Linear Elastic Fracture Mechanics Theory Research Articles

Three-Dimensional Probabilistic Semi-Explicit Cracking Model for Concrete Structures

This paper introduces a three-dimensional (3D) semi-explicit probabilistic numerical model for simulating crack propagation within the framework of the finite element method. The model specifically addresses macrocrack propagation using linear volume elements. The criteria governing the macrocrack propagation is based on the softening behavior observed in concrete under uniaxial tension. This softening behavior corresponds to a dissipated cracking energy that is equal to the mode I critical fracture energy (GIC) used in the Linear Elastic Fracture Mechanics theory (LEFM). The probabilistic nature of this model revolves around the random distribution of two mechanical properties: tensile strength (ft) and fracture energy, which varies based on the volume of finite elements. The scattering of the fracture energy increases as the volume of finite elements decreases in order to consider the strong heterogeneity of the material. This work primarily aims to estimate the relationship between the standard deviation of GIC and the volume of finite elements through the development of the numerical model. For this purpose, an inverse analysis is conducted based on a fracture mechanical test simulation. This test involves macrocrack propagation in a large Double Cantilever Beam (DCB) specimen with a crack length exceeding two meters. The proposed inverse analysis procedure yields highly significant results, indicating that the numerical model effectively evaluates both crack length and crack opening during propagation.

Assessment on flexural performance of monolithic glass considering spatial and depth characteristics of scratches

Glass products are widely used in building envelopes. Fracture can be triggered accidentally from the glass surface damage such as scratch. This study proposed an approach to assess the flexural performance of the aged monolithic glass panel, which has scratch with spatial and depth information detected via non-contact sensing. The coaxial double ring (CDR) test was conducted for experimentally evaluating the flexural strength on a total number of 160 scratched annealed (AN) and fully tempered (FT) glass specimens. Finite element (FE) analysis was performed to find the exact stress state by considering the geometric non-linear behaviour of glass panel under large deflection. Test and numerical results demonstrate that the scratch depth characterized by a confocal microscope and spatial location can both influence the failure load of FT glass, whereas AN glass is only sensitive to the latter. With combining linear elastic fracture mechanics (LEFM) theory, the relationship between the scratch depth and the residual strength of FT glass can be accurately determined. The spatial and depth characteristics of scratch can then be integrated for the failure load assessment of monolithic FT glass panel. The assessment framework can contribute to the accurate non-destructive assessment on aged architectural glass.

Evaluation Method of Fatigue Life for Asphalt Pavement on the Steel Bridge Deck Based on the Inhomogeneous Poisson Stochastic Process.

The paving layer on the steel box girder bridge deck is widely used when constructing pavements for steel bridges. Owing to the orthotropic feature of steel decks, a transverse clapboard and rib can lead to a concentration of stress. Consequently, fatigue cracks are often identified in asphalt concrete pavement layers due to re-compaction caused by heavy vehicles. This study aims to derive an evaluation method of fatigue life for asphalt pavement based on the inhomogeneous Poisson stochastic process in view of the highly random and uncertain working conditions of layered composite structures. According to the inhomogeneous Poisson stochastic process, along with Miner's fatigue damage accumulation theory and the linear elastic fracture mechanics theory, the fatigue life formula could be deduced. Meanwhile, fatigue experiments for asphalt concrete are designed to investigate the correlation between the theoretical formula and the actual fatigue damage life of the material. Compared with the test, the accuracy error is within 10%, which is better than other traditional methods. Therefore, the fatigue life prediction model could better reflect the loading order effect and the interaction between loads, providing a new path for the fatigue reliability design of steel bridge deck asphalt pavement.

Plane Stress Fracture Toughness Testing of Freestanding Ultra-Thin Nanocrystalline Gold Films on Water Surface.

Fracture toughness, which is the resistance of a material to crack propagation, is a critical material property for ensuring the mechanical reliability of damage-tolerant design. Recently, damage-tolerant design is introduced to flexible electronics by adopting micro-cracked ultra-thin nanocrystalline (NC) gold films as stretchable electrodes in a plane stress state. However, experimental investigation of the plane stress fracture toughness of those films remains challenging due to the intrinsic fragility from their sub-100nm thicknesses. Here, a quantitative method for systematically evaluating the plane stress fracture toughness of freestanding ultra-thin NC gold film on water surface platform is presented. After effectively fabricating single-edge-notched-tension samples with femtosecond laser, mode I stress intensity factors are measured in the plane stress state on water surface. Moreover, investigation regarding the effect of notch length, notch sharpness, and notch tip plasticity validates this method based on linear elastic fracture mechanics theory. As a demonstration, the thickness-dependent plane stress fracture toughness of ultra-thin NC gold films is qualitatively unveiled. It is revealed that the thickness confinement effect on grain boundary sliding induces a transition in fracture behavior. This method is expected to further clarify the fracture-related properties of various ultra-thin films for next-generation electronics.

