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

일반적인 응력 상태는 물체가 경험하는 다양한 힘과 압력을 나타냅니다.
한 면은 OA 선에 수직이고 다른 면은 좌표 평면에 평행한 사면체를 생각해 보십시오.
평행 면의 면적은 면 XYZ의 면적에 선 OA의 방향 코사인을 곱하여 결정됩니다.
다양한 응력 성분에 의해 정의되는 점 O에서의 응력 상태는 각 면에 가해지는 힘에 영향을 미칩니다.
면 XYZ에 가해지는 힘에는 법선력과 전단력이 모두 포함됩니다. OA를 따르는 모든 힘의 합은 0이며, 이는 법선 응력 방정식으로 이어집니다.
법선 응력 방정식은 코사인 방향의 2차 형식이며, 이를 통해 방정식을 단순화하는 좌표 축을 선택할 수 있습니다. 응력의 주축이라고 하는 좌표 축은 점 O에서의 응력 상태에 따라 달라집니다.
이에 따라 좌표 평면은 응력의 주 평면으로 알려져 있으며 수직 응력은 점 O에서의 주 응력입니다.

일반적인 스트레스 상태

General State of Stress

재료 내의 일반적인 응력 상태는 응력 텐서를 사용하여 정확하게 묘사할 수 있습니다. 이 텐서는 외부 힘이나 변형을 받는 재료 내에 분산된 내부 힘을 캡슐화합니다.

구체적으로, XYZ로 표시된 한 면이 OA 선에 수직이고 나머지 면은 점 O를 원점으로 하는 좌표축과 정렬되는 사면체 요소를 생각해 보세요. 점 O와 같은 임의의 지점에서 응력 텐서는 O를 통해 모든 평면의 응력 구성 요소를 결정하는 데 사용할 수 있습니다. 이 텐서는 힘을 면의 수직 구성 요소와 전단 구성 요소로 분석하여 다양한 하중 조건에서 재료가 반응하는 방식을 이해하는 데 중요합니다. 사면체의.

사면체의 좌표 정렬된 면의 면적은 면 XYZ의 면적에 선 OA의 방향 코사인 λ_x, λ_y 및 λ_z를 곱하여 계산됩니다. 이러한 코사인은 면의 방향을 좌표축에 연결하여 재료 및 구조 설계에 중요한 힘 분해능을 돕습니다. OA를 따른 모든 힘의 합이 0이라는 평형 조건은 방향 코사인을 사용하는 2차 형식으로 표현되는 수직 응력 방정식으로 이어집니다.

<img alt="Equation 1" src="/files/ftp_upload/15656/15656_Equation_1.svg" style="height: 30px;">

이 양식은 응력의 주요 축을 식별합니다. 방향 코사인을 기반으로 새 좌표계가 정의되면 전단 응력 항이 사라지고 응력 텐서가 단순화됩니다. 이러한 축은 전단 응력이 사라지고 주 응력으로 알려진 수직 응력이 최대화되는 주 평면을 정의합니다. 재료의 파손 모드를 예측하고 구조 설계를 향상하려면 이러한 응력 구성요소를 이해하는 것이 필수적입니다.

<img alt="Equation 2" src="/files/ftp_upload/15656/15656_Equation_2.svg" style="height: 19px;">

일반적인 응력 상태는 물체가 경험하는 다양한 힘과 압력을 나타냅니다.
선에 수직인 한면이 있는 사면체를 고려하십시오 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Transformations of Stress and Strain

응력과 변형률의 변환

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 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

Stress Tensor and Principal Stresses

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