An interface crack in a coating-substrate composite with mixed-mode Dugdale corrections

Considering both plane stress and plane strain conditions, the plastic zone size and the crack tip opening displacement of an interface crack between a coating and a semi-infinite substrate under a normal load on the crack surfaces are investigated by the mixed-mode Dugdale model. In the model, stresses applied in the plastic zones satisfy the Von Mises yield criterion. The plastic zone size can be calculated by satisfying the condition that the complex stress intensity factors vanish. After the plastic zone size is solved, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, a uniform load is considered, and the effects of the normalized elastic modulus (the ratio of the elastic modulus of the coating to the elastic modulus of the substrate) and the normalized crack depth (the ratio of the coating thickness to the interface crack length) on the normalized plastic zone size and the normalized crack tip opening displacement are examined. Numerical examples show in the case of thin coatings, the value of the normalized plastic zone size decreases with increasing the normalized elastic modulus.     

DETERMINATION OF THE MOST APPROPRIATE MESH SIZE FOR A 2-D FINITE ELEMENT ANALYSIS OF FATIGUE CRACK CLOSURE BEHAVIOUR

Abstract— A two-dimensional elastic-plastic finite element analysis is performed for plane stress conditions with 4-node isoparametric elements to examine closure behaviour of fatigue cracks, giving special attention to the determination of the most appropriate mesh sizes. It is found that a smaller mesh size does not always give more accurate simulation results in the fatigue crack closure analysis, unlike a conventional structural analysis. A unique, most-appropriate mesh size exists for a given loading condition that will provide numerical results which agree well with experimental data. The most appropriate mesh size can be determined approximately in terms of the theoretical reversed plastic zone size. In particular, the ratio of the most appropriate mesh size to the theoretical reversed plastic zone size is nearly constant for a given stress ratio in the so-called crack-length-fixed method proposed in this study. By using the concept of the most appropriate mesh size, the finite element analysis can predict fatigue crack closure behaviour very well.     

On the evaluation of monotonic and cyclic plastic zones

This paper reviews and compares the various approaches for finding monotonic and cyclic plastic zone sizes for elastic-perfectly plastic and strain hardening materials. All through the analysis Ramberg-Osgood relationship has been used representing the monotonic and cyclic stress-strain curves. The results for elastic-perfectly plastic materials are compared with those available using finite-element technique. The ratio of monotonic plastic zone sizes strain hardening and elastic perfectly plastic materials are found to be related to the Ramberg-Osgood index $\text{n}$.The values of cyclic plastic zone size are found to depend upon factors like crack opening stress, material properties and specimen geometry. The cyclic plastic zone size obtained in this paper compare well with those given by Antolovich.     

Cyclic crack opening displacement behavior during high-amplitude block loading

Load-crack opening displacement hysteresis behavior was monitored for fifty cycles of highamplitude loading which followed fatigue pre-cracking at low stress intensity factor levels. The material studied was quenched and tempered (400°C) AISI 4140 steel which showed pronounced cyclic softening. Despite this softening behavior, cycle-to-cycle decreases in load-COD hysteresis were observed during the initial cycles of high-amplitude loading. Steady state load-COD hysteresis behavior was attained by fifty loading cycles in each case and the fifty-cycle hysteresis loop widths agreed well with those for continuously cycled (non-pre-cracked) samples for equivalent loading conditions. The cycles during which the load-COD hysteresis decreased most dramatically represented fatigue crack growth distances equal to approximately 30% of the calculated plane strain monotonic plastic zone size. Greater percentage reductions in load-COD hysteresis were observed for lower stress intensity factor ranges. The observed behavior was in general agreement with that predicted by finite element fatigue crack closure models in the literature. In addition, the level of prior loading was found to have a pronounced effect on subsequently measured fracture toughness values for this material.     

