Correct Answer - Option 4 : E = 3K(1 - 2μ)
Explanation:
For linear elastic, isotropic, and homogeneous material, We have relations
E = 2G (1 + μ)
E = 3K (1 – 2μ)
If we know any of two variables from E, G, K, and μ then we can find other variables using these relations.
Hence, If We need to express the stress-strain relationships completely for this material, at least any two of the four must be known.
Elastic Modulus (E)
When the body is loaded within its elastic limit, the ratio of stress and strain is constant. This constant is known as Elastic modulus or Young's Modulus.
\({\rm{E}} = \frac{{{\rm{Stress}}}}{{{\rm{Strain}}}} = \frac{{\rm{\sigma }}}{\epsilon}\)
Rigidity modulus (G)
When a body is loaded within its elastic limit, the ratio of shear stress and shear strain is constant, this constant is known as the shear modulus.
\({\rm{G}} = \frac{{{\rm{Shear\;stress\;}}}}{{{\rm{Shear\;strain}}}} = \frac{{\rm{\tau }}}{\phi }\)
Bulk modulus (K)
When a body is subjected to three mutually perpendicular like stresses of the same intensity then the ratio of direct stress and the volumetric strain of the body is known as bulk modulus
\({\rm{K}} = \frac{{{\rm{Direct\;stress}}}}{{{\rm{Volumetric\;strain}}}} = \frac{{\rm{\sigma }}}{{\frac{{{\rm{\delta V}}}}{{\rm{V}}}}}\)
The relationship between E, K, and, G is:
\({\bf{E}} = \frac{{9{\bf{KG}}}}{{3{\bf{K}} + \;{\bf{G}}}}\)
