The evolution of the stress–strain fields near a fatigue crack tip and plasticity-induced crack closure revisited

The evolution of the stress–strain fields near a stationary crack tip under cyclic loading at selected R‐ratios has been studied in a detailed elastic–plastic finite element analysis. The material behaviour was described by a full constitutive model of cyclic plasticity with both kinematic and isotropic hardening variables. Whilst the stress/strain range remains mostly constant during the cyclic loading and scales with the external load range, progressive accumulation of tensile strain occurs, particularly at high R‐ratios. These results may be of significance for the characterization of crack growth, particularly near the fatigue threshold. Elastic–plastic finite element simulations of advancing fatigue cracks were carried out under plane‐stress, plane‐strain and generalized plane‐strain conditions in a compact tension specimen. Physical contact of the crack flanks was observed in plane stress but not in the plane‐strain and generalized plane‐strain conditions. The lack of crack closure in plane strain was found to be independent of the material studied. Significant crack closure was observed under plane‐stress conditions, where a displacement method was used to obtain the actual stress intensity variation during a loading cycle in the presence of crack closure. The results reveal no direct correlation between the attenuation in the stress intensity factor range estimated by the conventional compliance method and that determined by the displacement method. This finding seems to cast some doubts on the validity of the current practice in crack‐closure measurement, and indeed on the role of plasticity‐induced crack closure in the reduction of the applied stress intensity factor range.     

Thermal effect of plastic dissipation at the crack tip on the stress intensity factor under cyclic loading

Plastic dissipation at the crack tip under cyclic loading is responsible for the creation of an heterogeneous temperature field around the crack tip. A thermomechanical model is proposed in this paper for the theoretical problem of an infinite plate with a semi-infinite through crack under mode I cyclic loading both in plane stress or in plane strain condition. It is assumed that the heat source is located in the reverse cyclic plastic zone. The proposed analytical solution of the thermo-mechanical problem shows that the crack tip is under compression due to thermal stresses coming from the heterogeneous stress field around the crack tip. The effect of this stress field on the stress intensity factor (its maximum and its range) is calculated analytically for the infinite plate and by finite element analysis. The heat flux within the reverse cyclic plastic zone is the key parameter to quantify the effect of dissipation at the crack tip on the stress intensity factor.     

Plastic yielding at the tip of a blunt notch during static and fatigue loading

Rice's analytical Mode III solution for the relationship between anti-plane stress and anti-plane strain was used to determine the small scale plastic yielding at the tip of a two-dimensional blunt notch. The results were applied to fatigue loading. The plastic zone size and crack opening displacement derived in the present analysis were determined as functions of applied stress, geometric factors (notch radius and length) and material properties (yield stress and the work hardening rate). The minimum stress intensity required for plastic yielding at a blunt notch tip was postulated to be the experimentally observed threshold stress intensity for fatigue crack initiation. The threshold stress intensity so determined depends not only on the notch geometry but also on material properties. There is good agreement with calculated and measured values of the threshold stress intensity for fatigue crack initiation.     

Experimental study of heat generation at the vertex of a fatigue crack

The process of heat generation at the vertex of a fatigue crack in VT-6 titanium alloy under conditions of cyclic loading has been studied by infrared thermography. The spatial and temporal variations of temperature at the crack vertex have been measured, the shape of the zone of heat evolution has been determined, and the intensity of heat generation has been evaluated. Comparison of the obtained experimental data to the relations of the linear theory of elasticity shows that neither the observed shape of the plastic deformation zone nor the measured dynamics of heat evolution at the crack vertex is consistent with predictions of the linear theoretical models. The experimental results revealed a time delay between the moments of maximum applied stress and maximum intensity of heat evolution at the vertex of a fatigue crack.     

Finite deformation cohesive polygonal finite elements for modeling pervasive fracture

In the present study, mode I crack subjected to cyclic loading has been investigated for plastically compressible hardening and hardening–softening–hardening solids using the crack tip blunting model where we assume that the crack tip blunts during the maximum load and re-sharpening of the crack tip takes place under minimum load. Plane strain and small scale yielding conditions have been assumed for analysis. The influence of cyclic stress intensity factor range (\(\Delta \hbox {K})\), load ratio (R), number of cycles (N), plastic compressibility (\({\upalpha })\) and material softening on near tip deformation, stress–strain fields were studied. The present numerical calculations show that the crack tip opening displacement (CTOD), convergence of the cyclic trajectories of CTOD to stable self-similar loops, plastic crack growth, plastic zone shape and size, contours of accumulated plastic strain and hydrostatic stress distribution near the crack tip depend significantly on \(\Delta \hbox {K}\), R, N, \({\upalpha }\) and material softening. For both hardening and hardening–softening–hardening materials, yielding occurs during both loading and unloading phases, and resharpening of the crack tip during the unloading phase of the loading cycle is very significant. The similarities are revealed between computed near tip stress–strain variables and the experimental trends of the fatigue crack growth rate. There was no crack closure during unloading for any of the load cycles considered in the present study.     

