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

Lorsqu’un matériau est soumis à une charge axiale, il subit une contrainte normale, ce qui entraîne une génération d’énergie de déformation. Cette équation est valable pour des contraintes uniformément réparties, et la densité d’énergie de déformation est constante dans tout le matériau.
Pour un matériau ayant une distribution de contrainte non uniforme, la densité d’énergie de déformation est définie pour le petit volume du matériau.
Ici, la densité d’énergie de déformation est exprimée comme le produit de la contrainte appliquée et de la déformation produite.
L’intégration de la densité d’énergie de déformation sur l’ensemble du volume du matériau donne l’énergie de déformation totale stockée dans le matériau.
L’équation obtenue pour l’énergie de déformation ne s’applique qu’à la déformation élastique et est également appelée énergie de déformation élastique.
Lorsque le matériau est soumis à une charge axiale centrée, les contraintes normales sont supposées être uniformes sur toute section transversale. Ici, la contrainte normale s’écrit comme le rapport entre les forces internes et l’aire de la section transversale considérée.
L’énergie de déformation stockée dans de tels cas est écrite en termes de force interne et de module d’élasticité.

Énergie de déformation élastique pour les contraintes normales

Elastic Strain Energy for Normal Stresses

L'énergie de déformation quantifie l'énergie stockée dans un matériau en raison de la déformation sous des conditions de chargement, un concept fondamental en science et ingénierie des matériaux. L'énergie de déformation peut être modélisée lorsqu'un matériau est soumis à un chargement axial avec une contrainte uniformément répartie. Dans ce scénario, la contrainte subie par le matériau est la force interne divisée par la surface de la section transversale, et la déformation induite est directement proportionnelle à cette contrainte via le module d'élasticité.

Si la répartition des contraintes est uniforme, la densité d'énergie de déformation, définie comme le produit de la contrainte et de la déformation, peut être intégrée sur tout le volume du matériau pour produire l'énergie de déformation totale stockée.

<img alt="Equation 1" src="/files/ftp_upload/15684/15684_Equation_1v2.svg" style="text-align: center; height: 50px;">

Cependant, le calcul de l’énergie de déformation devient plus complexe pour les matériaux présentant des répartitions de contraintes non uniformes. Dans de tels cas, la densité d'énergie de déformation doit être définie pour les petits volumes afin de tenir compte des variations locales de contrainte et de déformation. L'énergie de déformation totale est la somme de ces densités dans le volume du matériau.

<img alt="Equation 2" src="/files/ftp_upload/15684/15684_Equation_2v2.svg" style="text-align: center; height: 40px;">

Cette considération suppose un comportement élastique, où la déformation est réversible et où le matériau reprend sa forme d'origine lorsque la charge est supprimée. Comprendre et calculer l'énergie de déformation est essentiel pour concevoir des matériaux et des composants capables de résister aux contraintes opérationnelles sans défaillance.

Lorsqu’un matériau est soumis à une charge axiale, il subit une contrainte normale, ce qui entraîne une génération d’énergie de déformation. Cette équation est valable pour uniformément ...

elastic-strain-energy-for-normal-stresses

Education

Enseignement

JoVE Core

Mechanical Engineering

Génie mécanique

Unit 1

Unité 1

Energy Methods

Méthodes énergétiques

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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

Tags