Analysis of accuracy of Williams series approximation of stress field in cracked body – influence of area of interest around crack-tip on multi-parameter regression performance

J Sobek,P Frantík,V Veselý

doi:10.3221/igf-esis.39.14

Abstract

A study on the accuracy of an approximation of the stress field in a cracked body is presented. Crack-tip stress tensor is expressed using the linear elastic fracture mechanics (LEFM) theory in this work, more precisely via its multi-parameter formulation, i.e. by Williams power series (WPS). Determination of coefficients of terms of this series is performed using a least squares-based regression technique known as over-deterministic method (ODM) for which results from finite element (FE) method computation are usually taken as inputs. Main attention is paid to a detailed analysis of a suitable selection of FE nodes whose results serve as the inputs to the employed method. Two different ways of FE nodal selection are compared – nodes selected from the crack tip vicinity lying at a ring of a certain radius versus nodes selected more or less uniformly from a specified part of the test specimen body. Comparison of these approaches is made with the help of procedures developed by the authors which enable both the determination of the coefficients of terms of the analytical WPS approximation of the stress field based on the FE results and the backward reconstruction of the field (again using WPS) from those determined terms’ coefficients/functions. The wedge-splitting test (WST) specimen with a crack is taken as example for the study.

Highlights

Determination of the higher order terms of Williams power series [1] approximating the fields of stress and displacements in a cracked body is of substantial interest of the presented work
Determination of coefficients of terms of this series is performed using a least squares-based regression technique known as over-deterministic method (ODM) for which results from finite element (FE) method computation are usually taken as inputs
Procedures enabling the multi-parameter description of the near crack-tip fields usually process results from FE nodes selected from the close vicinity of the crack tip

Summary

Introduction

Determination of the higher order terms of Williams power series [1] approximating the fields of stress and displacements in a cracked body is of substantial interest of the presented work. Published study [19] on WPS approximation with nodal selection from more than one ring around the crack tip resulted into studies Another type of nodal selection was considered in subsequent works by the authors [20,21,22]; the nodes were selected from specific parts of the test specimen body with specific distribution functions of their distance and angular position from the crack tip. This way was employed with expectations that the fields will be better (more accurately/efficiently) approximated. Main motivation of this study is to find the easiest way for the multi-parameter stress field description to obtain the sufficiently accurate solution valid for the as far as possible area from the crack tip

Methods

Results

Conclusion