidirectional composite such as the one shown in Fig Assume that plane sections of this composite remain plane after deformation. Let us apply a force Pc in the fiber direction. Now, if the two components adhere perfectly and if they have the same Poisson’s ratio, then each component will undergo the same longitudinal elongation, Δl. Thus, we can write for the strain in each component εf= εm= εcl= Δl/l ----------------(7) where εcl is the strain in the composite in the longitudinal direction. This is called the isostrain or action-in-parallel situation.

If both fiber and matrix are elastic, we can relate the longitudinal stress σ in the two components to the longitudinal strain εl by Young’s modulus E. Thus σf =Ef εcl and σm = Ef εcl Let Ac be the cross-sectional area of the composite, A m, that of the matrix, and Af, that of all the fibers. The applied load on the composite, P c is shared between the fiber and the matrix. We can write P c = P f + P m σcl Ac = σf Af + σm Am -------------------(8) From equation 7 & 8 we get σcl Ac =( Ef Af + E m Am ) εcl or Ecl = σcl /εcl = E f Af / Ac+ Em Am / Ac Now for a given length of a composite Af / Ac = Vf and Am / Ac = V m Then the preceding equation expression can be simplified to Ecl = EfVf + EmVm = E11 -----------------------(9) Equation (9) is called rule of mixtures for Young’s modulus in the fiber direction. A similar expression can be obtained for the composite longitudinal strength from equation (8) namely σcl = σf Vf + σm Vm -----------------------(10) For properties in the transverse direction, we can represent the simple unidirectional composite by what is called the action-in-series or isostress situation.

In this case, we group the fibers together as a continuous phase normal to the stress. Thus, we have equal stresses in the two components For loading transverse to the fiber direction, we have σct = σf =σm while the total displacement of the composite in the thickness direction tc, is the sum of displacements of the components that is, Δtc= Δtm+ Δtf dividing throughout by tc , the gage length of the composite we obtain Δtc/tc = Δtm/tc+ Δtf/tc now Δtc/tc = εct strain in the composite in the transverse direction , while Δtm and Δtf equal the strains in the matrix and fiber times their respective gage lengths that is Δtm = εm tm+ Δtf = εf tf then εct =Δtc/tc = Δ tm tm/tmtc+Δtm tm/tftc or εct =εm tm /tc+ εftf /tc -----------------------(11) For given cross-sectional area of the composite under the applied load , the volume fractions of fiber and matrix can be written as Vm= tm /tc and Vf= tf /tc Substitute the value of V m & V f in equation -(11) εct =εm Vm + εm Vf -----------------------(12) considering both components to be elastic and remembering that σct = σf =σm in this case , we can write equation (12) as σct /Ect= (σct / E m) Vm + (σct / Ef) Vf or 1/ Ect = Vm/Em + V f/Ef = 1/ E22 -----------------------(13) the relationship given by equations 5,9,10,12,13 are commonly referred as rule of mixtures....