1) demonstrate the mixture law for a UD composite in order to find its longitudinal young modulus El. start form the displacements and the forces. consider the true unidirectional material as an assembly of two plates perfectly glued together.2) plot El as a function of reinforcement volume ratio vf. data Ef=70Gpa; Em=3GPa3) what is the maximum possible value for Vf for a UD? what is the common minimum value for Vf for a UD (Vf: volumic ratio of reinforcement in the composite)?

1. The mixture law for a unidirectional (UD) composite to find its longitudinal Young's modulus (El) can be demonstrated as follows:

The UD composite can be considered as an assembly of two plates perfectly glued together, one being the matrix (m) and the other being the fiber (f).

The total force (F) acting on the composite is the sum of the forces acting on the matrix (Fm) and the fiber (Ff).

F = Fm + Ff

The displacement (d) experienced by the composite is the same for both the matrix and the fiber.

d = dm = df

The stress (σ) is the force divided by the cross-sectional area (A), and the strain (ε) is the displacement divided by the original length (L).

σ = F/A and ε = d/L

The Young's modulus (E) is the stress divided by the strain.

E = σ/ε

For the composite, the total cross-sectional area (Ac) is the sum of the cross-sectional areas of the matrix (Am) and the fiber (Af), and the total original length (Lc) is the same for both the matrix and the fiber.

Ac = Am + Af and Lc = Lm = Lf

Substituting these into the equations for stress and strain gives:

σc = F/Ac and εc = d/Lc

Substituting these into the equation for Young's modulus gives:

Ec = σc/εc = (F/Ac) / (d/Lc)

This can be rearranged to give:

Ec = (F/d) * (Lc/Ac)

Substituting the expressions for F and Ac gives:

Ec = (Fm + Ff) / d * Lc / (Am + Af)

Substituting the expressions for Fm and Ff (Fm = Em * Am * εm and Ff = Ef * Af * εf) gives:

Ec = (Em * Am * εm + Ef * Af * εf) / d * Lc / (Am + Af)

Since εm = εf = εc, this simplifies to:

Ec = (Em * Vm + Ef * Vf)

where Vm and Vf are the volume fractions of the matrix and the fiber, respectively.

2. To plot El as a function of the reinforcement volume ratio (Vf), you would use the equation derived above. Given that Ef = 70 GPa and Em = 3 GPa, the equation becomes:

El = 3 * (1 - Vf) + 70 * Vf

This is a linear equation, with El increasing as Vf increases.

3. The maximum possible value for Vf for a UD composite is 1, which would mean the composite is entirely made up of the fiber. The common minimum value for Vf for a UD composite is typically around 0.1, which would mean the composite is 10% fiber and 90% matrix.