Name: stress tensor and principal stresses
Uploaded: 2024-05-23T12:59:54.000Z
Duration: 1 min 21 s
Channel: JoVE

応力の一般的な状態とは、物体が受けるさまざまな力と圧力を指します。
一方の面が線 OA に垂直で、もう一方の面が座標平面に平行な四面体を考えます。
平行な面の面積は、面 XYZ の面積に線 OA の方向余弦を掛けることによって決定されます。
点Oでの応力の状態は、さまざまな応力成分によって定義され、各面に加えられる力に影響を与えます。
面XYZにかかる力には、法線力とせん断力の両方が含まれます。OAに沿ったすべての力の合計はゼロであり、垂直応力方程式につながります。
垂直応力方程式は、方向コサインを持つ二次形式であるため、方程式を簡略化する座標軸を選択できます。応力の主軸と呼ばれる座標軸は、点Oでの応力の状態によって異なります。
これに対応して、座標平面は応力の主平面と呼ばれ、垂直応力は点Oの主応力です。

一般的な応力状態

General State of Stress

材料内の応力の一般的な状態は、応力テンソルを使用して正確に表現できます。 このテンソルは、外部の力または変形を受ける材料内に分布する内部力をカプセル化します。

具体的には、XYZ とラベル付けされた 1 つの面が線 OA に垂直で、残りの面が点 O を原点とする座標軸に沿った四面体要素を考えます。 点 O などの任意の点で、 応力テンソルを使用して、O を通る任意の平面上の応力成分を決定できます。 このテンソルは、正四面体の面上の法線成分とせん断成分に分解することにより、さまざまな荷重条件下で材料がどのように反応するかを理解する上で重要です。

四面体の座標に沿った面の面積は、面 XYZ の面積に線分 OA の方向余弦 λ_x, λ_y,λ_z を乗じることによって計算されます。 これらの余弦は面の向きを座標軸に結び付け、 材料および構造設計にとって重要な力の解決に役立ちます。 OA に沿ったすべての力の合計がゼロに等しいという平衡条件により、 方向余弦を伴う二次形式で表される法線応力を求める方程式が導かれます。

<img alt="Equation 1" src="/files/ftp_upload/15656/15656_Equation_1.svg" style="height: 30px;">

この形式は応力の主軸を特定します。 新しい座標系が方向余弦に基づいて定義される場合、 せん断応力項が削除され、応力テンソルが単純化されます。これらの軸は、せん断応力が消失し、 主応力として知られる垂直応力が最大になる主平面を定義します。 これらの応力成分を理解することは、材料の破損モードを予測し、構造設計を強化するために不可欠です。

<img alt="Equation 2" src="/files/ftp_upload/15656/15656_Equation_2.svg" style="height: 19px;">

応力の一般的な状態とは、物体が受けるさまざまな力と圧力を指します。
1つの面が線に垂直な四面体を考えてみましょう...

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

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.

The areas of the coordinate-aligned faces come from the area of face XYZ and the direction cosines of line OA. These direction cosines are λx, λy, and λz. They link the orientation of the face to the coordinate axes and help resolve the forces.

Force balance along OA gives the equilibrium condition, where the sum of all forces in that direction is zero. This leads to the normal stress equation in quadratic form with direction cosines. That form identifies the principal axes of stress.

If a new coordinate system is chosen using those direction cosines, the shear stress terms disappear. The result is a simpler stress tensor. The new axes are the principal planes, where shear stress is zero and the normal stresses are the principal stresses. These ideas help predict failure modes and support structural design.

Stress Tensor and Principal Stresses

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

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

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...

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 a stress tensor resolves normal and shear stress on a tetrahedral element, and how principal axes and principal stresses are found.

Stress Tensor and Principal Stresses | 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

Stress Tensor and Principal Stresses

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