Friction Effects in Metalworking

[1350-F]

Homework Statement

Show how a constant interfacial shear stress mk < k decreases the forces required for uniaxial upsetting using mohr circles with decreasing m (<1). Assume a slab force balance.

Homework Equations

Mohr Circle Equations

dσ_x/dx = 2mk/h

σ_x + P = 2k

The Attempt at a Solution

Attached. What I have shows what the question asks but I'm not sure I did it right.

 

[KataKoniK]

Thank you for your post. I would like to provide a response to your question regarding the relationship between a constant interfacial shear stress and the required forces for uniaxial upsetting.

To begin, let us define some terms and equations that will be useful in our analysis. The Mohr circle is a graphical representation of stress states in a material, where the x-axis represents normal stress (σx) and the y-axis represents shear stress (τxy). The equations for the Mohr circle are as follows:

σx = (σ1 + σ2)/2 + (σ1 - σ2)/2 * cos(2θ)
τxy = (σ1 - σ2)/2 * sin(2θ)

where σ1 and σ2 are the principal stresses, and θ is the angle between the x-axis and the line connecting the center of the circle to a point on the circle.

Now, let us consider a slab of material that is being subjected to uniaxial upsetting, meaning that a force (P) is applied in one direction, causing the material to deform and increase in thickness (h). In this case, we can use the equation for the stress state in the x-direction (dσx/dx = 2mk/h) to determine the stress at any point in the material.

If we assume a constant interfacial shear stress (τxy = mk), we can use the equations for the Mohr circle to plot the stress states for different values of m. As m decreases (m < 1), the Mohr circle will shift downwards, indicating a decrease in the required force (P) for uniaxial upsetting. This can be seen in the attached figure.

To understand why this happens, let us consider the force balance on the slab. The total force acting on the slab is equal to the applied force (P) plus the force required to overcome the interfacial shear stress (τxy = mk) over the area of the interface (h). As m decreases, the interfacial shear stress decreases, resulting in a decrease in the required force and a lower Mohr circle.

In conclusion, we have shown that a constant interfacial shear stress (τxy = mk) will decrease the required forces for uniaxial upsetting as m decreases, as shown by the downward shift of the Mohr circle. This is due to the decrease in interfac