黏弹性功能梯度材料裂纹问题的有限元方法

针对组分材料体积含量任意分布的黏弹性功能梯度材料裂纹问题建立有限元分析途径. 通过Laplace变换,将黏弹性问题转化到象空间中求解,基于反映材料非均匀的梯度单元和裂纹尖端奇异特性的奇异单元计算象空间中的位移、应力和应变场,应用虚拟裂纹闭合方法得到应变能释放率,分别由应力和应变能释放率确定应力强度因子. 给出这些断裂参量在物理空间和象空间之间的对应关系,由数值逆变换求出其在物理空间的相应值. 文中分析两端均匀受拉的黏弹性边裂纹板条,首先针对松弛模量表示为空间函数和时间函数乘积的特殊梯度材料进行计算,结合对应原理验证方法的有效性. 然后分析组分材料体积含量具有任意梯度分布的情形,由Mori-Tanaka方法预测象空间中的等效松弛模量. 计算结果表明,蠕变加载条件下,应变能释放率随时间增加,其增大程度与黏弹性组分材料体积含量相关. 由于梯度材料的非均匀黏弹性性质,产生应力重新分布,导致应力强度因子随时间变化,其变化范围与组分材料的体积含量分布方式有关.

FINITE ELEMENT METHOD FOR CRACK PROBLEMS IN VISCOELASTIC FUNCTIONALLY GRADED MATERIALS

A finite element approach is developed to analyze the crack problems in viscoelastic functionally materials with arbitrary volume fraction distribution of constituents. By Laplace transform, the boundary problems are solved in phase domain based on graded element considering the heterogeneous of material and singular element describing the singularity of stresses near crack tips. The virtual crack closure technique modified by Rybicki et al. is applied to evaluate strain energy release rate, and the stress intensity factor is determined by means of nodal stress and strain energy release rate, respectively. The relationships of fracture parameters in time domain and phase domain are formulated, and the corresponding solutions in time domain are obtained by numerical Laplace inversion. The crack problem in viscoelastic functionally graded plate with edge crack parallel to graded direction is investigated. Two cases are involved in the analysis. The first one is for special functionally graded material with relaxation modulus expressed by the product of spatial variable function and time function. The second one is for general functionally graded materials of arbitrary volume fraction distribution of constituents. The validity of the finite element method proposed in the paper is verified in the first case on the basis of the elastic-viscoelastic correspondence principle. For the second case, Mori-Tanaka method is used to predict the effective relaxation modulus of functionally graded materials in phase domain. The results in creep loading condition show that the strain energy release rate increases with time elapsed, and the variation range depends on the volume content of viscoelastic constituent. The stress intensity factor may change over time due to the stress redistribution around crack tip originating from the heterogeneous viscoelasic property of graded material, and the time-dependent variation is influenced by the distribution pattern of volume fraction.