蜂窝拉胀复合材料非线性分析新方法研究及应用

An efficient method is proposed to analyze the mechanical properties of auxetic honeycombs under large deformation. The existing elliptic integral method is limited to some simple situations. For overcoming that limitation, based on the bending theory of beams in large deflection, an improved elliptic integral method is developed for the large deformation of honeycombs by considering the effect of axial deformation, torsional deformation of cell members and plastic hinges at the cell member ends. Next, the improved elliptic integral method mentioned above is applied to the unit cell of auxetic honeycombs. Then the associated computation method of elliptic integral solutions to every cell member is established. Based on these, the nonlinear constitutive relations of auxetic honeycombs can be obtained. By introducing the nonlinear modified factors, the nonlinear constitutive relation can be expressed as the product of linear constitutive relation and the nonlinear modified factors. The linear constitutive relation is associated with the shape and density of honeycombs, while the modified factors only related to the shape. As a result, the nonlinear constitutive relation of the honeycombs with same shape but in different densities can be described in a simple form by the same modified factors. Besides, the elastic/plastic bucking of auxetic honeycombs, designs of novel auxetic structure and the ways to enhance the stiffness of auxetic honeycombs would also be considered in this project. This research broadens the application scope of conventional elliptic integral method, enriches the methods and contents of analysis to the large deformation of honeycombs, which not only has important scientific significance, but also has wide application prospect.

针对蜂窝拉胀材料大变形问题，提出一套有效的力学分析方法。首先，基于梁大挠度弯曲理论，建立一种能综合考虑胞壁轴向变形、扭转变形和杆端塑性铰等因素的改进椭圆积分解法，以突破现有椭圆积分解仅能处理个别简单问题的局限；然后，将改进椭圆积分解应用于一般条件下的蜂窝拉胀材料，建立各胞壁椭圆积分解的联合求解方法，并在此基础上研究各类蜂窝拉胀材料的非线性本构关系。通过引入非线性修正因子，将蜂窝材料非线性本构关系转化为线性本构关系与非线性因子之积，由于前者一般具有解析表达式，后者与形状有关而与密度无关，故相同形状不同密度的蜂窝可用同一非线性因子描述，能有效地简化非线性本构关系。此外，本项目还将探索蜂窝拉胀材料的新型结构形式，以及多种提高蜂窝拉胀材料模量的增强方案。本项研究拓宽了传统椭圆积分解法的应用范围，丰富了蜂窝材料大变形分析的方法和内容，不仅具有重要的理论意义，同时还具有广泛的应用前景。

作为力学超材料的一种，蜂窝拉胀材料以其种种优良性能，在航空航天、机械交通和生物医学工程等众多领域显示出巨大的潜在应用价值。研发更多种类的新型蜂窝拉胀材料并对其性能进行优化，具有重要的理论意义和应用价值。本项目针对蜂窝拉胀材料及与此相关的负热膨胀材料和拉扭耦合材料的设计、性能分析方法展开研究，并取得以下成果。.一、改进并完善了平面梁弹性大挠度弯曲分析的椭圆积分方法。首先将椭圆积分法由经典的欧拉梁推广到铁摩辛柯梁，扩大了椭圆积分法的分析精度和适用范围；然后，针对基于铁摩辛柯梁椭圆积分法在固定边界近似程度大这一不足，给出了一种更精确的表达方法，进一步提高了分析的精度和适用范围。.二、利用上述椭圆积分法，研究了一些蜂窝拉胀材料的非线性行为。给出了内凹蜂窝、手性蜂窝等材料大变形时的正应力-正应变非线性曲线，以及剪应力-剪应变非线性曲线；给出了蜂窝壁板局部进入塑性铰时的材料强度公式。.三、针对蜂窝拉胀材料面内刚度低这一不足，提出了在晶格中内嵌增强的模式。这种方式实施简单，增强效果好，易于推广。此外，针对增强杆局部进入失稳后的材料双线性行为，提出了一种简便的“增量分析方法”，使复杂的材料非线性本构变得十分简单。.四、设计了7种新型的蜂窝拉胀材料(其中3种同时还具有负热膨胀效应）。.五、设计了一批具有拉扭耦合效应的蜂窝材料，这些材料的拉扭效果较已有工作有了大幅提升。.六、设计了具有热扭耦合效应的蜂窝材料。这种“热-扭效应”在已有研究中尚未见有报道。.本项目所取得的以上成果，丰富了力学超材料的设计思路和研究方法，不仅具有重要的理论意义，也具有广泛的应用前景。