EFFECT OF ELASTIC MISMATCH ON THE GROWTH OF A CRACK INITIALLY TERMINATED AT AN INTERFACE IN ELASTIC PLASTIC BIMATERIALS

Abstract A crack perpendicular to, and initially with the tip on, a bimaterial interface is studied. An asymptotic analysis is performed and crack growth proceeds straight ahead at constant remote load. Mode I conditions and plane strain are assumed. The materials on both sides of the interface are elastic perfectly-plastic with different elastic properties and the same yield stress. A finite element analysis is made and crack growth is simulated by an element relaxation technique. Because of the interface, the crack-tip driving force is not constant, which is reflected in the near-tip state. The development of the plastic zone and the crack opening displacements is presented for different elastic mismatches. Small scale yielding like results are obtained after a crack extension of about the plastic zone size from the interface, i.e. long before a square-root singular stress field may be expected to embed the plastic zone. An important observation is that the development of the crack opening displacement at the initial stage of growth is reversed when plasticity is introduced, as compared to the prediction by an elastic model. A region of stable crack growth is identified at the initial phase of growth into a stiffer material, solely due to elastic mismatch.     

Fatigue crack growth retardation under constant amplitude and variable mean stress

Fatigue crack growth retardation under stress spectra with constant amplitude and variable mean stress, respectively, was studied. Flat specimens with a central through crack were tested under tension-tension load. The specimens were made with low alloy steel 4 mm thick, with yield strength of 625 Mpa and ultimate tensile strength of 784 MPa. Overload affected crack length increments Δa¹ were studied. The best correlation was obtained between monotonic plane stress plastic zone size 2r_y and Δa¹. The cyclic plastic zone size 2r_pc correlated with crack length increment of minimum crack growth rate after overload. Forman's equation and Willenborg's model of fatigue crack growth retardation were used for theoretical prediction of fatigue crack propagation life. The best agreement between theoretical and experimental results was also obtained using monotonic plastic zone size instead of monotonic plastic zone radius or cyclic plastic zone size. The agreement is reasonably good, even though in the case of one spectrum, cracks were arrested for several thousand cycles.     

Transient behaviour in the numerical analysis of plasticity induced crack closure

A transient behaviour is observed in the numerical analysis of plasticity induced crack closure at the beginning of crack propagation, as the residual plastic field is being formed. The extent of crack propagation prior to plasticity induced crack closure measurement has a major influence on the accuracy of numerical prediction and on computation time. The objective here is to quantify and understand the minimum propagation, Δa_stb, required to obtain stabilized crack opening values. For plane stress state, Δa_stb was found to increase with ΔK. Under plane strain conditions, a peak of closure exists at the beginning of crack propagation for relatively low ΔK values, which promotes relatively large transient periods. Two driving forces explain the stabilization behaviour, the formation of residual plastic wake and the stabilization of plastic strain, but the second seemed to control the phenomenon. Finally, two strategies are proposed to accelerate convergence. The first, consisting of a progressive increase of maximum load, is relevant in plane strain and 3D studies, in order to eliminate the initial peak. The second strategy consists of an extrapolation model and is very effective for plane stress conditions.     

A numerical study of plasticity induced crack closure under plane strain conditions

The level of plasticity induced crack closure (PICC) is greatly affected by stress state. Under plane strain conditions, however, the level and even the existence of PICC still are controversial. The objective here is to study the influence of the main numerical parameters on plane strain PICC, namely the total crack propagation, the number of load cycles between crack increments, the finite element mesh and the parameter used to quantify PICC. The PICC predictions were included in a parallel numerical study of crack propagation, in order to quantify the impact of plane strain values on fatigue life. The results indicate that literature may be overestimating plane strain PICC due to incorrect numerical parameters. The number of load cycles usually considered is unrealistically small, and its increase was found to vanish crack closure, particularly for kinematic hardening. This effect was linked to the ratcheting effect observed at the crack tip. The total crack increment, Δa, must be large enough to obtain stabilized PICC values, but this may imply a huge numerical effort particularly for 3D models. The size of crack tip plastic zone may be overestimated in literature, which means that the meshes used may be too large. Additionally, the crack propagation study showed that the plane strain PICC has usually a dominant effect on fatigue life, and plane stress PICC is only relevant for relatively thin geometries.     

