Name: principal strain directions in 3d
Uploaded: 2024-05-23T12:59:55.000Z
Duration: 1 min 29 s
Channel: JoVE

L’analyse de déformation tridimensionnelle utilise les principaux axes de contrainte pour évaluer la déformation des matériaux élastiques et homogènes.
Un petit élément cubique, lorsqu’il s’étend autour de ces axes, se transforme en parallélépipède rectangulaire, illustrant la déformation.
L’analyse de déformation consiste à faire pivoter un élément autour d’un axe principal, comme l’axe n, et à évaluer les composantes de déformation sur des faces perpendiculaires à cet axe. Cette méthode est basée sur la transformation de la déformation plane.
À l'aide du cercle de Mohr, la transformation de déformation est examinée lorsque le cube tourne autour des axes principaux. Cette approche identifie la déformation de cisaillement maximale en un certain point, équivalente au diamètre du plus grand des trois cercles.
En déformation plane, l'axe n se transforme en un axe principal avec une déformation nulle à l'origine du diagramme circulaire de Mohr. Les déformations principales situées sur les côtés opposés de cette origine représentent les déformations normales maximales et minimales.
Dans les plaques minces de structures soumises à des contraintes planes, l’axe n se transforme en un axe de contrainte principal.
La déformation principale le long de l’axe n est associée aux déformations dans le plan. La rotation autour de l’axe m définit la déformation maximale par cisaillement.

analyse tridimensionnelle de la déformation

Three-Dimensional Analysis of Strain

L'analyse des déformations tridimensionnelles est cruciale pour comprendre comment les matériaux se déforment sous l'effet des contraintes, en particulier dans les matériaux élastiques et homogènes. Cette méthode utilise les principaux axes de contrainte pour simplifier les états de contrainte complexes en des formes plus compréhensibles. Soumis à une contrainte, un petit élément cubique dans un matériau se dilate ou se contracte le long de ces axes, se transformant en un parallélépipède rectangle. Cette transformation illustre bien la déformation du matériau. Les principaux axes de contrainte sont orthogonaux et représentent les directions dans lesquelles la contrainte n'induit pas de cisaillement au sein du matériau.

Le cercle de Mohr est un outil essentiel dans l'analyse des déformations. Il fournit une représentation graphique des états de contrainte en un point et évalue les composantes de déformation lorsque l'élément tourne autour d'un axe principal, tel que l'axe n. Cette analyse se concentre sur les transformations de déformations planes, où les déformations à l'origine de l'axe n sont nulles, simplifiant ainsi la détermination des déformations maximales et minimales représentées sur les côtés opposés du cercle de Mohr.

Dans des scénarios tels que des plaques minces soumises à une contrainte plane, l'axe n devient un axe de contrainte principal. Ici, la déformation principale le long de l’axe n est directement en corrélation avec les déformations dans le plan du matériau. La rotation autour d'un autre axe principal, tel que l'axe-m, permet d'identifier les emplacements et les ampleurs des contraintes de cisaillement maximales, qui sont cruciales pour prédire le comportement des matériaux sous charge et garantir la sécurité et la fiabilité des conceptions structurelles.

Analyse tridimensionnelle de la déformation

L’analyse de déformation tridimensionnelle utilise les principaux axes de contrainte pour évaluer la déformation des matériaux élastiques et homogènes.
Un petit élément cubique, lorsque...

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JoVE Core

Mechanical Engineering

Génie mécanique

Unit 1

Unité 1

Transformations of Stress and Strain

Transformations des contraintes et des déformations

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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 material. This simplifies a complex stress state into a form that is easier to analyze and compare.

Mohr's circle is a key graph used in strain analysis. It shows the stress state at a point and helps evaluate strain components as the element rotates about a principal axis, such as the n-axis. In plane strain transformations, the strain at the origin of the n-axis is zero. This makes it easier to find the maximum and minimum strains, which appear on opposite sides of Mohr's circle.

For thin plates under plane stress, the n-axis becomes a principal stress axis. In that case, the principal strain along the n-axis is directly related to the in-plane strains of the material. Rotation about another principal axis, such as the m-axis, helps locate the maximum shearing strain and its magnitude. These results are important for predicting how materials behave under load and for supporting safe, reliable structural design.

Principal Strain Directions in 3D

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...

principal stresses: problem solving

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

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...

three-dimensional analysis of strain

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 principal stress axes, Mohr's circle, and plane stress help show 3D strain, including maximum, minimum, and shearing strain in materials.

Principal Strain Directions in 3D | 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

Principal Strain Directions in 3D

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