Problem Set $3\mathrm{.}2$

$3\mathrm{.}2\mathrm{.}1$ Use a mechanics of materials approach to determine the apparent Young's modulus

for a composite material with an 'inclusion' of arbitrary shape in a cubic element of

equal unit$-$length sides as in the representative volume element $($RVE$)$ of Figure

$3-17.$ Fill in the details to show that the modulus is

$E=\frac{}{}=\frac{\frac{F}{A}}{\frac{}{L}}=\frac{\frac{F}{1[L]1[L]}}{(1[L])}=\frac{F}{[L]}$

where $[L]$ represents units of length and can be written as

$\frac{1}{E}={\displaystyle \underset{0}{\overset{1}{}}}\frac{dx}{E_1+(E_2-E_1)A_2(x)}$

where $A_2(x)$ is the distribution of the inclusion. Note that the slice $dx$ long of the RVE

represents a microscopic portion of the RVE, l$.$e$.,$ recognize the difference between

what happens for a slice and what happens for the entire RVE. Use this result in

Problems $3\mathrm{.}2\mathrm{.}2$ through $3\mathrm{.}2\mathrm{.}4.$

$3\mathrm{.}2\mathrm{.}2$ Verify that the general expression for the modulus of a dispersion$-$stiffened com$-$

posite material reduces to

$\frac{E}{E_m}=\frac{E_m+(E_d-E_m)V_d^{\frac{2}{3}}}{E_m+(E_d-E_m)V_d^{\frac{2}{3}}[1-V_d^{\frac{1}{3}}]}$

for a cubic particle of modulus $E_d$ in a matrix with modulus $E_m.$ The volume fraction

of the cubic particles is $V_d$ and that of the matrix is $V_m$ or $1-V_d.$ Hint: the repre$-$

sentative volume element is a cube within a cube. Figure $3-17$ Particulate Reinforcement $($After Paul $[3-4])$Use a mechanics of materials approach to determine

that the general expression for the modulus of a dispersion$-$

stiffened composite material is given by

$\frac{E}{E_m}=\frac{E_m+(E_d-E_m)V_d^{\frac{2}{3}}}{E_m+(E_d-E_m)V_d^{\frac{2}{3}}[1-V_d^{\frac{1}{3}}]}$

for a cubic particle of modulus $E_d$ in a matrix with modulus $E_m.$

The volume fraction of the cubic particles is $V_d$ and the that of

the matrix is $V_{dm}$ or $1-V_d.$ The representative volume element

$($RVE$)$ is assumed to be a cube within a cube.