A cylindrical specimen of a nickel alloy having an elastic...

A cylindrical specimen of a nickel alloy having an elastic modulus of $207 \mathrm{GPa}\left(30 \times 10^{6}\right.$ psi) and an original diameter of $10.2 \mathrm{mm}$ $(0.40 \text { in. })$ will experience only elastic deformation when a tensile load of $8900 \mathrm{N}\left(2000 \mathrm{lb}_{\mathrm{f}}\right)$ is applied. Compute the maximum length of the specimen before deformation if the maximum allowable elongation is $0.25 \mathrm{mm}$ $(0.010 \text { in. })$.

A cylindrical specimen of a nickel alloy having an elastic modulus of $207 \mathrm{GPa}\left(30 \times 10^{6}\right.$ psi) and an original diameter of $10.2 \mathrm{mm}$ $(0.40 \text { in. })$ will experience only elastic deformation when a tensile load of $8900 \mathrm{N}\left(2000 \mathrm{lb}_{\mathrm{f}}\right)$
is applied. Compute the maximum length of the specimen before deformation if the maximum allowable elongation is $0.25 \mathrm{mm}$ $(0.010 \text { in. })$.

Key Concepts

Tensile Stress

Tensile stress is the internal force per unit area that develops in a material subjected to a stretching or pulling load. It is calculated by dividing the applied tensile force by the cross-sectional area over which the load acts. Understanding tensile stress is crucial for determining whether a material will behave elastically under an applied load.

Elastic Modulus

The elastic modulus (or Young’s modulus) is a material property that characterizes the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain (deformation relative to original dimensions) when the material deforms elastically, meaning that it will return to its original shape upon unloading.

Hooke’s Law

Hooke’s Law is the fundamental principle that relates stress and strain in the elastic region of a material’s deformation. It states that the stress applied to a material is directly proportional to the strain produced, with the elastic modulus acting as the constant of proportionality. This law is essential for calculating deformations under loads that remain within the elastic limit.

Axial Strain

Axial strain is the measure of deformation defined as the change in length of an object divided by its original length when subjected to tensile or compressive forces. It is a unitless quantity and is directly linked to tensile stress through Hooke’s Law in the elastic deformation region.

Cross-Sectional Area of a Cylinder

For cylindrical specimens, the cross-sectional area plays a key role in determining stress. The area is calculated using the formula for the area of a circle (?r²), where r is the radius of the cylinder. The accurate calculation of this area is critical for converting an applied force into stress and, subsequently, determining the resultant strain and elongation.

Allowable Elongation

Allowable elongation is the maximum permissible change in length that a material can experience without permanent deformation or failure. It is used as a design constraint to ensure that the material remains within the elastic range of deformation under applied loads. This concept is important in design calculations to ensure that deformations remain serviceable and reversible.

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