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

재료에 축 방향 하중이 가해지면 수직 응력이 발생하여 변형 에너지가 생성됩니다. 이 방정식은 균일하게 분포된 응력에 유효하며 변형 에너지 밀도는 재료 전체에서 일정합니다.
응력 분포가 균일하지 않은 재료의 경우, 변형 에너지 밀도는 재료의 작은 부피에 대해 정의됩니다.
여기서, 변형 에너지 밀도는 적용된 응력과 생성된 변형률의 곱으로 표현됩니다.
재료의 전체 부피에 걸쳐 변형 에너지 밀도를 통합하면 재료에 저장된 총 변형 에너지를 얻을 수 있습니다.
변형 에너지에 대해 얻어진 방정식은 탄성 변형에만 적용되며 탄성 변형 에너지라고도 합니다.
재료에 중심 축 하중이 가해질 때 수직 응력은 모든 횡단면에서 균일한 것으로 간주됩니다. 여기서 수직 응력은 고려 중인 단면적에 대한 내부 힘의 비율로 작성됩니다.
이러한 경우에 저장된 변형 에너지는 내부 힘과 탄성 계수 측면에서 작성됩니다.

수직 응력에 대한 탄성 변형 에너지

Elastic Strain Energy for Normal Stresses

변형 에너지는 재료 과학 및 공학의 기본 개념인 하중 조건 하의 변형으로 인해 재료 내에 저장된 에너지를 정량화합니다. 재료가 균일하게 분포된 응력을 갖는 축 하중을 받을 때 변형 에너지를 모델링할 수 있습니다. 이 시나리오에서 재료가 겪는 응력은 내부 힘을 단면적으로 나눈 것이며, 유발된 변형은 탄성 계수를 통해 이 응력에 정비례합니다.

응력 분포가 균일한 경우 응력과 변형률의 곱으로 정의되는 변형 에너지 밀도를 재료의 전체 부피에 걸쳐 적분하여 저장된 총 변형 에너지를 얻을 수 있습니다.

<img alt="Equation 1" src="/files/ftp_upload/15684/15684_Equation_1v2.svg" style="height: 50px;">

그러나 응력 분포가 균일하지 않은 재료의 경우 변형 에너지를 계산하는 것이 더 복잡해집니다. 이러한 경우 응력과 변형률의 국지적 변화를 설명하기 위해 작은 부피에 대한 변형률 에너지 밀도를 정의해야 합니다. 총 변형 에너지는 재료 부피 전체에 걸쳐 이러한 밀도의 합입니다.

<img alt="Equation 2" src="/files/ftp_upload/15684/15684_Equation_2v2.svg" style="height: 40px;">

이러한 고려 사항은 변형이 가역적이며 하중이 제거되면 재료가 원래 모양으로 돌아가는 탄성 동작을 가정합니다. 변형 에너지를 이해하고 계산하는 것은 고장 없이 작동 응력을 견딜 수 있는 재료와 부품을 설계하는 데 필수적입니다.

재료에 축 방향 하중이 가해지면 수직 응력이 발생하여 변형 에너지가 생성됩니다. 이 방정식은 균일하게 유효합니다 ...

elastic-strain-energy-for-normal-stresses

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Energy Methods

에너지 방법

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

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