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

材料が軸方向の荷重を受けると、垂直応力がかかり、ひずみエネルギーが発生します。この方程式は、一様に分布する応力に対して有効であり、ひずみエネルギー密度は材料全体で一定です。
応力分布が不均一な材料の場合、ひずみエネルギー密度は材料の小さな体積に対して定義されます。
ここで、ひずみエネルギー密度は、加えられた応力と生成されたひずみの積として表されます。
材料の全体積にわたるひずみエネルギー密度を積分すると、材料に蓄積された全ひずみエネルギーが得られます。
得られたひずみエネルギーの方程式は、弾性変形にのみ適用され、弾性ひずみエネルギーとも呼ばれます。
材料が中心軸荷重を受けると、垂直応力は任意の横断面で均一であると仮定されます。ここで、垂直応力は、考慮対象の断面積に対する内力の比として記述されます。
このような場合に蓄積されるひずみエネルギーは、内力と弾性率で表されます。

垂直応力の弾性ひずみエネルギー

Elastic Strain Energy for Normal Stresses

ひずみエネルギーは、荷重条件下での変形によって材料内に蓄えられるエネルギーを定量化したもので、材料科学および材料工学の基本概念です。材料が応力が均一に分散された軸方向荷重を受けるときのひずみエネルギーをモデル化できます。このシナリオでは、材料が受ける応力は内部力を断面積で割ったもので、誘発されるひずみは弾性率を通じてこの応力に直接比例します。

応力分布が均一であれば、応力とひずみの積として定義されるひずみエネルギー密度を材料の体積全体にわたって積分して、蓄積された総ひずみエネルギーを求めることができます。

<img alt="Equation 1" src="/files/ftp_upload/15684/15684_Equation_1v2.svg" style="height: 50px;">

ただし、応力分布が不均一な材料の場合、ひずみエネルギーの計算はより複雑になります。このような場合、応力とひずみの局所的な変動を考慮して、小さな体積のひずみエネルギー密度を定義する必要があります。総ひずみエネルギーは、材料の体積全体にわたるこれらの密度の合計です。

<img alt="Equation 2" src="/files/ftp_upload/15684/15684_Equation_2v2.svg" style="height: 40px;">

この考察は、変形が可逆的であり、荷重が取り除かれると材料が元の形状に戻る弾性挙動を前提としています。ひずみエネルギーを理解して計算することは、故障することなく動作応力に耐えることができる材料や部品を設計するために不可欠です。

材料が軸方向の荷重を受けると、垂直応力がかかり、ひずみエネルギーが発生します。この方程式は、一様に...

elastic-strain-energy-for-normal-stresses

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Energy Methods

エネルギー法

KEY TERMS AND CONCEPTS

主要用語と概念

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

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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...

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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

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Dropped-Weight Bending of a Cantilever Beam

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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...

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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 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Strain Energy in Stressed Materials

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