  Nonlinear kinetic theory of cosmic ray (CR) acceleration in supernova
remnants is employed to calculate CR spectra. The magnetic field in SNRs is
assumed to be significantly amplified by the efficiently accelerating nuclear
CR component. It is shown that the calculated CR spectra agree in a
satisfactory way with the existing measurements up to the energy $10^{17}$ eV.
The power law spectrum of protons extends up to the energy $3\times 10^{15}$ eV
with a subsequent exponential cutoff. It gives a natural explanation for the
observed knee in the Galactic CR spectrum. The maximum energy of the
accelerated nuclei is proportional to their charge number $Z$. Therefore the
break in the Galactic CR spectrum is the result of the contribution of
progressively heavier species in the overall CR spectrum so that at $10^{17}$
eV the CR spectrum is dominated by iron group nuclei. It is shown that this
component plus a suitably chosen extragalactic CR component can give a
consistent description for the entire Galactic CR spectrum.
