Name: shear modulus and angular change
Uploaded: 2024-03-07T13:22:38.000Z
Duration: 1 min 20 s
Channel: JoVE

In einem kubischen Element mit allgemeinem Spannungszustand können die Schubspannungen einen Würfel zu einem schrägen Quader verformen, ohne die normalen Dehnungen zu beeinflussen. Die Winkeländerung unter Schubspannung ist die Schubdehnung.
Wenn ein Würfel nur einer Schubspannung ausgesetzt ist, verwandelt er sich in eine Raute, was die Auswirkungen der Schubdehnung hervorhebt.
Wenn der Winkel zwischen den Achsen des Würfels schrumpft, wird die Dehnung als positiv angesehen. Das Schubspannungs-Dehnungs-Diagramm folgt dem Hookeschen Gesetz, bei dem die anfängliche Gerade die Proportionalität zwischen Schubspannung und Dehnung angibt.
Die Konstante G im Hookeschen Gesetz für das Scheren wird als Schubmodul oder Steifigkeitsmodul des Materials bezeichnet. Sie wird in den gleichen Einheiten wie die Schubspannung ausgedrückt und beträgt weniger als die Hälfte, aber mehr als ein Drittel des Elastizitätsmoduls.
Für allgemeine Spannungsbedingungen wird ein verallgemeinertes Hooke'sches Gesetz unter Verwendung der Schub-Dehnungs-Relationen und des Hook'schen Gesetzes erhalten.
Die Konstanten in diesen Zusammenhängen werden experimentell bestimmt, anhand derer dann Materialverformungen aus verschiedenen Spannungskombinationen vorhergesagt werden können.

scherdehnung

Die Scherdehnung stellt die Winkeländerung eines kubischen Elements dar, wenn es einer Scherbeanspruchung ausgesetzt wird. Diese Art von Spannung kann einen Würfel in ein schiefes Parallelepiped verwandeln, ohne die Normaldehnungen zu beeinflussen. Das kubische Element erfährt bei alleiniger Scherbeanspruchung eine deutliche Transformation. Seine Form verändert sich von einem perfekten Würfel zu einem Rhomboid, was deutlich den Effekt der Scherbeanspruchung zeigt. Der Grad dieser Dehnung gilt als positiv, wenn dadurch der Winkel zwischen den Achsen verringert wird.

Die Beziehung zwischen Scherspannung und Dehnung kann durch ein Diagramm ähnlich einem Standard-Spannungs-Dehnungs-Diagramm visualisiert werden. Die Ausbeute und die Endfestigkeit sind jedoch unterschiedlich. Dieses Diagramm veranschaulicht das Hookesche Gesetz in Aktion, wobei die anfängliche gerade Linie eine direkte Proportionalität zwischen Scherspannung und Dehnung anzeigt.

<img alt="Equation1" src="/files/ftp_upload/15602/15602_Equation_1.svg" style="height: 22px;">

Die Konstante G im Hookeschen Gesetz für Scherspannung wird als Steifigkeitsmodul oder Schermodul des Materials bezeichnet. Diese Konstante wird in denselben Einheiten ausgedrückt wie die Scherspannung. Die Scherdehnung kann für eine Reihe von Spannungen definiert werden. Indem wir das Superpositionsprinzip nutzen, können wir die verallgemeinerte Form des Hookeschen Gesetzes ableiten. Dieses Gesetz beinhaltet drei Konstanten. Zwei Konstanten werden experimentell bestimmt, während die dritte rechnerisch ermittelt wird.

<img alt="Equation2" src="/files/ftp_upload/15602/15602_Equation_2.svg" style="height: 46px;">

Scherdehnung

In einem kubischen Element mit allgemeinem Spannungszustand können die Schubspannungen einen Würfel zu einem schrägen Quader verformen, ohne die normale ...

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Mechanical Engineering

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Unit 1

Einheit 1

Stress and Strain - Axial Loading

Spannung und Dehnung – axiale Belastung

KEY TERMS AND CONCEPTS

SCHLÜSSELBEGRIFFE UND KONZEPTE

Shearing Strain

shear modulus and angular change

Shearing strain is the angular change that happens in a cubic element when it is loaded by shearing stress. Under this stress, the cube can change into an oblique parallelepiped without creating normal strains. If the cube is acted on only by shearing stress, its shape can shift strongly into a rhomboid.

The amount of shearing strain is considered positive when the angle between the axes becomes smaller. This change in angle shows the direct effect of shear on the material. The deformation can be compared with the stress-strain behavior shown on a standard chart, although the yield strength and ultimate strength are different for this case.

The first straight part of the shear stress-strain graph shows Hooke's law in action. In this range, shearing stress and shearing strain are directly proportional. The constant G in this relation is called the modulus of rigidity, or shear modulus, and it has the same units as shearing stress.

Shearing strain can be defined over a range of stresses. Using the principle of superposition, the generalized form of Hooke's law can be derived. This law includes three constants. Two are found by experiment, and the third is calculated from them.

Shear Modulus and Angular Change

normal strain under axial loading

Axial Stress and Strain in a Rod

Axial stress and strain in a rod describe how a material changes length when a force is applied along its axis. The force may stretch the rod or compress it. In mechanics of materials, this deformation is called normal strain. Normal strain is found by dividing the change in length by the original length. This gives a unitless ratio that shows how much the material deforms under load. A load-deformation diagram can show elongation, but it cannot predict how another rod will deform. When a...

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hooke's law

Stress-Strain Limits in Materials

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fatigue

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deformation of member under multiple loadings

deformation-of-member-under-multiple-loadings

Calculating Rod Stretch from Mixed Loads

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statically indeterminate problem solving

Axial Rod Reactions with Redundant Support

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thermal strain

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poisson's ratio

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generalized hooke's law

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bulk modulus

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shearing strain

Shearing strain is the angular change that happens in a cubic element when it is loaded by shearing stress. Under this stress, the cube can change into an oblique parallelepiped without creating normal strains. If the cube is acted on only by shearing stress, its shape can shift strongly into a rhomboid. The amount of shearing strain is considered positive when the angle between the axes becomes smaller. This change in angle shows the direct effect of shear on the material. The deformation can...

relation between poisson's ratio, modulus of elasticity and modulus of rigidity

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Axial Load Effects on Shear Strain

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saint-venant's principle

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residual stresses

Residual Stresses After Welding and Cooling

Residual stresses are stresses that remain in a structure after the original load or heat source is removed. They often form when different parts of a material undergo uneven plastic deformation, which means permanent shape change. A rod stretched past its yield point is one example. It will not return to its original length, and some stress stays inside even after the load is gone. These stresses can also develop during welding. Heat changes the material during the weld and can cause plastic...

key terms and concepts

Learn how shearing stress changes a cube into a rhomboid, how shear strain is measured, and why the shear modulus G matters in Hooke's law.

Shear Modulus and Angular Change | JoVE K12

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mechanical engineering

Shear Modulus and Angular Change

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