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

軸方向の荷重により、ロッドが伸び、ひずみエネルギーが蓄積されます。ロッドのひずみエネルギー密度は、単位体積あたりのエネルギーです。
ここで、面積に対する軸力の比率が応力であり、ロッドの元の長さに対する伸びの比率が生成されるひずみです。
弾性領域内では、ひずみエネルギー密度は弾性率で表すことができます。
応力-ひずみ曲線は弾性領域で線形関係を示し、ひずみエネルギー密度がフックの法則に従うことを意味します。ここでは、応力を取り除くとひずみはゼロになります。このエネルギー密度の領域は、レジリエンス係数と呼ばれます。
塑性領域では、応力を取り除いた後でも、ある程度の永久ひずみが残ります。したがって、ひずみエネルギー密度の一部のみが回復され、残りのエネルギー密度は熱の形で物体を変形するために費やされます。 
曲線の下の総面積は靭性係数と呼ばれ、ロッドを破裂させるのに必要な総エネルギー密度を表します。

ひずみエネルギー密度

Strain-Energy Density

軸方向の荷重がかかった材料のひずみエネルギー密度を理解することは、材料の機械的挙動と耐久性を評価するために重要です。棒がそのような負荷を受けると、棒は伸びて、ひずみエネルギーとして知られるエネルギーを材料内の位置エネルギーとして蓄えます。このエネルギーは、単位体積あたりのエネルギーとして測定されます。

材料の弾性領域では、応力とひずみの関係は線形であり、フックの法則に従います。この領域のひずみエネルギー密度は、弾性限界までの応力-ひずみ曲線の下の面積から計算されます。 この蓄積されたエネルギーは回復可能であり、弾性率と呼ばれます。これは、材料がどれだけのエネルギーを吸収し、除荷時に元の形状に戻ることができるかを示します。

弾性限界を超えると、材料は塑性的に動作し、永久に変形します。この可塑性領域では、貯蔵されたエネルギーの一部のみが除荷時に回収可能です。残りは熱として失われるか、永久変形に使用されます。破断する前に材料が吸収できる総エネルギーは、靭性係数によって測定されます。

<img alt="Equation 1" src="/files/ftp_upload/15683/15683_Equation_1v2.svg" style="height: 44px;">

この値は、高い耐衝撃性や延性が必要な用途にとって重要であり、特定の用途向けの材料の選択や機械的負荷に耐える構造の設計に役立ちます。

軸方向の荷重により、ロッドが伸び、ひずみエネルギーが蓄積されます。ロッドのひずみエネルギー密度は、単位体積あたりのエネルギーです。
ここで、比率は...

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

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 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

Energy Stored in Loaded Rods

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