A mechanistic model for the influence of stress ratio on the LEFM fatigue crack growth behavior of metals and alloys—I. Crack-ductile materials

Concentrating on local behavior of a highly stressed zone ahead of the crack tip, a recent mechanistic approach to analyse LEFM fatigue crack growth behavior in three stages at stress ratio R = 0 is extended here to include the effect of a positive stress ratio. This paper is limited to analysing primarily the stages I and II of “crack-ductile” materials, characterised by a purely “reversed shear” (or ductile “striation”) growth mechanism in stage II. It is shown that in these materials stage I is R-sensitive and stage II is insensitive, and these can, without invoking crack closure arguments, be rationalised alternatively by considering the dominance of a K_max-controlled “Submicroscopic Cleavage” and a ΔK-controlled “ reversed shear ” fracture mechanism, respectively. Assuming Paris type power relations to hold, a predictive model is developed that contains separate growth equations with R-effect for stages I and II and shows the existence of a characteristic “master shear-curve” and a “moving pivot-point” on this curve for a class of materials. Good agreement was found between quantitatively predicted growth curves at selected R-values and a relatively large volume of available experimental data for low strength steels, aluminum alloys and titanium alloys. Besides providing more physical explanations for the observed growth behavior, the model may also be useful as a convenient alternative to crack closure for obtaining fairly accurate and conservative estimates of fatigue life for design applications.     

On the combined influences of Young''s modulus and stress ratio on the LEFM fatigue crack growth process: A new mechanistic approach

A new mechanistic approach (NMA) was used recently to examine the physical aspects of LEFM (long) fatigue crack growth (FCG) process in crack-ductile materials in stages I and II. In this paper, NMA is extended to examine both the physical and analytical aspects of the combined effects of Young's modulus, E and stress ratio, R, in the same stages of the same materials. It is shown that, (i) with submicroscopic cleavage or reversed shear mechanism operating in the pure form, E is the most influential intrinsic “material” property controlling FCG, (ii) E-dependence of da/dN is a natural consequence of near-crack-tip displacement control proposed previously, and (iii) the demonstrated similarity of FCG curves and the existence of characteristic “pivot points” on these curves for a “class of materials” results from E-influence which continues even at a higher R. A simple analytical model based on “strain intensity factor,” K₀, which contains E-influence implicitly and controls da/dN in all materials irrespective of class, is proposed. Model-predicted K₀-based theoretical values of threshold, “Idealised Master Growth Curves (IMGCs)” and mechanism transition point, all agreed excellently with experimental data for at least three classes of materials, i.e. steels, Al-alloys and Ti-alloys at extreme R-values of 0 and ≥ 0.6. The K₀-parameter concept is used here to raise the status of the analysis of the E-effect from a simple “normalisation” to that of direct data “representation”. Using NMA existing empirical relations are given some sound theoretical base. In addition to aiding in a clearer physical understanding of the FCG process, the unique IMGCs developed for different R-values are considered useful in quick, accurate and conservative life estimations, and performing failure analyses usually required in selection and design of materials.     

Modelling and measurement of crack closure and crack growth following overloads and underloads

Ignoring crack growth retardation following overloads can result in overly conservative life predictions in structures subjected to variable amplitude fatigue loading. Crack closure is believed to contribute to the crack growth retardation, although the specific closure mechanism is debatable. The delay period and corresponding crack growth rate transients following overload and overload/underload cycles were systematically measured as a function of load ratio (R) and overload magnitude. These responses are correlated in terms of the local “driving force” for crack growth, i.e. the effective stress intensity factor range (ΔK_eff). Experimental results are compared with the predictions of a Dugdale-type crack closure model and improvements in the model are suggested.     

