Li Xue, Sun Jun-Yi, Dong Jiao, He Xiao-Ting
School of Civil Engineering, Chongqing University, Chongqing 400045, China.
Key Laboratory of New Technology for Construction of Cities in Mountain Area (Chongqing University), Ministry of Education, Chongqing 400045, China.
Materials (Basel). 2018 May 17;11(5):830. doi: 10.3390/ma11050830.
The material considered in this study not only has a functionally graded characteristic but also exhibits different tensile and compressive moduli of elasticity. One-dimensional and two-dimensional mechanical models for a functionally graded beam with a bimodular effect were established first. By taking the grade function as an exponential expression, the analytical solutions of a bimodular functionally graded beam under pure bending and lateral-force bending were obtained. The regression from a two-dimensional solution to a one-dimensional solution is verified. The physical quantities in a bimodular functionally graded beam are compared with their counterparts in a classical problem and a functionally graded beam without a bimodular effect. The validity of the plane section assumption under pure bending and lateral-force bending is analyzed. Three typical cases that the tensile modulus is greater than, equal to, or less than the compressive modulus are discussed. The result indicates that due to the introduction of the bimodular functionally graded effect of the materials, the maximum tensile and compressive bending stresses may not take place at the bottom and top of the beam. The real location at which the maximum bending stress takes place is determined via the extreme condition for the analytical solution.
本研究中所考虑的材料不仅具有功能梯度特性，而且还表现出不同的拉伸和压缩弹性模量。首先建立了具有双模量效应的功能梯度梁的一维和二维力学模型。通过将梯度函数取为指数表达式，得到了双模量功能梯度梁在纯弯曲和横向力弯曲作用下的解析解。验证了从二维解到一维解的回归。将双模量功能梯度梁中的物理量与其在经典问题和无双模量效应的功能梯度梁中的对应物理量进行了比较。分析了纯弯曲和横向力弯曲下平截面假设的有效性。讨论了拉伸模量大于、等于或小于压缩模量的三种典型情况。结果表明，由于材料双模量功能梯度效应的引入，最大拉伸和压缩弯曲应力可能不会出现在梁的底部和顶部。通过解析解的极值条件确定最大弯曲应力发生的实际位置。
