Continuum mechanics and normal shear stress

[Niles]

Homework Statement

I am self-studying this note and I am stuck in the derivation of the normal shear stress. I can't see how the relations (23) and (24) come about, i.e. I don't understand

$$\tau'_{xx} = \frac{\tau_{xx}+\tau_{yy}}{2}+\tau_{yx}$$
and
$$\tau'_{yy} = \frac{\tau_{xx}+\tau_{yy}}{2}-\tau_{yx}$$

Can someone elaborate on the note to make it clearer? Thanks in advance.

 

[Chestermiller]

Homework Statement

I am self-studying this note and I am stuck in the derivation of the normal shear stress. I can't see how the relations (23) and (24) come about, i.e. I don't understand

$$\tau'_{xx} = \frac{\tau_{xx}+\tau_{yy}}{2}+\tau_{yx}$$
and
$$\tau'_{yy} = \frac{\tau_{xx}+\tau_{yy}}{2}-\tau_{yx}$$

Can someone elaborate on the note to make it clearer? Thanks in advance.

Let me guess. The primed stresses are the stresses for a set of axes oriented at 45 degrees to the x and y axes. Correct?

 

[Niles]

Let me guess. The primed stresses are the stresses for a set of axes oriented at 45 degrees to the x and y axes. Correct?

YES! How did you know that? And are we allowed to "add" stresses like this?

 

[Chestermiller]

YES! How did you know that? And are we allowed to "add" stresses like this?

This is just a transformation of the stress tensor components from one set of coordinate axes to another set of coordinate axes. It is analogous to the transformation of a vector in component form from one set of coordinates (i.e., using one set of unit vectors) to another set of coordinates (using another set of basis vectors). Do you know the general transformation relationship for transforming the coordinates of the stress tensor from one set of cartesian coordinates to another set of cartesian coordinates which are rotated through an angle θ relative to the first set?

Chet

 

[paranormal]

I can understand your confusion with the derivation of the normal shear stress. Continuum mechanics can be a complex subject and it requires a strong foundation in mathematics and physics to fully grasp its concepts.

To help you understand the relations (23) and (24), let's break them down step by step. In continuum mechanics, we use stress as a measure of forces acting within a material. The normal shear stress, denoted by \tau'_{xx} and \tau'_{yy}, represents the force acting in the x and y directions, respectively.

Now, in order to understand how these relations come about, we need to first look at the definitions of normal stress (\sigma_{xx} and \sigma_{yy}) and shear stress (\tau_{xy} and \tau_{yx}). Normal stress is defined as the force per unit area acting perpendicular to the surface, while shear stress is defined as the force per unit area acting parallel to the surface.

In continuum mechanics, we assume that the stress acting on a material is continuous and can be represented by a tensor. This means that the stress acting in one direction can affect the stress acting in another direction. In this case, the normal stress in the x-direction (\sigma_{xx}) can affect the shear stress in the y-direction (\tau_{yx}) and vice versa.

Now, if we consider a small element within a material, we can see that there are two components of stress acting on it - normal stress and shear stress. In order to simplify the equations, we can combine the normal and shear stresses in each direction to get the average stress acting in that direction. This is where the relations (23) and (24) come in.

By combining the normal stress in the x-direction (\sigma_{xx}) with the shear stress in the y-direction (\tau_{yx}), we get the average stress acting in the x-direction, which is represented by \tau'_{xx}. Similarly, by combining the normal stress in the y-direction (\sigma_{yy}) with the shear stress in the x-direction (\tau_{xy}), we get the average stress acting in the y-direction, which is represented by \tau'_{yy}.

I hope this explanation helps you understand the derivation of the normal shear stress better. Continuum mechanics can be a challenging subject, but with persistence and a solid understanding of the fundamentals, you will be able to grasp its concepts. Keep studying and don't hesitate to ask