Effective Stress Intensity Factor Research Articles

On the Link between Plastic Wake Induced Crack Closure and the Fatigue Threshold

This purpose of this paper is to examine the relationship between crack growth equations based on Elber’s original plastic wake induced crack closure concept and the fatigue threshold as defined by the American Society for Testing and Materials (ASTM) fatigue test standard ASTM E647-15el. It is shown that, for a number of conventionally manufactured metals, the function U(R), where R is the ratio of the minimum to maximum applied remote stress, that is used to relate the stress intensity factor ΔK to the effective stress intensity factor ΔKeff is inversely proportional to the fatigue threshold ΔKth(R). This finding also results in a simple closed form equation that relates the crack opening stress intensity factor Ko(R) to ΔK, Kmax, and the fatigue threshold terms ΔKth(R) and ΔKeff,th. It is also shown that plotting da/dN as function of ΔK/ΔKth(R) would appear to have the potential to help to identify the key fracture mechanics parameters that characterise the effect of test temperature on crack growth. As such, for conventionally manufactured metals, plotting da/dN as function of ΔK/ΔKth(R) would appear to be a useful addition to the tools available to assess the fracture mechanics parameters affecting crack growth.

Fatigue Crack Growth Rates and Crack Tip Opening Loads in CT Specimens Made of SDSS and Manufactured Using WAAM.

Additive manufacturing offers greater flexibility in the design and fabrication of structural components with complex shapes. However, the use of additively manufactured parts for load-bearing structural applications, specifically involving cyclic loading, requires a thorough investigation of material fatigue properties. These properties can be affected by many factors, including residual stresses and crack tip shielding mechanisms, which can be very different from those of conventionally manufactured materials. This research focuses on super duplex stainless steels (SDSSs) fabricated with wire arc additive manufacturing (WAAM) and investigates their fatigue crack growth rates and the net effect of crack tip shielding mechanisms. Using the compliance-based method, we measured crack tip opening loads in compact tension (CT) specimens with cracks propagating longitudinally and transversely to the WAAM deposition direction. It was found that fatigue crack growth rates were very similar in both directions when correlated by the effective stress intensity factor range. However, the differences in crack tip opening loads explain a quite significant influence of the deposition direction on the fatigue life.

Cutting-edge insights: Crack growth rates in 9% Ni steel from room to cryogenic conditions influenced by crack closure and effective stress intensity factors

This study investigates the fatigue crack growth rate of 9% Ni steel at varying stress ratios under both room and cryogenic temperatures, attributing different growth rates to the phenomenon of crack closure. Traditional methods of studying crack closure, including the opening force, compliance ratio, and adjusted compliance ratio techniques, are assessed. Our results are juxtaposed with previous formulae for the opening force technique, leading to the introduction of a new formula that incorporates temperature effects by normalising the stress intensity factor with the yield strength at the respective test temperature. We highlight the adjusted compliance ratio (ACR) technique's limitations at cryogenic temperatures due to challenges in measuring notch open-crack compliance. As a solution, we present an innovative method that determines compliance using the effective elastic modulus. The mechanics behind the ACR technique are elucidated through the lens of the wedging mechanism. Our insights reveal that the ACR technique precisely calculates the effective stress intensity factor range (ΔKeff) under specific conditions. We address existing limitations by introducing a novel method, the triple compliance ratio (TCR) technique. A comparative analysis of effective stress intensity factors from all four techniques reveals the TCR technique's superior ability to correlate crack closure with crack growth. By eliminating the crack closure effect, the TCR technique offers a more accurate prediction of fatigue crack growth rate across diverse load ratios.