Some comments on the effect of biaxial stress on fatigue crack growth and fracture

The influence of biaxial loading on plastic zone size and crack opening displacement has been examined. Loading parallel to the crack plane was accounted for in satisfying the yield criteria, proposed by Von Mises, along the crack plane. Comparing plastic zone sizes for biaxial and uniaxial loading, when the applied stress normal to the crack plane is the same in both cases, shows a reduction of up to 65 per cent in plastic zone size for the biaxial loading case. The crack opening displacement is also reduced up to 33 per cent for biaxial loading. Based on critical crack opening displacement, fatigue crack propagation rates would be lower than for uniaxial loading, and an apparent increase in fracture toughness would also occur. The theoretical predictions are supported by a limited amount of experimental data.     

A strain-based Dugdale model for cracks under generally yielding conditions

To develop an analytical method for quantifying the growth behaviour of short cracks embedded in notch plastic zones, a critical assessment of the Dugdale model is first made by comparison against finite element analysis for an edge-cracked plate subjected to an applied strain varying linearly along the crack path. It is shown that the conventional stress-based Dugdale model provides accurate estimates for the crack-tip opening displacement and the plastic zone size provided that the applied strain does not exceed one third of the yield strain. These estimates become significantly inaccurate at higher strain levels. To overcome this limitation of the conventional model, a strain-based implementation of the Dugdale model is proposed in which the conventional equilibrium equation is replaced by strain compatibility. Comparison with finite element results shows that this strain-based model provides accurate values for both the crack-tip-opening displacement and the plastic zone size for applied strains up to four times the yield strain and with no evidence of decreasing accuracy with increasing strain. Furthermore, it is shown that the relevant plastic constraint factor to be used for plane strain is that appropriate for the notch plastic zone in the absence of a crack, rather than the more usual choice which is appropriate only for small-scale yielding conditions. This provides a practical and physically plausible approach for extending the scope of current predictive software for fatigue crack growth based on the Dugdale model to include conditions of large-scale yielding.     

Closed-Form Solutions for the Mode ii Crack tip Plastic Zone Shape

Analytical closed-form solutions for the Mode II crack tip plastic zone shape have been derived for a semi-infinite crack in an isotropic elastic-plastic solid under both plane stress and plane strain conditions. Two yield criteria have been applied: the Von Mises and Tresca yield criteria. The results indicate that the Tresca zone is larger in size than the Von Mises zone. The results also reveal an intricate dependence on Poisson's ratio in plane strain conditions.     

Acoustic emission behaviour during stage II fatigue crack growth in an AISI type 316 austenitic stainless steel

Acoustic emission (AE) behaviour during fatigue crack growth (FCG) in a ductile AISI type 316 austenitic stainless steel is reported. The two substages in the stage II Paris regime of FCG could be distinguished by a change in the rate of acoustic activity with increase in crack growth rate. The transition point in the cumulative ringdown count plot coincides with that in the da/dn plot. The AE activity increases with increase in ΔK during stage IIa and decreases during stage IIb. The major source of AE during stage IIa is found to be the plastic deformation within the cyclic plastic zone (CPZ) as compared to the phenomena such as monotonic plastic zone (MPZ) expansion, ductile crack growth, crack closure, etc. The increase in AE activity with increase in ΔK during stage IIa is attributed to the increase in the size of the CPZ which is generated and developed only under plane strain conditions. The decrease in AE activity during stage IIb is attributed to the decrease in the size of the CPZ under plane stress condition. The high acoustic activity during the substage IIa is attributed to irreversible cyclic plasticity with extensive multiplication and rearrangement of dislocations taking place within the CPZ. The AE activity is found to strongly depend on the optimum combination of the volume of the CPZ, average plastic strain range and the number of cycles before each crack extension. Based on this, an empirical relationship between the cumulative RDC and ΔK has been proposed and is found to agree well with experimentally observed values.     

