The 50 -mm-diameter cylinder is made from Am 1004- T61...

The 50 -mm-diameter cylinder is made from Am $1004-$ T61 magnesium and is placed in the clamp when the temperature is $T_{1}=15^{\circ} \mathrm{C}$. If the two 304 -stainless-steel carriage bolts of the clamp each have a diameter of $10 \mathrm{mm}$ and they hold the cylinder snug with negligible force against the rigid jaws, determine the temperature at which the average normal stress in either the magnesium or the steel first becomes $12 \mathrm{MPa}$

Key Concepts

Constraint Conditions

This concept refers to the mechanical restrictions imposed by fixtures or contacts (such as clamps and bolts) that prevent free expansion or contraction of a component. These constraints lead to the development of internal forces and stresses when the material undergoes thermal expansion or contraction.

Elastic Modulus (Young’s Modulus)

This is a measure of a material's stiffness and is used in Hooke’s law to relate stress and strain. In the context of thermal loading, the elastic modulus helps in quantifying the stress developed when thermal strains are induced and the material is confined from expanding freely.

Thermal Stress

Thermal stress is the internal stress that a material develops when it is prevented from expanding or contracting freely due to temperature changes. It is determined by the product of the material's elastic modulus and the thermal strain, highlighting how constraints lead to mechanical forces in heated or cooled components.

Coefficient of Thermal Expansion

This property indicates the rate at which a material expands or contracts per degree change in temperature. It plays a critical role in problems involving thermal stresses, especially when different materials with varying coefficients are combined, as differential expansion can lead to significant internal stresses.

Thermal Expansion

This concept describes how a material's dimensions change in response to temperature variations, typically quantified by the linear expansion formula ?L = ?L?T. It is fundamental in understanding how the geometry of components changes with temperature and is directly related to the induced strains when a material is constrained.

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