Determination of the Probabilistic Properties of the Critical Fracture Energy of Concrete Integrating Scale Effect Aspects

This paper presents an extension of the validation domain of a previously validated three-dimensional probabilistic semi-explicit cracking numerical model, which was initially validated for a specific concrete mix design. This model is implemented in a finite element code. The primary objective of this study is to propose a function that enables the estimation of the critical fracture energy parameter utilized in the model and validate its effectiveness for various concrete mix designs. The model focuses on macrocrack propagation and introduces significant aspects such as employing volume elements for simulating macrocrack propagation and incorporating two key factors in governing its behavior. Firstly, macrocrack initiation is linked to the uniaxial tensile strength (ft). Secondly, macrocrack propagation is influenced by a post-cracking dissipation energy in tension. This energy is taken equal to the mode I critical fracture energy (GIC) based on the linear elastic fracture mechanics theory. Importantly, both ft and GIC are probabilistic properties influenced by the volume of concrete under consideration. Consequently, in the numerical model, they are dependent on the volume of the finite elements employed. To achieve this objective, numerical simulations of fracture mechanical tests are conducted on a large double cantilever beam specimen. Through these simulations, we validate the proposed function, which is a crucial step towards expanding the model’s applicability to all concrete mix designs.

Effect of in vitro ribosylation on the dynamic fracture behavior of mature bovine cortical bone

In this study, the fracture behavior of ribosylated bovine cortical bone is investigated under loading conditions simulating a fall event. Single edge notched specimens, separated into a control group (n = 11) and a ribosylated group (n = 8), were extracted from the mid-diaphysis of a single bovine femur harvested from a mature cow. A seven-day ribosylation process results in the accumulation of Advanced-Glycation End Products (AGEs) cross-links and AGE adducts. Specimens were subjected to symmetric three point bending (opening mode) and an impact velocity of 1.6 m/s using a drop tower. Near-crack displacement fields up to fracture initiation are determined from high-speed images post-processed using digital image correlation. A constrained over-deterministic least squares regression and orthotropic material linear elastic fracture mechanics theory are used to extract the in-plane critical stress intensity factors at fracture initiation (i.e., fracture initiation toughness values). Statistically significant differences were not observed when comparing the in-plane fracture initiation toughness values (p≥0.96) or energy release rate (p=0.90) between the control and seven-day ribosylated groups. The intrinsic variability of bone may require high sample numbers in order to achieve an adequately powered experiment when assessing dynamic fracture behavior. While there are no detectable differences due to the ribosylation treatment investigated, this is likely due to the limited sample sizes utilized.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

Prediction of the effect of shot peening residual stress on fretting fatigue behaviour

Fretting occurs at two contact bodies, which sustain oscillation loading. The fretting phenomenon decreases the fatigue lifetime dramatically. Shot peening is a commonly used surface treatment to enhance the resistance to damage because the residual stress is introduced to the surface of the material. Some studies showed that residual stress was helpful to improve fatigue lifetime. Under fretting fatigue condition, there are two main stages: crack initiation stage and the crack propagation stage. In this paper, the authors apply the Critical Plane (CP) method combined with the Theory of Critical Distance (TCD) to study the fretting fatigue crack initiation behaviour under the influence of shot peening residual stress, including the crack initiation position, orientation and the crack initiation lifetime. Meanwhile, the Linear Elastic Fracture Mechanics (LEFM) theory and the Extension Maximum Tangential Stress (EMTS) criterion are adopted to study crack propagation behaviour considering shot peening residual stress effect. The results show that the methods applied in this paper can make good predictions of fretting fatigue behaviour, including both initiation and propagation stages. Meanwhile, the influence of residual stress in these two stages are studied as well.