Experimental Data-Driven approach for the evaluation of crack tip opening loads under variable amplitude loading

In this paper, we present a new data-driven approach for the evaluation of crack tip opening load (Pop) values under variable amplitude loading. These values are analysed for CT specimens manufactured from common aircraft grade aluminium alloys and subjected to several load sequences, all of which have a relatively high R-ratio content without significant overloads and underloads. Direct experimental measurements and outcomes of numerical simulations indicate that the Pop change abruptly from cycle to cycle. For this type of fatigue loading (including a military transport aircraft load spectrum with more than 400,000 turning points) and materials, it was found that Pop is well described by a linear function of the preceding minimum and maximum values of the applied loading. This finding provides a new simple way to evaluate the effective stress intensity factor range, which is often considered as a fatigue crack driving force. It is also verified via the use of machine learning that the proposed approach is capable of predicting the mean stress (R-ratio) effect on Pop for block loading. Therefore, the developed data-driven approach is a promising alternative to the computational methods, which currently dominate advanced fatigue life assessment procedures.

The Two-Parameter Fracture Criterion Taking into Account Two-Dimensional Deformation Constraints at the Front of a Mixed-Type Crack

In this article, a new fracture criterion for a mixed-type crack (type I + type II) is proposed. It is based on the assumption that the maximum tangential stresses in the prefracture region are equal to the local strength of the material. In this case, the size of the prefracture area and the local strength are determined taking into account nonsingular Тхx and Тzz stresses included in the expansion of the Williams stress function. The use of Тхx and Тzz stresses in the calculations allows one to describe the two-dimensional local constraint of the deformation along the crack front in three-dimensional bodies. In addition to KI and KII, the expression for the effective stress intensity factor (SIF) includes the ratios of the Тхx and Тzz stresses to the yield strength. This makes it possible to take into account the constraint of deformations in the transverse and longitudinal directions of the crack front, respectively. An example of the implementation of the developed criterion in relation to the determination of the fracture load of a tensile-stressed plate with a through inclined crack is given. Dependences of the Тxx and Тzz stresses on the plate thickness are presented for various crack slopes and plate thicknesses. It is shown that an increase in the plate thickness and a decrease in the crack slope lead to a decrease in the fracture load.

Presentation of machine learning methods and multi-objective optimization of fracture indices for asphalt rubber mixtures containing wax-based warm mix additives modified by nano calcium carbonate

Warm mix asphalt (WMA) additives have been proposed to overcome the high viscosity problem of crumb rubber modified asphalt (CRMA) binders. Various additives have been used to improve the low-temperature cracking performance of asphalt mixtures, but no study has been conducted to present the optimum additive content in CRMA binders in different loading modes. Therefore, this research aims to present the optimum content of nano calcium carbonate (NCC) to improve the fracture toughness and energy of asphalt rubber mixtures containing WMA additives (slack wax (SW) and polypropylene wax (PPW)). The semi-circular bend (SCB) fracture test was applied under pure mode I, mixed mode I-II, mixed mode II-I and pure mode II loadings at subzero temperature. Machine learning methods, including multivariate regression (MVR) and artificial neural network (ANN) models of group method of data handling (GMDH) and multilayer perceptron (MLP), were used to provide the prediction models of effective stress intensity factor (Keff) and fracture energy (Gf). Finally, the multi-objective optimization of Keff and Gf was performed to obtain optimum NCC content. The results indicated that in MVR model, the outputs had a small correlation with laboratory values, so that R value of MVR was 0.8406 and 0.8011 for Keff and Gf, respectively. Also, it was revealed in MVR that NCC had the highest impact on Keff and Gf significantly. GMDH model results showed that the relationships between predicted and laboratory values of Keff and Gf are appropriately described with R value of 0.9546 and 0.9229 for Keff and Gf models, respectively. In MLP model, different layer structures of the feed-forward neural network were developed to obtain the most accurate structure. It was indicated that MLP with 4–22–1 and 3–19–1 structures had a higher accuracy for Keff and Gf prediction models with R value of 0.9951 and 0.9978, respectively. Finally, by the use of the best model relationships, the results of the multi-objective optimization indicated that 4.91% and 6.37% NCC were the design optimum contents resulting in a maximum of Keff and Gf simultaneously for SW and PPW-modified CRMA mixtures. Moreover, the optimum NCC contents in loading modes of mode mixity Me of 1, 0.8, 0.4 and 0 were 3.62%, 5.12%, 4.08% and 3.42% for SW-modified CRMA mixtures and 4.35%, 6.51%, 7.19% and 2.84% for PPW-modified CRMA mixtures, respectively.

