Name: load-deformation and stored strain energy
Uploaded: 2025-05-22T09:06:18.000Z
Duration: 1 min 13 s
Channel: JoVE

Рассмотрим стержень, закрепленный на одном конце и подверженный осевой силе на свободном конце, которая вызывает напряжение и приводит к удлинению стержня. 
С увеличением осевой силы удлинение увеличивается. График зависимости осевой силы от удлинения дает характерный график зависимости нагрузки и деформации.
Работа, выполняемая осевой силой, когда стержень удлиняется на небольшую величину, является произведением осевой силы и соответствующего удлинения.
Эта работа равна площади под графиком деформации, ширина которого равна малой деформации.
Общая проделанная работа может быть рассчитана путем интегрирования уравнения в пределах удлинения и равна площади под кривой нагрузки-деформации в тех же пределах удлинения.
Общая работа, проделанная на стержне, представляет собой энергию, поступающую в систему, известную как энергия деформации.
В пределах предела упругости осевая сила пропорциональна величине удлинения. Таким образом, энергия деформации пропорциональна квадрату удлинения.

Энергия деформации

Strain Energy

Энергия деформации это фундаментальное понятие в области материаловедения и строительной инженерии, описывающее энергию, поглощаемую материалом или конструкцией, когда она деформируется под нагрузкой.

Рассмотрим стержень, закрепленный на одном конце и подверженный осевой силе на свободном конце. Эта осевая сила вызывает напряжение внутри стержня, что приводит к его удлинению. По мере увеличения осевой силы увеличивается и удлинение стержня, что иллюстрирует прямую связь между приложенной силой и результирующей деформацией. Это поведение можно графически представить графиком нагрузки-деформации.

Работа, совершаемая осевой силой для удлинения стержня на небольшую величину, равна произведению силы и соответствующего удлинения. Графически эта работа соответствует площади под кривой нагрузка-деформация для заданного удлинения. Чтобы рассчитать общую работу, совершаемую стержнем при любом заданном удлинении, необходимо проинтегрировать уравнение силы-удлинения по предельным значениям удлинения. Эта общая работа представляет собой энергию, поступившую в систему, которая сохраняется в виде энергии деформации. В пределах упругости материала энергия деформации U, запасенная в материале вследствие упругой деформации, пропорциональна квадрату удлинения.

<img alt="Equation 1" src="/files/ftp_upload/15682/15682_Equation_1.svg" style="text-align: center; height: 52px;">

Концепция энергии деформации позволяет инженерам количественно оценить способность материала или компонента поглощать энергию, что важно для оценки его способности выдерживать нагрузки без необратимых деформаций или разрушений. Энергия деформации также играет ключевую роль в различных инженерных приложениях, таких как проектирование амортизаторов, устройств накопления энергии и при анализе ударных сил.

Рассмотрим стержень, закрепленный на одном конце и подверженный осевой силе на свободном конце, которая вызывает напряжение и приводит к удлинению стержня.
В качестве ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Energy Methods

Энергетические методы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 makes it easier to see how much the rod stretches at each load.

The work done by the axial force over a small elongation is the force multiplied by that elongation. On the graph, this work appears as the area under the load-deformation curve for a given elongation. To find the total work for any elongation, the force-elongation equation is integrated over the elongation limits. That total work is the energy put into the system, and it is stored as strain energy.

Within the elastic limit of the material, the strain energy stored in the rod is proportional to the square of the elongation. This means the stored energy rises as the deformation increases, as long as the material still behaves elastically. Knowing strain energy helps engineers judge how much energy a material or component can absorb before permanent deformation or failure. It is also useful in shock absorbers, energy storage devices, and impact force analysis.

Load-Deformation and Stored Strain Energy

load-deformation-and-stored-strain-energy

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

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

Explore how load creates deformation in a rod, how work becomes stored strain energy, and why this matters in shock absorbers and impact analysis.

Load-Deformation and Stored Strain Energy | 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

Load-Deformation and Stored Strain Energy

Description

Transcript