Original model of elastic contact interaction of ideally hard spherical body with a plate of anisotropic composite material was developed. The sizes of contact spot and and indention of spherical body where defined by iterations repeating the procedure of finite element analysis. Zero-contact-stress condition on the contact spot contour allows to obtain the solution of contact problem. Conclusions of numerical analysis of the elastic stress state of composite at low-speed impact are compared with results of microscopic investigation. The use of the general criteria of destruction can give the sufficient information for the approximate prediction of kind of partial destructions in a material and the sizes of a zone of damage. Conclusion on effect of stresses localness has been confirmed. Results of microscopic researches of a surface of a cut in a zone of impact confirm the described shape of destruction of fiberglass at impact with rather small energy. Really, the density of microcracks in epoxy matrix sharply decreases on edge of a zone of destruction. The density of breack of fibers within the limits of the same zone varies significantly less.