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

Общее состояние напряжения относится к различным силам и давлениям, которые испытывает объект.
Рассмотрим тетраэдр, одна грань которого перпендикулярна линии ОА, а другая грань параллельна координатным плоскостям.
Площади параллельных граней определяются путем умножения площади грани XYZ на направление косинусов линии ОА.
Состояние напряжения в точке О, определяемое различными компонентами напряжения, влияет на силы, действующие на каждую грань.
Силы, действующие на поверхность XYZ, включают в себя как нормальную, так и поперечную силу. Сумма всех сил вдоль OA равна нулю, что приводит к уравнению нормального напряжения.
Уравнение нормального напряжения имеет квадратичную форму с направлением косинусов, что позволяет выбрать оси координат, упрощающие уравнение. Координатные оси, называемые главными осями напряжения, зависят от состояния напряжения в точке О.
Соответственно, координатные плоскости известны как главные плоскости напряжений, а нормальные напряжения — как главные напряжения в точке О.

Общее состояние напряжения

General State of Stress

Общее состояние напряжения внутри материала можно точно отобразить с помощью тензора напряжений. Этот тензор отражает внутренние силы, распределенные внутри материала, подвергающегося внешним силам или деформациям.

В частности, рассмотрим четырехгранный элемент, в котором одна грань, обозначенная XYZ, перпендикулярна линии OA, а остальные грани совпадают с осями координат с точкой O в качестве начала координат. В любой точке, например в точке O, тензор напряжений можно использовать для определения компонентов напряжения в любой плоскости через O. Этот тензор имеет решающее значение для понимания того, как материалы реагируют на различные условия нагрузки, разделяя силы на нормальные и сдвиговые компоненты на гранях тетраэдра.

Площади граней тетраэдра, выровненных по координатам, вычисляются путем умножения площади грани XYZ на направляющие косинусы λ_x, λ_y, и λ_z линии OA. Эти косинусы связывают ориентацию грани с осями координат, помогая в определении силы, что имеет решающее значение для проектирования материалов и конструкций. Условие равновесия, когда сумма всех сил вдоль ОА равна нулю, приводит к уравнению нормального напряжения, выраженному в квадратичной форме с направляющими косинусами.

<img alt="Equation 1" src="/files/ftp_upload/15656/15656_Equation_1.svg" style="text-align: center; height: 30px;">

Эта форма определяет основные оси напряжения. Если новая система координат определяется на основе направляющих косинусов, то члены касательного напряжения выпадают, упрощая тензор напряжений. Эти оси определяют главные плоскости, в которых напряжения сдвига исчезают, а нормальные напряжения, известные как главные напряжения, максимальны. Понимание этих компонентов напряжения имеет важное значение для прогнозирования режимов разрушения материалов и улучшения проектирования конструкций.

<img alt="Equation 2" src="/files/ftp_upload/15656/15656_Equation_2.svg" style="text-align: center; height: 19px;">

Общее состояние напряжения относится к различным силам и давлениям, которые испытывает объект.
Рассмотрим тетраэдр с одной гранью, перпендикулярной линии...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Transformations of Stress and Strain

Преобразования напряжений и деформаций

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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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