Paris Law Research Articles

Numerical and Experimental Studies for Fatigue Damage Accumulation of CFRP Cross-Ply Laminates Based on Entropy Failure Criterion

The transverse cracking behavior of a carbon-fiber-reinforced plastic (CFRP) cross-ply laminate is investigated using a fatigue test and an entropy-based failure criterion in this study. The results of fatigue experiments show that the crack accumulation behavior depends on the cyclic number level and frequency, in which two obvious transverse cracks are observed after 104 cyclic loads and 37 transverse cracks occur after 105 cycles. The final numbers of transverse cracks decrease from 29 to 11 when the load frequency increases from 5 Hz to 10 Hz. An entropy-based failure criterion is proposed to predict the long-term lifetime of laminates under cyclic loadings. The transverse strength of 90° ply is approximated by the Weibull distribution for a realistic simulation. Progressive damage and transverse cracking behavior in CFRP ply can be reproduced due to entropy generation and strength degradation. The effects of stress level and load frequency on the transverse cracking behavior are investigated. It is discovered that, at the edge, the stress σ22 + σ33 that is a dominant factor for matrix tensile failure mode is greater than the interior at the first cycle load, and as stress levels rise, a transverse initial crack forms sooner. However, the initial transverse crack initiation is delayed as load frequencies increase. In addition, transverse crack density increases quickly after initial crack formation and then increases slowly with the number of load cycles. The proposed method's results agree well with those of the existing experimental method qualitatively. In addition, the proposed entropy-based failure criterion can account for the effect of load frequency on transverse crack growth rate, which cannot be addressed by the well-known Paris law.

Application of the J-Integral and Digital Image Correlation (DIC) to Determination of Multiple Crack Propagation Law of UHPC under Flexural Cyclic Loading

This study investigated the fatigue crack propagation behavior of ultra-high-performance concrete (UHPC) incorporated with different steel fiber lengths of 6, 13, and 20 mm under flexural cyclic loading, based on the Paris law and nonlinear fracture mechanics. In addition, multiple crack covering areas and fatigue J-integral amplitudes were employed to quantitatively evaluate the fatigue crack propagation rate and predicate the fatigue life of the UHPC during the steady development stage. The results indicated that the maximum crack opening displacement (COD) values were 0.312, 0.673, and 1.265 mm and the minimum crack growth rates were -3.05, -4.48 and -4.62 for SF6, SF13, and SF20, respectively. The critical crack length was approximately 65 mm for UHPC specimens containing different fiber length at a given fiber volume fraction (2.0%), indicating that the critical crack length was simply related to the fiber length. Interestingly, when the fatigue crack area of all the tested series reached approximately 35 mm2, fracture failure occurred. There were very small predictions between the actual tested and predicated fatigue lives, all less than 7.21%. Hence, it was reasonable to predict the fatigue life of the UHPC based on the J-integral according to the DIC technique.

Service Life Modelling of Single Lap Joint Subjected to Cyclic Bending Load

Bonded joints used in wing sections and frames of aircraft structures are mostly exposed to cyclic loadings instead of static ones during their services. Bending types of dynamic loadings are mostly encountered. In this study, the fatigue response of a single lap joint (SLJ) exposed to bending loading was studied with the developed advanced finite-element (FE) model. The cohesive zone model describing the behaviour of the adhesive layer used the damage mechanism, where static and fatigue damages were linked to each other; i.e., the total damage was accumulated because of material deterioration and cyclic plastic separation. This enabled us to predict the fatigue characteristics including the finite fatigue life, crack propagation rate using Paris law. The model was implemented via a user-defined UMAT subroutine offered in ABAQUS-Standard. The numerical model was validated by experiments available in the literature. The fatigue performance of an SLJ subjected to bending loading was investigated for different lap joint configurations. A smaller bending load, a thicker adherend or a longer overlap length (OL) led to enhanced fatigue life. For instance, the fatigue life was observed to increase up to 50 times for a 66% increase in OL.

