A cylindrical bar of aluminum 19 mm(0.75 in. ) in diameter is to be deformed elastically by app

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A cylindrical bar of aluminum 19 mm(0.75 in. ) in diameter...

A cylindrical bar of aluminum $19 \mathrm{mm}(0.75 \text { in. })$ in diameter is to be deformed elastically by application of a force along the bar axis. Using the data in Table 6.1 , determine the force that will produce an elastic reduction of $2.5 \times$ $10^{-3} \mathrm{mm}\left(1.0 \times 10^{-4} \mathrm{in.}\right)$ in the diameter.

Key Concepts

Poisson’s Ratio

Poisson’s Ratio is the ratio of the transverse strain (lateral contraction or expansion) to the axial strain, under uniaxial loading conditions. This concept explains how an axial load, which causes elongation or compression, leads to a proportional change in dimensions perpendicular to the applied load.

Young’s Modulus

Young’s Modulus is a material constant that measures the stiffness of a solid material, defined as the ratio of axial stress to axial strain in the elastic region. It is essential in determining how much a material will deform under a given load.

Cylindrical Geometry and Cross-Sectional Area

Understanding the geometry of a cylindrical bar is key to calculating its cross?sectional area, which is necessary for converting applied force into stress. The area, calculated from the diameter, directly influences the material’s response to the axial load in terms of stress distribution and deformation.

Elastic Deformation

Elastic deformation refers to the reversible change in shape of a material when an external force is applied, ensuring that once the force is removed, the material returns to its original dimensions. This concept is crucial in problems that involve small strains where the material's response remains in the elastic regime.

Hooke’s Law

Hooke’s Law states that within the elastic limit, the deformation (strain) of a material is directly proportional to the applied stress. This linear relationship is fundamental for calculating the stress or force required to achieve a particular strain in a component.

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