Name: principal strain directions in 3d
Uploaded: 2024-05-23T12:59:55.000Z
Duration: 1 min 29 s
Channel: JoVE

3次元ひずみ解析では、主応力軸を使用して、弾性で均質な材料の変形を評価します。
小さな立方体要素は、これらの軸を中心に展開すると、直方体に変換され、変形を示します。
ひずみ解析では、n軸などの主軸を中心に要素を回転させ、この軸に垂直な面上のひずみ成分を評価します。この方法は、平面ひずみ変換に基づいています。
モールの円を使用して、立方体が主軸を中心に回転するときのひずみ変換が調べられます。このアプローチでは、3 つの円のうち最大の円の直径に相当する、特定の点における最大せん断ひずみが特定されます。
平面ひずみでは、n軸はモールの円図の原点でひずみがゼロの主軸に変換されます。この原点の反対側にある主ひずみは、最大および最小の正規ひずみを表します。
平面応力を受ける構造物の薄いプレートでは、n軸が主応力軸に変わります。
n軸に沿った主ひずみは、面内ひずみに関連付けられています。m軸を中心とした回転は、最大せん断ひずみを定義します。

ひずみの3次元解析

Three-Dimensional Analysis of Strain

3次元ひずみ解析は、特に弾性のある均質な材料において、応力下で材料がどのように変形するかを理解するために重要です。この方法では、主応力軸を使用して、複雑な応力状態をよりわかりやすい形式に単純化します。応力を受けると、材料内の小さな立方体要素がこれらの軸に沿って膨張または収縮し、直方体に変形します。この変換は、材料の変形を効果的に示します。主応力軸は直交しており、応力が材料内にせん断を引き起こさない方向を表しています。

モール円はひずみ解析に不可欠なツールです。これは、点での応力状態をグラフィック表示し、要素が n 軸などの主軸の周りを回転するときのひずみ成分を評価します。この解析は、n 軸の原点でのひずみがゼロである平面ひずみ変換に焦点を当てており、モール円の反対側に描かれた最大ひずみと最小ひずみの決定を簡素化します。

平面応力下の薄板のようなシナリオでは、n 軸が主応力軸になります。ここで、n 軸に沿った主ひずみは、材料の面内ひずみと直接相関します。 m 軸などの別の主軸を中心とした回転は、最大せん断ひずみの位置と大きさを正確に特定するのに役立ちます。これは、荷重下での材料の挙動を予測し、構造設計の安全性と信頼性を確保するために重要です。

3次元ひずみ解析では、主応力軸を使用して、弾性で均質な材料の変形を評価します。
小さな立方体の要素、...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Transformations of Stress and Strain

応力とひずみの変換

KEY TERMS AND CONCEPTS

主要用語および概念

principal strain directions in 3d

Three-dimensional strain analysis shows how a material changes shape when it is loaded. It is especially useful for elastic, homogeneous materials. A small cubic element inside the material can expand or contract along principal stress axes. That change turns the cube into a rectangular parallelepiped, which makes the deformation easier to picture.

The principal stress axes are orthogonal, or at right angles to each other. Along these directions, the stress does not create shear inside the material. This simplifies a complex stress state into a form that is easier to analyze and compare.

Mohr's circle is a key graph used in strain analysis. It shows the stress state at a point and helps evaluate strain components as the element rotates about a principal axis, such as the n-axis. In plane strain transformations, the strain at the origin of the n-axis is zero. This makes it easier to find the maximum and minimum strains, which appear on opposite sides of Mohr's circle.

For thin plates under plane stress, the n-axis becomes a principal stress axis. In that case, the principal strain along the n-axis is directly related to the in-plane strains of the material. Rotation about another principal axis, such as the m-axis, helps locate the maximum shearing strain and its magnitude. These results are important for predicting how materials behave under load and for supporting safe, reliable structural design.