Anisotropic high cycle fatigue property estimation for laser additive manufactured Ti6Al4V alloy dependence on tomographic imaging of defect population

Improvement of fatigue properties of additive manufactured metals is subject to the regulation of process-induced microstructure and defect population. Here, the high cycle fatigue properties of the annealed Ti6Al4V alloy manufactured by two types of laser powder bed fusion (L-PBF) processes, in orthogonal sample orientations, were studied. The fatigue performance of this alloy was also estimated dependence on the investigation of microstructure and defects using optical microscopy, X-ray diffraction, scanning electron microscopy and micro computed tomography. The results demonstrated that the fatigue properties were degraded by the negative effect of prior-β columnar grain boundaries and defects, wherein the surface keyhole governed the pronounced fatigue scatter and anisotropy due to its geometrical parameters and anisotropy severity along the building direction. The projection size √area of keyholes was well applied in the fatigue property estimation. The theorical fatigue limits were estimated individually due to the keyhole anisotropy, and provided conservatively in the improved Kitagawa-Takahashi diagram based on the modified El-Haddad model and the extreme size of keyholes. A relation derived from Paris law was proposed for fatigue life prediction, and to well establish a linear fitting correlating fatigue life with the projection size of surface keyholes and stress amplitude range with the achievement of the alleviation of fatigue anisotropy and scatter. The research provides a reference for the non-destructive estimation of fatigue properties in terms of tomographic imaging of defect population.

Prediction of rolling contact fatigue crack propagation in bearing steels using experimental crack growth data and linear elastic fracture mechanics

Rolling contact fatigue (RCF) is a major life limiting factor for machine elements that employ non-conformal, rolling sliding, lubricated contacts such as rolling bearings and gears. This paper explores the application of linear elastic fracture mechanics (LEFM) principles, as commonly used in structural fatigue, for prediction of RCF crack propagation. A triple-disc contact fatigue machine is used to generate RCF cracks of varying lengths in AISI 52100 bearing steel roller specimens. Crack propagation rates across the surface are measured using optical inspection of test specimens and the final crack geometry is established through specimen sectioning. A numerical finite element model of surface breaking RCF cracks based on LEFM methods is devised to predict the evolution of stress intensity factors (SIFs) during over-rolling of the contact over the experimentally observed crack geometries. The model employs a suitable fracture mechanics mesh to resolve stresses at the crack tip and accounts for Hertzian contact stresses, contact friction and crack face friction. Potential effects of lubricant pressurisation within the crack are not modelled. The predicted SIFs are then related to the experimentally measured crack propagation rates to establish the applicability of the LEFM principles to RCF crack propagation. Results show that LEFM can be used to predict the growth of surface braking RCF cracks. For cracks longer than about 100 μm, a Paris law relationship with the stress intensity exponent of about 4 is derived. Mode II was seen to be the dominant mode of propagation for surface braking RCF cracks. Mode I SIFs are much smaller but can exhibit significant values when the contact is located just ahead of the crack mouth. Decreasing the crack face friction significantly increases mode II stress intensity suggesting that this is one important mechanism by which lubricant entry into the surface braking crack can accelerate its propagation. The findings can help in improving the reliability of mechanical systems by supporting the development of new tools for prediction of remaining useful life of machine components such as bearings and gears.

Prediction of Fatigue Cracks Using Gamma Function

Abstract In the present study it has been endeavored to estimate the fatigue crack propagation in V-notch Charpy specimens of 2024 T351 Al-alloy. For this purpose, a new application of fatigue crack growth (FCG) is developed based on the “Gamma function.” Experimental fatigue tests are conducted for stress ratios from 0.1 to 0.5 under constant amplitude loading. The empiric model depends principally on physical parameters and materials’ properties in non-dimensional form. Deviation percentage, prediction ratio, and band error are used for validation of the performance of the fatigue life. The results determined from Gamma application are in good agreement with experimental FCG rates and those obtained from using Paris law.

Fatigue Threshold Analysis of Adhesives: Displacement Control vs. Load Control Strategy

The aim of this paper is to study the response of a two-part polyurethane-resin adhesive under quasi-static and fatigue loading conditions, to compare load control and displacement control approaches for the mode I fatigue threshold analysis of the tested material. To achieve this, double cantilever beam (DCB) joints were manufactured and tested. For the post-processing of the raw data, a compliance-based beam method was used. Out of this analysis, R curves and Paris law curves were obtained. Both approaches showed a similar Paris law slope meaning a low sensibility of the crack growth rate between them. As the displacement control load decreases gradually during the test, it can give more precise threshold energy than the load control technique.