Fatigue crack propagation behavior in Ti−6Al−4V alloy with surface gradient structure fabricated by high-energy shot peening

A gradient structure was successfully fabricated on the surface of Ti−6Al−4V alloy by high-energy shot peening (HESP), and its effect on fatigue crack propagation was investigated. Optical microscope, scanning electron microscope, transmission electron microscope and X-ray diffractometer were used to characterize the microstructure and residual stress evolution during HESP. The results show that a gradient nanostructure with residual compressive stress layer of 220 μm in depth is formed. The generation of gradient nanostructures improves the strength−ductility combination of the alloy. Maximum residual compressive stress is generated on the subsurface, which gradually increases with the increase of shot peening time. HESP treatment effectively reduces the crack propagation rate and increases the fatigue crack propagation life. The residual compressive stress reduces the effective stress intensity factor range at crack tip, thereby generating the crack closure effect and delaying the crack propagation. At the same time, the synergy effects of increase in grain boundaries, decrease in effective slip length and the plastic zone at the crack tip caused by the refinement of the surface grains can increase the crack propagation resistance as well.

Prediction of the durability of a plate with a through crack taking into account biaxial constraints of deformations along the front of a normal rupture crack

A methodology for evaluating the durability of plate elements of structures taking into account biaxial constraints of deformations along the front of a normal rupture crack (Mode I crack) is presented. The absence of the available literature data in which the prediction of the crack growth is carried out using Txx- and Tzz-stresses which are non-singular terms in the Williams expansion for stresses at the crack tip is noted. The calculation of the fatigue crack growth rate is based on the Paris equation in which the range of the effective SIF is used instead of the range of the usual stress intensity factor (SIF). In this case, the expression for the effective SIF includes Txx- and Tzz-stresses in addition to the usual SIF. This approach provides taking into account, for example, the thickness of the plate for predicting the durability, which is impossible when only the SIF and Txx-stresses are used. The formula for the effective SIF is derived proceeding from the assumption that tangential stresses in the pre-fracture zone are equal to the local strength of the material. In this case, the size of the pre-fracture zone and the local strength of the material are determined taking into account Txx- and Tzz-stresses. The numerical simulation is based on the proprietary finite-element program which allows calculating Txx- and Tzz-stresses at the front of a through crack in a plate subjected to cyclic uniaxial and biaxial tension. It is shown that nonsingular Txx-stresses primarily describe the effect of biaxial loading on the survivability, whereas Tzz-stresses describe the effect of the plate thickness on the survivability. It is shown that with increasing thic kness of the plate the value of the effective SIF increases due to the increased constraint along the crack front, thus increasing the crack growth rate and decreasing the survivability. With an increase in the stress ratio R, under the condition of a constant stress range, the maximum effective SIF reaches the critical value equal to the fracture toughness much faster thus reducing the durability. It is shown that for uniaxial cyclic tension, the durability predicted by the proposed methodology is higher than that in the classical approach, when the conventional SIF is used in the Paris equation. For biaxial cyclic tension of a plate, an increase in stresses directed parallel to the crack banks leads to an increase of crack front constraints and therefore to a decrease in the durability compared to the classical approach. In other words, the classical theory does not always provide a conservative estimate of the durability, which indicates the expediency of using the developed method for calculating the durability taking into account biaxial constraints of deformations along the crack front.

Enhanced strength-ductility synergy of laser additive manufactured stainless steel/Ni-based superalloy dissimilar materials characterized by bionic mechanical interlocking structures

