Crack Closure Model Research Articles

Damage Tolerance and Reliability Assessment under Random Markovian Loads

Accounting for load uncertainties plays an important role in the design of safe structural components of aircrafts under damage tolerance requirements. The purpose of this paper is to develop a reliability assessment technique for cracked structures submitted to non-stationary random fatigue loads modeled by first-order Markov chains with discrete state space and identified from in-flight measurements. The strategy based on a multi-level version of the cross-entropy method consists in progressively updating the tran- sition probability matrix in order to generate load sequences of increasing severity which are likely to cause failure. The proposed method is applied to a cracked M(T) specimen under the defined random fatigue loads. Load cycle interactions and retardation effects are accounted for by means of the PREFFAS crack closure model. The efficiency of the proposed approach in terms of computational cost is clearly observed for rare failure events in comparison with direct Monte Carlo simulations. In addition to the failure probability estimate, the multi-level cross-entropy method provides the analyst with information on the most probable load sequences at failure.

A Numerical Strip-Yield Model for the Creep Crack Incubation in Steels

Abstract A numerical strip-yield model was developed to simulate creep crack incubation in heat-resistant steels. The model is based on a formulation proposed by Newman (Newman, J. C., Jr., “A Crack-Closure Model for Predicting Fatigue Crack Growth under Aircraft Spectrum Loading,” Methods and Models for Predicting Fatigue Crack Growth under Random Loading, ASTM STP 748, J. B. Chang and C. M. Hudson, Eds., ASTM International, West Conshohocken, PA, 1981, pp. 53–84) for fatigue crack growth under variable amplitude loading. The time evolution of the plastic deformation ahead of a crack loaded in tension is modeled using the Norton law for secondary creep stage, and the primary and tertiary creep stages are neglected. The model assumes a pre-existing crack in a specimen and models the behavior of the material prior to the beginning of crack propagation due to creep loading. The evolution with time of the crack-tip plastic zone, crack-tip opening displacement, and yield strength in the plastic zone are computed at constant temperature for center crack panels. Comparison with two previous strip-yield models and experimental data is performed, and good correlation is obtained for several Cr-Mo-V steels. This approach to modeling creep crack incubation has the potential to be applied to other types of cracked specimens under constant or variable amplitude loading.

Fatigue-Life Prediction Method Based on Small-Crack Theory in an Engine Material

Plasticity effects and crack-closure modeling of small fatigue cracks were used on a Ti-6Al-4V alloy to calculate fatigue lives under various constant-amplitude loading conditions (negative to positive stress ratios, R) on notched and un-notched specimens. Fatigue test data came from a high-cycle-fatigue study by the U.S. Air Force and a metallic materials properties handbook. A crack-closure model with a cyclic-plastic-zone-corrected effective stress-intensity factor range and equivalent-initial-flaw-sizes (EIFS) were used to calculate fatigue lives using only crack-growth-rate data. For un-notched specimens, EIFS values were 25-μm; while for notched specimens, the EIFS values ranged from 6 to 12 μm for positive stress ratios and 25-μm for R = −1 loading. Calculated fatigue lives under a wide-range of constant-amplitude loading conditions agreed fairly well with the test data from low- to high-cycle fatigue conditions.

A short summary on finite element modelling of fatigue crack closure

This paper presents a short summary pertaining to the finite element modelling of fatigue crack closure. Several key issues related to finite element modelling of fatigue crack closure are highlighted: element type, mesh refinement, stabilization of crack closure, crack-tip node release scheme, constitutive model, specimen geometry, stress-states (i.e., plane stress, plane strain), crack closure monitoring. Reviews are presented for both straight and deflected cracks.

