Rule of Mixtures
turesRule of Mixtures Composite stiffness can be predicted using a micro-mechanics approach termed the rule of mixtures. Assumptions 1. 2. 3. 4. 5. 6. Fibers are uniformly distributed throughout the matrix. Perfect bonding between fibers and matrix. Matrix is free of voids. Applied loads are either parallel or normal to the fiber direction. Lamina is initially in a stress-free state (no residual stresses). Fiber and matrix behave as linearly elastic materials.
Longitudinal Modulus Equal strain assumption: εc = εf = εm 2
Matrix Fiber
σ1
1
σ1
Matrix L Representative Volume Element (RVE) ΔL
Static equilibrium requires that the total resultant force on the element must equal the sum of the forces acting on the fiber and matrix.
σ C1 Ac = σ F 1 AF + σ M 1 AM σ C1 = σ F 1
AF A + σ M1 M AC AC
Where, AC, AF, AM are composite, fiber, and matrix cross sections. Then, we can also say that: VF = AF/AC and VM = AM/AC where VF and VM are volume fractions and not volumes of fiber and matrix.
Invoking Hooke’s Law, we get EC1εC1 = EF1
εF1VF + EM1 εM1VM
Then, to have strain compatibility, we have to assume that the average strains in the composite, fiber, and matrix along the 1-direction are equal. Therefore, EC1 = EF1 VF + EM1 VM = EF1 VF + EM1 (1-VF) -- Parallel combination rule of mixtures
Therefore, the fraction of load carried by fibers in a unidirectional continuous fiber lamina is
EF VF EM
σ F VF PF E F VF = = = PC σ F V F + σ M (1 − V F ) E F V F + E M (1 − V F ) E F VF + (1 − VF ) EM
If you assume wood-plastic composite and that EF is 1,000,000 psi and Em is 175,000 psi (HDPE for example), then
PF 6V F = PC 6V F + (1 − V F )
Then, PF/PC = 0.6 or 60% Strength – Unidirectional Continuous Fiber Lamina: In general, fiber failure strain is lower than the matrix failure strain. Assuming all fibers have the same strength, the tensile rupture of fibers will determine the rupture in the composite. Therefore, estimation of longitudinal
Longitudinal Modulus Equal strain assumption: εc = εf = εm 2
Matrix Fiber
σ1
1
σ1
Matrix L Representative Volume Element (RVE) ΔL
Static equilibrium requires that the total resultant force on the element must equal the sum of the forces acting on the fiber and matrix.
σ C1 Ac = σ F 1 AF + σ M 1 AM σ C1 = σ F 1
AF A + σ M1 M AC AC
Where, AC, AF, AM are composite, fiber, and matrix cross sections. Then, we can also say that: VF = AF/AC and VM = AM/AC where VF and VM are volume fractions and not volumes of fiber and matrix.
Invoking Hooke’s Law, we get EC1εC1 = EF1
εF1VF + EM1 εM1VM
Then, to have strain compatibility, we have to assume that the average strains in the composite, fiber, and matrix along the 1-direction are equal. Therefore, EC1 = EF1 VF + EM1 VM = EF1 VF + EM1 (1-VF) -- Parallel combination rule of mixtures
Therefore, the fraction of load carried by fibers in a unidirectional continuous fiber lamina is
EF VF EM
σ F VF PF E F VF = = = PC σ F V F + σ M (1 − V F ) E F V F + E M (1 − V F ) E F VF + (1 − VF ) EM
If you assume wood-plastic composite and that EF is 1,000,000 psi and Em is 175,000 psi (HDPE for example), then
PF 6V F = PC 6V F + (1 − V F )
Then, PF/PC = 0.6 or 60% Strength – Unidirectional Continuous Fiber Lamina: In general, fiber failure strain is lower than the matrix failure strain. Assuming all fibers have the same strength, the tensile rupture of fibers will determine the rupture in the composite. Therefore, estimation of longitudinal