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

일반적인 응력 상태를 가진 입방체 요소에서 전단 응력은 정육면체를 수직 변형에 영향을 주지 않고 비스듬한 평행 육면체로 변형할 수 있습니다. 전단 응력 하에서의 각도 변화는 전단 변형률입니다.
정육면체는 전단 응력만 가하면 마름모꼴로 변형되어 전단 변형의 영향을 강조합니다.
큐브의 축 사이의 각도가 줄어들면 변형률은 양수로 간주됩니다. 전단 응력-변형률 다이어그램은 초기 직선이 전단 응력과 변형률 사이의 비례를 나타내는 Hooke의 법칙을 따릅니다.
전단에 대한 Hooke의 법칙에서 상수 G는 재료의 전단 계수 또는 강성 계수라고합니다. 전단 응력과 동일한 단위로 표시되며 탄성 계수의 절반 미만이지만 1/3 이상입니다.
일반적인 응력 조건의 경우 전단 변형률 관계와 Hook의 법칙을 사용하여 일반화된 Hooke의 법칙을 얻습니다.
이러한 관계의 상수는 실험적으로 결정되며, 다양한 응력 조합으로 인한 재료 변형을 예측하는 데 사용할 수 있습니다.

전단 변형

전단 변형률은 전단 응력을 받을 때 입방체 요소의 각도 변화를 나타냅니다. 이러한 유형의 응력은 일반 변형에 영향을 주지 않고 입방체를 경사 평행육면체로 변환할 수 있습니다. 입방체 요소는 전단 응력에만 노출될 때 상당한 변형을 경험합니다. 그 모양은 완벽한 입방체에서 마름모꼴로 바뀌며 전단 변형의 효과를 명확하게 보여줍니다. 이 변형 정도는 축 사이의 각도를 줄이면 양수로 간주됩니다.

전단 응력과 변형률 간의 관계는 표준 응력-변형률 차트와 유사한 다이어그램을 통해 시각화할 수 있습니다. 그러나 수율과 최종 강도는 다릅니다. 이 다이어그램은 훅의 법칙이 작용하는 모습을 보여주며 초기 직선은 전단 응력과 변형률 사이의 정비례를 나타냅니다.

<img alt="Equation1" src="/files/ftp_upload/15602/15602_Equation_1.svg" style="height: 22px;">

전단 응력에 대한 훅의 법칙에서 상수 G는 재료의 강성 계수 또는 전단 계수라고 합니다. 이 상수는 전단 응력과 동일한 단위로 표현됩니다. 전단 변형률은 다양한 응력에 대해 정의될 수 있습니다. 중첩의 원리를 활용함으로써 우리는 훅의 법칙의 일반화된 형태를 유도할 수 있습니다. 이 법칙에는 세 가지 상수가 포함됩니다. 두 개의 상수는 실험적으로 결정되고 세 번째 상수는 계산적으로 얻어집니다.

<img alt="Equation2" src="/files/ftp_upload/15602/15602_Equation_2.svg" style="height: 46px;">

일반적인 응력 상태를 가진 입방 요소에서 전단 응력은 입방체를 법선에 영향을 주지 않고 비스듬한 평행 육면체로 변형할 수 있습니다.

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Stress and Strain - Axial Loading

응력과 변형률 - 축 하중

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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...

stress-strain diagram

Tensile Test Data and Material Behavior

A stress-strain diagram shows how a material responds during a tensile test. It records the mechanical behavior of a cylindrical specimen as it is pulled by an increasing centric load. The specimen is carefully prepared before testing. Two gauge marks are inscribed on the central part of the specimen. The distance between these marks is the gauge length. As the load increases, the gauge length also increases. This change in length is called elongation. The elongation is measured for each load...

stress-strain diagram - ductile materials

Ductile Material Stress-Strain Stages

Ductile materials such as structural steel and aluminium show a multi-stage stress-strain response. When a specimen is loaded, its length first increases linearly at a slow rate. This produces a steep straight line on the stress-strain diagram. That initial part shows elastic deformation, which means the material returns to its original shape after unloading. When the stress reaches a critical value, plastic deformation begins. In this stage, the material changes shape a lot while the applied...

stress-strain diagram - brittle materials

Brittle Material Failure Under Tension

Brittle materials such as glass, cast iron, and stone fail with very little change in elongation. Their breaking strength and ultimate strength are nearly the same, and the strain at rupture is low. In these materials, failure is usually driven by normal stress, so the break surface is typically perpendicular to the applied load. Brittle materials also show little or no necking. Necking is the local narrowing of a material’s cross section under stress. Another important feature is that many...

true stress and true strain

Stress-Strain Curves Beyond Yield

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...

