In this paper, we construct a new theory of nanobeams taking into account the dependence of the material properties on the stress state. The theory is based on the Euler–Bernoulli kinematic model in the first approximation. The beam material is isotropic but heterogeneous. For the first time, the physical nonlinearity and the dependence of the material properties on the temperature are taken into account in the study of nanobeams, and the theory is developed for arbitrary materials. It is based on the theory of small elasticplastic strains and on the the modified torque theory of elasticity. The stationary temperature field is determined by solving the three-dimensional Poisson equation with boundary conditions of orders 1–3. The initial equations are derived from the Hamilton–Ostrogradskii principle. The desired system of partial differential equations is reduced to the Cauchy problem by the finite difference method of the second order of accuracy, and the Cauchy problem is solved by the Runge–Kutta or Newmark method. At each time step, an iterative procedure is developed by the Birger method of variable elasticity parameters. The stationary solution follows from the dynamic solution of the problem obtained by 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 to have almost infinite number of degrees of freedom. Numerical examples are given for a beam rigidly clamped at the ends with the stress-strain diagram for aluminum. Accounting for the size-dependent parameter in the nanobeam theory significantly affects the load-carrying capacity of nanobeams.

中文翻译：

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考虑物理非线性的纳米束理论

在本文中，我们构建了一种新的纳米梁理论，考虑到材料特性对应力状态的依赖性。该理论基于 Euler-Bernoulli 运​​动学模型的一阶近似。梁材料是各向同性但异质的。在纳米梁的研究中首次考虑了物理非线性和材料特性对温度的依赖性，并为任意材料开发了该理论。它基于小弹塑性应变理论和修正的弹性扭矩理论。固定温度场通过求解具有 1-3 阶边界条件的三维泊松方程来确定。初始方程源自 Hamilton-Ostrogradskii 原理。通过二阶精度的有限差分法将所需的偏微分方程组简化为柯西问题，并通过龙格-库塔或纽马克方法求解柯西问题。在每个时间步长，一个迭代过程由可变弹性参数的 Birger 方法开发。平稳解来自于通过确定方法（参数位置的方法）获得的问题的动态解。根据有限差分法中沿梁长度和厚度划分的点数以及求解柯西问题的方法和尺寸相关参数，研究解的收敛性，即解的问题被认为具有几乎无限数量的自由度。用铝的应力-应变图给出了在端部刚性夹紧的梁的数值示例。考虑纳米梁理论中与尺寸相关的参数会显着影响纳米梁的承载能力。