A New Mathematical Model of Functionally Graded Porous Euler–Bernoulli Nanoscaled Beams Taking into Account Some Types of Nonlinearities

A new mathematical model of flexible physically (FN), geometrically (GN), and simultaneously physically and geometrically (PGN) nonlinear porous functionally graded (PFG) Euler–Bernoulli beams was developed using a modified couple stress theory. The ceramic phase of the functionally material was considered as an elastic material. The metal phase was considered as a physically non-linear material dependent on coordinates, time, and stress–strain state, which gave the opportunity to apply the deformation theory of plasticity. The governing equations of the beam as well as boundary and initial conditions were derived using Hamilton’s principle and the finite difference method (FDM) with a second-order approximation. The Cauchy problem was solved by several methods such as Runge–Kutta from 4-th to 8-th order accuracy and the Newmark method. Static problems, with the help of the establishment method, were solved. At each time step, nested iterative procedures of Birger method of variable elasticity parameters and Newton’s method were built. The Mises criterion was adopted as a criterion for plasticity. Three types of porosity-dependent material properties are incorporated into the mathematical modeling. For metal beams, taking into account the geometric and physical nonlinearity, the phenomenon of changing the boundary conditions, i.e., constructive nonlinearity (CN) was found.