plastic behavior

Yielding and Strain-Hardening in Metals

Yielding and strain-hardening describe how a material changes when stress goes beyond the elastic range. Under elastic behavior, the stress disappears when the load is removed, and the material returns to its original shape. Once stress exceeds the yield point, yielding begins and permanent deformation, or plastic set, starts. The switch from elastic to plastic behavior depends on the peak stress and on how long the load stays in place before removal. A specimen that is loaded, unloaded, and...

fatigue

Fatigue Failure in Repeated Loading

Fatigue failure happens when materials rupture after repeated or fluctuating loads. The stress can be far below the material’s static breaking strength. Even ductile materials can fail in a brittle way under fatigue. This behavior matters in machines and structural parts that face repetitive or changing loads. One example is an unbalanced pump impeller that creates vibrations. Another is a thin steel rod or wire bent back and forth in the same spot until it breaks. The number of loading...

deformation of member under multiple loadings

deformation-of-member-under-multiple-loadings

Calculating Rod Stretch from Mixed Loads

Calculating rod stretch from mixed loads starts with breaking the rod into parts. This step is needed when the rod is made of different materials or has changes in cross-section. Each part has its own internal force, cross-sectional area, length, and modulus of elasticity. These values are used to find the deformation of the full rod. For a member with a variable cross-section, the strain is not constant. Instead, it changes with position along the rod. The deformation of a small element is...

statically indeterminate problem solving

Axial Rod Reactions with Redundant Support

Axial rod reactions with a redundant support are found by combining equilibrium with deformation. The structure here uses two cylindrical rods, one steel and one brass, joined at point B and held by rigid supports at points A and C. The goal is to determine the reactions at A and C and the deflection at B. This setup is statically indeterminate because the supports create more unknown reactions than the equilibrium equations can solve. Statics alone is not enough to find every internal force...

thermal strain

Thermal Expansion of a Free Rod

Thermal strain describes how a structure changes length when temperature changes. It is important in real-world design because structures do not always stay at one temperature. Temperature shifts can affect how they behave and how long they last. A simple example is a homogeneous rod with a uniform cross-section resting freely on a flat horizontal surface. When the rod is heated, it elongates. The amount of elongation depends on the temperature change and the rod’s original length. This...

temperature dependent deformation

Thermal Stress in a Composite Rod

A steel-and-brass composite rod can build stress when temperature changes while both ends are restrained. The rod is nonhomogeneous, so its two sections respond differently. To find the compressive load, the problem is treated as statically indeterminate, which means the force cannot be found from equilibrium alone. The analysis begins by disconnecting one support so the rod can expand or contract freely under the temperature change. An unknown force is then applied at the free end. That force...

poisson's ratio

Material Strain Under Axial Loading

Poisson's ratio describes how a material changes shape when it is pulled or compressed along one direction. It links the strain in the direction of the force to the strain that happens sideways. When a slender bar is loaded axially, it stretches along the load and contracts laterally. Poisson's ratio is defined as the negative ratio of lateral contraction to axial elongation. The negative sign is used because the two strains usually have opposite signs. This keeps the ratio positive and makes...

generalized hooke's law

Multiaxial Stress and Strain Relation

Multiaxial stress and strain describe how an isotropic material responds when forces act along more than one direction. A cube-shaped sample under normal stresses on the three coordinate axes can change into a rectangular parallelepiped. Even after this deformation, the sides remain equal, and normal strain appears along the coordinate axes. The strain components are found from the stress components by studying each stress effect one at a time. The separate effects are then combined using the...

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

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

Axial load effects on shear strain are shown by how a slender bar deforms under tension or compression. When the bar is loaded along its axis, it changes in both the axial direction and the transverse direction. A small cubic element inside the bar can turn into a rectangular parallelepiped or a rhombus, depending on how it is oriented. This change in shape creates shearing strain, which is the distortion caused when angles in a material change. Axial loading produces both normal strain and...

saint-venant's principle

Load Effects Away from the Ends

Saint-Venant's principle describes how stress changes in a structural member when a load is applied. It shows that the exact way a load is placed matters most near the point of application. Farther away from that point, the stress pattern becomes much less dependent on how the load was applied. A simple example uses a member with plates on both ends. When the loads are applied at the center of the plates, the plates move toward each other without rotating. The member shortens in length and...

stress concentrations

Stress at Holes and Fillets

Stress can become much higher near holes and abrupt changes in shape within a structural member. These discontinuities create localized stress that may exceed the average stress in the part. For flat bars, the stress distribution can be measured experimentally with a photoelastic method. This approach is used for bars with a circular hole or with changing widths joined by fillets. The results depend on geometric ratios, such as the hole radius to the smaller width for a circular hole. For...

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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