Help with understanding stress tensors

In summary, the stress tensor \sigma_{ij} represents the force per unit area in the i-direction on a surface with normal in the j-direction. When the indices are the same, it represents a pressure, and when the direction of the first index is normal to that of the second, it represents a shear. Pn, Pnn, and Pnt refer to specific components of the stress tensor, with Pn representing the force per unit area in the i-direction on the normal surface, Pnn representing the normal force on the normal surface, and Pnt representing the force in the normal direction on the tangent surface.
  • #1
MaxManus
277
1
I'm taking a continuum mechanics course and we use the 3*3 stress tensor a lot. The problem is that I do not understand what each component mean.

What does the tension(Pn normal(Pnn and tangent(Pnt tension mean, or just Pxy?
 
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  • #2
For a stress tensor [itex]\sigma[/itex], [itex]\sigma_{11}[/itex] is the normal stress in the 1-direction and [itex]\sigma_{12}[/itex] is the shear stress in the 1-2 plane. Does this answer your question?
 
  • #3
The component [itex]\sigma_{ij}[/itex] of the stress tensor [itex]\sigma [/itex] is the force per unit area in the i-direction on a surface with normal in the j-direction. When the indices are the same, the force is normal to the surface so that it represents a pressure. When the direction of the first index is normal to that of the second, it represents a shear.
 
  • #4
Thanks, does that mean that the Pn is the force per unit area in the i-direction on the normal surface and Pnn is the the normal force on the normal surface and Pnt is the the force in the normal direction on the tangent surface?

Pn = P dot n
pnn = P dot n dot n
Where P is the stress tensor and n is the normal.
 
  • #5


The stress tensor is a mathematical representation of the distribution of forces within a material. It is a 3x3 matrix that describes the magnitude and direction of the stresses at each point within the material. The three components, Pn, Pnn, and Pnt, represent the normal and tangential stresses in the x, y, and z directions, respectively.

The normal stress, Pn, is the force per unit area acting perpendicular to the surface of the material. This can be thought of as the force pushing or pulling on the material in a specific direction.

The normal normal stress, Pnn, is the normal stress acting in the same direction as the surface itself. It is a measure of the resistance to deformation of the material in that direction.

The tangential stress, Pnt, is the force per unit area acting parallel to the surface of the material. This can be thought of as the force trying to shear or twist the material.

The Pxy component represents the tangential stress in the xy plane, or the shear stress in the x direction acting on a surface perpendicular to the y direction.

Overall, the stress tensor provides a comprehensive view of the internal forces within a material, allowing for better understanding and analysis of its mechanical behavior. I hope this helps in your understanding of stress tensors.
 

What is a stress tensor?

A stress tensor is a mathematical representation of stress, which is a measure of the internal forces within a material that cause deformation. It is a mathematical quantity that describes the distribution of forces acting inside a material.

What do the components of a stress tensor represent?

The components of a stress tensor represent the magnitude and direction of the internal forces acting on a material in three dimensions. Each component represents the force acting in a particular direction (x, y, and z), and the combination of these components describes the overall stress state of the material.

How is a stress tensor used in material science?

A stress tensor is used in material science to understand and predict the behavior of materials under different loading conditions. It helps engineers and scientists design and analyze structures, determine material properties, and predict failure points.

What is the difference between a stress tensor and a strain tensor?

A stress tensor describes the internal forces within a material, while a strain tensor describes the resulting deformation caused by those forces. In other words, stress is the cause, and strain is the effect.

How is a stress tensor calculated?

A stress tensor is calculated using equations based on the laws of mechanics and physics. The most common method is using the Cauchy stress tensor, which calculates stress at a specific point in a material. Other methods include using the Piola-Kirchhoff stress tensor or the Hencky stress tensor, which take into account the deformation of the material.

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