Prediction of fatigue crack propagation based on dynamic Bayesian network

To address the problem of low prediction accuracy in the current research on fatigue crack propagation prediction, a prediction method of fatigue crack propagation based on a dynamic Bayesian network is proposed in this paper. The Paris Law of crack propagation and the extended finite element method (XFEM) are combined to establish the state equation of crack propagation. The uncertain factors of crack propagation are analyzed, and the prediction model of fatigue crack propagation based on the dynamic Bayesian network is constructed. A Bayesian inference algorithm based on the combination of Gaussian particle filter and firefly algorithm is proposed. The fatigue experiment of the specimen with the pre-cracks is carried out to test the correlation between the fatigue load cycles and the crack propagation depth. The experimental results show that the crack propagation prediction method proposed in this paper can effectively improve the prediction accuracy of crack propagation depth.

A progressive damage model for composite laminates based on non-linear continuum mechanics—Static and fatigue loadings

Efficient yet reliable predictive modelling tools for damage tolerance analysis became an aerospace industry requirement as composite materials provided the potential for design performance. As damage can be separated into interlaminar and intralaminar, different approaches were developed over the years. For fracture analysis of intralaminar damage, a physical link to the Linear Elastic Fracture Mechanics (LEFM) provides a reliable framework for Continuum Damage Mechanics (CDM) models. These models, however, may exhibit pathological problems related to mesh dependence, objectivity and convergence issues. Differently from Cohesive Zone Modelling (CZM) for interlaminar damage, most intralaminar models are based on phenomenological approaches for fatigue analysis and life prediction. This work provides a description of some of the current intralaminar CDM models shortcomings, related to large strain analysis and finite element topology. A novel progressive damage model is proposed where damage variables are linked to the deformation gradient. Additionally, a fatigue damage behaviour under the assumption of a Paris law for crack evolution is implemented in the proposed methodology. Both static and fatigue analysis were performed on a Compact Tension (CT) specimen geometry and compared to experimental data available on the scientific literature. Model predictions and experimental data were confronted allowing for conclusions to be drawn.

The prediction of residual life of liquid-storage tank considering the tank wall surface state

PurposeThe purpose of the paper is to predict the residual life of liquid-storage tank to ensure safety and long-term service life of the structure.Design/methodology/approachThe paper carried out the stress analysis of the wall plate and bottom plate of the liquid-storage tank, and the influence of circumferential stress on the tank is considered. On the other hand, considering the influence of the tank wall surface on the tank life, based on the Paris law, the surface processing coefficient and surface roughness coefficient are introduced to improve the Paris law.FindingsThe effectiveness of the improved model is verified by comparing with theoretical and experimental data, which provide a new method for the prediction of the remaining service life of the tank. Combined with the fatigue crack data in the test report and the calculated circumferential stress, the residual life of the storage tank is predicted.Originality/valueThe improved model provides a new method for the prediction of the remaining service life of the tank.

Numerical simulation of pre-cracked thin aluminum alloy skin repaired with a FRP patch: Pre-crack length studies

The fatigue life of a thin pre-cracked aluminium-alloy panel, repaired with Carbon Fiber Reinforced Plastics (CFRP) patch on one side, is investigated using a numerical model. CFRP plies of different lengths are used to form patch which in turn repair pre-cracked aluminium skin. Numerical studies are performed to determine variation in stress intensity factor (SIF) range and bulging out deflection of specimen subjected to minimum and maximum fatigue load. Patch width to pre-crack and length to pre-crack ratio are kept constant for different pre-crack studies. Numerical simulations are carried out for three different pre-crack length. Three different patch configurations are studied for each pre-crack length. Maximum stress considered is at 50 % yield strength of aluminium skin. The numerical model is simulated by using ANSYS 15.0, a Finite Element Analysis package. The interface behaviour between the skin and the patch is simulated by using cohesive zone material model (CZM). Stress intensity factor (SIF) is determined from J-integral results. Relation between SIF range and crack length can be used to determine fatigue crack propagation by invoking the Paris law. It is found that with increasing the patch length, ΔKI decreased significantly which may result into the increase of the fatigue life.

