Name: multiaxial stress and strain relation
Uploaded: 2025-05-22T09:06:18.000Z
Duration: 1 min 22 s
Channel: JoVE

广义胡克定律将胡克定律推广至所有应力类型和方向，有助于理解材料在多轴应力状态下的行为。
考虑一个各向同性的立方体，其受到多轴加载，其中法向应力沿三个坐标轴方向作用。
立方体变形为长方体，其各边相等，且沿坐标轴方向存在正应变。
通过分别考虑每个应力分量的影响，并利用叠加原理将这些影响进行组合，即可将应变分量表示为应力分量的函数。该方法假设每个影响与其载荷呈线性关系，且变形极小。
对于多轴加载，若应力不超过材料的比例极限，且任一给定表面上的应力不会引起显著变形从而影响其他表面的应力计算，则满足条件。
各个方向的应力分量会在其对应方向上产生应变，同时也会在另外两个方向上产生应变。
通过综合各个单独效应，可推导出对应于多轴加载的应变分量，这被称为广义胡克定律。

广义胡克定律

Generalized Hooke's Law

广义胡克定律是胡克定律的扩展版本，它适用于所有类型中的应力和各个方向。思考能够形成立方体的各向同性材料，它们是能够承受多轴载荷的。在这种情况下，法向应力沿着三个坐标轴进行力的施加。由于这些应力，立方体的形状就会变为长方体。尽管出现这种变形，新的形状仍然保持着其相等的边，并且在坐标轴的方向上存在着法向应变。应变分量是从应力分量推导出来的。此过程涉及到单独考虑每个压力分量的影响，然后对这些影响进行整合。

该方法使用了叠加原理，假设每个效应与其载荷是线性相关的，并且其产生的变形很小。如果其中的应力没有超过材料的比例极限，那么这些条件便适用于多轴载荷。此外，施加在任何给定面上的应力不会引起可能影响到其他面上的应力所计算得出的显著变形。因此每个应力分量都会在其各自的方向上引起应变，并在其他的两个方向上引起应变。与多轴载荷相对应的应变分量可以通过合并这些单独的效应来得出。这些派生分量便表示了广义胡克定律。

<img alt="Equation 1" src="/files/ftp_upload/15600/15600_Equation_1.svg" style="text-align: center; height: 160px;">

广义胡克定律是胡克定律的扩展形式，适用于各种类型的应力及其在所有方向上的作用。考虑一个……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Stress and Strain - Axial Loading

应力与应变 - 轴向加载

KEY TERMS AND CONCEPTS

关键术语与概念

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 superposition principle. This approach assumes that each strain changes linearly with its load and that the deformation stays small.

These conditions are valid for multiaxial loading when the stresses do not go beyond the material's proportional limit. The stress on one face should also not create large deformations that would affect stress calculations on the other faces. Each stress component causes strain in its own direction and also strain in the other two directions. When these individual effects are added together, they give the strain components described by the generalized Hooke's law.

Multiaxial Stress and Strain Relation

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 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

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...

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 multiaxial loading changes an isotropic cube and how stress components combine to produce normal strain using the generalized Hooke's law.

Multiaxial Stress and Strain Relation | 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

Multiaxial Stress and Strain Relation

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