Fatigue fracture mechanics in gold-based MEMS notched specimens: experimental and numerical study

The characterization of fatigue fracture mechanics in gold-MEMS notched specimens is presented in this work. A test microstructure with a central notched specimen is specifically designed and built to perform on-chip fatigue test. The central specimen undergoes cyclic loading due to the application of alternating voltage. The variation in the microstructure deflection is measured using an optical profilometer and is attributed to the crack growth in the gold material, causing the variation in the specimen stiffness. The occurrence of pull-in condition is used as a fracture detector, then the fracture of the specimen can be recognized without performing scanning electron microscope inspections during the fatigue test. Crack propagation in the test specimen is simulated through a coupled-field electromechanical fracture finite element model and the resulting crack path is compared to the experimental measurments performed with scanning electron microscope analyses. Finally, Paris’ law is applied and the number of cycles to failure is computed by exploiting the results of the fracture model and experimental measurements. Both experimental and numerical results demonstrate that the notch acts as a stress and strain raiser, fostering crack nucleation, and that the linear elastic fracture mechanics theory is still valid to describe crack propagation in micro-size samples.

Fretting fatigue crack propagation under out-of-phase loading conditions using extended maximum tangential stress criterion

In this paper, a numerical crack growth analysis is conducted by using the Maximum Tangential Stress (MTS) criterion and its extension, which considers the contact stress between the crack surfaces. Moreover, fretting fatigue is a multiaxial non-proportional loading problem, and most components are subjected to asynchronous loads. Therefore, this paper mostly focuses on the influence of loading phase difference (Φ) on the crack propagation behavior. Meanwhile, for the fretting fatigue problem, as there is a relationship between the crack initiation and propagation, the effect of initiation characteristics, including crack initiation position and direction is studied under different loading phase differences using Linear Elastic Fracture Mechanics (LEFM) theory. It is observed that the predicted crack path and lifetime by using the extended MTS criterion and Paris’ law is in good agreement with the experimental results in the literature. It is found that the phase difference has a strong effect on the crack propagation behavior. Furthermore, we found that the crack initiation position has a greater influence on crack propagation behavior than the crack initiation direction.

Effect of double crack on fatigue crack growth life of 3D printing compressor impeller

A compressor impeller with pressure ratio less than 3 is taken as the research object. The 3D printing residual stress was introduced into damage tolerance assessment of blades, and the effect of double crack on Fatigue Crack Growth Life (FCGL) was investigated. The impeller was printed by selective laser melting technology and then heat treated. Finite element method was applied to numerically simulate 3D printing process, heat treatment process and actual working process of impeller, and mechanical quantities such as stress and deformation were completely analyzed. Numerical simulation results agreed well with the experimentally observed crack location in the middle of the blade. Forman–Newman–de Koning model and linear elastic fracture mechanics theory were applied to give the growth law of impeller under single and double crack conditions. The result shows that crack initiation locations of impeller during printing and operating are respectively middle part and root part of blade. The stress intensity factor at the initial crack tips of single and double cracks exhibits a quadratic relationship with rotational speed when only external load is considered. The crack growth paths of single crack in middle and root are radial and axial respectively, and the FCGL of single crack in root is lower than that of single crack in middle. With the increase of rotational speed, the critical crack length decreases linearly and the cycles decreases nonlinearly in both single and double crack conditions. The FCGL of double crack is between two single cracks at low speed and lower than two single cracks at high speed. The critical crack length of double crack is lower than that of corresponding single crack. At 60000rpm, the dominant crack changes from middle crack to root crack.

Numerical Analysis on Fatigue Crack Growth at Negative and Positive Stress Ratios.

The finite element method was used to investigate the effect of the stress ratio on fatigue crack propagation behavior within the framework of the linear elastic fracture mechanics theory. The numerical analysis was carried out using ANSYS Mechanical R19.2 with the unstructured mesh method-based separating, morphing, and adaptive remeshing technologies (SMART). Mixed mode fatigue simulations were performed on a modified four-point bending specimen with a non-central hole. A diverse set of stress ratios (R = 0.1, 0.2, 0.3, 0.4, 0.5, -0.1, -0.2, -0.3, -0.4, -0.5), including positive and negative values, is employed to examine the influence of the load ratio on the behavior of the fatigue crack propagation, with particular emphasis on negative R loadings that involve compressive excursions. A consistent decrease in the value of the equivalent stress intensity factor (ΔKeq) is observed as the stress ratio increases. The observation was made that the stress ratio significantly affects both the fatigue life and the distribution of von Mises stress. The results demonstrated a significant correlation between von Mises stress, ΔKeq, and fatigue life cycles. With an increase in the stress ratio, there was a significant decrease in the von Mises stress, accompanied by a rapid increase in the number of fatigue life cycles. The results obtained in this study have been validated by previously published literature on crack growth experiments and numerical simulations.

