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What is the purpose of the constitutive equations?
What is the purpose of the constitutive equations?
What is the difference between engineering strain and true strain?
What is the difference between engineering strain and true strain?
What is the mathematical representation of the strain tensor?
What is the mathematical representation of the strain tensor?
What is the type of stress that is perpendicular to a surface?
What is the type of stress that is perpendicular to a surface?
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What is the type of strain that is a measure of the change in volume?
What is the type of strain that is a measure of the change in volume?
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How many components does the strain tensor have?
How many components does the strain tensor have?
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What is the mathematical representation of the stress tensor?
What is the mathematical representation of the stress tensor?
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What is the type of stress that is parallel to a surface?
What is the type of stress that is parallel to a surface?
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Quel est le type de déformation décrit par le tenseur de déformation ?
Quel est le type de déformation décrit par le tenseur de déformation ?
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Quel est le composant du tenseur de déformation qui décrit le changement de longueur ?
Quel est le composant du tenseur de déformation qui décrit le changement de longueur ?
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Quelle est la propriété du tenseur de déformation qui signifie que εxy = εyx ?
Quelle est la propriété du tenseur de déformation qui signifie que εxy = εyx ?
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Quel est le type de stress qui décrit les forces perpendiculaires à une surface ?
Quel est le type de stress qui décrit les forces perpendiculaires à une surface ?
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Quel est le théorème qui décrit la relation entre le tenseur de contrainte et le vecteur normal à une surface ?
Quel est le théorème qui décrit la relation entre le tenseur de contrainte et le vecteur normal à une surface ?
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Quel est le composant du tenseur de contrainte qui décrit les forces parallèles à une surface ?
Quel est le composant du tenseur de contrainte qui décrit les forces parallèles à une surface ?
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Quel est le type de déformation qui décrit le changement de volume d'un matériau ?
Quel est le type de déformation qui décrit le changement de volume d'un matériau ?
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Quel est le but principal du tenseur de déformation et du tenseur de contrainte ?
Quel est le but principal du tenseur de déformation et du tenseur de contrainte ?
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Study Notes
Strain Tensor
- Definition: A measure of the deformation of a material, describing the change in shape and size of an object.
- Mathematical representation: A second-order tensor, often denoted as ε (epsilon), which describes the strain at a point in a material.
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Components: The strain tensor has 9 components, which can be represented as a 3x3 matrix:
- εxx, εxy, εxz
- εyx, εyy, εyz
- εzx, εzy, εzz
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Types of strain:
- Linear strain: A measure of the change in length per unit length.
- Shear strain: A measure of the change in shape, without a change in volume.
- Volumetric strain: A measure of the change in volume.
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Strain measures: There are different ways to measure strain, including:
- Engineering strain: A measure of strain based on the change in length.
- True strain: A measure of strain based on the instantaneous change in length.
Stress Tensor
- Definition: A measure of the internal forces that are distributed within a material, describing the forces that cause deformation.
- Mathematical representation: A second-order tensor, often denoted as σ (sigma), which describes the stress at a point in a material.
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Components: The stress tensor has 9 components, which can be represented as a 3x3 matrix:
- σxx, σxy, σxz
- σyx, σyy, σyz
- σzx, σzy, σzz
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Types of stress:
- Normal stress: A measure of the force perpendicular to a surface.
- Shear stress: A measure of the force parallel to a surface.
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Stress measures: There are different ways to measure stress, including:
- Cauchy stress: A measure of stress based on the force per unit area.
- Piola-Kirchhoff stress: A measure of stress based on the force per unit area of the undeformed material.
- Relationship between stress and strain: The stress tensor is related to the strain tensor through the constitutive equations, which describe the behavior of the material.
Strain Tensor
- Definition: Measures deformation of a material, describing changes in shape and size.
- Mathematical Representation: A 2nd-order tensor, denoted as ε (epsilon), describing strain at a point in a material.
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Components: 9 components, represented as a 3x3 matrix:
- εxx, εxy, εxz
- εyx, εyy, εyz
- εzx, εzy, εzz
-
Types of Strain:
- Linear Strain: Measures change in length per unit length.
- Shear Strain: Measures change in shape, without a change in volume.
- Volumetric Strain: Measures change in volume.
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Strain Measures:
- Engineering Strain: Measures strain based on change in length.
- True Strain: Measures strain based on instantaneous change in length.
Stress Tensor
- Definition: Measures internal forces distributed within a material, causing deformation.
- Mathematical Representation: A 2nd-order tensor, denoted as σ (sigma), describing stress at a point in a material.
-
Components: 9 components, represented as a 3x3 matrix:
- σxx, σxy, σxz
- σyx, σyy, σyz
- σzx, σzy, σzz
-
Types of Stress:
- Normal Stress: Measures force perpendicular to a surface.
- Shear Stress: Measures force parallel to a surface.
-
Stress Measures:
- Cauchy Stress: Measures stress based on force per unit area.
- Piola-Kirchhoff Stress: Measures stress based on force per unit area of the undeformed material.
- Relationship between Stress and Strain: Stress tensor related to strain tensor through constitutive equations, describing material behavior.
Strain Tensor
- A measure of material deformation, describing shape and size changes of a small material cube
- Represented as a second-order tensor, denoted by ε (epsilon)
- Comprised of normal strains (εxx, εyy, εzz) and shear strains (εxy, εxz, εyz)
- Types of strain include:
- Linear strain (ε = ΔL / L): change in length per unit length
- Shear strain (γ = Δx / y): change in angle between two lines
- Volumetric strain (εv = ΔV / V): change in volume per unit volume
- Strain tensor is symmetric, meaning εxy = εyx, εxz = εzx, and εyz = εzy
- Fundamental in continuum mechanics, describing material deformation
Stress Tensor
- A measure of internal material forces, describing force distribution within the material
- Represented as a second-order tensor, denoted by σ (sigma)
- Comprised of normal stresses (σxx, σyy, σzz) and shear stresses (σxy, σxz, σyz)
- Types of stress include:
- Tensile stress (σ > 0): stretching force
- Compressive stress (σ < 0): compressing force
- Shear stress (τ): force parallel to the surface
- Stress tensor is symmetric, meaning σxy = σyx, σxz = σzx, and σyz = σzy
- Fundamental in continuum mechanics, describing internal material forces
- Cauchy's stress theorem: The stress vector at a point is proportional to the normal vector of the surface at that point
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Description
Understand the definition, mathematical representation, and components of strain tensor, a measure of material deformation.
