功能梯度多孔梁弯曲问题再研究

Re-research on bending problem of functionally graded porous beams

  • 摘要: 一维梁理论作为三维弹性理论的简化模型,在处理具有显著非均质特性的梯度多孔材料时,需要谨慎考量其计算精度并明确其适用边界。为此,本文以弯曲问题为例,系统评估文献中被广泛采用的经典梁(CBT)、一阶剪切梁(FSDBT)、三阶剪切梁(TSDBT)、三角剪切梁理论(BSDBT)对梯度多孔梁力学响应问题的有效性和准确性。考虑三种常见的孔隙梯度分布(FGX、FGO、UD),并假设材料物性参数沿厚度方向连续变化。根据最小势能原理,推导了梯度多孔梁弯曲问题的控制方程,采用微分求积法得到方程的精确数值解。以ABAQUS三维有限元分析结果为参考,衡量了四种梁理论预测结果之间的差异。通过数值算例和参数分析,讨论了孔隙梯度、孔隙率系数、长厚比以及边界条件等对不同梁理论及有限元结果之间差异的影响,并分析了产生差异的原因。结果表明:对于高孔隙率FGX多孔梁,不同梁理论与三维有限元分析结果存在显著差异;这些差异随着孔隙率系数和约束刚度的增大而增大,随着长厚比的增大而减小;当孔隙率系数e0>0.9或长厚比L/h<20时,上述梁理论均无法准确描述FGX多孔梁横截面的变形特征,预测结果将不再可靠!对于这类情况,建议采用三维弹性理论/三维有限元仿真获得精确解,或基于三维弹性理论发展出精确适用于梯度多孔材料的梁理论模型,以确保计算结果的可靠性。

     

    Abstract: As a simplified model of three-dimensional elasticity theory, the predictive accuracy and applicable boundary of one-dimensional beam theory must be rigorously examined and clarified when dealing with functionally graded porous materials (FGPMs) with significant inhomogeneous properties. This study systematically evaluates the effectiveness and accuracy of four widely-adopted beam theories in the literature, namely the classical beam theory (CBT), first-order shear deformation beam theory (FSDBT), third-order shear deformation beam theory (TSDBT), and trigonometric shear deformation beam theory (BSDBT), for the bending problem of functionally graded porous beams. Three common porosity gradient distributions (FGX, FGO, UD) were considered, and it was assumed that material properties vary continuously along the thickness direction. According to the principle of minimum total potential energy, the governing differential equations for different beam theories were derived, and the numerical solutions for the bending problem of functionally graded porous beams were obtained using the differential quadrature method. By comparing the calculated results with finite element simulation ones, the effectiveness and accuracy of different beam theories in studying the bending problem of functionally graded porous beams were examined. Parametric studies were then conducted to discuss the influences of porosity gradient and coefficient, aspect ratio, and boundary conditions on the difference between theoretical predictions and simulation results, and the underlying reasons for the differences were analyzed. Results show that for the FGX porous beams of high porosity, the difference increases significantly as the porosity coefficient and constraint stiffness increase, while it decreases as aspect ratio increases. When the porosity coefficient exceeds 0.9 or the aspect ratio is less than 20, the four beam theories are all unable to accurately describe the deformation of the cross-section of FGX porous beams, and the theoretical predictions are no longer reliable. For such cases, it is recommended to employ 3D elasticity theory/finite element simulation to obtain precise solutions, or alternatively develop a more refined beam theory specifically adapted for functionally graded porous materials to ensure the computational reliability.

     

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