The two-level structural model of magnetodeformation effect of a fibrous core under the influence of an external quasistatic magnetic field is offered and new exact analytical decisions for arising magnetoelastic fields at micro and macro levels on the basis of generalization of known polydisperse models of mechanics of composites are received on a new class of problems of magnetooperated elastomers with is focused by the magnetized fibers. New analytical decisions for effective residual magnetization and for a deformation field in an elastomer of a compound cylindrical cell of a composite from action on fiber of the rotating magnetic moment are given. The analysis of influence of a volume fraction of the magnetized fibers on a tortional deformativnost of a composite core in an external magnetic field is carried out. Nonmonotonic nature of this dependence is established, optimum values of a volume fraction of fibers are defined and the conclusion is drawn on need of a choice of an elastomer with high magnetic permeability and small rigidity on stretching. Coordinates of macropoints of hollow core and internal (in macropoint) local coordinates of points of interphase surface of fiber with the maximum values of district tangent tension for case of highly magnetopermeable elastomer and rigid fibers are determined. The analysis of influence of the relative size of cavity of core and size of volume fraction of rigid fibers on the maximum values of interphase tangent tension is carried out. Nonmonotonic nature of this dependence is established and values of volume fraction of fibers at which the smallest values of the maximum tension are reached are defined.