Question about the critical stress for fracture in abaqus 1

Can't comment much on your failure criteria, you will have to figure that out. Reason for negative principal stress is because your beam is bending, and you have axial (along the length of the beam) compression on the top surface.

Your mesh is not suitable for this, both element type and refinement. Use C3D8R for this, with much smaller element size especially near the impact location. Also be sure to use spherical contact smoothing for the ball.

The general meaning of the different principal stresses and their sign is explained on many websites.

Actually, the mesh shows in this picture is not what I used, because there is a version limits of nodes number on my own computer. I use C3D10M(only can use Tet for the beam).