Fully plastic analyses of plane strain single-edge-cracked specimens subject to combined tension and bending

Accurate yield surfaces of plane strain single-edge-cracked specimens having shallow as well as deep cracks are developed using finite element limit analyses and monotonic interpolation functions. Fully plastic shallow crack configurations are classified based on certain aspects of the yield surfaces. Relationships between incremental plastic crack tip and crack mouth opening displacements and incremental load point displacement/rotation are obtained for a wide range of relative crack depths and loading ratios. Fully plastic crack-tip fields for a sufficiently deep crack in a single-edge cracked specimen are examined to provide the stress triaxiality and the angular orientation of flow line at the crack tip in terms of the remotely applied tension-to-bending ratio. Evidence for fully plastic crack-tip stress fields consisting of an incomplete Prandtl fan and a crack plane constant state region is discussed.     

Influence of state of stress on mixed mode stable crack growth through D16AT aluminium alloy

The stable crack growth through three-point bend (TPB) and stiffened and unstiffened compact tension (CT) specimens of D16AT aluminium alloy has been studied both theoretically and experimentally. The specimen thickness is 8 mm. The variation of load with crack opening displacement, the extent of stable crack growth, the cumulative plastically deformed zone and crack edge profiles have been obtained experimentally. These are also predicted theoretically under the assumption of either a state of plane stress or plane strain using a finite element scheme and the COA criterion. Generally, the experimental results agree well with the predictions based on the plane stress condition. There appears to be no significant variation in size of the experimental cumulative plastic zone across the specimen thickness, thereby indicating that the constraint on the plastic zone does not develop near the mid-thickness region.     

X-ray diffraction studies on fatigue crack plastic zones developed under plane strain state conditions

An overview of the X-ray fractography technique, as performed on fatigue crack surfaces of several steels and Al-alloys under different loading conditions, is presented. The plastic zone sizes of fatigue cracks, for plane strain conditions, are measured from the in-depth distribution of residual stresses and X-ray diffraction peak broadening. In addition to the usual monotonic plastic zone size determination methodology, a model for the estimation of the reverse plastic zone size was established in the case of fatigue softening materials. Monotonic and cyclic plastic zone sizes are related to the stress intensity by, respectively, r_{pm = α (Kmax} /σ_ys )² and r_{pc = α} (ΔK/2σ′ys )2. The α-value, in the monotonic plastic zone size equation, increases as the yield strength of the material increases, following the relationship α = 0.196 [σ_ys /(129 + 0.928σ_ys )]2. The α-value versus σ_ys evolution has been understood through the influence of the hardening rate of materials on the plastic zone size. X-ray fractography has been applied to actual failure analyses to predict some aspects of the actual loadings.     

Effects of non-singular stresses on crack-tip fields for pressure-sensitive materials,Part 1: Plane strain case

Mode I near-tip stress fields for elastic perfectly plastic pressure-sensitive materials under plane strain and small-scale yielding conditions are presented. A Coulomb-type yield criterion described by a linear combination of the effective stress and the hydrostatic stress is adopted in the analysis. The finite element computational results sampled at the distance of a few crack opening displacements from the tip show that, as the pressure sensitivity increases, the magnitudes of the normalized radial and hoop stress ahead of the tip decrease, the total angular span of the singular plastic sectors decreases, and the angular span of the elastic sectors bordering the crack surfaces increases. When non-singular T stresses are considered along the boundary layer of the small-scale yielding model, the near-tip stresses decrease as the T stress decreases. The plastic zone shifts toward the crack surfaces as the T stress increases. When the discontinuities of the radial stress and the out-of-plane normal stress along the border between the plastic sector and the elastic sector are allowed, the angular variations of the asymptotic crack-tip fields agree well with those of the finite element computations. Variation of the Q stresses for pressure-sensitive materials can be found from the asymptotic solutions when the plastic zone size ahead of the tip is relatively larger than the crack opening displacement. In addition the T stress is shown to have strong effects on the plastic zone sizes and shapes which could affect the toughening of pressure-sensitive materials.     

