Shear deformation mathematical modeling of functional graded clamped beams under bending

In this paper, high order bending theories are used to develop an analytical model considering shear strains in displacement fields which have not been taken into account by other theories for functionally graded material (FGM) symmetric beams under bending. A new polynomial shear function satisfied the stress-free boundary conditions, is used in this investigation.These theories do not require a shear correction factor and consider a hyperbolic shape function. Material properties are assumed to vary in the thickness direction, a simple power-law distribution in terms of volume fractions of constituents is considered. The mathematical model is established by differential equations which are derived by the principle of virtual work. Equilibrium equations and boundary conditions are introduced. The solution model is based on a variation approach (integrals) to predict the field component of displacements and the basic constitutive laws. The solution of the analytical model for the beam case with clamped ends represents the originality of this research work presented in this paper. Results in terms of displacement fields including rotation of the section, deformations, and stresses, predicted from the proposed model and compared with those of simply supported end beams found in the literature, are presented. This contribution helps to understand the FGM beams and the use of the proposed mathematical model.本文针对受弯工况下的功能梯度材料（Functionally Graded Material, FGM）对称梁，采用高阶弯曲理论构建考虑位移场中剪切应变的解析模型——现有相关理论均未纳入该剪切应变因素。本研究采用满足无应力边界条件的新型多项式剪切函数，所提理论无需引入剪切修正系数，且采用双曲型形状函数。本文假设材料属性沿厚度方向变化，并采用基于组分体积分数的简单幂律分布形式。通过虚功原理推导微分方程以构建数学模型，并引入平衡方程与边界条件。求解模型基于积分形式的变分法，用于预测位移场分量，并结合基本本构定律。针对固支端梁工况的解析模型求解结果，体现了本文研究工作的创新性。本文给出了由所提模型预测的位移场相关结果，包括截面转角、变形与应力，并与文献中简支端梁的相关结果进行对比。本研究有助于加深对功能梯度材料梁的理解，以及所提数学模型的应用。