Digital image correlation measurement of near‐tip fatigue crack displacement fields: constant amplitude loading and load history effects

Recent work by de Matos and colleagues employed digital image correlation to measure near tip displacement fields for fatigue cracks in 6082 T6 aluminium alloy. The main focus of this work was to directly measure fatigue crack closure, but the measurements can also be used to examine conditions at and ahead of the crack tip. In this paper, the results are re‐analysed and compared to two crack‐tip deformation models. The first assumes simple elastic deformation (according the Westergaard solution). This allows the history of crack‐tip stress intensity to be examined. Reasonable agreement with the elastic model is obtained, although there is a residual stress intensity caused by the plastic wake, which gives rise to crack closure. The second model examined is a simple elastic–plastic assumption, proposed by Pommier and colleagues. This can be applied to constant amplitude loading, although the results obtained here are very similar to the elastic case. A slightly more complex load case (a single overload in an otherwise constant amplitude variation of load) gives a much more complicated crack‐tip history. Here, the importance of crack‐tip plastic displacement, represented by the second term in Pommier's model becomes much clearer. Load history effects are captured by the residual value of this term and its associated displacement fields as well as by stress intensity factor. The implications for further modelling and experimental work are discussed.     

Analysis of the plastic zone of a circle crack under very high cycle fatigue

Metals used in industry for structures and aero-engine components are sometimes subjected to very high cycle fatigue (VHCF) damage during their working service. In this paper, a novel method is presented to determine the size and location of a circular crack located within a metal specimen under 20 kHz VHCF loading conditions. The method is based on an analysis of the temperature rise on the surface of the specimen and correlation of this temperature rise to the energy dissipation in the plastic zone of the crack. The approach taken is to first determine the heat source location and strength using an inverse heat transfer calculation based on the surface temperature measurements. Next, the relationship between the heat and the material hysteresis loop in the plastic zone, which is a function of stress intensity factor and vibration amplitude cyclic loading of the specimen, is found. The calculation of the stress intensity factor under vibrational loading is often an obstacle in VHCF research because there is currently no standard or existing formula. In this paper a general polynomial formula for the stress intensity factory under 20 kHz loading conditions is obtained using a finite element modeling approach as a function of the specimen’s material properties and position and size of the internal crack.     

Plastic zone formation and fatigue crack extension during high cyclic bending of steels

In repeated high cyclic bending, with constant load amplitude, the size and the shape of the plastic zone preceding the propagating crack is controlled by local structural conditions near the tip rather than by stress intensity. No significant correlations were found between the experimentally determined sizes of plastic zone and the theoretically predicted values of Liu and Rice. The plastic zone sizes ahead of the propagating crack cannot be simply expressed as proportional to the rate of fatigue crack propagation, though a simple relationship exists between the rate and the stress intensity factor. The relationship given by Paris, dl/dN = QΔKⁿ, describes the rate of crack propagation only in a limited range of relative crack length, x < 0.5. The extent of this range depends on the structure and on the level of applied cyclic stress. Beyond this range, the Paris equation could not be applied and the crack propagation cannot be related to the stress intensity factor.     

The effect of T‐stress on crack‐tip plastic zones under mixed‐mode loading conditions

In this paper, the influence of T‐stress on crack‐tip plastic zones under mixed‐mode I and II loading conditions is examined. The crack‐tip stress field is defined in terms of the mixed‐mode stress intensity factors and the T‐stress using William's series expansion. The crack‐tip stress field is incorporated into the Von Mises yield criteria to develop an expression that determines the crack‐tip plastic zone. Using the resultant expression, the plastic zone is plotted for various combinations of mode II to mode I stress intensity factor ratios and levels of T‐stress. The properties of the plastic zone affected by T‐stress and mixed‐mode phase angle are discussed. The observations obtained on plastic zones variations are important for further fatigue and fracture analyses for defects in engineering structures under mixed‐mode loading conditions.     