A dynamic measure of order in structural glasses

Although some patterns of physical behavior are common in the glass transition and in the properties of supercooled liquids and glasses (characteristic viscoelasticity, temperature dependence of viscosity and relaxation times, property evolution through “physical aging”, difficulties in performing equilibrium measurements or simulations, etc.), it is difficult to arrive at a definition of the glass transition which distinguishes it from other phenomena exhibiting similar features. The present paper addresses this problem by defining a dynamical measure of order involving the average “shape” of particle trajectories in supercooled liquids. This dynamic order parameter should provide a measure of “closeness” to the glass transition and some indirect insights into the physical nature of supercooled liquids and glasses. Arguments are given that the proposed dynamic measure of order [“generalized capacity”, C(T)] is related to the temperature-dependent “effective hydrodynamic radius” R_H(T) measured in supercooled liquids and model numerical calculations are given to support this view. Some consequences of the intermittent particle motion at low temperatures for stress relaxation are also discussed.     

The effect of load ratio on the threshold stresses for fatigue crack growth in medium carbon steels

A study of the effect of load ratio, R, on the low crack growth rates and the threshold stress conditions exhibited by five medium carbon steels has been conducted. It was found that decreased values of R retarded crack growth to an increasing degree as a defined threshold for crack growth was approached, such that this threshold showed a marked dependence on K max rather than ΔK. Comparisons of the five materials showed that the threshold increased as yield strength increased for a given R, but this effect could be normalised in terms of a constant value of the maximum plastic zone size at the crack tip.     

The effect of the stress ratio on fatigue crack growth in a 3% NaCl solution

Corrosion fatigue crack growth tests have been carried out at various stress ratios for a low alloy steel SNCM 2 and type 304 stainless steel.

Measurements of the effective stress intensity factor range ratio U were performed to explain the effect of stress ratio R.

The corrosive environment decreased da/dN at R = 0.1, 0.4 and little affected da/dN at R = 0.9 for SNCM 2 and increased da/dN at all R ratios for SUS 304.

It was confirmed that there exists a threshold stress intensity factor ΔK_thCF in 3% NaCl solution for both materials tested.

The corrosive environment decreased ΔK_thCF for all conditions tested except at R = 0.1 and 0.4 for SNCM 2, where ΔK_thCF-values were nearly equal to ΔK_th-values in air. ΔK_thCF/ΔK_th was 0.6 at R = 0.9 for SNCM 2 and 0.8, 0.5 and 0.7 at R = 0.1, 0.7 and 0.9 for SUS 304, respectively.

It was shown that the complicated effect of stress ratios on crack growth for SNCM 2 can be explained using effective stress intensity factor ΔK_eff.     

Assessment of the dynamics of corrosion fatigue crack initiation applying recurrence plots to the analysis of electrochemical noise data

Electrochemical current oscillations generated during the early stages of corrosion fatigue damage (CFD) were analysed applying recurrence plots. “This novel analysis tool allowed us to assess changes in the dynamics of the CFD process, differentiating pure electrochemical process like of pitting corrosion (PC) from the corrosion fatigue crack formation and initial growth”. The dynamics of CFD initiation was characterized by determining changes in the selected recurrence quantification analysis (RQA) parameter: the percentage of determinism (%D). A significant contribution of this work is that it was possible to separate through changes in %D as a function of the number of cycles (N), the electrochemical process of pitting corrosion from the corrosion fatigue crack initiation and growth, which has a random nature and involved low values of %D of around 5%. A subsequent augment of the %D to values from 75% to 95% with the increase of N could be related to the short fatigue crack arrest. The increment of %D indicates that the electrochemical pitting corrosion process was the predominant contribution to the current oscillations.     

A modified effective crack-length formulation in elastic-plastic fracture mechanics

In examining the performance of standard effective crack-length formulations, the authors noted quantitative accuracy up to “high” fractions of limit load under loading conditions for which the elastic T-stress was non-negative, while a pronounced deviation from the corresponding continuum elastic-plastic plane-strain finite-element solutions was seen in shallow-cracked geometries having negative T-stress. This trend can be rationalized by noting that, under modified boundary layer (K_I and T) loading, the maximum plastic zone radius strongly increases as the T-stress decreases from zero (J.R. Rice (1974), J. Mech. Phys. Solids 22, 17–26; S.G. Larsson and A.J. Carlsson (1973), J. Mech. Phys. Solids 21, 263–277; N.P. O'Dowd and C.F. Shih (1991), J. Mech. Phys. Solids 39(8), 989–1015.) Accordingly, we formulate a modified effective crack length to account for the effects of the elastic T-stress.

