Stress Tensor - simple problem (I think)

In summary: I'll post back to this thread if I get stuck.In summary, the stress tensor at point Q in a body with components pij is used to calculate the stress force f across a small area of surface at Q normal to n = (2,1,-1). The normal stress is found by finding the component of f in the direction of n, while the shear stress is found by subtracting the normal stress from the total stress. This calculation does not involve the tensor and can be done using basic algebra techniques.
  • #1
rarkup
9
0

Homework Statement



Stress tensor at a point Q in a body has components:

pij:
| 1 -1 0 |
|-1 2 1 |
| 0 1 3 |

(i) Calculate components of the stress force f across a small area of surface at Q normal to n = (2,1,-1).
(ii) The component of f in the direction of n is called the normal stress while the component of f tangential to the surface is called the shear stress. Find the normal stress and the shear stress at Q.

Homework Equations



f = fi = pijnj

The Attempt at a Solution



(i) I take the stress force at Q across the surface normal to n to be the product of the stress tensor pij and the normal vector n and arrive at:

f = (1x2 + -1x1 + 0x-1)i + (-1x2 + 2x1 + 1x-1)j + (0x2 + 1x1 +3x-1)k
= ( 1, -1, -2 )

(ii) I think this is a simple algebraic exercise - ie what components of ( 1, -1, -2) project onto n to give the normal stress, and which components project onto the plane normal to n to give the shear stress.

Am I reading this correctly, and what is the most efficient way of computing these components? I can't help but think the answer might be staring at me from within the tensor pij.
 
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  • #2
Welcome to PF!

Hi rarkup! Welcome to PF! :smile:
rarkup said:
(i) I take the stress force at Q across the surface normal to n to be the product of the stress tensor pij and the normal vector n and arrive at:

f = (1x2 + -1x1 + 0x-1)i + (-1x2 + 2x1 + 1x-1)j + (0x2 + 1x1 +3x-1)k
= ( 1, -1, -2 )

Yup, that's fine! :biggrin:
(ii) I think this is a simple algebraic exercise - ie what components of ( 1, -1, -2) project onto n to give the normal stress, and which components project onto the plane normal to n to give the shear stress.

Am I reading this correctly, and what is the most efficient way of computing these components? I can't help but think the answer might be staring at me from within the tensor pij.

Just find the component of ( 1, -1, -2) in the direction of ( 2, 1, -1) in the usual way (and then subtract that from the whole stress to get the shear stress) …

this has nothing to do with the tensor any more. :wink:
 
  • #3
Thanks for the welcome and your response tiny-tim.

Now it's time to dust off the Algebra books and relearn how to compute components of a vector in the direction of another.
 

1. What is a stress tensor?

A stress tensor is a mathematical representation of the distribution of forces acting on a material, typically in three dimensions. It is commonly used in engineering and physics to analyze the stresses and strains in a material under different conditions.

2. How is a stress tensor calculated?

A stress tensor is typically calculated by taking the derivative of the strain energy density with respect to the strain components. This can be done using various mathematical techniques, such as differential calculus or matrix operations.

3. What is the significance of the stress tensor in materials science?

The stress tensor is a crucial tool in materials science as it allows for the prediction and analysis of how materials will behave under different loading conditions. By understanding the distribution of forces within a material, engineers can design structures and materials that are more resistant to failure.

4. What is a simple problem that can be solved using a stress tensor?

A common simple problem that can be solved using a stress tensor is determining the maximum stress and strain in a material under uniaxial tension. This can be done by using the stress tensor to calculate the stress and strain components in the direction of the applied force.

5. How is the stress tensor used in continuum mechanics?

In continuum mechanics, the stress tensor is a fundamental tool for analyzing the behavior of continuous materials, such as fluids and solids. It is used to calculate the stress and strain at any point in a material, allowing for the prediction of material behavior under different conditions.

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