Nonlinear models of viscoelastic plates and shells

The dynamics of nonlinear viscoelastic plates and shells is a crucial area of study in modern mechanics, materials science, and engineering. This importance stems from the increasing demand for accurate modeling and analysis of structures subjected to complex loads, as well as the advancement of new materials and technologies. Modern materials, including carbon composites, polymers, and multilayer coatings, possess complex viscoelastic properties. Under dynamic loads, such as vibrations or impacts, viscoelastic materials exhibit time-dependent responses to these loads, necessitating careful consideration of their relaxation and creep characteristics. The unique viscoelastic properties allow these materials to adapt to applied loads, making them highly desirable for the design of sophisticated devices, such as sensors, membranes, and adaptive structures. Furthermore, interactions with external fields – such as electromagnetic or thermal forces – enhance the effects of nonlinearities and require the development of new modeling approaches. The paper presents the equations of dynamics of geometrically and physically nonlinear thin-walled elements. An operator approach based on Rabotnov’s hereditary kernels is proposed, which makes it possible to correctly account for relaxation processes. The novelty of the work lies in the consideration of the combined effect of geometric and physical nonlinearities. To demonstrate the applicability of the model, a numerical example of the deflection of a rectangular plate under uniform loading is examined. Graphs of the deflection evolution and the influence of thickness and relaxation parameters are presented.