$(30$ pts$)$

Consider the following composite materials: tungsten phase in a copper phase and carbon fiber phase in an epoxy phase. You create a $80\%$ carbon$-$fiber composite. Calculate the carbon$-$fiber composite's longitudinal and transverse elastic modulus $($or stiffness, in GPa$).$

Can you create a tungsten$-$copper composite with an upper$-$limit stiffness equal to the longitudinal stiffness of the carbon$-$fiber composite? If so$,$ what are the volume fractions in the tungsten$-$copper composite? Also, what is the tungsten$-$copper composite's lower limit stiffness?

Calculate both composite densities $($in $\frac{g}{m}l).$ Calculate the stiffness$/$density ratio for both elastic moduli of each composite $($stiffness divided by density$).$ Which composite has the highest stiffness$/$density ratio in each direction? Compare the longitudinal stiffness$/$density ratio to the upper$-$limit stiffness$/$density ratio, and the transverse stiffness$/$density ratio to the lower$-$limit stiffness$/$density ratio.

Reminder: In a particle$-$matrix composite, the upper$-$limit stiffness is the same equation as the longitudinal stiffness equation and the lower$-$limit stiffness is the same equation as the transverse stiffness equation $($replace "fiber $($f$)"$ with "particle $($p$)").$

$\backslash$table$[[,,,,,$Modulus of$],[,$Density $($g$/$ml$),$Modulus of Elasticity $($GPa$),,$Density $(\frac{g}{m}l),\backslash$table$[[$Elasticity$],[[$GPa $($psi$)]]]],[$Copper$,8\mathrm{.}9,110,$Carbon fiber,$1\mathrm{.}80,260(37{10}^6)$