$3\mathrm{.}21.$ A member whose material properties remain unchanged $($invariant$)$ under rotations of $90$ about axes $(x,y,z)$ is called a cubic material relative to axes

$(x,y,z)$ and has three independent elastic coefficients $(C_1,C_2,C_3).$ Its stress$-$strain relations relative to axes $(x,y,z)$ are $($a special case of Eq$.3\mathrm{.}50$

${}_{}=C_{1}lo{n}_{}+C_{2}lo{n}_{yy}+C_{2}lo{n}_{zz}$

${}_{yy}=C_{2}lo{n}_{}+C_{1}lo{n}_{yy}+C_{2}lo{n}_{zz}$

${}_{zz}=C_{2}lo{n}_{}+C_{2}lo{n}_{yy}+C_{1}lo{n}_{zz}$

${}_{xy}=C_3{}_{xy}$

${}_{xz}=C_3{}_{xz}$

${}_{yz}=C_3{}_{yz}$

Although in practice aluminum is often assumed to be an isotropic material $(E=72$GPa and $v=0\mathrm{.}33),$ it is actually a cubic material with $C_1=103$GPa,$C_2=55$

GPa $,$ and $C_3=27\mathrm{.}6$GPa. At a point in an airplane wing, the strain components are $lo{n}_{}=0\mathrm{.}0003,lo{n}_{yy}=0\mathrm{.}0002,lo{n}_{zz}=0\mathrm{.}0001,lo{n}_{xy}=0\mathrm{.}00005,$ and $lo{n}_{xz}=lo{n}_{yz}=0.$

a$.$ Determine the orientation of the principal axes of strain.

b$.$ Determine the stress components.

c$.$ Determine the orientation of the principal axes of stress.

d$.$ Calculate the stress components and determine the orientation of the principal axes of strain and stress under the assumption that the aluminum is

isotropic.