Dislocation-based model of plasticity and roughness-induced crack closure

Along with external forces and the macro-geometry of cracked bodies, the local stress intensity factors Δk and kmax at fatigue crack fronts are also determined by internal stress fields and the crack front microgeometry (extrinsic shielding). This means that a description of the crack-tip stress field by two external (remote) parameters ΔK (Kmax) and ΔT (Tmax) is not sufficient. The paper presents a discrete dislocation model of contact shielding effects in the case of small-scale yielding under plane-strain conditions. The model is physically transparent and, unlike continuum-based models for plane stress, it enables us to directly assess the magnitude of both plasticity and roughness-induced components of crack closure. Moreover, it reflects an influence of microstructure on the roughness-induced term. The closure components can be simply extracted from experimentally measured values of the remote ΔK using standard data on mechanical properties and microstructure. Thus, the effective threshold ΔKeff,th can be obtained as nearly independent of microstructure coarseness and applied cyclic ratio as shown for several important engineering materials.

Fracture toughness and growth of short and long fatigue cracks in ductile cast iron EN‐GJS‐400‐18‐LT

ABSTRACTHeavy components of ductile cast iron frequently exhibit metallurgical defects that behave like cracks under cyclic loading. Thus, in order to decide whether a given defect is permissible, it is important to establish the fatigue crack growth properties of the material. In this paper, results from a comprehensive study of ductile cast iron EN‐GJS‐400‐18‐LT have been reported. Growth rates of fatigue cracks ranging from a few tenths of a millimetre (‘short’ cracks) to several millimetres (‘long’ cracks) have been measured for load ratios R=−1, R= 0 and R= 0.5 using a highly sensitive potential‐drop technique. Short cracks were observed to grow faster than long cracks. The threshold stress intensity range, ΔKth, as a function of the load ratio was fitted to a simple crack closure model. Fatigue crack growth data were compared with data from other laboratories. Single plain fatigue tests at R=−1 and R= 0 were also carried out. Fracture toughness was measured at temperatures ranging from −40 °C to room temperature.

Random load sequences and stochastic crack growth based on measured load data

This paper addresses the modeling of random fatigue load sequences based on real measured loads, which represents a key step in stochastic damage tolerance. Discrete-time Markov chains and hidden Markov chains are investigated to model sequences of max–min stresses. The parameters of these models are inferred from in-flight measured loads of a given fleet of fighter aircrafts. The accuracy of the two models is assessed based on the statistical properties of a parameter representative of the crack growth under variable amplitude loads, which accounts for load interactions and retardation/acceleration effects. This parameter obtained with the PREFFAS crack closure model represents the maximum stress of a constant amplitude load sequence supposedly equivalent in terms of crack extension to the variable amplitude sequence. The statistical properties of this parameter corresponding to variable amplitude load sequences generated with these models are compared to those of sequences randomly sampled from the measured loads. The Hidden Markov Model appears to be the most appropriate and accurate model. It is then used in a stochastic analysis of a crack initiating at a hole of a skin-stringer panel in a fighter aircraft. This illustrative example explores the effects of the scatter in both loads and material properties (here through the crack growth rate) on the stochastic crack growth.

Application of a strip-yield model to predict crack growth under variable-amplitude and spectrum loading – Part 2: Middle-crack-tension specimens

In previous work, fatigue-crack-growth tests were conducted on middle-crack tension, M( T), and compact, C( T), specimens made from the same D16Cz (clad) aluminum alloy sheet. These tests were conducted over a wide range of stress ratios ( R = P min/ P max = −0.5 to 0.75) to generate crack-growth-rate data from threshold to near fracture. These tests were used to generate the effective stress-intensity factor range (Δ K eff ) against rate curve using a crack-closure model. The analyses collapsed the rate data from both specimen types into a fairly narrow band over many orders of magnitude in rates using proper constraint factors. Constraint factors were established from single-spike overload and the constant-amplitude tests. Herein, the life-prediction code, FASTRAN, which is based on the strip-yield model concept, was used to calculate the crack-length-against-cycles under constant-amplitude (CA) loading and the single-spike overload (OL) tests; and to predict crack growth under variable-amplitude (VA) loading and simulated aircraft loading spectrum tests on the M( T) specimens. The calculated crack-growth lives under CA and an OL tests were generally within ±20% of the test results, but slower crack growth under the double-shear fatigue mode, rather than single shear, may be the reason for some of the larger differences. The predicted crack-growth lives for the VA and Mini-Falstaff spectrum tests were also short by 25–45%. A modified model with some assumed notch constraint effects matched the spectrum tests quite well. Issues on the crack-starter-notch effects under spectrum loading are discussed, and recommendations are suggested on avoiding these notch effects.

