Mechanics of Materials: Hoop stress, the relationship between circumferential stress and internal forces, thermal expansion, and the difference between expansion due to forces and expansion under stress
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Dear teachers, I have a basic question regarding mechanics of materials. As shown in Figure 1, when a tank is under internal pressure P, any infinitesimal element in the wall thickness of the tank experiences tensile stress distributed along the circumferential direction; this circumferential tensile stress exists when there is internal pressure P, but it is absent in the absence of such pressure. When the same tank is not subjected to internal pressure and only experiences thermal expansion, there is no circumferential tensile stress within its wall thickness. Then why, when subjected to internal pressure P, is there tensile stress inside the wall thickness? Moreover, when P=0, the internal tensile stress is also 0. And during thermal expansion, is there never any circumferential tensile stress? My understanding is that when expanding under internal pressure P, the atoms move away from their equilibrium distance; in other words, the interatomic spacing increases as a result of P, becoming greater than the equilibrium distance L0. The force acting between the atoms is attractive, and at the macroscopic level this manifests as tensile stress. As for thermal expansion, from 50°C to 200°C, the distance between atoms expands freely from its equilibrium value at 50°C to its equilibrium value at 200°C; the distance between atoms remains constant throughout this process, meaning that the net force acting on the atoms is zero. Consequently, no elastic deformation occurs on a macroscopic scale, and no tensile stress is generated. This is my personal understanding; I’m not sure if it’s correct, so I hope the teachers can give me some guidance
Your understanding is correct. In mechanics of materials, hoop stress or circumferential stress is the force distribution resulting from internal pressure acting on the walls of a container. When an internal pressure P is present, the inner wall of the container is subjected to pressure, causing the material to expand outward and generating tension or tensile stress; this is a process of physical elastic deformation. The atomic spacing increases due to internal pressure, thereby creating tensile stress within the material. In the case of thermal expansion, the expansion of the material is caused by temperature changes. As the temperature rises, the motion of atoms or molecules intensifies and the average distance between them increases. However, this increase is uniform and occurs without any external forces acting; as a result, no additional attractive or repulsive forces arise between the atoms, and therefore no tensile stresses such as internal pressure are generated on a macroscopic scale. This expansion is a natural response based on the inherent properties of the material and thermodynamic principles, rather than a forced deformation caused by external forces. Therefore, in the case of only thermal expansion, no tensile stress is generated within the material, as it is in the case of internal pressure. .
After thermal expansion, what if the atomic spacing does not reach the equilibrium distance you mentioned? Also, material mechanics does not consider things from a microscopic perspective.
If the equilibrium distance is not achieved, internal stresses will still exist. These internal stresses are essentially related to the distances between atoms as well as the attractive and repulsive forces between them. For example, during quenching, limitations in heat conduction, along with the temperature differences between the inner and outer parts of thick workpieces, lead to asynchronous expansion and contraction; similarly, when areas are heated or cooled, the differences in expansion and contraction due to varying distances from the heat source also result in thermal stresses. Such stresses arise from the restriction on the thermal deformation of adjacent micro-elements. Ultimately, it can be said that the equilibrium distance does not fully capture the issues related to the attractive and repulsive forces between atoms. This topic is not covered in material mechanics, but it is discussed in the introductory basic courses for materials science.