Nanobeam theory taking into account the physical nonlinearity

In this paper, we construct a new theory of the study nano beam taking into account the dependence of material properties from the stress state. The theory is based on the kinematic model of the first approximation (Euler-Bernoulli). The beam material is isotropic but inhomogeneous. For the first time, the physical nonlinearity and dependence of material properties on temperature are taken into account for the study of nano beams, and the theory is constructed for any materials. The theory is based on the theory of small elastic-plastic deformations and modified torque theory of elasticity. The stationary temperature field is determined from the solution of the three-dimensional Poisson equation for boundary conditions of 1 - 3 kinds. The initial equations are derived from the Hamilton-Ostrogradskiy principle. The required system of partial differential equations is reduced to the Cauchy problem by the finite differences method of the second order of accuracy, and the Cauchy problem is solved by Runge-Kutta and Newmark methods. At each time step, an iterative procedure is constructed by the Birger’s method variable parameters of the elasticity. The stationary solution follows from the dynamic solution of the problem by means of the method of determination (the method of the parameter position). The convergence of the solution is investigated depending on the number of points of partition along the length and thickness of the beam in the finite difference method, as well as on the method of solving the Cauchy problem, and the size-dependent parameter, i.e. the solution of the problem is considered as a solution with "almost" infinite number of degrees of freedom. Numerical examples are given for a beam rigidly clamped at the ends with a deformation diagram for aluminum. Account dimension-dependent parameter in the theory nanobeam significantly affects its load-bearing capacity.

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