Problem $1.$

Please provide short answers.

$($a$)$ How many independent elastic constants are required to describe an $($i$)$ isotropic, linearly elastic material and $($ii$)$ a general anisotropic, linearly elastic material?

$($b$)$ A structural member made of a high strength steel alloy with tensile yield strength of $700$ MPa is loaded in tension. The principal stresses developed in a critical location at the root of a notch are $150$MPa,$750$MPa and $-150$ MPa $.$ Will the component yield at this critical location? Show your work and explain your answer.

$($c$)$ Derive the number of independent elastic constants for a material with a monoclinic unit cell. Hint: Consider that a monoclinic unit cell has one symmetry plane.