Towards a new model of crack tip stress fields

This work introduces a novel mathematical model of the stresses around the tip of a fatigue crack, which considers the effects of plasticity through an analysis of their shielding effects on the applied elastic field. The ability of the model to characterize plasticity-induced effects of cyclic loading on the elastic stress fields is assessed and demonstrated using full-field photoelasticity. The focus is on determining the form of the shielding stress components (induced by compatibility requirements at the elastic–plastic interface along the crack flank and via the crack tip plastic zone) and how they influence the crack tip elastic stress fields during a load cycle. The model is successfully applied to the analysis of a fatigue crack growing in a polycarbonate CT specimen.     

Mechanisms and modeling of low cycle fatigue crack propagation in a pressure vessel steel Q345

The fatigue process near crack is governed by highly concentrated strain and stress in the crack tip region. Based on the theory of elastic–plastic fracture mechanics, we explore the cyclic J-integral as breakthrough point, an analytical model is presented in this paper to determine the CTOD for cracked component subjected to cyclic axial in-plane loading. A simple fracture mechanism based model for fatigue crack growth assumes a linear correlation between the cyclic crack tip opening displacement (ΔCTOD) and the crack growth rate (da/dN). In order to validate the model and to calibrate the model parameters, the low cycle fatigue crack propagation experiment was carried out for CT specimen made of Q345 steel. The effects of stress ratio and crack closure on fatigue crack growth were investigated by elastic–plastic finite element stress–strain analysis of a cracked component. A good comparison has been found between predictions and experimental results, which shows that the crack opening displacement is able to characterize the crack tip state at large scale yielding constant amplitude fatigue crack growth.     

Effect of single and multiple overloading on the residual fatigue life of a structural steel

This work was carried out to determine the effect of overload cycles on the fatigue life of a structural steel used for offshore applications. Single and multiple overloads were adopted and the corresponding fatigue crack growth retardation was evaluated. Residual stress fields were measured in the vicinity of the crack tip using an X‐ray diffraction technique and their size compared with that of the overload cyclic plastic zone. In regard to crack growth retardation, the results indicated that the extension in fatigue life increases with an increase in overload, as a consequence of the generation of higher compressive residual stress levels over a larger distance ahead of the crack tip. The effect of two equal and consecutive overloads, with the second one applied at different intervals of crack propagation from the first, was also considered. Larger intervals were shown to lead to a longer residual fatigue life.     

Plastic zone size estimation under cyclic loadings using in situ optical microscopy fatigue testing

An in situ optical microscopy fatigue testing is proposed in this paper to investigate the forward and reversed plastic zone size under cyclic loadings for Al‐7075‐T6. This experimental study is used to verify the hypotheses in a recently developed small time scale formulation of fatigue crack growth. During the testing, the entire cyclic loading cycle is divided into a certain number of steps. Images of the crack tip are taken at each step. The full strain field around the crack tip is determined using the digital image correlation (DIC) technique. The plastic zone size is obtained by combining the DIC results and the material constitutive relationship. Experimental measurements from the proposed study are compared with theoretical predictions. It is observed that the crack closure has a large effect on the reversed plastic zone size. The plastic zone size remains almost constant when the unloading path is below a certain stress level, which is one of the hypotheses used in a previous crack growth model. Discussions are given for the modelling of plastic zone size variation under cyclic loadings and several conclusions are drawn based on the current investigation.     

The Influence of the Cyclic Loading Rate on the Plastic Zone Depth in VNS-25 Alloy

We have studied the plastic zone depth at the fatigue crack tip in VNS-25 (03Kh12N10MT) alloy specimens, previously subjected to the cyclic fracture toughness tests under symmetrical push-pull loading with a frequency of 20, 170, 600 Hz, 3 and 10 kHz. For the same values of the stress intensity factor, an increase in the loading frequency is shown to slow down the fatigue crack growth and reduce the plastic zone depth under the fracture surface. However, the dependence of the plastic zone depth on the crack growth rate is invariant with respect to the loading frequency.     

