Eventually, we find the mechanical properties of our single crystal quartz. We contact the company and they sent us the paper includes all the properties. On their website they just upload Cij and Sij not others. So here is the properties:

tensile strength: 80 perpendicular-110 parallel (MPa); Bending: 90 perpendicular-135 parallel (MPa) .

Once I used the average value of 85 MPa as for \sigma_c, I saw very soon fracture in the sample and the simulation failed. But later when I increased the magnitude of till \sigma_c=300-350 Mpa, the fracture initiated at the time close to the peak of the applied force and capture the full crack propagation through the depth of the specimen.

So at this moment I conclude that either the code has some issues in calculating the correct stress matrix or the experimental data is overestimated.