A steel wire is stretched with a stress of \(70 \mathrm{MPa}(10,000 \mathrm{psi})\) at \(20^{\circ} \mathrm{C}\left(68^{\circ} \mathrm{F}\right) .\) If the length is held constant, to what temperature must the wire be heated to reduce the stress to \(17 \mathrm{MPa}(2500 \mathrm{psi}) ?\) v

The thermal expansion coefficient is a fundamental property that describes how the size of a material changes with temperature. Different materials will expand or contract at different rates when heated or cooled, which is quantified by this coefficient, usually denoted as \( \alpha \).

For many solids, the expansion is linear, meaning it's directly proportional to the change in temperature. It is expressed in units of inverse temperature (usually per degree Celsius \( ^\circ\text{C}^{-1} \) or per Kelvin \( \text{K}^{-1} \) ). When a material is heated and its length is restrained, thermal stress can develop because the material cannot expand freely. This property helps us understand and calculate such stress under varying thermal conditions, as was necessary in the exercise with the steel wire.

Young's modulus, often represented by \( E \), is a measure of a material's stiffness and is a fundamental characteristic in the field of materials science and engineering. It describes the stress-strain relationship for materials exhibiting elastic behavior—at least within certain limits.

Defined as the ratio of stress (force per unit area) to strain (proportional deformation), Young's modulus has the units of pressure, usually expressed in pascals \( \text{Pa} \). In simple terms, a high value of \( E \), which steel possesses, signifies that a lot of force is needed to change its shape; thus, it is considered a \textbf{stiff} material. This concept is critical when calculating the necessary changes in temperature to alter the stress within a material, as it influences how much force is transferred into dimensional changes rather than deformation.

Stress is defined as the force applied over a cross-sectional area, and the resulting distortion or strain is how much a material deforms under that force. The relationship between stress and strain is often linear for small deformations, allowing for the calculation of changes in material length, area, or volume with formulas that apply Young's modulus and the thermal expansion coefficient.

An outstanding point to remember is that the relationship assumes the deformation is elastic—meaning the material will return to its original shape once the force is removed. If the stress exceeds a material's elastic limit, permanent deformation, or even failure, can occur. The exercise demonstrated this relationship by showing the calculations needed to determine the stress reduction in a wire through heating, under the assumption that the wire's deformation remained within its elastic range.