Chapter 27: Energy Methods

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27.2:

Strain-Energy Density

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Mechanical Engineering

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JoVE Core Mechanical Engineering

Strain-Energy Density

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27.1: Strain Energy

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27.3: Elastic Strain Energy for Normal Stresses

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An axial load causes a rod to elongate, and store strain energy in it. The strain energy density in the rod is the energy per unit volume. Here, the ratio of the axial force to the area is stress, and the ratio of elongation to the original length of the rod is the strain produced. Within the elastic region, the strain energy density can be expressed in terms of the modulus of elasticity. The stress-strain curve exhibits a linear relationship in the elastic region, implying that the strain energy density follows Hooke's Law. Here, upon removing the stress, the strain becomes zero. This region of energy density is called the modulus of resilience. In the plastic region, some permanent strain remains even after removing the stress. So, only part of the strain energy density is recovered, and the remaining energy density is spent in deforming the object in the form of heat. The total area under the curve is called the modulus of toughness and represents the total energy density required to rupture the rod.

27.2:

Strain-Energy Density

Understanding the strain energy density in materials under axial load is crucial for evaluating their mechanical behavior and durability. When a rod is subjected to such a load, it elongates and stores energy, known as strain energy, as potential energy within the material. This energy is measured in terms of energy per unit volume.

In the elastic region of a material, the relationship between the stress and the strain is linear and follows Hooke's Law. The strain energy density in this region is calculated from the area under the stress-strain curve up to the elastic limit. This stored energy is recoverable and is referred to as the modulus of resilience, which indicates how much energy the material can absorb and still return to its original shape upon unloading.

Beyond the elastic limit, the material behaves plastically, deforming permanently. In this plastic region, only part of the stored energy is recoverable upon unloading; the rest is lost as heat or used in permanent deformation. The total energy a material can absorb before rupturing is measured by the modulus of toughness.

This value is crucial for applications that require high impact resistance or ductility, aiding in the selection of materials for specific applications and designing structures to withstand mechanical loads.