The assembly has the diameters and material indicated. If it...

The assembly has the diameters and material indicated. If it fits securely between its fixed supports when the temperature is $T_1=70^{\circ} \mathrm{F}$, determine the average normal stress in each material when the temperature reaches $T_2=110^{\circ} \mathrm{F}$.

Key Concepts

Thermal Stress

Thermal stress is the stress developed in a material when its temperature changes while its expansion or contraction is restrained. When the free thermal expansion is prevented by fixed supports or connections, the resulting internal forces generate a stress within the material. This concept is critical in analyzing the performance of bonded or fixed assemblies subjected to temperature variations, as the induced stresses can impact the structural integrity and service life of the component.

Modulus of Elasticity

The modulus of elasticity, or Young's modulus, is a material property that describes the relationship between stress and strain in the elastic deformation region of the material. It is used to calculate the average normal stress that develops when a material is constrained and subjected to thermal expansion or contraction. A higher modulus indicates that a material is stiffer and will develop higher stress for a given level of strain.

Thermal Expansion

Thermal expansion refers to the tendency of a material's dimensions to change in response to a change in temperature. As a material is heated, its particles vibrate more and tend to move apart, leading to an increase in size unless constrained. This concept is central to understanding how temperature changes can affect the fit of mechanical assemblies and induce stresses when parts are prevented from expanding freely.

Coefficient of Thermal Expansion

The coefficient of thermal expansion is a material property that quantifies the rate at which a material expands per degree change in temperature. It is used to calculate the thermal strain, or the change in length per unit length, experienced by a material when its temperature changes. This parameter is fundamental in predicting how different materials will react under thermal loads and in designing structures that can tolerate temperature variations.

Average Normal Stress

Average normal stress is defined as the internal force per unit area acting perpendicular to a given cross-section of a material. In the context of thermal expansion, when an assembly is fixed between supports, the thermal strain is converted to stress, and the average normal stress can be calculated using the product of the modulus of elasticity and the constrained thermal strain (which accounts for the temperature change and the coefficient of thermal expansion).

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