基于欧拉 伯努利梁和铁木辛柯梁理论的功能梯度材料模量测定

为测定功能梯度材料的弹性模量和剪切模量,引入梁理论并将梁沿长度方向离散,建立单元平衡方程后可得到弹性模量和剪切模量分布;假设弹性模量为沿长度方向的线性函数或指数函数,用有限元软件仿真计算功能梯度材料梁单元节点处的挠度和转角,然后用插值法构造变形特征函数,并计算得出弹性模量和剪切模量,且计算值与理论值的误差较小.计算结果还表明,采用铁木辛柯梁理论不仅可以得到弹性模量,还可以计算剪切模量,且弹性模量计算结果比用欧拉-伯努利梁计算结果更接近真实值,但铁木辛柯梁理论中需测定转角,对测定过程的要求会更加严格。

Modulus determination of functionally graded materials based on theories of Euler Bernoulli beam and Timoshenko beam

To determine the elastic modulus and shear modulus of functionally graded materials, the beam theories are applied and the beam is discretized along the length direction, and the element equilibrium equations are established by which the distribution of elastic modulus and shear modulus can be obtained; assuming that the elastic modulus is linear function or exponential function along the length direction, the deflection and rotation angle of functionally graded materials at element nodes are simulated using finite element software. The deformation characteristic functions are constructed by interpolation method and the elastic modulus and shear modulus are calculated. The errors of the calculation values are smaller than the theoretical values. The calculation results also indicates that not only the elastic modulus but also the shear modulus can be obtained by Timoshenko beam, and the value of elastic modulus is closer to the actual one compared with the value calculated by Euler-Bernoulli beam, but rotation angle must be determined in Timoshenko beam that would strengthen the determination process.