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PTCUL AE E&M 2017 Official Paper (Set B)

Option 4 : None of these

**Explanation:**

**Anisotropic Material:**

- In Anisotropic materials
**properties in different directions are different**and the normal strain depends on the shear strain. - The number of independent elastic constants for Anisotropic material is 21.

**Isotropic Material:**

- In Isotropic materials properties in different directions at a point do not vary (e.g. metals & glasses).
- For Isotropic materials number of Independent elastic constants in the elastic constant matrix is 2.
- In such materials, Ex, Ey, and Ez are all same. Similarly, G and μ along all directions are also the same.

Hence the independent elastic constants are either E and G or E and μ or G and μ and the third always depends on the other two defined by the relation :

\(G = \frac{E}{{2\left( {1 + \mu } \right)}}\)

Elastic constant Matrix for isotropic materials:

\(\left[ {\begin{array}{*{20}{c}} {{\varepsilon _x}}\\ {{\varepsilon _y}}\\ {{\varepsilon _z}}\\ {{\gamma _{xy}}}\\ {{\gamma _{yz}}}\\ {{\gamma _{zx}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{1}{E}}&{ - \frac{\mu }{E}}&{ - \frac{\mu }{E}}&0&0&0\\ { - \frac{\mu }{E}}&{\frac{1}{E}}&{ - \frac{\mu }{E}}&0&0&0\\ { - \frac{\mu }{E}}&{ - \frac{\mu }{E}}&{\frac{1}{E}}&0&0&0\\ 0&0&0&{\frac{1}{G}}&0&0\\ 0&0&0&0&{\frac{1}{G}}&0\\ 0&0&0&0&0&{\frac{1}{G}} \end{array}} \right]\;\left[ {\begin{array}{*{20}{c}} {{\sigma _x}}\\ {{\sigma _y}}\\ {{\sigma _z}}\\ {{\tau _{xy}}}\\ {{\tau _{yz}}}\\ {{\tau _{zx}}} \end{array}} \right]\)

Glass is an example of isotropic material. Hence the number of independent elastic constants is 2.

Orthotropic materials:

- Orthotropic materials properties in different directions are different and the normal strain does not depend on the shear strain. e.g. Wood
- For Orthotropic materials number of Independent elastic constant in the elastic constant matrix is 9.
- These are \({E_x},\;{E_y},\;{E_z},\;{\mu _x},\;{\mu _y},\;{\mu _z},\;{G_{xy}},\;{G_{xz}},\;{G_{yz}}\).