Name: energy stored in loaded rods
Uploaded: 2024-05-23T13:00:11.000Z
Duration: 1 min 20 s
Channel: JoVE

轴向载荷会导致杆件伸长，并在其内部储存应变能。杆件中的应变能密度是指单位体积内的能量。
此处，轴向力与面积的比值为应力，伸长量与杆原始长度的比值为所产生的应变。
在弹性区域内，应变能密度可以用弹性模量来表示。
应力-应变曲线在弹性区域内呈线性关系，表明应变能密度遵循胡克定律。在此区域内，当去除应力后，应变变为零。这一区域的能密度称为弹性模量。
在塑性区域，即使去除应力后，仍会残留部分永久应变。因此，只有一部分应变能密度得以恢复，其余的能量密度则以热的形式消耗在物体的变形过程中。
曲线下的总面积称为韧性模量，表示使杆件断裂所需的总能量密度。

应变能密度

Strain-Energy Density

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 了解材料在轴向荷载下的应变能密度对于评估其机械行为和耐久性来说是至关重要的。当杆承受这样的荷载时，它会伸长并存储能量（即应变能），并将其作为材料内的势能。这种能量能够以单位体积的能量来进行衡量。
 
 在材料的弹性区域中，应力和应变之间呈线性关系，并遵循胡克定律。位于该区域的应变能密度是根据应力-应变曲线的下面积计算得出的，并直至弹性极限。这种储存的能量是可恢复的，并且被称为弹性模量，它表明了材料在卸载时能够吸收多少能量并且恢复到其原始形状。
 
 当超过弹性极限后，材料就会表现出相应的塑性，即永久变形。在这个塑性区域中，只有部分存储的能量可以在卸载时恢复；其余部分的能量则会以热量的形式损失或用于永久变形。材料在发生断裂前可以吸收的总能量则用韧性模量来进行衡量。
 
 <img alt="Equation 1" src="/files/ftp_upload/15683/15683_Equation_1v2.svg" style="text-align: center; height: 44px;">
 这一数值对于需要高抗冲击性或延展性的应用是至关重要的，这有助于为特定的应用场景选择材料，并设计出能够承受机械载荷的结构。
 
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轴向载荷会导致杆件伸长，并在其内部储存应变能。杆件中的应变能密度是指单位体积内的能量。
此处，比值……

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Mechanical Engineering

机械工程

Unit 1

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Energy Methods

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KEY TERMS AND CONCEPTS

关键术语与概念

energy stored in loaded rods

Energy stored in a rod under axial load helps show how a material will respond to stretching and breaking. When the rod elongates, it stores strain energy as potential energy inside the material. Strain energy density is the amount of that energy per unit volume.

In the elastic region, stress and strain have a linear relationship that follows Hooke's Law. The strain energy density in this region is found from the area under the stress-strain curve up to the elastic limit. This recoverable energy is called the modulus of resilience. It shows how much energy a material can absorb and still return to its original shape when the load is removed.

After the elastic limit, the material enters the plastic region. In this stage, the material deforms permanently, so only part of the stored energy comes back during unloading. The rest is lost as heat or used in permanent deformation. The total energy a material can absorb before rupture is called the modulus of toughness. This value is important when choosing materials for impact resistance and ductility, and for designing structures that must handle mechanical loads.

Energy Stored in Loaded Rods

strain energy

Load-Deformation and Stored Strain Energy

Strain energy is the energy stored in a material when it deforms under load. It is a key idea in materials science and structural engineering. A loaded rod fixed at one end shows this clearly. When an axial force pulls on the free end, the rod stretches and stress builds inside it. As the axial force increases, the rod elongates more. This creates a direct link between the applied force and the amount of deformation. Engineers can show this relationship with a load-deformation plot. The plot...

strain-energy density

Energy stored in a rod under axial load helps show how a material will respond to stretching and breaking. When the rod elongates, it stores strain energy as potential energy inside the material. Strain energy density is the amount of that energy per unit volume. In the elastic region, stress and strain have a linear relationship that follows Hooke's Law. The strain energy density in this region is found from the area under the stress-strain curve up to the elastic limit. This recoverable...

elastic strain energy for normal stresses

elastic-strain-energy-for-normal-stresses

Strain Energy in Stressed Materials

Strain energy in stressed materials describes the energy stored when a material deforms under load. It is an important idea in materials science and engineering. For axial loading with a uniform stress distribution, the stress is the internal force divided by the cross-sectional area, and the strain follows from the stress through the modulus of elasticity. When the stress is uniform, the strain energy density can be found from the product of stress and strain. That density is then integrated...

elastic strain energy for shearing stresses

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Shear Energy Stored in Twisted Shafts

Shear energy stored in twisted shafts depends on elastic strain energy in a material. This energy is stored when a solid is deformed within its elastic limit. It comes from internal work done against the cohesive forces inside the material. When a material carries shearing stress, it also develops shearing strain. The strain energy density is the energy stored per unit volume. Within the elastic limit, stress is proportional to strain through the modulus of rigidity, so the strain energy...

impact loading

Impact Loading on a Fixed Rod

Impact loading on a fixed rod happens when a moving object strikes a stationary rod that has a uniform cross-sectional area and is fixed at one end. The impact makes the rod deform and begin to develop stress. This setup shows how a sudden hit can store energy in a structure. As the rod moves back toward its original shape, the stress rises to a maximum. At that point, the energy from the moving object has changed from kinetic energy into strain energy. Kinetic energy is the energy of motion,...

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Dropped-weight bending of a cantilever beam shows how impact loading changes a beam’s shape and stress. The beam has a circular cross-section and is fixed at one end. An object is dropped on the free end from a height h, so its potential energy from gravity becomes kinetic energy at impact. After the object hits the beam, the beam bends and develops a bending moment along its length. The bending is not uniform. The fixed end carries the highest strain energy, which is the energy stored in the...

castigliano&#39;s theorem

Using Strain Energy to Find Deflection

Castigliano's theorem uses strain energy to find deflection and rotation in elastic structures. Strain energy is the work done by external loads during elastic deformation. The theorem links that energy to the movement caused by the load. The key idea is simple. The partial derivative of the total strain energy with respect to a specific force gives the displacement at the point where that force acts. The same method also works for a moment, where the derivative gives the rotation at the point...

castigliano&#39;s theorem: problem solving

Beam Deflection from Central Load

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam. For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support. Castigliano's theorem says the...

key terms and concepts

Learn how axial load, strain energy density, modulus of resilience, and modulus of toughness describe how materials store and release energy.

Energy Stored in Loaded Rods | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Energy Stored in Loaded Rods

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