In a pickup basketball game, your friend cracked one of his...

In a pickup basketball game, your friend cracked one of his teeth in a collision with another player while attempting to make a basket. To correct the problem, his dentist placed a steel band of initial internal diameter $4.4 \mathrm{~mm},$ and a crosssectional area of width $3.5 \mathrm{~mm},$ and thickness $0.45 \mathrm{~mm}$ on the tooth. Before placing the band on the tooth, he heated the band to $70 .{ }^{\circ} \mathrm{C}$. What will be the tension in the band once it cools down to the temperature in your friend's mouth $\left(37^{\circ} \mathrm{C}\right) ?$

Key Concepts

Stress-Strain Relationship and Young's Modulus

The stress-strain relationship defines how a material deforms under applied forces and is characterized by Young's modulus for linear elastic materials. Young's modulus provides a measure of the stiffness of the material, indicating how much stress is needed to produce a given strain. In thermal stress problems, the strain resulting from temperature changes is converted into stress using Young's modulus, thereby allowing for the calculation of forces such as tension in the material.

Thermal Expansion and Contraction

This concept deals with how materials change their dimensions in response to temperature variations. When a material is heated, it expands; when it is cooled, it contracts. This behavior is quantified by the material's coefficient of thermal expansion, which relates the rate of dimensional change to a change in temperature. In many practical applications, such as fitting a metal band onto a tooth or other components, accounting for thermal expansion or contraction is crucial to ensure a proper fit and secure assembly.

Thermal Stress

Thermal stress arises when there is a constraint that limits the natural expansion or contraction due to temperature changes. If a component is forced to maintain a dimension that differs from its free thermal expansion or contraction, internal stresses will develop. In the context of the problem, the steel band contracting upon cooling but remaining tight around the tooth results in tension, which is a form of thermal stress.

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