Fatigue crack growth model for constant amplitude loading

A fatigue crack growth model under constant amplitude loading has been developed considering energy balance during growth of the crack. The plastic energy dissipated during growth of a crack within cyclic plastic zone and area below cyclic stress–strain curve was used in the energy balance. The near crack tip elastic–plastic stress and strain were calculated on the basis of Hutchinson, Rice and Rosengren (HRR) formulations. Fatigue crack growth rate in linear and near threshold region of da/dN versus ΔK curve can be determined on the basis of the proposed model in terms of low cycle fatigue (LCF) properties determined on smooth specimen. The predictions of the model have been compared with the experimental and theoretical results available in the literature using mechanical and fatigue properties. The model compares well in the threshold and intermediate region of the da/dN versus ΔK curve for wide range of material tested.     

Temperature fields near a running crack tip

Near a running crack tip, the plastic work rate is high. According to the theory of irreversible thermodynamics, the plastic work will be almost completely converted into heat which may lead to high temperature rise at the running crack tip. The plastic zone is regarded as the zone of the heat source, and the plastic work rate as the strength of the heat source. In this paper, the plastic work rate is derived from the solution of stress and strain fields obtained by Chitaley and McClintock[1] for a steady state crack growth under anti-plane shear in an elastic perfectly-plastic material. The dependence of the thermal conductivity on temperature has been considered and a non-linear model for temperature fields has been proposed. The numerical results for glass have been given and compared with other papers.     

Prediction of fatigue crack growth based on low cycle fatigue properties

Theoretical models of the fatigue crack growth without artificial adjustable parameters were proposed by considering the plastic strain energy and the linear damage accumulation, respectively. The crack was regarded as a sharp notch with a small curvature radius and the process zone was assumed to be the size of cyclic plastic zone. The near crack tip elastic–plastic stress and strain were evaluated in terms of modified Hutchinson, Rice and Rosengren (HRR) formulations. Predicted results from two established models have been soundly compared with open reports for frequently used materials. It is found that experimental results agree well with theoretical solutions.     

THE EFFECT OF INTERMEDIATE HEAT TREATMENTS ON OVERLOAD INDUCED RETARDATIONS DURING FATIGUE CRACK GROWTH IN AN Al-ALLOY

Abstract— Crack growth fatigue tests were carried out on 2024-T3 specimens. Constant-amplitude loading was periodically interrupted by 10 overload cycles. Intermediate heat treatments (T4) were applied to remove the residual stress in the crack tip zone and the crack closure wake behind the crack tip. Retardation effects induced by crack closure due to the previous load history were fully erased by the heat treatments. Overload effects were easily introduced again by new overload cycles afterwards. Crack growth rate results and fractographic observations indicate that primary crack tip plastic deformation (in virgin material) is more effective for crack extension than secondary plastic deformation in an existing plastic zone. This conclusion is significant for cycle-by-cycle crack growth prediction models for variable-amplitude loading.     

Effect of adhesive thickness on fatigue and fracture of toughened epoxy joints – Part II: Analysis and finite element modeling

The effect of bondline thickness on the fatigue and fracture of aluminum adhesive joints bonded using a rubber-toughened epoxy adhesive was studied using finite element analysis. The fatigue data of Part I examined the dependence of the fatigue threshold and cyclic crack growth rate on the adhesive thickness under both mode-I and mixed-mode loading. The fracture data of Part I illustrated the relation between the adhesive thickness and the quasi-static crack initiation and steady-state critical strain energy release rates. These experimental trends are explained in terms of the effects of the adhesive thickness and the applied strain energy release rate on the stress distribution in the bondline, the stress triaxiality at the crack tip, and the plastic zone size in the adhesive layer.     

Effects of mechanical heterogeneity on the tensile and fatigue behaviours in a laser-arc hybrid welded aluminium alloy joint

The effects of mechanical heterogeneity on the tensile and high cycle fatigue (10⁴–10⁷ cycles) properties were investigated for laser-arc hybrid welded aluminium alloy joints. Tensile–tensile cyclic loading with a stress ratio of 0.1 was applied in a direction perpendicular to the weld direction for up to 10⁷ cycles. The local mechanical properties in the tensile test and the accumulated plastic strain in the fatigue test throughout the weld’s different regions were characterized using a digital image correlation technique. The tensile results indicated heterogeneous tensile properties throughout the different regions of the aluminium welded joint, and the heat affected zone was the weakest region in which the strain localized. In the fatigue test, the accumulated plastic strain evolutions in different subzones of the weld were analyzed, and slip bands could be clearly observed in the heat affected zone. A transition of fatigue failure locations from the heat affected zone caused by accumulated plastic strain to the fusion zone induced by fatigue crack at pores could be observed under different cyclic stress levels. The welding porosity in the fusion zone significantly influences the high cycle fatigue behaviour.