The band in Figure P 10.23 is stainless steel (coefficient...

The band in Figure $\mathrm{P} 10.23$ is stainless steel (coefficient of linear expansion $=17.3 \times 10^{-6 \circ} \mathrm{G}^{-1}$; Young's modulus $\left.=18 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}\right) .$ It is essentially circular with an initial mean radius of $5.0 \mathrm{~mm}$, a height of $4.0 \mathrm{~mm}$, and a thickness of $0.50 \mathrm{~mm}$. If the band just fits snugly over the tooth when heated to a temperature of $80^{\circ} \mathrm{C}$, what is the tension in the band when it cools to a temperature of $37^{\circ} \mathrm{C}$ ?

The band in Figure $\mathrm{P} 10.23$ is stainless steel (coefficient of linear expansion $=17.3 \times 10^{-6 \circ} \mathrm{G}^{-1}$; Young's modulus $\left.=18 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}\right) .$ It is
essentially circular with an initial mean radius of $5.0 \mathrm{~mm}$, a height of $4.0 \mathrm{~mm}$, and a thickness of $0.50 \mathrm{~mm}$. If the band just fits snugly over the tooth when heated to a temperature of $80^{\circ} \mathrm{C}$, what is the tension in the band when it cools to a temperature of $37^{\circ} \mathrm{C}$ ?

Key Concepts

Geometric Relationships in Circular Structures

Circular structures, such as rings or bands, have a circumference directly proportional to their radius. Understanding this geometrical relationship is key when considering thermal expansion or contraction, as changes in temperature alter the circumference and, when constrained, produce tension or stress in the material.

Hooke's Law

Hooke's Law describes the linear relationship between stress and strain in the elastic deformation regime of a material. It provides the basis for relating the thermal strain (due to temperature changes) to the resulting mechanical stress when the material's deformation is restrained.

Thermal Expansion

Thermal expansion is the increase or decrease in the dimensions of a material as its temperature changes. In mechanics of materials, this concept is described by the coefficient of linear expansion, which quantifies how much a unit length of the material expands per degree of temperature change.

Thermal Strain

Thermal strain is the relative change in size or shape of a material due to a temperature change, calculated by multiplying the coefficient of thermal expansion by the change in temperature. When a component is constrained from expanding or contracting freely, this thermal strain can lead to the development of internal stresses.

Young's Modulus

Young's modulus is a fundamental property that measures a material's stiffness. It is defined as the ratio of stress to strain in the elastic region of the material's behavior. This modulus is used to calculate the stress induced by deformations, including those arising from thermal effects.

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