Bulk Stress Due to a Temperature Increase. (a) Prove that,...

Bulk Stress Due to a Temperature Increase. (a) Prove that, if an object under pressure has its temperature raised but is not allowed to expand, the increase in pressure is $$\Delta p=B \beta \Delta T$$ where the bulk modulus $B$ and the average coefficient of volume expansion $\beta$ are both assumed positive and constant. (b) What pressure is necessary to prevent a steel block from expanding when its temperature is increased from $20.0^{\circ} \mathrm{C}$ to $35.0^{\circ} \mathrm{C} ?$

Bulk Stress Due to a Temperature Increase. (a) Prove that, if an object under pressure has its temperature raised but is not allowed to expand, the increase in pressure is
$$\Delta p=B \beta \Delta T$$
where the bulk modulus $B$ and the average coefficient of volume expansion $\beta$ are both assumed positive and constant. (b) What pressure is necessary to prevent a steel block from expanding when its temperature is increased from $20.0^{\circ} \mathrm{C}$ to $35.0^{\circ} \mathrm{C} ?$

Key Concepts

Bulk Modulus

The bulk modulus is a material property that quantifies a substance's resistance to uniform compression; it represents how much pressure is needed to achieve a given fractional decrease in volume. In problems involving thermal stresses, it serves as the proportionality constant relating pressure changes to relative volume changes when the object is prevented from expanding.

Coefficient of Thermal Expansion

The coefficient of thermal expansion describes how the size of a material changes with temperature. Specifically, the volume expansion coefficient (often denoted ?) indicates the fractional change in volume per unit temperature change, assuming the material is free to expand. This concept is crucial for understanding thermal deformation in materials under heating.

Thermal Stress

Thermal stress refers to the internal stress induced in a material when its temperature changes while its dimensions are constrained. Because the material cannot expand or contract freely, the imposed temperature change leads to the build-up of internal forces. This stress can be quantified in terms of the material's elastic properties (such as the bulk modulus) and its thermal expansion characteristics.

Constrained Thermal Expansion

When an object's temperature increases but its expansion is mechanically restricted, the natural tendency of the material to expand is opposed by the constraints. This scenario is central to the derivation showing that the induced pressure is proportional to both the bulk modulus and the thermal expansion coefficient, as the resulting pressure is effectively the reaction force per unit area required to maintain the original volume.

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