Analysis of a Notched Four-Point Bending Sample Containing Various Residual Stresses
Authors: Kapp JA , Stacey A 
Published: February 2009

An analytical Round Robin to predict the fatigue life of notched four-point bend samples has been conducted by ASTM Subcommittee E09.02 on Residual Stress Effects in Fatigue. The notched samples with no residual stresses or shot peened residual stresses (compressive) or mechanically induced overstrain residual stresses (tensile) were subjected to cyclic pure bending. Material properties including low cycle fatigue data, fatigue crack propagation data, cyclic stress-strain curves, fracture toughness, and tensile properties were given. Measured and calculated residual stresses as well as the applied elastic stress distribution were also presented. From these, the overall fatigue life of the three different residual stress conditions was predicted.
Fatigue crack initiation was predicted using a “local strain” approach. Using the elastic solution for the applied stress and the elastic modulus from the cyclic stress-strain curve, the strains at and near the notch tip were determined. The ratio of minimum to maximum strain during the loading cycle was calculated for each residual stress case. For the case of no residual stress, the minimum and maximum loads were used to determine the strain ratio; for the shot peened specimens the measured residual stress was added to both the minimum and maximum applied stresses before the local strains were calculated; and for the mechanically induced residual stresses, the calculated residual stresses were added to the applied stresses to calculate the strain ratios. For these calculations and the low cycle fatigue data the life to initiate a small fatigue crack of length ao is predicted. Once a crack is present, it must be decided if the crack is a propagating or non-propagating crack. This was done by either of two methods. We calculate the total life to initiate a crack to a depth a small amount deeper than previously predicted (ao + Δa) and also the number of loadings necessary to grow the crack that same small increment Δa via fatigue crack propagation, which is added to the number of cycles to initiate the crack to ao. These two predictions are compared and that which produces the fewer total cycles is assumed to be the life necessary to propagate the crack to the overall length (ao + Δa). This constant competition between crack initiation and crack propagation was thought to eliminate the problem of determining the fatigue crack growth threshold. The second method was to arbitrarily decide that some fatigue crack growth threshold exists and that once this threshold is exceeded during the cyclic loading, the crack is fully initiated and grows further only by fatigue crack propagation. Since no fatigue crack thresholds were given, this property was assumed to be either 10 or 20 MPa ∙ m1/2 when the load ratio is zero. For nonzero load ratios, crack propagation was assumed to commence when the maximum stress intensity factor during the fatigue cycle Kmax exceeded either 10 or 20 MPa ∙ m1/2. When calculating initiation lives or crack propagation rates, linear interpolation was used.
In order to predict fatigue crack propagation, accurate stress intensity factors must be known. These solutions were obtained using the boundary integral equation (BIE) method. The notched specimen was modeled by cubic boundary elements, and cracks of various lengths were introduced. The resulting solutions are considered to be accurate within 3% by comparison with the solution for a pure bending sample with deep cracks. Also, with no cracks present, the elastic stress solution agreed within 5% of the finite-element solution for the specimen that was presented with the round robin handout. The stress intensity solutions for the mechanically induced residual stresses were obtained by application of the negative of the calculated residual stresses on the crack face, this procedure having been demonstrated to be accurate by superposition arguments. Stress intensity factors were not developed for the shot peened samples. It was assumed that these residual stresses affect only crack initiation.
Wide range expressions were fitted to all of the stress intensity factor solutions. Mathematical expressions were also fitted to the applied stress and calculated residual stress distributions as well as the low cycle fatigue data and the fatigue crack propagation data.
The above outlined method of fatigue life prediction assumes that there is no effect of the applied loading on the residual stresses. A simple analysis of the elastic solution of the applied loads shows that plastic deformation should occur to a little less than 10% of the remaining thickness. Using a very simple rigid plastic model and assuming that recovery occurs elastically, an estimate of an additional residual stress distribution was made. This distribution was very similar to the negative of the calculated mechanically induced residual stress distribution, but its magnitude was somewhat smaller. Realizing this, the stress intensity factor caused by this additional residual stress is estimated and used this in calculating fatigue lives as well as including its effects in crack initiation. The resulting calculated fatigue lives including this additional residual stress are substantially increased.