Experimental investigation and fatigue life prediction for 7475-T7351 aluminum alloy with and without shot peening-induced residual stresses

The effect of shot peening on small crack growth and on the fatigue life of 7475-T7351 aluminum alloy has been investigated. The experimental results show that cracks initiate at second phase particles in the alloy and that fatigue crack growth rates are greatly reduced after shot peening treatment. Stress intensity factors for small cracks subjected both to external loads and to shot peening-induced residual stresses have been determined using weight function methods. Small crack growth rates and fatigue lives of naturally occurring small cracks in shot peened and unpeened specimens were calculated using small crack theory and a crack closure model. The predicted results agree well with the experimental data. The study demonstrates that fatigue life extension by shot peening can be attributed to beneficial compressive residual stresses in the surface layer, that the principle of superposition is applicable to fracture mechanics, and that the effect of residual stress on crack growth can be quantitatively analyzed using weight function methods. It is shown that the total fatigue lives for materials/structures containing residual stresses can be predicted by the crack closure-based small crack theory, provided that residual stresses are properly taken into account.

Crack-surface displacements for cracks emanating from a circular hole under various loading conditions

The purpose of this paper is to calculate and develop equations for crack–surface displacements for two-symmetric cracks emanating from a circular hole in an infinite plate for use in strip-yield crack-closure models. In particular, the displacements were determined under two loading conditions: (1) remote applied stress and (2) uniform stress applied to a segment of the crack surface (partially loaded crack). The displacements were calculated by an integral-equation method based on accurate stress–intensity factor equations for concentrated forces applied to the crack surfaces and those for remote applied stress or for a partially loaded crack surface. A boundary-element code was also used to calculate crack–surface displacements for some selected cases. Comparisons made with crack–surface displacement equations previously developed for the same crack configuration and loading showed significant differences near the location where the crack intersected the hole surface. However, the previous equations were fairly accurate near the crack-tip location. Herein an improved crack–surface displacement equation was developed for the case of remote applied stress. For the partially loaded crack case, only numerical comparisons were made between the previous equations and numerical integration. A rapid algorithm, based on the integral-equation method, was developed to calculate these displacements. Because cracks emanating from a hole are quite common in the aerospace industry, accurate displacement solutions are crucial for improving life-prediction methods based on the strip-yield crack-closure models.

Application of digital image correlation to the investigation of crack closure following overloads

This paper describes a set of fatigue experiments carried out on 6082 Aluminium CCT specimens. Crack propagation rates and crack opening loads were measured before and after a single overload in an otherwise constant amplitude loading sequence. Crack retardation was observed after the overload, and this was more significant in thin (3 mm) specimens than in thick (25 mm) specimens of otherwise similar geometry. Digital image correlation and conventional strain gauges were used to monitor crack closure, and it was found that surface closure levels were similar in the two thicknesses of specimen. However, the thick specimens showed lower closure levels as detected by the strain gauges. This observation is consistent with the lower levels of crack retardation observed. A plane stress strip-yield model of crack closure was employed to simulate the experimental loading cycle and this showed qualitatively similar behaviour to that observed on the surface of the specimens.

Fatigue-life calculations on pristine and corroded open-hole specimens using small-crack theory

This paper uses a plasticity-induced crack-closure model and small-crack theory to calculate the fatigue lives of 2024-T3 aluminum alloy sheet specimens with open holes subjected to cantilever ( R = −1) bending loads. The bending specimens had three drilled holes that were either pristine or exposed to outside weather conditions at various locations for 3–12 months. The exposed specimens developed various levels of corrosion pits in and around the holes. These pre-corroded specimens were then returned to the laboratory and fatigue tested under laboratory-air conditions with cantilever bending loads. The present paper uses fatigue-crack growth with an equivalent-initial-flaw-size (EIFS) to fit the fatigue behavior of the exposed specimens. Improved stress-intensity factor equations for a corner crack at the edge of an open hole under remote bending were used to make the fatigue-life calculations. The EIFS values agreed well with the median corrosion-pit depths measured on the exposed specimens with moderate corrosion. For one case of severe corrosion, the EIFS value was much larger than the measured pit depth.