The unique mechanism of toughness enhancement found in natural materials has served as inspiration for the development of a new generation of dissimilar metal materials (DMMs). Here we present a bionic mechanical interlocking structure strategy that stainless steel 316L and Inconel 625 DMMs with an enhanced strength-conductivity trade-off processed by laser additive manufacturing (AM), possessing an ultimate tensile strength of 675.64 MPa, and a uniform elongation of 33.6%. The incompatible deformation between SS316L (plastic deformation) and IN625 (non-deformation) contributes to overcoming the strength-ductility trade-off. The integration of alternating soft and hard non-uniform structures plays a pivotal role in effectively absorbing and dissipating the energy introduced by external loads during crack propagation. This geometric arrangement and multilevel structure at the interface notably reduces the effective stress intensity factor along the crack path, leading to added resistance against loading. The multi-strengthening mechanism coupled with the heterogeneous deformability gives the material excellent strength-ductility synergy, especially the bionic mechanical interlocking structure extending the crack expansion path, which helps to avoid premature failure of weak interface layers. These findings will provide new insights into the controlled design of high-performance DMMs, which highlights the potential to inherit merits from constitutive materials for better strength-ductility combinations.

Study of Effective Stress Intensity Factor through the CJP Model Using Full-Field Experimental Data.

In this work, the Christopher-James-Patterson crack tip field model is used to infer and assess the effective stress intensity factor ranges measured from thermoelastic and digital image correlation data. The effective stress intensity factor range obtained via the Christopher-James-Patterson model, which provides an effective rationalization of fatigue crack growth rates, is separated into two components representing the elastic and retardation components to assess shielding phenomena on growing fatigue cracks. For this analysis, fatigue crack growth tests were performed on Compact-Tension specimens manufactured in pure grade 2 titanium for different stress ratio levels, and digital image correlation and thermoelastic measurements were made for different crack lengths. A good agreement (~2% average deviation) was found between the results obtained via thermoelastic stress analysis and digital image correlation indicating the validity of the Christopher-James-Patterson model to investigate phenomena in fracture mechanics where plasticity plays an important role. The results show the importance of considering crack-shielding effects using the Christopher-James-Patterson model beyond considering an exclusive crack closure influence.

The Improved Irwin’s Model for Crack Problems in Power-Law Hardening Materials

In this paper, an improved Irwin’s model for power-law hardening materials is established through the static equivalence between the linear elastic stress solution and the stress redistribution in the plasticity-affected zone (PAZ) under small-scale yielding (SSY) conditions. HRR solution is adopted to describe the stress field within the crack-tip plastic zone (CTPZ). Analytical expressions of the effective stress intensity factor (SIF) and the size of PAZ are derived to illustrate the effect of CTPZ on the fracture behavior for power-law hardening materials. Influences of external load and hardening exponent on fracture behavior are explored. The proposed model can assess the toughening effect of CTPZ accurately using the SIF ratio, which is always smaller than 1. This model can be applied to various materials obeying power-law hardening constitutive equations. It will reduce to the extended Irwin’s model for elastic-perfectly plastic materials in the limiting case.

Mixed mode crack growth behaviour considering plasticity-induced and roughness-induced closure

A theoretical analytical solution of the Christopher-James-Patterson (CJP) model in mixed mode is presented in this paper, where the influence of roughness-induced crack closure (RICC) on fatigue crack growth is considered and a new effective stress intensity factor range (ΔKP-R) is proposed as the driving force for crack propagation. The solution showed that both the maximum tangential stress criterion (MTS) and the minimum strain energy density criterion (S) can reasonably well predict the crack deflection angle. Experimental testing on EA4T axle steel samples showed that both the 8 % compliance offset criterion in the conventional ASTM procedure and the parabolic-line method were suitable for the determination of the crack-opening force. The solution includes plasticity-induced crack closure (PICC) and RICC effects, therefore the crack propagation behaviour can be described in a more effective way. Compared with the traditional mixed model crack growth models with higher fitting correlation coefficient, the fitting effect of the CJP model has been improved by 12 % in this solution, and the residual stress in the pre-crack stage increased the crack-closure level. Due to the influence of residual stress, the rate of roughness value (RA) showed a fluctuation of 4.5 %, which indirectly affected the fatigue crack growth rate (FCGR) of the test.