Elastic and plastic stress analysis of an interface crack between two dissimilar layers

In the present paper, the mixed-mode Dugdale model is applied to investigate the plastic zone size and the crack tip opening displacement of an interface crack between two dissimilar layers. In the analysis, both normal and shear stresses are assumed to exist in the plastic zones and satisfy the Von Mises yield criterion. The plastic zone sizes can be determined on condition that the stress intensity factors caused by the stresses in the plastic zones and applied loading are zero. Then, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, the plane stress condition is considered. The plastic zone size and the crack tip opening displacement of an interface crack between two dissimilar layers under a uniform load are examined. The effects of Dundurs’ parameters and the thickness of materials on the plastic zone size and the crack tip opening displacement are investigated in detail. Numerical results show that in the case of small thickness, the values of the normalized plastic zone size and the normalized crack tip opening displacement decrease with increasing Dundurs’ parameters, α and β, while, in the case of infinite thickness, the value of the normalized plastic zone size is independent of α, and the value is symmetric about the axis on which β = 0.     

Effect of geometry and finite root radius on plastic zone and tip opening displacement

The present paper describes a simple mathematical model to predict tip opening displacement in notches and removes anomalies that arise when the earlier theories of sharp cracks are applied to notches. The evaluation of plastic zone size, stress intensity factor and tip opening displacement in notches under plane strain mode of loading has been presented for two different geometries and the results have been discussed vis-a-vis the sharp crack case.     

Fractographic and numerical study of hydrogen–plasticity interactions near a crack tip

This paper offers a fractographic and numerical study of hydrogen–plasticity interactions in the vicinity of a crack tip in a high-strength pearlitic steel subjected to previous cyclic (fatigue) precracking and posterior hydrogen-assisted cracking (HAC) under rising (monotonic) loading conditions. Experiments demonstrate that heavier cyclic preloading improves the HAC behaviour of the steel. Fractographic analysis shows that the microdamage produced by hydrogen is detectable through a specific microscopic topography: tearing topography surface or TTS. A high resolution numerical modelling is performed to reveal the elastoplastic stress–strain field in the vicinity of the crack tip subjected to cyclic preloading and subsequent monotonic loading up to the fracture instant in the HAC tests, and the calculated plastic zone extent is compared with the hydrogen-assisted microdamage region (TTS). Results demonstrate that the TTS depth has no relation with the active plastic zone dimension, i.e., with the size of the only region in which there is dislocation movement, so hydrogen transport cannot be attributed to dislocation dragging, but rather to random-walk lattice diffusion. It is, however, stress-assisted diffusion in which the hydrostatic stress field plays a relevant role. The beneficial effect of crack-tip plastic straining on HAC behaviour might be produced by the delay of hydrogen entry caused by residual compressive stresses and by the enhanced trapping of hydrogen as a consequence of the increase of dislocation density after cyclic plastic straining.     

FATIGUE CRACK TIP PLASTIC STRAIN IN HIGH-STRENGTH ALUMINUM ALLOYS

Abstract— Plastic zone size and shape and the distribution of strain within the plastic zone are determined for the high-strength aluminium alloys 2024-T4, 6061 T6, and 7075-T6 using the technique of selected area electron channeling. Plastic zone size is found to correlate with the work done in creating a unit of new crack surface and the yield stress, rather than with the stress intensity factor and yield stress. Plastic strain distribution is found to be a logarithmic function of distance from the crack tip, in agreement with the mathematical analysis for a moving crack in plane strain.