A mixed-mode energy-based elastoplastic fatigue induced damage model for the peridynamic theory

In recent years a considerable effort has been dedicated to the development of analysis tools that enable the design of damage tolerant structures, particularly in aerospace applications where weight reduction is a crucial requirement. One of these tools developed in recent years is the peridynamic theory, employed to solve numerically complex elastodynamics problems. One of its advantages reported in the literature is the natural ability to simulate the initiation and crack growth without the need for additional numerical procedures commonly employed in other numerical approaches, like in the conventional finite element formulation. Within this context, this paper presents a novel elastoplastic fatigue-induced damage model whose formulation is based on a strain energy framework combined with a smeared crack approach to simulate the damage process without the need of knowing the location of the crack tip and its length within the domain. The proposed model also can predict the mixed-mode damage propagation in ductile materials without knowing a priori the mixity mode ratio. This approach incorporates an analytical methodology based on the material properties that correlate the strain energy calculated away from the crack tip to the expected propagation rate predicted by the Paris Law. The accuracy of the proposed model is verified by comparing the results obtained using an in-house peridynamic FORTRAN code with the experimental results available in the open literature. Some improvements for the peridynamic material parameters are also presented in this paper, which is also verified by using the in-house code.

Mixed-mode I+II fatigue/fracture characterisation of bi-material Aluminium/CFRP bonded joints

The fatigue/fracture behaviour of bi-material Aluminium/CFRP (Carbon-Fibre Reinforced Polymer) bonded joints was addressed in this work using the modified Paris law. Five types of tests were employed to cover appropriately the entire mode I – mode II range. Data reduction schemes based on equivalent crack length concept were developed for all types of tests. Following this methodology, the compliance versus number of cycles relation becomes sufficient to perform post-processing fatigue data analysis. The coefficients of the modified Paris law for each type of test were obtained by fitting a power law to the crack growth rate versus the ratio of total strain energy release rate and its critical value. The objective was to obtain relations representative of the evolution of those coefficients as function of mode ratio. These relations were subsequently input in a cohesive zone model at the local level, i.e., at each integration point. It was verified that the compliance versus number of cycles curves and the fatigue lives predicted numerically are in agreement with the experimental values.

Crack initiation and growth simulations of functionally graded dental implant subjected to cyclic axial loads

AbstractTwo essential characteristics in designing dental implants are having high fracture resistance and being compatible with jawbone. One of the best implant choices to achieve this aim is titanium‐hydroxyapatite functionally graded (FG) material. Also, fracture of implants under cyclic loads is one of the most common causes of destruction of dental implants. However, the crack initiation location and growth path as well as the optimum material distribution for the FG implants are questions that have not been answered yet. The aim of this study was introducing a new and powerful numerical approach to predict the crack initiation behavior and progressive damage in the FG dental implants, subjected to cyclic axial loadings. The approach is developed by employing extended finite element method (XFEM) in ABAQUS software, Visual Studio, and FORTRAN Compiler. Also, the Python code was modified for crack growth modeling in fatigue fracture analysis based on Paris law. The result showed that the micro‐cracks below the outer surface of FG implant are created, due to the application of cyclic load, which is invisible. This research shows that FG implants with power factor less than 1 show much more resistance to crack growth under pre‐tightening and cyclic loadings.

Effect of aging on the fatigue crack growth properties of carbon-polyamide 6 thermoplastic composites using the multi ΔG-control method

This paper focuses on the fatigue crack growth behavior of unidirectional C/PA6 composites subjected to hydrolytic aging at temperatures ranging from 100 to 140 °C. The fatigue crack growth properties were determined using a multi-ΔG control test method. The latter is fully automatic and allows measurement of crack growth rates at various ΔG levels. It was shown that the fatigue crack growth properties were highly affected by aging. Whatever the aging temperature, the crack growth rate was increased by about a decade after aging, and the fatigue crack growth properties are highly dependent on the molar mass. Finally, it was shown that the traditional Paris law does not allow a full description of the behaviour so the Hartman-Schijve relationship was applied to characterize the fatigue response.

Face/core disbond fatigue growth in honeycomb cored aircraft sandwich elements under mixed mode flatwise tension loading

Disbond damage growth in honeycomb cored sandwich structures due to static and fatigue mixed mode loading is investigated numerically and experimentally. A two dimensional finite element model was generated using core homogenization and the Crack Surface Displacement Extrapolation mode separation method, integrated into a fracture mechanics based analysis sub-routine to predict face/core interface fatigue crack propagation. The Cycle Jump technique was furthermore applied to accelerate fatigue analysis. Mixed mode fatigue characterization testing was conducted using Double Cantilever Beam specimens loaded with Uneven Bending Moments, generating a relationship between crack propagation rates and energy release rate amplitudes as a modified Paris Law, measured at three mode-mixity phase angles. The measured Paris laws were subsequently used as input data for the numerical fatigue model. The numerical model was validated against CFRP/Nomex® Sandwich Tearing Test specimen tests with a propagating face/core interface crack yielding varying mode-mixities. The results from the validation showed good agreement between numerical predictions and experimental measurements.

