A complex potential method for the asymptotic solution of wedge problems using first-order shear deformation plate theory
A complex potential method for the asymptotic solution of wedge problems using first-order shear deformation plate theory
复制标题
DOI：
10.1016/j.euromechsol.2016.09.016
复制
发表时间：
2017
影响因子：
4.1
通讯作者：
J. Felger;W. Becker
中科院分区：
文献类型：
--
作者：
J. Felger;W. Becker
在目前的工作中，导出了一种复杂的势方法，以便使用一阶剪切变形板理论研究楔形或尖锐凹口中的应力奇异性。重点是作为断裂力学基本量的奇点指数的计算。考虑各向同性均质楔块和双材料楔块。详细讨论了沿缺口面的不同边界条件的影响以及弹性对比度对奇点指数的影响。在渐近分析中，通过引入三个全纯势来求解控制 PDE 系统。必须满足边界和连续性条件的要求导致了确定奇异指数和场变量的角度分布的特征值问题。基本特征值方程是一个高度非线性的超越方程，通常通过数值求解。对于特定的双材料配置，可以获得闭合形式的解析解。研究结果与文献结果和有限元计算进行了比较。结果表明，所提出的复势方法是研究渐近解行为的非常有效的方法。与基于实值本征函数展开的方法相比，由于复杂势形式主义的性质，所有场变量只能取实值的物理要求会自动满足。获得的近场允许进一步的应用，例如嵌入数值方法中。
In the present work a complex potential approach is derived in order to investigate stress singularities in wedges or sharp notches using first-order shear deformation plate theory. The focus is on the calculation of the singularity exponent as a fundamental quantity in fracture mechanics. Isotropic homogeneous and bi-material wedges are considered. The effect of different boundary conditions along the notch faces and the influence of the elastic contrast on the singularity exponent are discussed in detail. Within an asymptotic analysis, the governing PDE-system is solved introducing three holomorphic potentials. The requirement that the boundary and continuity conditions must be fulfilled leads to an eigenvalue problem determining the singularity exponent and the angular distribution of the field variables. The underlying eigenvalue equation is a highly non-linear transcendental equation which in general is solved numerically. For specific bi-material configurations closed-form analytical solutions are obtained. The findings are compared with results from literature and with finite element calculations. It is shown, that the proposed complex potential method is a very efficient approach to study the asymptotic solution behaviour. In contrast to methods based on real-valued eigenfunction expansions the physical requirement that all field variables can take only real values is fulfilled automatically due to the nature of the complex potential formalism. The obtained near fields allow for further applications such as an embedding in numerical methods.