### Elastic Constants

• Summarize Hooke’s Law.

• Discuss engineering/elastic constants and their relationship to each other.

E, G, K and $\mu$ are called elastic constants because these are applicable only within elastic limit and their values are constant for a particular material within elastic limits.

## Modulus of Elasticity or Young’s Modulus:

A tensile test is generally conducted on a standard specimen to obtain the relationship between the stress and the strain which is an important characteristic of the material. In the test, the uniaxial load is applied to the specimen and increased gradually. The corresponding deformations are recorded throughout the loading. Stress-strain diagrams of materials vary widely depending upon whether the material is ductile or brittle in nature. If the material undergoes a large deformation before failure, it is referred to as ductile material or else brittle material.
Initial part of the loading indicates a linear relationship between stress and strain, and the deformation is completely recoverable in this region for both ductile and brittle materials.
This linear relationship, i.e., stress is directly proportional to strain, is popularly known as Hooke’s law. $E = \frac{\mathrm{stress}}{\mathrm{strain}}$ The co-efficient E is called the modulus of elasticity or Young’s modulus.
Most of the engineering structures are designed to function within their linear elastic region only.

## Relation between Elastic Constants:

$E = 2G (1 + \nu)$ $E = 3K (1 - 2\nu)$ $E = \dfrac{9KG}{(3K+G)}$

#### Solved Examples

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