Young's Modulus

In 1807 Thomas Young developed a theory for characterizing material bodies deformed by external forces. Two decades later, C. Navier established a mathematical form of Young's theory in 1826. In one dimension, the stress ? is the force applied to a body divided by its area ? = F/A. Strain ?, a dimensionless factor, is the change in length ? L divided by the original length L ₀ or ? = ? L/ L ₀. Young's modulus, often referred to as the elastic modulus, is defined as the ratio of the stress divided by the strain

(12.1)

for a body in tension or compression. Young's modulus is an intrinsic property of a given material that is independent of its shape. The units of E are force divided by area N/m ² or Pa. The relation between Young's modulus and Hooke's law F = k ? L relating the linear stretch of a body with force constant k under a given load for sufficiently small ? L may be obtained considering

(12.2)

Solving for F gives

(12.3)

so that the familiar spring constant is Young's modulus times the cross-sectional area divided by the initial length.

Shear Modulus

The shear modulus S is defined by the shear stress ? _? = F _?/ A divided by the shear strain ? _? = ? in radians