Integrating SMA and CFRP for fatigue strengthening of edge-cracked steel plates

Inclined fatigue cracks of U-ribs and diaphragms in orthotropic steel-box girders are challenging to be repaired using prestressing techniques. Integrating shape memory alloys (SMAs) and carbon fibre reinforced polymers (CFRPs) can achieve a hybrid self-prestressing strengthening approach for such fatigue cracks. To examine the strengthening effect of this approach on inclined cracks, 39 specimens were prepared in four groups and tested under static and fatigue tensioning in this study. Both static and fatigue test results showed that SMA/CFRP composite strengthening effectively retard steel cracking. Compared to control specimens, SMA/CFRP composite strengthening enhanced the notch tip yield load by 128% and 111% for specimens with a notch inclination of 0° and 30°, and the corresponding fatigue life by 7.62 and 8.69 times, respectively. Besides, the strengthening effect of combining SMA and CFRP (SMA/CFRP composite) on cracked steel plates was greater than that of either SMA or CFRP alone, particularly for the steel plates with an inclined crack. Finally, the calculation of mixed-mode I/II effective stress intensity factors confirmed the superior strengthening effect of SMA/CFRP composite on steel plates with an inclined crack. The findings of this study can provide a reference for the reinforcement of orthotropic steel-box girders.

A modified formula for cyclic compression crack growth model considering compressive load effect

Fatigue crack growth under cyclic compression has been confirmed by experiments and widely exists in engineering applications, with the expression of stress intensity factor often serving as milestones. A typical difficulty in calculating the stress intensity factor under cyclic compression is the effect of compressive load on fatigue crack growth. Compressive load effect, resulting in the change of plastic zone and residual stress near crack tip, is a long-term challenge for predicting fatigue crack growth life. We modify the expression of effective stress intensity factor to predict fatigue crack growth life based on the distribution of residual stress, and analyze the effect of load characteristics. It considers the characteristic of material and load, and calculates the driving force of crack growth by integrating the residual stress in plastic zone. In the study of fatigue crack growth experiments under cyclic compression, the modified formula is proved to improve, by comparative analysis, the predictive ability on the whole range of experiment data. Consequently, the modified formula is conducive to calculate the fatigue crack growth.

Mixed-mode fracture model to quantify local toughness in nacre-like alumina

With the progress of ceramic engineering, tough technical ceramics such as nacre-like alumina now exhibit complex fracture patterns with high degrees of deflection, branching and bridging. If these mechanisms highly contribute to improving crack resistance, the crack geometries go beyond the hypotheses assumed in standard techniques to quantify toughness, making it difficult to understand the mechanisms responsible for crack propagation. We report a robust local solution based on a combination of analytical formulas and finite element analysis to accurately estimate the effective stress intensity factors at the tips of multiple deflected cracks. The influence of sample size on local R-curves is analysed and shown to be less important relative to standard methods. We use this method to evaluate the direct influence of zirconia nanoparticles in the nacre-like structure and highlight the role of these findings in the understanding of mechanical response and future optimisation of highly textured tough ceramics.

Effect of thermal degradation on mechanical properties of cast Al-Si alloys for cylinder heads

The effect of thermal degradation on mechanical properties of cast Al-Si alloys for cylinder heads is investigated. Thermal degradation has a significant deteriorating effect on the tensile strength caused by the coarsening and coalescence of the θ phase (Al2Cu) due to precipitation reaction. The dominant factor for degradation at 250 ℃ is exposure temperature, while exposure time has a greater effect at 200 ℃. Thermal degradation increases the long crack growth rate significantly due to the increase in plasticity scale ahead of the crack tip. The effective stress intensity factor range (ΔKeff) is modified for thermally exposed alloys by adding the plasticity zone size to the crack length and considering the reduction in crack resistance from the precipitates.

Numerical modelling of fatigue crack closure and its implication on crack front curvature using ΔCTODp

