Elastic Moduli

Introduction

When you apply a force to any solid, the solid will change shape under the force. We call this the deformation of solids.

But why, you ask, do solids like sponges seem to "bounce back" after being deformed? The answer lies in the elastic property of solids. Elasticity is a tendency of an object to return to its original shape after a deformation.

Formulas for the deformation of solids resemble the general formula:

stress = elastic modulus x strain

Young's Modulus

Young's Modulus relates the tensile stress on a solid to its tensile strain. It measures elasticity in length.

$F/A = Y * (\bigtriangleup L) /L_0$

where $F/A$ is the tensile stress, $Y$ is the Young's modulus of the solid, and $(\bigtriangleup L) / L_0$ is the tensile strain.

Shear Modulus

Shear Modulus happens when two forces are exerted on parallel surfaces in opposite directions. It measures elasticity of shape. Essentially, an object goes from becoming a rectangle to a parallelogram with no right angles. An object undergoes shear stress which results in shear strain.

$F_s / A = S * (\bigtriangleup X )/ L = S tan \theta$

where $F_s$ is the force exerted on one of the surfaces, $S$ is the shear modulus, and $\theta$ is the angle between the vertical plane and the slanted height of the sheared object.

Bulk Modulus

Bulk Modulus is elasticity in volume. It can also be applied to fluids, not only solids. When outward pressure pushes in on a solid/fluid, the solid undergoes volume stress, and with the bulk modulus one can calculate volume strain.

$\bigtriangleup P = - B * (\bigtriangleup V )/ V$

The negative sign is there to ensure a positive bulk modulus for each solid/fluid ( $\bigtriangleup P$ is positive, while $(\bigtriangleup V) /V$ is negative)

All three moduli are constant for a certain type of solid/fluid. For example, the Young's modulus of steel never changes.