Failure analysis of an aero-engine inter-shaft bearing due to clearance between the outer ring and its housing

The inter-shaft bearing is a critical but vulnerable component of dual rotors, because of the severe operational conditions and loads. This paper investigates the failure mechanism of a fractured inter-shaft bearing belonging to a high thrust-weight ratio turbofan engine. Fractographic and microscopic analyses of the fracture surfaces were conducted, followed by vibration signal processing. Finite element analyses of the outer ring were also performed considering the centrifugal, thermal, and roller-raceway loads. What’s more, using the linear elastic fracture mechanics theory, the crack growth path simulation was obtained, whose result showed good agreement with the real fracture morphologies. The results indicated that due to the clearance between the outer ring and its housing during the operation, the anti-rotation tabs of the outer ring were subjected to non-uniform load, thus several tabs might suffer large stress concentrations at the root fillets, which could lead to the crack initiation. During the following operation of the aero-engine, the crack propagated through the outer ring and cause the fatigue fracture of tabs. The investigation provided herein suggests that the outer ring and its housing should be redesigned to eliminate the clearance during the operation, an outer ring formed integrally with the bearing housing flange is one of the failure preventions of this failure mode.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

Estimation of leak area-pressure relationship for cracks on water pipes using models based on linear-elastic fracture mechanics

The assessment and control of the real losses from water distribution systems require the accurate estimation of the flow rate from an individual leak as a function of the internal pressure. The lack of analytical models able to accurately describe the relationship between the area of the leak and the pressure head is the key problem. This paper utilized the linear-elastic fracture mechanics (LEFM) theory for thin shells to derive models for both longitudinal and circumferential cracks. The models were validated by both finite element (FE) simulations and laboratory experiments under varying crack and pipe parameters. Both fluid-structure interaction (FSI) and traditional FE simulations were performed, and the results were compared to quantify the effect of leakage hydraulics on leak area. In the laboratory experiments, an image analysis technology was utilized to measure the leak area and flow rate simultaneously, so that the effect of the discharge coefficient could be excluded. In addition, the leak area was systematically measured under the effect of different parameters. The results revealed that the values predicted by the derived models were in good agreement with the experimental and FE simulation values for both types of cracks. The LEFM theory and the phenomena observed in this study can improve our understanding of the leak behavior and enable the development of effective pressure management strategies for water distribution systems.

Influence of Different Surfactants on Carbon Fiber Dispersion and the Mechanical Performance of Smart Piezoresistive Cementitious Composites

This experimental study presents the effect of different surfactants on micro-scale carbon fiber (CFs) distribution into carbon fiber reinforced cement-based composites (CFRC) in terms of flexural and compressive strength, stiffness, flexural toughness, and strain-sensing ability. Conducting a narrative review of the literature focusing on the fibers’ separation, this paper follows a methodology introducing a combination of mechanical and chemical carbon fibers dispersion, as well as the different mixing processes (wet or dry). Three types of surfactants: Carboxymethyl cellulose (CMC), cellulose nanocrystal (CNC), and superplasticizer (SP), were applied to evaluate the CFs distribution in the cement paste matrix. Compressive and flexural strength, modulus of elasticity, and ductility of the cement-based composites (CFRC) reinforced with 0.5 wt.% CFs were investigated by three-point bending and compressive tests; flexure tests were also conducted on notched 20 × 20 × 80 mm specimens using the Linear Elastic Fracture Mechanics (L.E.F.M.) theory. Moreover, the electrical conductivity and the piezoresistive response were determined by conducting electrical resistance measurements and applying compressive loading simultaneously. The results clearly reveal that the CFs/SP solution or the CFs’ dry incorporation led to a significant enhancement of flexural strength by 32% and 23.7%, modulus of elasticity by 30% and 20%, and stress-sensing ability by 20.2% and 18.2%, respectively. Although the wet mixing method exhibits improved mechanical and electrical conductivity performance, constituting an adequate strain and crack sensor, the authors propose dry mixing as the most economical method, in addition to the enhanced mechanical and electrical responses. The authors recommend an effective method for structural health monitoring systems combining an economical CFs insertion in cementitious smart sensors with great mechanical and self-sensing responses.

