What is the relationship between force lines and the stress tensor field?

em3ry · Dec 10, 2020

Force lines method is used in Solid Mechanics for visualization of internal forces in a deformed body. A force line represents graphically the internal force acting within a body across imaginary internal surfaces. The force lines show the maximal internal forces and their directions.

But stress in a solid body is a tensor not a single vector. What is the relationship between force lines and the stress tensor field? If I know the force lines can I calculate the stress tensor field? If I know the stress tensor field can I calculate the force lines?

Chestermiller · Dec 10, 2020

em3ry said:

Force lines method is used in Solid Mechanics for visualization of internal forces in a deformed body. A force line represents graphically the internal force acting within a body across imaginary internal surfaces. The force lines show the maximal internal forces and their directions.

But stress in a solid body is a tensor not a single vector. What is the relationship between force lines and the stress tensor field? If I know the force lines can I calculate the stress tensor field? If I know the stress tensor field can I calculate the force lines?

View attachment 274095

If you know the stress tensor, you can determine everything there is to know about the internal force distribution. Google Cauchy stress relationship.

em3ry · Dec 10, 2020

So are the force lines just the maximum principal stress at each point?

And I notice that the force lines never go in circles. I am guessing that when you remove the maximum principle stress then what's left over just goes in circles. ~~And I guess that stress at a point is actually a vector plus a bivector.~~

Chestermiller · Dec 10, 2020

em3ry said:

So are the force lines just the maximum principal stress at each point?

And I notice that the force lines never go in circles. I am guessing that when you remove the maximum principle stress then what's left over just goes in circles. And I guess that stress at a point is actually a vector plus a bivector.

I really don’t know what force lines are. With regard to stress, it is a second order tensor.

em3ry · Dec 10, 2020

I am still not sure what the answer to the original question is but I think it probably has to do with this:

The stress tensor

$\sigma _{ij}$

can be expressed as the sum of two other stress tensors:

a mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor,
$\pi \delta _{ij}$
, which tends to change the volume of the stressed body; and
a deviatoric component called the stress deviator tensor,
$s_{ij}$
, which tends to distort it.

So:

$\sigma _{ij}=s_{ij}+\pi \delta _{ij},\,$

where

$\pi$

is the mean stress given by

$\pi ={\frac {\sigma _{kk}}{3}}={\frac {\sigma _{11}+\sigma _{22}+\sigma _{33}}{3}}\,$

Chestermiller · Dec 11, 2020

em3ry said:

I am still not sure what the answer to the original question is but I think it probably has to do with this:

The stress tensor
$\sigma _{ij}$
can be expressed as the sum of two other stress tensors:

a mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor,
$\pi \delta _{ij}$
, which tends to change the volume of the stressed body; and

a deviatoric component called the stress deviator tensor,
$s_{ij}$
, which tends to distort it.

So:

$\sigma _{ij}=s_{ij}+\pi \delta _{ij},\,$

where
$\pi$
is the mean stress given by

$\pi ={\frac {\sigma _{kk}}{3}}={\frac {\sigma _{11}+\sigma _{22}+\sigma _{33}}{3}}\,$

This is all correct, but I don't see how it relates to your original question.

Unless the body is in hydrostatic equilibrium or is experiencing a purely volumetric change, the principal stresses and their directions can always be determined at each spatial location (i.e., the principal stresses are determined solely by the deviatoric stress tensor). Tracing out the directions of the principal stresses is what you might mean by the force lines.

em3ry · Dec 11, 2020

Chestermiller said:

This is all correct, but I don't see how it relates to your original question.

Unless the body is in hydrostatic equilibrium or is experiencing a purely volumetric change, the principal stresses and their directions can always be determined at each spatial location (i.e., the principal stresses are determined solely by the deviatoric stress tensor).

OK. I see what you mean. Thanks

Chestermiller said:

Tracing out the directions of the principal stresses is what you might mean by the force lines.

All I know about force lines is what I read at wikipedia

https://en.wikipedia.org/wiki/Force_lines

But there are 3 principal stress lines and only one force line. I am trying to figure out why there is this discrepancy. The article seems to imply that the force line is just the maximum principle stress vector. But I can't be sure.

Whats so special about the maximum principle stress? If you remove the maximum principle stress then what is left?

Chestermiller · Dec 11, 2020

em3ry said:

OK. I see what you mean. Thanks

All I know about force lines is what I read at wikipedia

https://en.wikipedia.org/wiki/Force_lines

But there are 3 principal stress lines and only one force line. I am trying to figure out why there is this discrepancy. The article seems to imply that the force line is just the maximum principle stress vector. But I can't be sure.

Whats so special about the maximum principle stress? If you remove the maximum principle stress then what is left?

What do you mean by removing the maximum principal stress?

em3ry · Dec 11, 2020

Chestermiller said:

What do you mean by removing the maximum principal stress?

If the force lines are the maximum principle stress then what's the rest?

Chestermiller · Dec 11, 2020

em3ry said:

If the force lines are the maximum principle stress then what's the rest?

Still don't understand. As best I can tell from the article, somehow they trace out the maximum principal stress directions. You do understand how the principal stresses and their directions are determined, right?

em3ry · Dec 11, 2020

I don't know how to be any clearer

Assuming that the force lines are just the maximum principle stress then:

principle stresses = maximum principle stress vector + ?
principle stresses = force lines + ?

I am trying to understand what the force lines represent and what the components of the stress tensor represent

Chestermiller · Dec 11, 2020

Are you familiar with the Cauchy stress relationship, which uses the stress tensor to determine the traction vector on an area of arbitrary orientation at a location within a body? This is key to understanding how the stress tensor works and how the principal directions and stresses are determined.

em3ry · Dec 11, 2020

Chestermiller · Dec 11, 2020

The you know that the stress tensor dotted with a unit normal to an internal area gives the traction vector on that area. This traction vector (force per unit area) has components normal to the area (normal component) and tangent to the area (shear component). There are exactly 3 directions of orientation for the area in which the shear component will be zero, and only the normal component will exist. These three directions are the three principal directions of stress at that location. The largest of these three normal tractions is called the maximum principal stress.

To get the principal stresses and principal directions, you set the dot product of the stress tensor and unit normal equal to a multiplier times the unit normal, and solve for the directions of the unit normal that satisfy this constraint.

em3ry · Dec 11, 2020

I know the math. I am trying to understand what the math is saying. I am trying to understand the ideas.

I am trying to understand what force lines are and how they differ from the principal stresses.

Chestermiller · Dec 11, 2020

em3ry said:

I know the math. I am trying to understand what the math is saying. I am trying to understand the ideas.

I am trying to understand what force lines are and how they differ from the principal stresses.

As best as I can understand from your article, the "force lines" are loci created by identifying the maximum principal stress direction at given location, extending a small segment of it to a neighboring location, and then determining the maximum principal stress at that location. In that way a continuous locus of maximum principal stress directions can be traced out.

em3ry · Dec 11, 2020

Whats so special about the maximum principal stress?

Chestermiller · Dec 11, 2020

em3ry said:

Whats so special about the maximum principal stress?

The maximum principal stress determines the tensile failure criterion of the material and differences between the principal stresses determine the yield and shear failure criteria of the material.

em3ry · Dec 11, 2020

You have been extremely helpful. I appreciate you taking the time to listen to me and to give me your extremely helpful feedback.

What is the relationship between force lines and the stress tensor field?

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