Based on the model of a perfectly rigid-plastic body and a structural model of the composite, the solution is constructed for dynamic bending of two-connected curvilinear composite plates with hinge-supported or clamped contours in a viscous medium under the action of explosive loads. The plates are hybrid, multilayered and fibrous with a symmetrical distribution of layers with respect to the middle surface. In each layer the reinforcing fibers of different materials are located in directions parallel or normal to the inner plate contour. It is shown that, depending on the load amplitude, two deformation schemes of the plate’s deformation in the form of a set of several linear surfaces separated by curved plastic hinges are possible. For each of the schemes, the equations of the dynamic deformation of the plates are obtained on the basis of the virtual power principle in combination with the d’Alembert principle. The operating conditions of these deformation mechanisms are analyzed. To simplify the calculation of double integrals over curvilinear areas with time-varying boundaries, a curvilinear orthogonal coordinate system associated with the equation of the inner contour of the plates is introduced. The analysis of the dynamic deformation is carried out for the plates under the action of arbitrary loads of explosive type. The location of the linear surfaces and the plastic hinges separating them are determined at the moment of the beginning of the plate movement depending on the value of the applied load at the initial time. The limit load, final deflections and time of deformation of the plates are determined. Numerical examples are given for a two-connected plate with an internal contour in the form of an ellipse with different variants of reinforcement, viscous foundation and supporting of the contours under the condition of the same consumption of reinforcing fibers of constant thickness. The number of parameters of the developed mathematical model allows for a wide range to change the structure of reinforcement plates, volume content of reinforcing fibers, layer thickness and physical characteristics of the composite, viscous foundation, also the geometric shape of double-connected plates and conditions of support of the contours.