Name: strain energy in stressed materials
Uploaded: 2024-05-23T13:00:12.000Z
Duration: 1 min 22 s
Channel: JoVE

当材料受到轴向载荷时，会产生正应力，从而导致应变能的产生。该公式适用于应力均匀分布的情况，且材料内的应变能密度保持恒定。
对于应力分布不均匀的材料，应变能密度定义为材料微小体积内的量。
此处，应变能密度表示为所施加的应力与产生的应变的乘积。
对材料整个体积内的应变能密度进行积分，可得到材料中储存的总应变能。
所获得的应变能方程仅适用于弹性变形，也称为弹性应变能。
当材料受到轴向对称载荷作用时，可假设其横截面上的正应力均匀分布。此处，正应力表示为内力与所考虑横截面积之比。
此类情况下储存的应变能用内力和弹性模量来表示。

法向应力的弹性应变能

Elastic Strain Energy for Normal Stresses

<div>
 应变能量化了由于在荷载条件下所产生的变形而存储在材料内的能量，这是材料科学和工程学中的基本概念。当材料受到均匀分布应力的轴向荷载时，便可以对应变能进行建模。在这种情况下，材料承受的应力是内力与横截面积进行相除所得到的结果，通过弹性模量所产生的应变与该应力是成正比的。
 
 如果应力的分布是均匀的，则应变能密度（定义为应力和应变的乘积）则可以对材料的整个体积进行积分，以此来得出其中所存储的总应变能。
 
 <img alt="Equation 1" src="/files/ftp_upload/15684/15684_Equation_1v2.svg" style="text-align: center; height: 50px;">
 然而，对于应力分布不均匀的材料，应变能的计算则会变得更加复杂。在这种情况下，必须为小体积是应变能密度进行定义，以此来分析应力和应变的局部变化。总应变能是在材料整个体积内这些密度的总和。
 
 <img alt="Equation 2" src="/files/ftp_upload/15684/15684_Equation_2v2.svg" style="text-align: center; height: 40px;">
 这种情况则假设了其中的弹性状态，即变形是可逆的，并且当荷载移除时，材料则会恢复到原始形状。了解和计算应变能对于设计能够承受运行应力而不发生故障的材料和部件是至关重要的。
 
</div>

当材料受到轴向载荷时，会产生正应力，从而导致应变能的生成。该方程适用于均匀的……

elastic-strain-energy-for-normal-stresses

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Energy Methods

能量法

KEY TERMS AND CONCEPTS

关键术语与概念

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 over the full volume of the material to calculate the total strain energy stored. This approach works for elastic behavior, where the deformation is reversible and the material returns to its original shape after the load is removed.

The calculation becomes more complex when stress is not distributed evenly. In those cases, strain energy density must be defined for small volumes so local changes in stress and strain are included. The total strain energy is then the sum of these small energy contributions across the material. This idea is useful for understanding how materials and components can handle operating stresses without failure.

Strain Energy in Stressed Materials

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 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...

elastic strain energy for normal stresses

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

elastic-strain-energy-for-shearing-stresses

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,...

impact loading on a cantilever beam

Dropped-Weight Bending of a Cantilever Beam

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 strain energy is stored in materials under load, how uniform stress is used in calculations, and why local stress changes matter.

Strain Energy in Stressed Materials | 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

Strain Energy in Stressed Materials

Description

Transcript