Using damage evolution for fiber reinforced materials leads to different results when changing the mesh size

We use ABAQUS for simulation of fiber reinforced matarials. We also tried to evaluate the boundaries of the damage evolution model for our CFRP material. To determine the right fracture energy we found out that by choosing the same fracture energy, but a different mesh size, there will be different results. Please find attached plots of the results that compare the results. Here are four different mesh sizes choosen 0.4mm, 0.5mm, 0.6mm and 0.75mm. When the same fracture energy is used in the every model, only with different mesh sizes, you can see that the results of a 4-point-bending simulation leads to different froce-displacement curves. The results a shown in the file 4-point-bending_same_fracture_enery_different_mesh_size. When we normalize the fracture energy to the characteristic lenght of 1 and then calculate for the different mesh sizes the fracture energy the picture looks not so mesh dependend (see file 4-point-bending_normalized_fracture_enery_different_mesh_size). Is this a known problem or is this the right behaviour of the damage evolution model? Because we thought that the fracture energy would be normalized to the characteristic element lenght of the model that is currently simulated.