A probabilistic-based airframe integrity management model

This paper proposes a lognormal distribution model to relate crack-length distribution to fatigue damage accumulated in aging airframes. The fatigue damage is expressed as fatigue life expended (FLE) and is calculated using the strain-life method and Miner's rule. A two-stage Bayesian updating procedure is constructed to determine the unknown parameters in the proposed semi-empirical model of crack length versus FLE. At the first stage of the Bayesian updating, the crack closure model is used to simulate the crack growth based upon generic but uncertain physical properties. The simulated crack-growth results are then used as data to update the uninformative prior distributions of the unknown parameters of the proposed semi-empirical model. At the second stage of the Bayesian updating, the crack-length data collected from field inspections are used as evidence to further update the posteriors from the first stage of the Bayesian updating. Two approaches are proposed to build the crack-length distribution for the fleet based on individual posterior crack distribution of each aircraft. The proposed distribution model of the crack length can be used to analyze the reliability of aging airframes by predicting, for instance, the probability that a crack will reach an unacceptable length after additional flight hours.

A crack closure model of fatigue crack growth in plates of finite thickness under small-scale yielding conditions

Crack growth rates are significantly affected by the thickness of the specimen when all other parameters are kept constant. A quantitative estimation of the thickness effect is thus necessary to make predictions of crack growth rates more accurate and reliable. For this purpose a theoretical model was developed based on the strip-yield assumption and first-order plate theory. No empirical or fitting parameters were used in this work unlike some previous studies. The theoretical values obtained for the normalised load ratio parameter, U, were employed to describe experimental data, obtained under small-scale yielding conditions, at various load ratios and plate thicknesses. Such a representation considerably narrowed the scatter in the crack growth rates versus the effective stress intensity factor range, Δ K eff, demonstrating the potential of the theoretical model.

Damage tolerance based design optimisation of a fuel flow vent hole in an aircraft structure

This paper investigates the design optimisation of a fuel flow vent hole (FFVH) located in the wing pivot fitting (WPF) of an F-111 aircraft assuming a damage tolerance design philosophy. The design of the vent hole shape is undertaken considering the basic durability based design objectives of stress, residual (fracture) strength, and fatigue life. Initially, a stress based optimised shape is determined. Damage tolerance based design optimisation is then undertaken to determine the shape of the cutout so as to maximise its residual strength and fatigue life. For stress optimisation, the problem is analysed using the gradient-less biological algorithm and the gradient-based nonlinear programming methods. The optimum designs predicted by the two fundamentally different optimisation algorithms agree well. The optimum shapes of the vent hole are subsequently determined considering residual strength and fatigue life as the distinct design objectives in the presence of numerous 3D cracks located along the vent hole boundary. A number of crack cases are considered to investigate how the crack size affects the optimal shapes. A semi-analytical method is employed for computation of the stress intensity factors (SIF), and an analytical crack closure model is subsequently used to evaluate the fatigue life. The 3D biological algorithm is used for designing the cutout profiles that optimise residual strength and fatigue life of the component. An improved residual strength/fatigue life (depending on the optimisation objective) is achieved for the optimal designs. The variability in SIF/fatigue life around the cutout boundary is reduced, thereby making the shape more evenly fracture/fatigue critical. The vent hole shapes optimised for stress, residual strength, and fatigue life are different from each other for a given nature and size of the flaws. This emphasises the need to consider residual strength and/or fatigue life as the explicit design objective. The durability based optimal vent hole shapes depend on the initial and final crack sizes. It is also shown that a damage tolerance optimisation additionally produces a reduced weight WPF component, which is highly desirable for aerospace industries. The design space near the ‘optimal’ region is found to be flat. This allows us to achieve a considerable enhancement in fatigue performance without precisely identifying the local/global optimum solution, and also enables us to select a reduced weight ‘near optimal’ design rather than the precise optimal shape.

