Young’s modulus, numerical constant, named for the 18th-century English physician and physicist Thomas Young, that describes the elastic properties of a solid undergoing tension or compression in only one direction, as in the case of a metal rod that after being stretched or compressed lengthwise returns to its original length. Young’s modulus is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression. Sometimes referred to as the modulus of elasticity, Young’s modulus is equal to the longitudinal stress divided by the strain. Stress and strain may be described as follows in the case of a metal bar under tension.

If a metal bar of cross-sectional area A is pulled by a force F at each end, the bar stretches from its original length L₀ to a new length L_n. (Simultaneously the cross section decreases.) The stress is the quotient of the tensile force divided by the cross-sectional area, or F/A. The strain or relative deformation is the change in length, L_n − L₀, divided by the original length, or (L_n − L₀)/L₀. (Strain is dimensionless.) Thus Young’s modulus may be expressed mathematically as

Young’s modulus = stress/strain = (FL₀)/A(L_n − L₀).

This is a specific form of Hooke’s law of elasticity. The units of Young’s modulus are newtons per square metre (N/m²). The value of Young’s modulus for aluminum is about 7.0 × 10¹⁰ N/m². The value for steel is about three times greater, which means that it takes three times as much force to stretch a steel bar the same amount as a similarly shaped aluminum bar.

The work W done in stretching the bar is stored as elastic potential energy U and is found by integrating the force F from L₀ to L_n. Setting L_n - L₀= ΔL,$W = \underset{0}{\overset{\Delta L}{\int }}\frac{YA\Delta L}{L_0}d\text{l} = \frac{YA{\left(\Delta L\right)}^2}{2L_0}$.The volume of the bar is AL, and from the original equation for Young’s modulus,W = 1/2 * stress * strain * volume.The work W is equal to the elastic potential energy U, and the elastic potential energy per unit volume is just ½ * stress * strain.

Young’s modulus is meaningful only in the range in which the stress is proportional to the strain, and the material returns to its original dimensions when the external force is removed. As stresses increase, the material may either flow, undergoing permanent deformation, or finally break.

When a metal bar under tension is elongated, its width is slightly diminished. This lateral shrinkage constitutes a transverse strain that is equal to the change in the width divided by the original width. The ratio of the transverse strain to the longitudinal strain is called Poisson’s ratio and has dimensionless units. The average value of Poisson’s ratio for steels is 0.28, and for aluminum alloys, 0.33. The volume of materials that have Poisson’s ratios less than 0.50 increase under longitudinal tension and decrease under longitudinal compression.

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