Stress Concentration: Hole in infinite thin plate loaded in biaxial tension vs biaxial compression

I think you have two questions here,

1. "Is there a difference in the stress concentration of a hole in an infinite thin plate loaded in biaxial tension vs biaxial compression?"
2. "What happens during biaxial compression?"

The first question: No, there is no difference between the biaxial and the uniaxial case. If you look at the equations, for $K_t$ what happens is that the uniaxial case is a special case of the biaxial for $sigma_2=0$

What you have to keep in mind is that the formulation you are using requires principal stresses ($sigma_1>sigma_2>sigma_3$ algebraically), while the $K_t$ requires that the ratio between them is between -1 and 1

The second question: You are right that in most books, biaxial compression does not get explicitly mentioned. However, the expression should still hold. So if you have the following biaxial states of stress, you might have to do a bit of juggling

• $0>sigma_1>sigma_2rightarrow K_t= 3 - color{red}{frac{sigma_1}{sigma_2}}$,
• $sigma_1>0>sigma_2$, and $|sigma_1|>|sigma_2|rightarrow K_t= 3 - frac{sigma_2}{sigma_1}$
• $sigma_1>0>sigma_2$, and $|sigma_1|<|sigma_2| rightarrow K_t= 3 - color{red}{frac{sigma_1}{sigma_2}}$