Fracture mechanics analysis for residual stress and crack closure corrections

Abstract

A new methodology is proposed to determine (1) stress intensity factors due to residual stress, K_res, when samples with high residual stress are tested, and (2) effective stress intensity factors, K_eff, from corresponding applied stresses when closure contributions are significant. Residual stress is acknowledged to cause changes in the applied stress intensity factors, and these changes are dependent on the residual stress nature (compressive or tensile). Due to the magnified effects of residual stress on long crack growth behavior, corrections for residual stress are needed for appropriate fatigue life evaluations and realistic behavior comparisons. Crack closure, on the other hand, is considered to be the main mechanism leading to differences between applied and effective stress intensity factor ranges at a crack tip. The effective stress intensity factor ranges are important because they represent the major physical cause of fatigue crack growth. Various methods for closure correction have been proposed, but due to their empirical nature and lack of analytical basis, their applicability and general acceptance is limited. The proposed method is an analytical formulation based on load–displacement record differences for incremental changes in crack length. Examples of residual stress and crack closure corrections are provided to validate the method.

Introduction

Macro-residual stresses are found in most alloy systems (wrought and cast aluminum alloys, superalloys, titanium alloys, wrought and P/M steels, etc.), and they are usually an undesired result of various processing (casting, welding, forming operations such as forging and extrusion, etc.) or post-processing (heat treatment, mainly quenching) conditions. Residual stresses alter the values of the applied stress intensity factors compared to the residual stress free values; compressive residual stresses result in decreases while tensile residual stresses induce increases in stresses as well as stress intensity factors. As a result, when compressive residual stresses are present, higher applied stress intensity factors are necessary to achieve a given growth rate; oppositely, when tensile residual stresses are present, lower applied stress intensity factors are necessary to achieve the same growth rate. More importantly, the effect of residual stress on long crack growth data (i.e. data obtained from specimens such as compact tension) can be significantly amplified compared to the effect on small crack growth data due to reduced applied stresses for given stress intensities. To understand the effects of residual stress on long cracks and further assess its corresponding implications on naturally initiated (or small) cracks, appropriate residual stress corrections need to be developed. Moreover, when fatigue crack growth data generated from long cracks are used as comparative measures, residual stress corrections need to be applied first in order to compare intrinsic material properties rather than properties affected by externally-induced, random effects of residual stress.

A method to determine stress intensity factors due to residual stress in edge–notched specimens was introduced by Schindler [1] under the name of “cut compliance technique”, and it was based on a procedure introduced by Cheng and Finnie [2], [3] used to measure the distribution of residual stresses. The method is based on successive extensions of a slot or notch in a specimen containing residual stress while measuring the resulting changes in strain (or displacement) at a reference location on the specimen. The measured changes in strain (or displacement) are further used to compute stress intensity factors due to residual stress. The method was successfully applied by Prime [4] on compact tension samples (preloaded beyond yielding) using strain measurements at the back face. The method was further modified by Lados and Apelian [5] to evaluate stress intensity factors due to residual stress, K_res, using notch displacement measurements at the front face. Lados and Apelian [5] also extended the applicability of the method, using K_res measurements to correct actual fatigue crack growth data for residual stress. In [5], K_res were computed in a simplified manner using the final notch length of the compact tension specimen and the corresponding front face induced displacement; this has the advantage that the notch can be machined in one step rather than several successive steps. In all previous studies, K_res measurements were made off-line (not during an actual fatigue crack growth test), and if residual stress corrections to fatigue crack growth data were done, they were done after the test was completed [5].

In this study, both situations (“off-line cut compliance measurements” and “on-line crack compliance measurements” performed during an actual crack growth test) are considered, and the appropriate mathematical formulations for residual stress intensity factors computation and residual stress correction are presented in Sections 2.1 Residual stress corrective model: “off-line” computation of stress intensity factors due to residual stresses using the “cut compliance technique”, 2.2 Residual stress corrective model: “on-line” computation of stress intensity factors due to residual stresses using the “crack compliance technique”. When applied to residual stress, the method does not consider the additional effects of surface interactions behind the crack tip during the crack advance regardless whether the measurements are done off-line or on-line.

A further application of the method is related to crack closure. Crack closure under positive loading was first observed by Christensen [6], and it was later well established by Elber [7], [8] for cyclic tension. Newman [9], [10] used plasticity-induced crack closure in a finite element program based on a strip yield model attempting to predict crack growth under spectrum loading. It was subsequently recognized that other sources of closure in addition to plasticity-induced closure have significance on crack growth behavior especially in the near-threshold regime, e.g. Suresh and Ritchie [11].

To account for closure effects on fatigue crack growth, ASTM standard E647 [12] introduced a procedure for using load displacement records to determine the crack opening load under cyclic loading. The method identifies the upper linear portion of the load–displacement record within a defined degree of accuracy, which was regarded as the load range for the effective stress intensity. The method was introduced by Donald [13] and explored in a round-robin test program [14]. It has been found [15], [16] that the method does not identify adequately the load ranges for effective stress intensity factor ranges in the near-threshold regime. Earlier, Vecchio and Hertzberg [17], [18] also recognized the inadequacy of the linear load–displacement range as the sole contributor to the stress intensity factor range. To establish appropriate stress intensity factor ranges, various alternative closure corrective methods have been proposed [15], [16], [19]. From the proposed methods, the one provided by Donald et al. [15] most nearly agrees with the mathematical analysis developed in the present study. A procedure for determining effective stress intensity factors after closure correction is presented in Section 2.3.