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

Une charge axiale provoque l’allongement d’une tige et y stocke l’énergie de déformation. La densité d’énergie de déformation dans la barre est l’énergie par unité de volume.
Ici, le rapport entre la force axiale et la surface est la contrainte, et le rapport entre l’allongement et la longueur d’origine de la tige est la déformation produite.
Dans la région élastique, la densité d’énergie de déformation peut être exprimée en termes de module d’élasticité.
La courbe contrainte-déformation présente une relation linéaire dans la région élastique, ce qui implique que la densité d'énergie de déformation suit la loi de Hooke. Ici, après avoir éliminé la contrainte, la déformation devient nulle. Cette région de la densité d’énergie s’appelle le module de résilience.
Dans la région du plastique, une certaine tension permanente subsiste même après l’élimination de la contrainte. Ainsi, seule une partie de la densité d’énergie de déformation est récupérée, et la densité d’énergie restante est dépensée pour déformer l’objet sous forme de chaleur. 
L’aire totale sous la courbe est appelée module de ténacité et représente la densité d’énergie totale nécessaire pour rompre la barre.

densité de déformation-énergie

Strain-Energy Density

Comprendre la densité d'énergie de déformation dans les matériaux soumis à une charge axiale est crucial pour évaluer leur comportement mécanique et leur durabilité. Lorsqu'une tige est soumise à une telle charge, elle s'allonge et stocke de l'énergie, appelée énergie de déformation, sous forme d'énergie potentielle dans le matériau. Cette énergie est mesurée en termes d’énergie par unité de volume.

Dans la région élastique d'un matériau, la relation entre la contrainte et la déformation est linéaire et suit la loi de Hooke. La densité d'énergie de déformation dans cette région est calculée à partir de l'aire sous la courbe contrainte-déformation jusqu'à la limite élastique. Cette énergie stockée est récupérable et est appelée module de résilience, qui indique la quantité d'énergie que le matériau peut absorber tout en revenant à sa forme originale lors du déchargement.

Au-delà de la limite élastique, le matériau se comporte plastiquement en se déformant de façon permanente. Dans cette région plastique, seule une partie de l’énergie stockée est récupérable lors du déchargement ; le reste est perdu sous forme de chaleur ou utilisé en déformation permanente. L'énergie totale qu'un matériau peut absorber avant sa rupture est mesurée par le module de ténacité.

<img alt="Equation 1" src="/files/ftp_upload/15683/15683_Equation_1v2.svg" style="text-align: center; height: 44px;">

Cette valeur est cruciale pour les applications qui nécessitent une résistance aux chocs ou une ductilité élevée, facilitant la sélection de matériaux pour des applications spécifiques et la conception de structures capables de résister aux charges mécaniques.

Densité de déformation-énergie

Une charge axiale provoque l’allongement d’une tige et y stocke l’énergie de déformation. La densité d’énergie de déformation dans la barre est l’énergie par unité de volume.
Ici, le ratio ...

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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

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 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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