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

In a cubic element with a general state of stress, the shearing stresses can deform a cube into an oblique parallelepiped without affecting normal strains. The angular change under shearing stress is the shearing strain.
A cube, when subjected to only shearing stress, morphs into a rhomboid, highlighting the impact of shearing strain.
When the angle between the axes of the cube shrinks, the strain is deemed positive. The shearing stress-strain diagram follows Hooke&#39;s law, where the initial straight line indicates proportionality between shearing stress and strain.
The constant G in Hooke&#39;s law for shearing is termed the shear modulus or modulus of rigidity of the material. It is expressed in the same units as the shearing stress and is less than half but more than one-third of the modulus of elasticity.
For general stress conditions, a generalized Hooke&#39;s law is obtained using the shearing strain relations and Hook&#39;s law.
The constants in these relationships are determined experimentally, which can then be used to predict material deformations from various stress combinations.

shearing strain

The shearing strain represents a cubic element's angular change when subjected to shearing stress. This type of stress can transform a cube into an oblique parallelepiped without influencing normal strains. The cubic element experiences a significant transformation when exposed solely to shearing stress. Its shape alters from a perfect cube into a rhomboid, clearly demonstrating the effect of shearing strain. The degree of this strain is considered positive if it reduces the angle between the axes.
The relationship between shearing stress and strain can be visualized through a diagram similar to a standard stress-strain chart. However, the yield and ultimate strengths differ. This diagram illustrates Hooke's law in action, with the initial straight line indicating a direct proportionality between shearing stress and strain.
<img alt="Shear stress equation &#964;xy=G&#947;xy for elasticity, formula in mechanics." src="/files/ftp_upload/15602/15602_Equation_1.svg" style="text-align: center; height: 22px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
The constant G in Hooke's law for shearing stress is referred to as the modulus of rigidity or shear modulus of the material. This constant is expressed in the same units as shearing stress. Shearing strain can be defined for a range of stresses. By leveraging the principle of superposition, we can derive the generalized form of Hooke's law. This law involves three constants. Two constants are determined experimentally, while the third is obtained computationally.
<img alt="Shear modulus equation \(G=\frac{E}{2(1+\nu)}\) formula; mechanical properties analysis." src="/files/ftp_upload/15602/15602_Equation_2.svg" style="text-align: center; height: 46px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />

Shearing Strain

In a cubic element with a general state of stress, the shearing stresses can deform a cube into an oblique parallelepiped without affecting normal ...

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JoVE Core

Mechanical Engineering

Unit 1

Stress and Strain - Axial Loading

KEY TERMS AND CONCEPTS

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...

stress-strain diagram

Tensile Test Data and Material Behavior

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Ductile Material Stress-Strain Stages

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stress-strain diagram - brittle materials

Brittle Material Failure Under Tension

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Stress-strain curves beyond yield show how ductile materials behave after they start to deform permanently. Engineering stress is found by dividing the load by the original, undeformed cross-sectional area. It is a useful approximation, but it can lose accuracy when strain becomes large. True stress gives a more exact picture of the same specimen under load. It is calculated by dividing the applied load by the instantaneous cross-sectional area during deformation. As the specimen stretches,...

hooke's law

Stress-Strain Limits in Materials

Stress-strain limits in materials explain how solids respond when force is applied. Stress is the applied force per area, and strain is the change in shape or size that results. In the elastic range, strain stays directly proportional to stress. The constant that connects them is called the modulus of elasticity, or Young's modulus. This linear behavior continues only up to the proportional limit. After that point, the stress-strain relationship becomes nonlinear. For ductile materials, this...

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Yielding and Strain-Hardening in Metals

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fatigue

Fatigue Failure in Repeated Loading

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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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poisson's ratio

Material Strain Under Axial Loading

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generalized hooke's law

Multiaxial Stress and Strain Relation

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bulk modulus

Volume Change Under Hydrostatic Pressure

Bulk modulus describes how much a material resists a uniform squeeze. It links a change in pressure to the relative change in volume. For high school students, it is easiest to think of it as a measure of how hard it is to compress a material evenly from all sides. The idea becomes clearer when an isotropic material element is treated as a cube with unit volume. When normal stresses act on the cube, it deforms into a rectangular parallelepiped and its volume changes. The difference between the...

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

relation-between-poisson-s-ratio-modulus-elasticity-modulus

Axial Load Effects on Shear Strain

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saint-venant's principle

Load Effects Away from the Ends

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stress concentrations

Stress at Holes and Fillets

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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

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

Shear Modulus and Angular Change

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