A new mathematical model of flexible physically (FN), geometrically (GN), and simulta-
neously 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-depend-
ent 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.