One of the widely used approaches to characterize fatigue crack propagation is the use of the effective stress intensity factor range ΔKeff, which relies on determination of crack closure value Kcl. The used models of crack closure are most frequently-two-dimensional, however, for real cracks, the 3D effects should be taken into account. The paper presents 3D finite element analyses of the influences of crack front shape, inserted cycles (loading cycles between node releases) and crack closure on the crack driving force in terms of ΔKeff and ΔCTODp (plastic part of crack tip opening displacement range). Numerically obtained crack closure depended on the simulation strategy. In the case of inserted cycles, crack closure disappeared in the internal part of the specimen and remained only near the free edges. The use of ΔCTODp had the advantage of a well-defined parameter in situations where ΔKeff was problematic, namely at the corner points, which did not allow finding of equalized crack driving force along the whole crack front using ΔKeff. Equalized crack driving force in terms of ΔCTODp was found for crack front curvature with the edge angle 15.4° in simulation with crack closure, which was in good agreement with the experimentally measured value of 16°. Loops produced by loading and unloading branches of the force vs CTOD diagrams helped to describe the crack closure process and magnitude. Actual values of CTOD did not agree with the classical idea of 2D solutions under plane strain and plane stress. CTOD was larger in the internal part of the specimen than at the free edges, even in simulations with no crack face contact.

Mixed-mode dynamic fracture parameters for soda-lime glass

Dynamic mixed-mode fracture of soda-lime glass (SLG) was experimentally investigated using a full-field optical method - Digital Gradient Sensing (DGS) - in conjunction with ultrahigh-speed photography. Single edge-notched specimens were subjected to reverse impact loading using a modified Hopkinson pressure bar. The specimens were eccentrically loaded at different offset distances relative to the initial notch to achieve a wide range of mode-mixities, from mode-I to nearly mode-II condition, at initiation. Two time-resolved orthogonal angular deflection fields of light rays proportional to the respective full-field stress gradients were optically measured during experiments on different geometries. Mode-I and -II stress intensity factor histories spanning pre- and post-initiation behaviors were evaluated via over-deterministic least-squares error minimization of optically measured full-field data. By considering the critical stress intensity factors at different crack initiation modes, a fracture envelope for SLG encompassing various mode-mixities was developed for the first time for SLG and compared with predictions from prevailing approaches. An empirical fit of data seems to capture the overall trend whereas the prevailing methods do not. Measured crack kink angles are compared with the popular MTS and SED fracture criteria; both predict the kink angles reasonably well. Unlike brittle polymers, the critical effective stress intensity factors for SLG are independent of mode-mixity over a large range of values but show a decreasing trend as mode-II conditions become dominant and an extrapolated critical mode-II stress intensity factor of 0.37 MPa√m, approx. one-half of its mode-I counterpart, is seen.

Novel Characterizations of Effective SIFs and Fatigue Crack Propagation Rate of Welded Rail Steel Using DIC

The fatigue crack growth-rate test of rail head, waist, and bottom material for U71Mn welded rail was carried out. Digital image correlation (DIC) was used to capture the full-field displacement. The crack-tip position was accurately obtained based on the full-field displacement data, and an accurate crack-tip opening displacement (CTOD) measurement point was found. The CTOD values of the welded rail head under overloaded and unloaded condition were extracted, and the area size of elastic CTOD and plastic CTOD was obtained. According to COD data under different experimental conditions, the corresponding crack opening force was extracted, the crack opening function introduced based on the Elber model, and a calculation method of effective stress-intensity factors (SIFs) considering the plasticity-induced crack closure proposed. The results in this paper provide some references for accurately assessing the fatigue life of welded rail.

Prediction of surface crack growth life for AA7075-T6 under nonproportional loading

The growth direction and growth rate of fatigue cracks initiated from the surface of an aluminum alloy 7075-T6 plate were investigated under six sets of nonproportional loadings produced by combined axial-torsion loading. A digital image correlation method was simultaneously applied to measure the crack opening displacements around the crack tip during the fatigue tests. The effective stress intensity factor ranges were determined on the basis of the measured crack opening displacements and applied to arrange the crack growth rate. Results showed that fatigue crack tended to propagate orthogonally to the direction in which the resultant normal stress in the nonproportional loading cycle was maximized, suggesting that the maximum resultant normal stress dominates crack growth under nonproportional loadings. When the effective stress intensity factor range based on the maximum resultant normal stress was used to arrange the crack growth rates, all data collapsed into a single narrow scatter band. The growth lifetime until the surface crack penetrates the plate could be estimated within a relative error of about ± 20 % based on a modified Paris-type equation through numerical integration.