Name: principal and shear strains on mohr’s circle
Uploaded: 2025-05-22T09:06:18.000Z
Duration: 1 min 18 s
Channel: JoVE

Der Mohr-Kreis analysiert die ebene Dehnung, indem er Punkte mit der Abszisse gleich der Normaldehnung und der Ordinate gleich der halben Schubdehnung darstellt.
Der Mittelpunkt O des Mohrschen Kreises ist definiert, wobei die Abszisse von O und der Radius gleich der durchschnittlichen Dehnung bzw. der Radiusgleichung sind.
Die Drehrichtung der Seiten, die den Dehnungskomponenten zugeordnet sind, gibt an, wo die entsprechenden Punkte auf dem Mohrschen Kreis eingezeichnet sind.
Die Schnittpunkte des Mohrschen Kreises mit der horizontalen Achse entsprechen den maximalen und minimalen Hauptdehnungen, berechnet als Summe und Differenz der mittleren Dehnung bzw. des Radius.
Bei der elastischen Verformung in homogenen, isotropen Materialien fallen die Hauptdehnungsachsen mit den Spannungsachsen nach Anwendung des Hookeschen Gesetzes für Schubspannung und -dehnung zusammen.
Die maximale Schubdehnung in der Ebene entspricht dem Durchmesser des Mohrschen Kreises. Die Dehnungskomponenten, die einer Drehung der Koordinatenachsen um einen Winkel θ entsprechen, erhält man, indem man den Durchmesser des Mohrschen Kreises um einen Winkel von 2θ dreht.

mohrs kreis für ebene dehnung

Mohr&#39;s Circle for Plane Strain

Der Mohrsche Kreis ist eine wichtige grafische Methode zur Analyse der ebenen Dehnung durch Auftragen der Dehnung auf einem Satz kartesischer Koordinaten, wobei die Abszisse die Normaldehnung ε und die Ordinate die Scherdehnung γ ist. Ähnlich wie beim Mohrschen Kreis für ebene Spannung werden zwei Punkte X und Y eingezeichnet. Ihre Koordinaten sind (ε_x, -γ_XY) bzw. (ε_Y, γ_XY).  

Der Mohrsche Kreis stellt die Dehnungszustände unter verschiedenen Bedingungen visuell dar, was für das Verständnis des Materialverhaltens von wesentlicher Bedeutung ist. Der Mittelpunkt des Mohrschen Kreises mit der Bezeichnung O entspricht der durchschnittlichen Normaldehnung, wobei der Kreisradius aus der Beziehung zwischen Normal- und Scherdehnung abgeleitet wird. Dadurch lässt sich visualisieren, wie sich Dehnungen unter verschiedenen Belastungsbedingungen verändern, indem die Drehungen und Verschiebungen im Kreis dargestellt werden, wenn sich die Koordinatenachsen drehen.

Besonders bedeutsam sind die Punkte, an denen der Mohrsche Kreis die horizontale Achse schneidet und die maximalen und minimalen Hauptdehnungen darstellen. Diese Hauptdehnungen werden aus der durchschnittlichen Dehnung plus bzw. minus dem Kreisradius berechnet und geben die Dehnungsgrenzen an, die ein Material unter einer bestimmten Belastung aushalten kann. In homogenen, isotropen Materialien, die einer elastischen Verformung unterliegen, richten sich die Hauptdehnungsachsen nach den Spannungsachsen aus, eine Korrelation, die durch das Hookesche Gesetz für Scherspannung und -dehnung festgelegt wird. Diese Ausrichtung hilft bei der Vorhersage materieller Reaktionen unter Stress.

Darüber hinaus stellt der Durchmesser des Mohrschen Kreises die maximale Scherbeanspruchung in der Ebene dar. Bei der Analyse mit gedrehten Koordinatenachsen werden durch Drehen des Durchmessers XY des Mohrschen Kreises um einen Winkel 2θ effektiv die Dehnungskomponenten in dieser Ausrichtung für um den Winkel θ gedrehte Koordinatenachsen bestimmt.

Mohrs Kreis für ebene Dehnung

Der Mohr-Kreis analysiert die ebene Dehnung, indem er Punkte mit der Abszisse gleich der Normaldehnung und der Ordinate gleich der halben Scherung ...

Education

Lehre

JoVE Core

Mechanical Engineering

Maschinenbau

Unit 1

Einheit 1

Transformations of Stress and Strain

Spannungs- und Dehnungstransformationen

KEY TERMS AND CONCEPTS

ZENTRALE BEGRIFFE UND KONZEPTE

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 summarizes the full strain state in one picture.

As the coordinate axes rotate, the circle shifts and rotates with them. This makes it easier to see how strain changes under different loading conditions. For a new axis orientation, rotating the diameter XY by 2θ gives the strain components for axes rotated by θ.

The points where the circle crosses the horizontal axis are important because they show the maximum and minimum principal strains. These principal strains equal the average strain plus or minus the circle’s radius. They mark the strain limits a material can sustain under a given load.

For homogeneous, isotropic materials in elastic deformation, the principal strain axes line up with the stress axes. This link follows from Hooke’s law for shear stress and shear strain. It helps predict how the material will respond under stress.

The diameter of Mohr’s circle also shows the maximum in-plane shearing strain. Together, the circle’s center, radius, and axis intersections give a clear way to read strain values. This graphical method is especially useful for understanding material behavior in plane strain.

Principal and Shear Strains on Mohr’s Circle

principal-and-shear-strains-on-mohrs-circle

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...

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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

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 Mohr’s circle shows normal strain, shear strain, principal strains, and maximum in-plane shear for plane strain in a clear graphical form.

Principal and Shear Strains on Mohr’s Circle | JoVE K12

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Principal and Shear Strains on Mohr’s Circle

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