带裂纹三维二十面体准晶平面弹性的无摩擦接触问题

Frictionless contact problem of three-dimensional icosahedral quasicrystals plane elasticity with cracks

利用平面弹性复变方法,通过求解边值问题,研究了单个刚性压头作用在带任意形状裂纹的三维二十面体准晶下的无摩擦接触问题,求得了应力函数封闭解的表达式,同时得到了裂纹左右端点处应力强度因子和压头下方任意点处声子场接触应力的显式表达式。理论结果表明,声子场接触应力在压头边缘具有-1/2阶奇异性。如果忽略相位子场作用,本文得到的结果可退化为已有文献中弹性材料相应结论。数值结果用于分析带水平直裂纹三维二十面体准晶下半平面与单个平底刚性压头无摩擦接触时量纲为一的应力强度因子和量纲为一的接触应力的分布规律;量纲为一的应力强度因子在裂纹距边界垂直距离和压头半宽度之比约为0.3和1.5处取得最大值;量纲为一的接触应力呈对称分布,在压头中心处最小,在压头边缘处达到峰值且具有奇异性;相位子场弹性常数、耦合系数与Lamé常数的比值对量纲为一的声子场接触应力的大小和分布规律几乎无影响。

By applying the complex variables method and solving the boundary value problems,the frictionless contact problem in three-dimensional icosahedral quasicrystals half plane with arbitrary shape cracks is discussed under the action of a single rigid punch.The stress functions are obtained in closed form.The explicit expressions of the stress intensity factors at crack tips and the contact stress distribution under the punch are also derived.From the expressions of contact stress,it can be seen that contact stress has an order of-1/2 singularity at the edge of contact zone.Without considering the effect of the phason field,the results obtained match those classic conclusions for elastic materials.Numerical example used to analyze the frictionless contact problem in icosahedral quasicrystals half plane elasticity with a horizontal straight crack under the action of a rigid flat punch:(1)When the distance between the crack and the lower half-plane boundary of quasicrystal increased gradually,the symbol of dimensionless stress intensity factors are changed and the maximum value is obtained at the ratio of vertical distance between crack and boundary and the half width of the punch is about 0.3 or 1.5.(2)As the ratio of phase field elastic constants and Laméconstant take different values,the variations of the dimensionless contact stress are changed between 0.001~0.013 in the contact surface,so the variations have little influence on the dimensionless contact stress.(3)The dimensionless contact stress is distributed symmetrically and has a minimum value(about 0.3)at the punch center.The closer to the edge of the punch,the greater the contact stress.The dimensionless contact stress is reached to peak(about 0.8)and has singularity at the edge of the punch.The conclusions obtained may provide a theoretical basis for the application of quasicrystalline materials in modern industrial production.