Fatigue analysis in rotor of a prototype bulb turbine based on fluid-structure interaction

Bulb turbine units are widely used in run-of river or dam projects with relatively low head, and can also be used for tidal energy utilization. Some types of bulb turbines have excessive vibration and fatigue failure problems, which threats their fatigue life. The model analyzed in this paper is a prototype bulb turbine rotor which suffered cracks, probably due to erosion and welding defects. The dynamic stress characteristics are analyzed under not only the non-cracked shaft condition but also the cracked one, to find possible causes of the failure.The computational fluid dynamics (CFD) simulation is performed to obtain reliable hydraulic load on the runner for performing the FEM analysis. The CFD results are verified by comparing with the site test for the prototype one. The linear elastic fracture mechanics theory combined with the fluid-structural interaction theory is applied to developing a cracked FEM model, to obtain the dynamic stress characteristics for the runner, shaft and rotor. The results reveal that the change in dynamic stress level on the shaft flange root significantly depends on the operating conditions, and the stress amplitude on the shaft mainly depends on the structural weight. The stress at the cracked zone could be up to 376.5 MPa in the best efficiency point, with only 1.5 years of fatigue life.Based on the numerical calculations and site inspection, it could be concluded that the insufficient sealing could lead to leakage flows with specific flow rates in flange root, which resulted in the occurrence of erosion; in addition, the welding at the flange root probably generated tiny defects. These two factors are likely lead to fatigue cracks and major failure of the shaft. Suggestions for solutions for the problems of the shaft are also discussed. Results of this study helps to find the reasons of the shaft fatigue failure. Furthermore, the analysis method could be used for an accurate calculation of fatigue life and providing guidance in future designs and operations of bulb turbine units.

Crack propagation modeling of strengthening reinforced concrete deep beams with CFRP plates

Fracture analysis of reinforced concrete deep beam strengthened with carbon fiber-reinforced polymer (CFRP) plates was carried out. The present research aimed to discover whether crack propagation in a strengthened deep beam follows linear elastic fracture mechanics (LEFM) theory or nonlinear fracture mechanics theory. To do so, a new energy release rate based on nonlinear fracture mechanics theory was formulated on the finite element method and the discrete cohesive zone model (DCZM) was developed in deep beams. To validate and compare with numerical models, three deep beams with rectangular cross-sections were tested. The code results based on nonlinear fracture mechanics models were compared with the experimental results and the ABAQUS results carried out based on LEFM. The predicted values of initial stiffness, yielding point and failure load, energy absorption, and compressive strain in the concrete obtained by the proposed model were very close to the experimental results. However, the ABAQUS software results displayed greater differences from the experimental results. For instance, the predicted failure load for the shear-strengthened deep beam using the proposed model only had a 6.3% difference from the experimental result. However, the predicted failure load using ABAQUS software based on LEFM indicated greater differences (25.1%) compared to the experimental result.

Numerical Failure Analysis and Fatigue Life Prediction of Shield Machine Cutterhead.

This paper presents numerical failure analysis on cracking of shield machine cutterhead structure during a metro-tunnel construction. The stress intensity factors (SIFs) of surface cracks with different shapes and location angles were analyzed by a finite element simulation method based on linear elastic fracture mechanics (LEFM) theory. The ratios of variation in stress intensity factors of cracks with different shapes were analyzed. The maximum allowable crack depth of the cutterhead panel is 50.23 mm by dynamic stress calculation, and the damage tolerance criterion of the cutterhead panel was proposed. The influence of the Paris model parameter values was analyzed based on mathematical methods. It is proven that the location of the cutterhead cracking angle is mainly determined by the mixed-mode SIF. In practice, the crack section basically expanded into the semi-elliptical shape. The cutterhead structure may directly enter the stage of crack propagation due to welding defects during tunneling. The research results provide a theoretical basis and important reference for crack detection in the key parts of the cutterhead, as well as maintenance cycle determination and life prediction of the cutterhead mileage, both of which have important engineering value.

Analysis of crack on the labyrinth part of the gas turbine engine

The main objective of this study is to analyze the crack developed in the Labyrinth seal of a gas turbine engine and to determine the number of cycles required for the crack to grow to various lengths. Here, ANSYS Mechanical APDL is the tool to carry out the crack analysis. The crack is modelled and analyzed according to the principles of Linear Elastic Fracture Mechanics (LEFM) theory by Griffith and Paris law because of the negligible plasticity found at the crack tip. Here, the model is analyzed with and without the crack for the temperature and stress profile. The stress intensity factor is obtained using the ANSYS software. The information is derived from the graph of crack propagation rate as a function of stress intensity factor of Haynes 25 material which was used to analytically determine the number of cycles the labyrinth seal could sustain. The maximum and minimum temperatures were 1132 °C and 250 °C over a span of 10 mm length, which was the region of interest. The maximum thermal stress generated was around 1430 MPa at crack tip region for full crack model. The stress intensity factor was obtained was found to be 18.7 MPa√mm. The crack propagation rate was calculated analytically over a range of 2–4 mm with an increment of 0.5 mm. The number of cycles required for the crack to propagate over the successive 0.5 mm intervals was 97, 82, 71 and 63 cycles respectively.