一维准晶体Ⅲ型切口的应力与位移

Stress and displacement of Ⅲ-notch problem in one-dimensional quasi-crystal

为研究一维准晶的V型切口裂纹问题,采用解析函数方法将一维准晶的基本方程转化为极坐标形式;由切口面边界条件得出Ⅲ型切口问题的本征方程,求出声子场和相位子场的应力和位移渐进解;引入应力强度因子,计算切口尖端处的J积分.计算结果表明,声子场和相位子场通过本构方程耦合,切口尖端处声子场和相位子场位移也具有耦合性;Laue类5,7和9的位移分布与Laue类8不同,也与一般材料不同,但J积分仍具有相同形式;当切口变为裂纹时,J积分给出能量释放率.

In order to solve the problem of V-notch crack in one-dimensional quasi-crystal, the basic functions of one-dimensional quasi-crystal are transformed into polar form with the method of analytic function; the eigenvalue equation of the Ⅲ-notch problem is concluded from the boundary condition of the crack face, and the asymptotic solutions of stress and displacement in both phonon field and phason field are found; the J-integral at the notch tip is given by introducing the stress intensity factor. The results show that the phonon field couples with the phason field by the constitutive equations, and the stress and displacement in the phason field also couples with those in the phonon field; the displacement distribution of Laue class 5,7,9 is different from that of Laue class 8 and other general materials, while the J- integral is still of the same form; the J-integral gives the energy release rate when the notch changes into crack.