The new formulation consistently extends the load range for which accurate predictions of compliance, J-integral, and crack-tip constraint are obtained in several plane strain specimen geometries. The magnitude of influence of the T-stress varies with specimen type and relative crack depth. The greatest “improvement” to standard effective crack length approximations occurs in specimens of “moderately” negative T-stress.     

Influence of stress ratio at baseline loading on delayed retardation phenomena of fatigue crack growth in A553 steel and A5083 aluminium alloy

The delayed retardation phenomena of fatigue crack growth following a single application of tensile overload were investigated under the baseline loading with the stress ratio, R = σ_min/σ_max, ranging from −1 to 0.5 for A553 steel and A5083 aluminium alloy. Two different overload cycles were applied; the one is the case that the ratio of peak stress range to baseline stress range, r = Δσ₂/Δσ₁, is equal to two and the other is the case that the ratio of maximum peak stress to maximum baseline stress, σ_2max/σ_1max, is equal to two. The retardation took place stronger in aluminium than in steel. Under the condition of r = 2 the normalized number of cycles, N_D/N_C, (N_D: the number of cycles during retardation, N_C: the number of cycles required for propagation through the overload-affected-zone size) decreased slightly as the R ratio increased from −1 to 0.5, while under the condition of σ_2max/σ_1max = 2 the N_D/N_C-values increased drastically as the R ratio increased from −1 to 0 (or the overload ratio, r, increased from 1.5 to 2) in both the materials. These retardation behaviors were expressed theoretically according to the model proposed by Matsuoka and Tanaka [1, 3] by using four parameters: the overload ratio, r, the exponent in Paris equation, m, the overload-affected-zone size, ω_D, and the distance at the inflection point, ω_B.     

A fracture mechanics analysis of fatigue crack growth in a complex cross section

This paper describes research conducted to determine the fatigue crack growth behavior of corner cracks which occur in a beam with a complex “double-T” shaped cross section. Particular objectives were to determine the changes in crack shape as they grow from one leg of the double-T cross section into the wider portion of the specimen, and to determine whether crack growth could be delayed or arrested in this area. Since it is difficult, if not impossible, to obtain details of crack growth inside metal specimens, it was decided to test transparent polymer models which allow in situ photographs of the crack plane. Fatigue crack growth rates measured at various positions along the crack front were used to compute cyclic stress intensity factors from the baseline fatigue crack growth behavior of the test material. The results of these experiments do indicate that cracks may be delayed once they sever one leg of the double-T cross section, although crack arrest may be compromised if a second crack develops in the opposing leg.     

Fatigue crack initiation from notch root (local-strain damage accumulation process on crack initiation)

Local deformation (i.e. local-strain behavior) at the notch root in a crack initiation process in annealed 0.48% carbon steel was investigated by the real-time fine-grid method. The fatigue crack initiation cycle was controlled by local-strain damage accumulation. For a quantitative expression of cumulative fatigue damage, we propose a new parameter, the “average local-strain accumulation value,” , which is defined by the integration of local-strain histories until crack initiation. The relationship between average local-strain accumulation range, , and crack initiation cycles, N_c, showed a line whose slope was nearly −0.5 on a log-log coordinate graph. This line we term the “local-strain damage accumulation curve.” The mean stress effect in cases of R = −1, 0 on this line ( vs N_c) was very small or negligible. From the results of variable-loading tests, the linear cumulative damage law based on the local-strain value was also confirmed.     