Principal Strain Directions in 3D

transformation of plane stress

Plane Stress Changes with Rotation

Plane stress changes with rotation in a material element. The analysis looks at a prismatic element inside a cube under plane stress. As the element turns, the normal stress and shearing stress on its faces change in both size and direction. These changes are described with trigonometric functions of the rotation angle. The forces on the rotated faces are compared with the forces on the original faces that are perpendicular to each principal axis. The equilibrium equations use only the forces...

principal stresses

Stress Circle for Principal and Shear Stress

The stress circle shows how normal stress and shearing stress change with angle on a transformed plane. Its center, C, lies on the vertical axis and represents the average normal stress. The radius shows the amount of stress variation as the plane rotates. Points A and B, where the circle crosses the horizontal axis, mark the maximum and minimum normal stresses. At these points, shearing stress is zero. These points define the principal stress planes, where only normal stress, called principal...

principal stresses: problem solving

Finding Principal Planes and Stress Limits

Principal planes and principal stresses are found by analyzing two planes that meet at right angles under shearing, tensile, and compressive stresses. The goal is to use the given stress values to describe how the material is loaded and to locate the planes where the stress state changes. To find the principal planes, use the formula that relates the plane angle to twice the shearing stress divided by the difference between tensile and compressive stresses. After substituting the known stress...

mohr&#39;s circle for plane stress

Reading Stress Components with Mohr's Circle

Mohr's circle is a graphical way to read the stress state at a point in a material under plane stress. It turns normal stress and shearing stress into a two-dimensional picture. This makes stress transformation easier to analyze. The method uses Cartesian coordinates. The horizontal axis shows normal stress, written as σ, and the vertical axis shows shearing stress, written as τ. Two points, A and B, are plotted from the stresses on the element. Point A is at (σx, -τxy), and point B is at (σx,...

general state of stress

Stress Tensor and Principal Stresses

The stress tensor describes the internal forces inside a material. It is used when a material is under external force or deformation. At a point such as O, the tensor can show the stress components on any plane through that point. A tetrahedral element helps show how these forces act. One face, XYZ, is perpendicular to the line OA. The other faces line up with the coordinate axes, and point O is the origin. The stress on the tetrahedron is split into normal and shear components on each face. <br...

yield criteria for ductile materials under plane stress

yield-criteria-for-ductile-materials-under-plane-stress

Tresca and Von Mises Yield Criteria

Ductile materials can yield when they are loaded beyond a safe stress level. Engineers use yield criteria to check whether structural elements and machine parts can handle the expected load without permanent deformation. The yield point is first found with a tensile test, which measures how the material responds to uniaxial stress. A tensile test is not enough when a part faces biaxial or plane stress conditions. In those cases, more advanced criteria are needed to predict failure. One common...

transformation of plane strain

Plane Strain Under Rotated Axes

Plane strain changes when the coordinate axes are rotated. This matters for long bars and similar elongated structures that are loaded uniformly on their sides. In these members, one dimension is much larger than the others, so deformation is mainly studied in a single plane. At a point such as O, the key strain components lie along the x and y axes. The strain along the z-axis is negligible in plane strain. Together, the x and y strain components describe the deformation state in the xy-plane.

mohr&#39;s circle for plane strain

Principal and Shear Strains on Mohr’s Circle

Mohr’s circle for plane strain shows how normal strain and shear strain change on a graph. The horizontal axis is normal strain, written as ε, and the vertical axis is shear strain, written as γ. It uses two plotted points, X and Y, to represent the strain state. Point X has coordinates (εx, -γXY), and point Y has coordinates (εY, γXY). The center of the circle, O, gives the average normal strain. The radius comes from the relationship between normal strain and shear strain, so the circle...

three-dimensional analysis of strain

Three-dimensional strain analysis shows how a material changes shape when it is loaded. It is especially useful for elastic, homogeneous materials. A small cubic element inside the material can expand or contract along principal stress axes. That change turns the cube into a rectangular parallelepiped, which makes the deformation easier to picture. The principal stress axes are orthogonal, or at right angles to each other. Along these directions, the stress does not create shear inside the...

measurements of strain

How Strain Gauges Measure Deformation

Strain describes how a material deforms when a force is applied. Normal strain compares the change in length with the original length. Engineers use this measure to track stretching or compression in materials. For higher accuracy, they often use electrical strain gauges. A strain gauge has a conductive wire on a paper backing that sticks to the surface of the material. It works through the piezoresistive effect, which means the wire’s electrical resistance changes when the material is bent,...

key terms and concepts

Learn how principal stress axes, Mohr's circle, and plane stress help show 3D strain, including maximum, minimum, and shearing strain in materials.

Principal Strain Directions in 3D | 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

Principal Strain Directions in 3D

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