Elastic Constants and Relationships

Flashcards covering the definitions, concepts, numerical parameters, and mathematical formulas for elastic constants from Mechanics of Solids (BCLE203L) Module 2.

Elastic Constants

Parameters used to describe the behavior of a material's stress and strain relationship, including Young's Modulus ($E$), Bulk Modulus ($K$), Shear Modulus ($C$, $G$, or $N$), and Poisson's ratio ($\mu$).

Bulk Modulus ($K$)

The ratio of direct stress to volumetric strain when a body is subjected to 3 mutually perpendicular stresses of equal intensity, expressed mathematically as $K = \frac{\sigma}{\frac{\delta V}{V}}$.

Shear Stress

The stress induced when a section is subjected to two equal and opposite forces acting tangentially across the resisting section, causing the body to tend to shear off across the section.

Shear Modulus

The constant of proportionality equal to the ratio of shear stress to shear strain, also known as the Modulus of Rigidity ($C$, $G$, or $N$).

Volumetric Strain Formula

For a cube of side $l$ subjected to mutually perpendicular tensile stresses $\sigma$, the volumetric strain is expressed as $\frac{\delta V}{V} = 3\frac{\sigma}{E}(1 - 2\mu)$.

Relationship between Young's Modulus ($E$) and Bulk Modulus ($K$)

The formula relating Young's Modulus and Bulk Modulus, given by $E = 3K(1 - 2\mu)$.

Distortion under Shear Stress

The structural deformation in a body of length $l$ subjected to shear stress $\tau$, causing diagonal $BD$ to elongate to $BD_1$ and diagonal $AC$ to shorten to $AC_1$.

Linear Tensile Strain of Diagonal $BD$

The strain on diagonal $BD$ due to shear stress $\tau$, which is equal to half of the shear strain ($\frac{\tau}{2C}$).

Relationship between Young's Modulus ($E$) and Shear Modulus ($C$)

The formula relating Young's Modulus and Shear Modulus, given by $E = 2C(1 + \mu)$.

Poisson's Ratio in Terms of $K$ and $C$

The formula for Poisson's ratio ($\mu$) derived by equating the relations of Young's Modulus, expressed as $\mu = \frac{3K - 2C}{2C + 6K}$.

Combined Moduli Equation

The single equation relating Young's Modulus ($E$), Bulk Modulus ($K$), and Shear Modulus ($C$), given by $E = \frac{9KC}{C + 3K}$.

Example 7.9 Parameters

A problem scenario involving an alloy body with a modulus of elasticity of $150\,GPa$ and a Poisson's ratio of $0.25$ to determine its bulk modulus.

Example 7.10 Parameters

A problem scenario involving a material with Young's modulus of $120\,GPa$ and modulus of rigidity of $40\,GPa$, applied to a round bar of $50\,mm$ diameter and $2.5\,m$ length stretched by $2.5\,mm$ with Poisson's ratio $0.25$.

Example 7.12 Parameters

An experiment involving a bar of $30\,mm$ diameter subjected to a pull of $60\,kN$, with a measured extension of $0.09\,mm$ over a $200\,mm$ gauge length and a change in diameter of $0.0039\,mm$.