Surface singularity and crack propagation

This review article~Part II! is a sequel to an earlier one ~Part I! that dealt with means of removal and interpretation of stress singularities in elasticity, as well as their asymptotic and numerical analysis. It reviews contributions to the literature that have actually effected asymptotic identifications of possible stress singularities for specific configurations. For the most part, attention is focused on 2D elastostatic configurations with constituent materials being homogeneous and isotropic. For such configurations, the following types of stress singularity are identified: power singularities with both real and complex exponents, logarithmic intensification of power singularities with real exponents, pure logarithmic singularities, and log-squared singularities. These identifications are reviewed for the in-plane loading of angular elastic plates comprised of a single material in Section 2, and for such plates comprised of multiple materials in Section 3. In Section 4, singularity identifications are examined for the out-of-plane shear of elastic wedges comprised of single and multiple materials, and for the out-of-plane bending of elastic plates within the context of classical and higher-order theory. A review of stress singularities identified for other geometries is given in Section 5, axisymmetric and 3D configurations being considered. A limited examination of the stress singularities identified for other field equations is given as well in Section 5. The paper closes with an overview of the status of singularity identification within elasticity. This Part II of the review has 227 references. @DOI: 10.1115/1.1767846 #

Stress singularities in classical elasticity—II: Asymptotic identification

The solution of the three-dimensional Dirichlet problem for the Laplacian in a polyhedral domain has Special singular forms at corners and edges. The main result of this paper is a “tensor-product” decomposition of those singular forms along the edges. Such a decomposition with both edge singularities, additional corner singularities and a smoother remainder refines known regularity results for the solution where either the edge singularities are of non-tensor product form or the remainder term belongs to an anisotropic Sobolev space for data given in an isotropic Sobolev space.

Decompositions in Edge and Corner Singularities for the Solution of the Dirichlet Problem of the Laplacian in a Polyhedron

The approximation theory developed in [11] is used to determine the piecewise polynomial approximability of solutions of elliptic problems on polygonal domains in $\mathbb{R}^2 $ and polyhedra in $\mathbb{R}^3 $. From these estimates, convergence orders for the p-version of the finite element method applied to such problems are readily obtained. The critical issue is the approximation of the singularities which occur at the nonsmooth parts of the domain boundaries. It is seen that the estimates of [11] involving the weighted Sobolev spaces $Z_l^s $ are well suited for treating such singular functions, yielding directly the optimal approximation degree, up to an arbitrarily small $\varepsilon $.Numerical results for two problems from two-dimensional linear elasticity are also presented. The computations show that the predicted order of convergence is achieved even for low values of p. Moreover, in contrast to the usual h-version of the finite element method, the point at which the p-version enters the asym...

The approximation of solutions of elliptic boundary-value problems via the p -version of the finite element method

The previously reported neBEM solver has been used to solve electrostatic problems having three-dimensional edges and corners in the physical domain. Both rectangular and triangular elements have been used to discretize the geometries under study. In order to maintain very high level of precision, a library of C functions yielding exact values of potential and flux influences due to uniform surface distribution of singularities on flat triangular and rectangular elements has been developed and used. Here we present the exact expressions proposed for computing the influence of uniform singularity distributions on triangular elements and illustrate their accuracy. We then consider several problems of electrostatics containing edges and singularities of various orders including plates and cubes, and L-shaped conductors. We have tried to show that using the approach proposed in the earlier paper on neBEM and its present enhanced (through the inclusion of triangular elements) form, it is possible to obtain accurate estimates of integral features such as the capacitance of a given conductor and detailed ones such as the charge density distribution at the edges/corners without taking resort to any new or special formulation. Results obtained using neBEM have been compared extensively with both existing analytical and numerical results. The comparisons illustrate the accuracy, flexibility and robustness of the new approach quite comprehensively.

https://msvlab.hre.ntou.edu.tw/eabe2008-corner.pdf

A study of three-dimensional edge and corner problems using the neBEM solver

In the present work we calculate the strength of the singularity in surface charge density at the vertex of a sectorial conducting plate in an electrostatic field, including the case in which the media occupying the half-spaces above and below the plate have different dielectric constants. The singularity strength is calculated numerically for different vertex angles $\chi $. In addition, for a “spike” $( {\chi \ll 1} )$, a “nearly straight edge” $( {| {\chi - \pi } | \ll 1} )$ and a “slit” $( {2\pi - \chi \ll 1} )$ , analytic expressions for the singularity strength are found by singular perturbation techniques. The results are relevant to the calculation of circuit board stray capacitances, where one must determine the charge densities on rectilinear sheet conductors, mounted on a dielectric board, for constant conductor potentials. For this case, it is found that the charge singularity, inversely proportional to the square root of the distance from the edge for points far from the vertex, is proportion...

J. A. Morrison

Papers

Charge Singularity at the Corner of a Flat Plate