Static bending and elastic buckling of Euler–Bernoulli beam made of functionally graded (FG) materials along thickness direction is studied theoretically using stress-driven integral model with bi-Helmholtz kernel, where the relation between nonlocal stress and strain is expressed as Fredholm type integral equation of the first kind. The differential governing equation and corresponding boundary conditions are derived with the principle of minimum potential energy. Several nominal variables are introduced to simplify differential governing equation, integral constitutive equation and boundary conditions. Laplace transform technique is applied directly to solve integro-differential equations, and the nominal bending deflection and moment are expressed with six unknown constants. The explicit expression for nominal deflection for static bending and nonlinear characteristic equation for the bucking load can be determined with two constitutive constraints and four boundary conditions. The results from this study are validated with those from the existing literature when two nonlocal parameters have same value. The influence of nonlocal parameters on the bending deflection and buckling loads for Euler–Bernoulli beam is investigated numerically. A consistent toughening effect is obtained for stress-driven nonlocal integral model with bi-Helmholtz kernel.

中文翻译：

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使用双亥姆霍兹核的应力驱动非局部积分模型对功能梯度 Euler-Bernoulli 梁进行弯曲和屈曲分析

采用双亥姆霍兹核应力驱动积分模型，理论研究了功能梯度（FG）材料欧拉-伯努利梁沿厚度方向的静态弯曲和弹性屈曲，其中非局部应力与应变的关系表示为Fredholm型积分第一类方程。利用最小势能原理推导出微分控制方程和相应的边界条件。引入了几个名义变量来简化微分控制方程、积分本构方程和边界条件。直接应用拉普拉斯变换技术求解积分微分方程，用六个未知常数表示标称弯曲挠度和弯矩。静态弯曲名义挠度的显式表达式和屈曲载荷的非线性特征方程可以通过两个本构约束和四个边界条件确定。当两个非局部参数具有相同值时，本研究的结果与现有文献的结果得到验证。数值研究了非局部参数对欧拉-伯努利梁弯曲挠度和屈曲载荷的影响。对于具有双亥姆霍兹核的应力驱动非局部积分模型，获得了一致的增韧效果。数值研究了非局部参数对欧拉-伯努利梁弯曲挠度和屈曲载荷的影响。对于具有双亥姆霍兹核的应力驱动非局部积分模型，获得了一致的增韧效果。数值研究了非局部参数对欧拉-伯努利梁弯曲挠度和屈曲载荷的影响。对于具有双亥姆霍兹核的应力驱动非局部积分模型，获得了一致的增韧效果。