含V型缺口分数阶黏弹性复合材料反平面界面断裂的辛方法

A symplectic method for the anti-plane fracture analysis of an interfaceV-notch in fractional viscoelastic media

本文提出求解含V型缺口的分数阶黏弹性复合材料反平面界面断裂问题的辛方法.分数阶Kelvin-Zener模型用于描述材料的黏弹性特征;借助Laplace变换,将时域内黏弹性反平面断裂问题的基本方程转换到频域空间;通过引入位移的对偶变量广义应力,建立问题的哈密顿求解体系.在该体系下,对偶方程的本征值和本征解可以利用分离变量法求解,本征解级数展开的系数通过本征解的辛共轭正交关系和外边界条件确定.这样将得到含缺口黏弹性复合材料反平面应力/应变强度因子的解析表达式.最后通过Laplace逆变换,得到时域空间内的应力/应变强度因子.数值算例验证本文方法的准确性,并揭示了分数阶参数、缺口角度和外载荷对应力/应变强度因子的影响.

This paper presents a symplectic method for the anti-plane fracture analysis of an interface V-notch in fractional viscoelastic composite media.The fractional Kelvin Zener model is used to describe the viscoelastic characteristics of materials.With the help of Laplace transform,the fundamental equations of an anti-plane viscoelastic fracture problem in time domain are transformed into frequency domain.By introducing the dual generalized stress variables,the Hamiltonian system is established.Then the eigenvalues and eigensolutions of the Hamiltonian dual equation are obtained by the method of separation of variables,and the unknown coefficients of the symplectic series are determined by the symplectic adjoint orthogonal relationship and the outer boundary conditions.In this way,the analytical expression of the anti-plane stress/strain intensity factor of the viscoelastic media with a V-notch is derived obtained.Finally,the intensity factor in time domain is found by inverse Laplace transform.In numerical examples,the accuracy of the presented method is verified,and the effects of fractional order parameters,notch angle and external load on the stress/strain intensity factor are revealed.