Finite element and analytical modelling of crack closure due to repeated overloads

In this paper, investigations of crack closure due to repeated overloads are presented using finite element (FE) and simplified analytical modelling approaches. In particular, attempts are made to study the effects of overload spacing on closure levels and the underlying physical mechanisms involved. Overload closure behaviour is functionally similar for both FE and analytical approaches used and is seen to be in reasonable accord with relevant available experimental observations. In the first instance, it is noted that for double overload interactions, crack closure influence of an initial overload upon the unloading conditions of a second overload can explain much of the observed enhancement in crack growth retardation; however, for successive (repeated) overloads in plane strain, a critical influence of in-plane constraint arises to attenuate closure interactions.

An Evaluation of Fatigue Crack Growth in a Reactor Steel in Air and Water Environments Considering Closure Effects

The paper contains results of an experimental programme aimed at an evaluation of fatigue crack growth rate and threshold conditions in a reactor pressure vessel steel. Though the main target of the work was to gain a data basis for possible future needs of defect and risk assessment, an emphasis was put on an evaluation of crack growth mechanisms, too. It was shown that despite some recent works infirming crack closure phenomenon itself or methods of its evaluation, crack closure explained near-threshold fatigue crack behaviour in the specific case of the reactor steel in air conditions and was in a direct consistency with results of fractographical analyses. A fairly recent model of partial crack closure was very suitable for an explanation of an unexpected fatigue crack growth behaviour in water environment, when fatigue crack growth rates were rather irregular and significantly lower that in air.

Surface fatigue crack growth suppression in cylindrical rods by artificial infiltration

Infiltrating foreign materials into a fatigue crack has been attempted in a number of works to suppress the progress of the crack. Previous works have been limited to through-the-thickness cracks in standard laboratory specimens. By forcing in from one surface and letting out from the other, it is relatively easier to transport foreign materials into a through-thickness crack. Part-through surface cracks are more common in practice. It will be more difficult to bring about a good infiltrant penetration to ensure effective crack growth suppression. In the current work, a special fixture has been developed to induce polymeric resin into a surface fatigue crack in a cylindrical rod. A good infiltrant penetration and crack growth retardation of more than one order of magnitude have been achieved. The relative merits of using the conventional crack closure concept and the one based on Paris et al.’s partial crack closure model to predict the resulting retardation have been investigated.

Prediction of fatigue crack growth rates using crack closure finite element analysis

A three-dimensional finite element fatigue crack closure model of a corner crack and of a through thickness crack has been developed to evaluate the range of effective stress intensity factor from the distribution of the range of stress ahead of the crack tip. The corresponding fatigue crack growth rate was evaluated from a Paris law fit to experimental data from high stress ratio tests. The point of origin for the range of stress distribution was adjusted in accordance with Irwin’s plastic zone correction. Encouraging comparisons of finite element predictions of fatigue crack growth rate incorporating closure effects with experimental measurements were obtained.

Comparison of a New Equation Describing Fatigue Crack Growth Curves with the Nasgro Equation

The so-called NASGRO equation allows a very good description of fatigue crack growth curves, i.e. the dependence of crack length increase per one fatigue cycle da/dN on the stress intensity factor range K. In 1999 the first author of this paper published quite similar equation, which written for given loading cycle asymmetry with positive stress ratio R contains the same parameters having similar meaning as in the NASGRO equation. In most cases studied by the authors this equation leads to a better fit than the NASGRO equation, above all when the experimental curve contains relatively long Paris straight line and/or relatively sharp bend from this line to the threshold stress intensity factor range. The generalization of the NASGRO equation for various values of stress ratio R was made in a quite complicated way. The new equation was generalized using the Walker model based on the relation K(R) K(0)(1 R)m which is valid also for threshold value Kth but not for critical stress intensity Kc being a constant independent of R. Then the shift of the growth curves with the change of R ratio is described only by one parameter m (0 m 1) both for positive and for negative values of R. It means that the crack closure models are very important for explanation and deep study of fatigue crack growth mainly in negative R region but play no crucial role in a phenomenological description of the growth in the case of short-term tests in non-aggressive media.