Effects of notch radius and anisotropy on the crack tip plastic zone

Factors which influence the shape and size of the plastic zone in the immediate vicinity of a crack tip in isotropic materials at small loads are investigated. The plastic zone dimensions for the opening mode (Mode I) have been calculated over a range of values for the crack tip radius. An increase in tip radius results in an increase in the plastic zone dimension. In anisotropic materials, the orientation of crack slit and the anisotropic yield constants are other factors that affect the plastic zone size and shape. In this paper, typical curves for the shape and size of plastic zone are given to illustrate the influence of normal or shear anisotropic yield constants. For sheet metals the effects of anisotropy on the plastic zone dimensions can be evaluated in terms of R values. Suggested values of constant b for isotropic materials are given if the “radius” approximation is employed for small applied stresses.     

An equivalent domain integral method for computing crack-tip integral parameters in non-elastic, thermo-mechanical fracture

The crack-tip parameters, such asJ; T^*, ΔT^* etc, which quantify the severity of the stress/strain fields near the crack-tip in elastic-plastic materials subject to thermo-mechanical loading, are often expressed as integrals over a path that is infinitesimally close to the crack-tip (front). The integrand in such integrals involves the stress-working density, stress, strain and displacement fields arbitrarily close to the crack-tip. In a numerical analysis, such data near the crack-tip are not expected to be very accurate. This paper describes simple approaches and attendant computational algorithms, wherein, the “crack-tip integral” parameters may be evaluated through “equivalent domain integrals” (EDI) alone. It is also seen that the present (EDI) approaches form the generic basis for the popular “virtual crack extension” (VCE) methods. Several examples of thermo-mechanical fracture, including: (i) thermal loading of an elastic material, (ii) arbitrary loading/unloading/reloading of an elastic-plastic material, containing a single dominant crack, are presented to illustrate the present approach and its accuracy.     

Classification of environmentally assisted fatigue crack growth behavior

Fatigue crack growth is represented using fracture mechanics parameters, ΔK and K_max. Environmental effects that depend on time and stress affect the fatigue behavior predominantly through K_max parameter. The superimposed effects of environment and stress are seemingly complex. We have developed a methodology for classifying and separating the effects of environment on fatigue crack growth. A “crack growth trajectory map” is constructed from the behavior of ΔK versus K_max for various constant crack growth rate curves. A “pure fatigue” behavior is defined, in terms of environment-free behavior, such as in high vacuum. Deviation from this “pure fatigue” reference of the trajectory map is associated with either monotonic mode of fracture or to the superimposed environmental effects on crack growth. Using such an approach, called “Unified Damage Approach”, we classify the environmental effects in almost all materials into only five types. Each of these types shows the combination of time and stress affecting the crack tip driving force, and thus ΔK and K_max. The trajectory map depicts the changing material resistance due to the changing crack growth mechanisms with increasing crack growth rate, as reflected in terms of the applied stress intensities, ΔK and K_max. Thus the trajectory map provides a useful tool to separate the contributions from pure fatigue and superimposed monotonic modes and the governing crack growth mechanisms as a function of load-ratio, crack growth rate and environment. Understanding and quantification of the governing mechanisms would help in developing a more fundamental and reliable life prediction method.     

Energy-density concept in fracture mechanics

A theory of fracture mechanics is proposed in which attention is focused on the intensity of the energy field in the crack tip region. This energy field possesses a 1/r-type of singularity for both elastic and plastic materials. The strength or amplitude of this field will be referred to as the “energy-density factor”, S. Unlike the stress-intensity factor k in classical fracture mechanics which is only a measure of the local stress amplitude, the energy-density factor is also direction sensitive. The difference between k and S is analogous to the difference between a scalar and vector quantity. In this sense, the critical value S_cr specifies the direction of crack initiation as well as the fracture toughness of the material.     