THE ROMANIAN DIPLOMAT OF EUROPEAN STATURE NICOLAE TITULESCU IN THE VISION OF SOME CONTEMPORARIES

The Romanian diplomat of European stature, born on March 4, 1882 (died abroad, on March 17, 1941 in Cannes, France) on the coast of France, in a family of Oltenian owners; left without a father (former Deputy and Prefect) at just one year old, Nicolae Titulescu will study law in Paris, and when he returns to the country he will enter political life, in Take Ionescuʼs party (the Conservative-Democratic Party). He will be a Deputy, Minister of Finance (1917-1918), Minister of Foreign Affairs (1927-1928, 1932-1936), Romaniaʼs Envoy Extraordinary and Minister Plenipotentiary to London (1921-1927, 1928-1932), Romaniaʼs representative in the League of Nations and its President (twice: 1931 and in 1932). He was member of the Romanian Academy (elected in 1935). This paper presents some opinions (with more recent detailed bibliographic references) excerpted from the book – in 3 volumes, published (under the auspices of the European Titulescu Foundation) in 2012, Pro și Contra Titulescu, edited by George G. Potra – of some people politicians, diplomats, and journalists, Romanians and foreigners – contemporaries of him –, relative to the personality of the greatest diplomat of Romania, and one of the most famous of interwar Europe, whose birth marks, in 2022, 140 years.

Fatigue failure analysis of spur gear under moving load

This paper is prepared to analyze the crack propagation by developing a pointed moving load condition. The moving load from lower point tooth contact (LPTC) to high point tooth contact (HPTC) is applied on the gear flank, where the maximum load is acted on the pitch-point of the tooth flank. Different models of gear, taking a three teeth sector model (TTSM), are developed for different backup ratio (BR) of 0.3, 0.5, 0.8, 1.5 & 3.3 and initiating the initial crack (IC) length of 0.1 mm, 0.25 mm & 0.5 mm at orientation of critical planes for shear model oriented at 180° at the root region of the tooth. The finite element analysis (FEA) software - ANSYS is used to study the problem statement. The arbitrary crack method is used to initiate the crack presence in gear flank. The Paris law method is applied in separating morphing and adaptive remeshing technology (SMART) crack growth setup in static structural module of ANSYS, which helps in evaluation of number of cycles with crack extension length. The number of life cycles, crack extension length and crack propagation path are analyzed and their graphs are plotted to analyze the situation. From current study, the effect of rim thickness on the life cycle is studied when the point load is applied movable from LPTC to HPTC. It is observed that by having thin rims, the number of cycles of gear increases, but its failure is catastrophic that it can fracture other parts, mounted with gear. Although, higher backup ratios or larger rims have lower number of life cycles, but it will never lead to the catastrophic failure of the gear.

Neural cohesive model for delamination simulation in composite laminates under cyclic loadings

This work proposes a novel fatigue cohesive model that utilizes a single-hidden-layer neural network as its fatigue constitutive law for simulation of fatigue delamination under mixed-mode loadings. The neural network takes variables of the cohesive zone as inputs and outputs the accumulation rates for a fatigue damage variable to enforce fatigue delamination propagation, which is then embedded in Abaqus to verify the model under mode-I, mode-II and mixed-mode loadings with Paris law and Hartman-Schijve equation as references. On top of the good correlations, it is found that the proposed model is also insensitive to mesh sizes and very computationally efficient.

Effect of material heterogeneity and testing temperature on fatigue behaviour of Cr-Mo steel welded joints

Successful application of Cr-Mo steel depends crucially on behaviour of all zones in welded joints, i.e. base metal (BM), heat-affected-zone (HAZ) and weld metal (WM), both at room and working temperature. Different amplitude loadings were applied for testing of specimens without a crack, made of welded joint as a whole and of BM, in order to obtain Woehler’s curves and compare their dynamic strength. Amplitude loading was also applied for testing of specimens cracked in BM, HAZ and WM, in order to get fatigue crack growth curves and parameters C and m in Paris law. Results are analysed to get deeper insight into behaviour of welded joints, made of steel SA 387 Gr. 91, under amplitude loading both for room and working temperature.