Effects of corrosive environments on near-threshold fatigue crack growth behavior of T1-6A1-4V

The near-threshold fatigue crack growth behavior of Ti-6A1-4V alloy has been investigated in low O₂ steam (< 1 ppm), high O₂ steam (40ppm), and boiling water with various concentrations of Nad and/or Na₂₂SO₄. At load ratio (R) of 0.5, high O₂ steam increased the crack propagation rates in the threshold region, relative to low O₂ steam. However, at R = 0.8, the near-threshold crack growth rates in low and high O₂ steam were comparable. Values of threshold stress intensity range, ΔK_th, slightly increased with an increase in the concentration of NaCl in the solution. Varying solution pH from 5.0 to 10.0 in a 0.1 g NaCl plus 0.1 g Na₂SO₄ per 100ml H₂O solution had no effect on the rates of near-threshold crack propagation. Increasing the hydrazine level from 30 to 10⁷ ppb in the same salt solution also did not change the resistance to crack growth. Comparing the present results with the previous data on 403 stainless steel, the near-threshold crack propagation rate performance in Ti-6Al-4V alloy is superior to that in 403 steel in both the steam and salt solution environments.     

Mechanical behaviour of a two-phase material from the behaviour of its components: Interface modelling by finite element method

In this paper, the principle of a solution for “thermal” connection between two solids is analyzed. We shows results given by the solution applied to the mechanical behaviour of a γ/γ′ two-phase material and to “artificial” structures obtained from modern techniques for epitaxial deposit. It appears that the use of a true or fictitious thermal loading constitutes a simple “connection” procedure, but is particularly coherent with the mechanics of two-phase crystalline materials with different lattice parameters. It would be interesting to apply the model to real structures, with misfit and interfacial dislocations.     

Sustained Mode I and Mode II fatigue crack growth in flat plates

This study was conducted to contribute to the understanding of fatigue crack growth under mixed mode loading. This was accomplished by developing and analyzing a flat plate specimen capable of maintaining crack growth on a plane oblique to the direction of the applied load. Several specimens were built and exposed to controlled fatigue loading in the laboratory. These specimens were then modeled using finite elements to determine the stress intensity factors (SIF). For the “Mode I/Mode II” specimens developed, the crack was forced to grow in a direction other than perpendicular to the load. The resulting crack front did not remain straight and flat, but stabilized into a curved or warped shape. Based on finite element analyses of these curved specimen cracks, it is concluded that the SIR were predominantly Mode I, with the Mode II and III SIR being negligible.     

Role of plastic work in fatigue crack propagation in metals

The plastic work required for a unit area of fatigue crack propagation U was measured by cementing tiny foil strain gages ahead of propagating fatigue cracks and recording the stress-strain curves as the crack approached. Measurements of U and plastic zone size in aluminum alloys 2024-T4, 2219-T861, 2219 overaged, and A1-6.3 wt% Cu-T4, and a binary Ni-base alloy with 7.2 wt% A1 are herein reported. The results are discussed along with previously reported measurements of U in three steels and 7050 aluminum alloy. When U is compared to the fatigue crack propagation rate at constant ΔK along with strength and modulus, the conclusion is drawn that U is one of the parameters which determines the rate of fatigue crack propagation. The relation of U to microstructure is also discussed. “Homogeneous” plastic deformation in the plastic zone ahead of the crack seems desirable.     

A model for high cycle fatigue

Failure due to fatigue consists of such macroscopic events as crack initiation and propagation. Microscopic events including microcrack nucleation, microcrack growth and coalescence of some of the microcracks are also important in that such crack interactions can be considered to contribute to the development of a critical defect, i.e. a defect which can self propagate and lead to failure. Since crack initiation is important at high cycles, this paper considers a microcrack and computes its growth to advance a high cycle stress-life (S-N_f) formulation for metals and alloys based on crack initiation. In addition to a material's Burgers' vector and grain size, an indirect effort is also made to include the role of its propensity to cracking through a ratio S(σ_I)/S(σ_f), where S(σ_I) and S(σ_f) simply represent steady state crack spacings at stress amplitudes σ_I and σ_f (endurance limit), respectively. Conceptually, a decrease of this ratio suggests an increased tendency for cracking and vice versa. As shown in the text, the model predicts that the crack initiation period varies increasingly with this ratio, and the value of the steady state crack spacing ratio is in and of itself quite sufficient to model the experimental stress-life data in instances where life is controlled by crack initiation period and not the stage II crack propagation. Because of this limitation, extreme care must thus be taken with regard to its application.